TPTP Problem File: SLH0203^1.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_07145_291731__14298414_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1452 ( 296 unt; 169 typ; 0 def)
% Number of atoms : 5003 (1574 equ; 0 cnn)
% Maximal formula atoms : 19 ( 3 avg)
% Number of connectives : 21407 ( 687 ~; 46 |; 665 &;17677 @)
% ( 0 <=>;2332 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 1295 (1295 >; 0 *; 0 +; 0 <<)
% Number of symbols : 158 ( 155 usr; 12 con; 0-4 aty)
% Number of variables : 3856 ( 23 ^;3605 !; 228 ?;3856 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:48:24.698
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
produc1473018763691903991list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
produc7709606177366032167list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_M_062_Itf__a_M_Eo_J_J_Mt__List__Olist_Itf__a_J_J,type,
produc5032551385658279741list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
produc8685980395799941037list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
produc2579390645249093025st_a_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_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__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_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (155)
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_If_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
if_list_list_a: $o > list_list_a > list_list_a > list_list_a ).
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_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_Obind_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
bind_list_a_list_a: list_list_a > ( list_a > list_list_a ) > list_list_a ).
thf(sy_c_List_Obind_001t__List__Olist_Itf__a_J_001tf__a,type,
bind_list_a_a: list_list_a > ( list_a > list_a ) > list_a ).
thf(sy_c_List_Obind_001tf__a_001t__List__Olist_Itf__a_J,type,
bind_a_list_a: list_a > ( a > list_list_a ) > list_list_a ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( a > list_a ) > list_a ).
thf(sy_c_List_Obutlast_001t__List__Olist_Itf__a_J,type,
butlast_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Ocan__select_001t__List__Olist_Itf__a_J,type,
can_select_list_a: ( list_a > $o ) > set_list_a > $o ).
thf(sy_c_List_Ocan__select_001tf__a,type,
can_select_a: ( a > $o ) > set_a > $o ).
thf(sy_c_List_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olast_001t__List__Olist_Itf__a_J,type,
last_list_a: list_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_It__List__Olist_Itf__a_J_J,type,
cons_list_list_a: list_list_a > list_list_list_a > list_list_list_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_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_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_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_Itf__a_J,type,
tl_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_Itf__a_J,type,
list_ex1_list_a: ( list_a > $o ) > list_list_a > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Omap__tailrec__rev_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
map_ta5969768091587575669list_a: ( list_a > list_a ) > list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Omap__tailrec__rev_001t__List__Olist_Itf__a_J_001tf__a,type,
map_ta9088930921017855855st_a_a: ( list_a > a ) > list_list_a > list_a > list_a ).
thf(sy_c_List_Omap__tailrec__rev_001tf__a_001t__List__Olist_Itf__a_J,type,
map_ta1754794639992346235list_a: ( a > list_a ) > list_a > list_list_a > list_list_a ).
thf(sy_c_List_Omap__tailrec__rev_001tf__a_001tf__a,type,
map_tailrec_rev_a_a: ( a > a ) > list_a > list_a > list_a ).
thf(sy_c_List_Omaps_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
maps_list_a_list_a: ( list_a > list_list_a ) > list_list_a > list_list_a ).
thf(sy_c_List_Omaps_001t__List__Olist_Itf__a_J_001tf__a,type,
maps_list_a_a: ( list_a > list_a ) > list_list_a > list_a ).
thf(sy_c_List_Omaps_001tf__a_001t__List__Olist_Itf__a_J,type,
maps_a_list_a: ( a > list_list_a ) > list_a > list_list_a ).
thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
maps_a_a: ( a > list_a ) > list_a > list_a ).
thf(sy_c_List_Orotate1_001t__List__Olist_Itf__a_J,type,
rotate1_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
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_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
produc8111569692950616493list_a: ( a > a > $o ) > list_a > produc5032551385658279741list_a ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc8643929849434629545list_a: ( a > a ) > produc9164743771328383783list_a > produc1473018763691903991list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
produc8696003437204565271list_a: list_list_a > list_list_a > produc7709606177366032167list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001tf__a,type,
produc4781227316648555537st_a_a: list_a > a > produc2579390645249093025st_a_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__List__Olist_Itf__a_J,type,
produc6670463072477821725list_a: a > list_a > produc8685980395799941037list_a ).
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_001tf__a,type,
cohere6072184133013167079_rts_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_001t__List__Olist_Itf__a_J,type,
normal6589580540804570479list_a: ( list_a > list_a > list_a ) > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_001tf__a,type,
normal_sub_rts_a: ( a > a > a ) > set_a > $o ).
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_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_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_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_001tf__a,type,
normal8595587647932138008lass_a: ( a > a > a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts__axioms_001t__List__Olist_Itf__a_J,type,
normal2939518615156061708list_a: ( list_a > list_a > list_a ) > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts__axioms_001tf__a,type,
normal7698203753654205830ioms_a: ( a > a > 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_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_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_001tf__a,type,
paths_8083246678931295059_rts_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_001t__List__Olist_Itf__a_J,type,
paths_in_rts_list_a: ( list_a > list_a > list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_001tf__a,type,
paths_in_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_001tf__a,type,
paths_in_Arr_a: ( a > a > a ) > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OIde_001t__List__Olist_Itf__a_J,type,
paths_in_Ide_list_a: ( list_a > list_a > list_a ) > list_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_001t__List__Olist_Itf__a_J,type,
paths_1777230443808135851list_a: ( list_a > list_a > list_a ) > list_a > list_list_a > list_a ).
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_OResid1x__rel_001tf__a,type,
paths_6492648068886854876_rel_a: ( a > a > a ) > produc8685980395799941037list_a > produc8685980395799941037list_a > $o ).
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_001tf__a,type,
paths_in_Resid_a: ( a > a > a ) > list_a > list_a > list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResid__rel_001t__List__Olist_Itf__a_J,type,
paths_3409276745531958875list_a: ( list_a > list_a > list_a ) > produc7709606177366032167list_a > produc7709606177366032167list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResid__rel_001tf__a,type,
paths_in_Resid_rel_a: ( a > a > a ) > produc9164743771328383783list_a > produc9164743771328383783list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResidx1_001t__List__Olist_Itf__a_J,type,
paths_3541054012941122297list_a: ( list_a > list_a > list_a ) > list_list_a > list_a > list_list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResidx1_001tf__a,type,
paths_in_Residx1_a: ( a > a > a ) > list_a > 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_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_001tf__a,type,
paths_in_Trgs_a: ( a > a > a ) > list_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts__with__coherent__normal_001t__List__Olist_Itf__a_J,type,
paths_9180082184051307706list_a: ( list_a > list_a > list_a ) > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts__with__coherent__normal_001tf__a,type,
paths_6252547876539050804rmal_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts__with__coherent__normal_OPCong_001t__List__Olist_Itf__a_J,type,
paths_9002432012371690887list_a: ( list_a > list_a > list_a ) > list_list_a > list_list_a > $o ).
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_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_001tf__a,type,
paths_3208254328100665313Path_a: ( a > a > a ) > set_a > list_a > $o ).
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_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_001tf__a,type,
arr_a: ( a > a > a ) > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
con_list_list_a: ( list_list_a > list_list_a > list_list_a ) > list_list_a > list_list_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_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_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_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_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_001tf__a,type,
composable_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposite__of_001t__List__Olist_Itf__a_J,type,
composite_of_list_a: ( list_a > list_a > list_a ) > list_a > list_a > list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposite__of_001tf__a,type,
composite_of_a: ( 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_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_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_001tf__a,type,
joinable_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
seq_list_list_a: ( list_list_a > list_list_a > list_list_a ) > list_list_a > list_list_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_001tf__a,type,
seq_a: ( a > a > a ) > a > a > $o ).
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_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_001tf__a,type,
targets_a: ( a > a > a ) > a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001t__List__Olist_Itf__a_J,type,
rts_axioms_list_a: ( list_a > list_a > list_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001tf__a,type,
rts_axioms_a: ( a > a > a ) > $o ).
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_001tf__a,type,
rts_wi3777564303360811894ites_a: ( a > a > a ) > $o ).
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_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a2: a > set_a > set_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
accp_P8911511002227036208list_a: ( produc7709606177366032167list_a > produc7709606177366032167list_a > $o ) > produc7709606177366032167list_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
accp_P7377042638478740784list_a: ( produc9164743771328383783list_a > produc9164743771328383783list_a > $o ) > produc9164743771328383783list_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
accp_P3213725926765619766list_a: ( produc8685980395799941037list_a > produc8685980395799941037list_a > $o ) > produc8685980395799941037list_a > $o ).
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_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_Ba____,type,
ba: list_a ).
thf(sy_v_T,type,
t: list_a ).
thf(sy_v_Ta____,type,
ta: list_a ).
thf(sy_v__092_060NN_062,type,
nn: set_a ).
thf(sy_v_b____,type,
b: a ).
thf(sy_v_resid,type,
resid: a > a > a ).
% Relevant facts (1277)
thf(fact_0_Trgs_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [T: a] :
( X
!= ( cons_a @ T @ nil_a ) )
=> ~ ! [T: a,V: a,Va: list_a] :
( X
!= ( cons_a @ T @ ( cons_a @ V @ Va ) ) ) ) ) ).
% Trgs.cases
thf(fact_1_append1__eq__conv,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a,Y: list_a] :
( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) )
= ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_2_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_3_append_Oright__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ A @ nil_list_a )
= A ) ).
% append.right_neutral
thf(fact_4_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_5_append__Nil2,axiom,
! [Xs: list_list_a] :
( ( append_list_a @ Xs @ nil_list_a )
= Xs ) ).
% append_Nil2
thf(fact_6_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_7_append__self__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_a ) ) ).
% append_self_conv
thf(fact_8_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_9_self__append__conv,axiom,
! [Y: list_list_a,Ys: list_list_a] :
( ( Y
= ( append_list_a @ Y @ Ys ) )
= ( Ys = nil_list_a ) ) ).
% self_append_conv
thf(fact_10_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_11_append__self__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_a ) ) ).
% append_self_conv2
thf(fact_12_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_13_self__append__conv2,axiom,
! [Y: list_list_a,Xs: list_list_a] :
( ( Y
= ( append_list_a @ Xs @ Y ) )
= ( Xs = nil_list_a ) ) ).
% self_append_conv2
thf(fact_14_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_15_Nil__is__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( nil_list_a
= ( append_list_a @ Xs @ Ys ) )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% Nil_is_append_conv
thf(fact_16_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_17_append__is__Nil__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= nil_list_a )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% append_is_Nil_conv
thf(fact_18_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_19_rev__induct,axiom,
! [P: list_list_a > $o,Xs: list_list_a] :
( ( P @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_list_a @ Xs2 @ ( cons_list_a @ X2 @ nil_list_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_20_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X2: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_21_rev__exhaust,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ~ ! [Ys2: list_list_a,Y2: list_a] :
( Xs
!= ( append_list_a @ Ys2 @ ( cons_list_a @ Y2 @ nil_list_a ) ) ) ) ).
% rev_exhaust
thf(fact_22_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y2: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_23_Cons__eq__append__conv,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X @ Xs )
= ( append_list_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_a )
& ( ( cons_list_a @ X @ Xs )
= Zs ) )
| ? [Ys3: list_list_a] :
( ( ( cons_list_a @ X @ Ys3 )
= Ys )
& ( Xs
= ( append_list_a @ Ys3 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_24_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys3: list_a] :
( ( ( cons_a @ X @ Ys3 )
= Ys )
& ( Xs
= ( append_a @ Ys3 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_25_list_Oinject,axiom,
! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( ( cons_list_a @ X21 @ X22 )
= ( cons_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_26_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_27_same__append__eq,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_28_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_29_append__same__eq,axiom,
! [Ys: list_list_a,Xs: list_list_a,Zs: list_list_a] :
( ( ( append_list_a @ Ys @ Xs )
= ( append_list_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_30_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_31_append__assoc,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( append_list_a @ ( append_list_a @ Xs @ Ys ) @ Zs )
= ( append_list_a @ Xs @ ( append_list_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_32_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_33_append_Oassoc,axiom,
! [A: list_list_a,B: list_list_a,C: list_list_a] :
( ( append_list_a @ ( append_list_a @ A @ B ) @ C )
= ( append_list_a @ A @ ( append_list_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_34_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_35_not__Cons__self2,axiom,
! [X: list_a,Xs: list_list_a] :
( ( cons_list_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_36_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_37_append__eq__append__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,Ts: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Zs @ Ts ) )
= ( ? [Us: list_list_a] :
( ( ( Xs
= ( append_list_a @ Zs @ Us ) )
& ( ( append_list_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_list_a @ Xs @ Us )
= Zs )
& ( Ys
= ( append_list_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_38_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 ) )
= ( ? [Us: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us ) )
& ( ( append_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us )
= Zs )
& ( Ys
= ( append_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_39_append__eq__appendI,axiom,
! [Xs: list_list_a,Xs1: list_list_a,Zs: list_list_a,Ys: list_list_a,Us2: list_list_a] :
( ( ( append_list_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_a @ Xs1 @ Us2 ) )
=> ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_40_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us2: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us2 ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_41_list__nonempty__induct,axiom,
! [Xs: list_list_a,P: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X2: list_a] : ( P @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [X2: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_42_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_43_list__induct2_H,axiom,
! [P: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( P @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys2: list_list_a] : ( P @ nil_a @ ( cons_list_a @ Y2 @ Ys2 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_44_list__induct2_H,axiom,
! [P: list_list_a > list_a > $o,Xs: list_list_a,Ys: list_a] :
( ( P @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a] : ( P @ ( cons_list_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_list_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_45_list__induct2_H,axiom,
! [P: list_list_a > list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a] : ( P @ ( cons_list_a @ X2 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys2: list_list_a] : ( P @ nil_list_a @ ( cons_list_a @ Y2 @ Ys2 ) )
=> ( ! [X2: list_a,Xs2: list_list_a,Y2: list_a,Ys2: list_list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_46_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_47_neq__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
= ( ? [Y3: list_a,Ys4: list_list_a] :
( Xs
= ( cons_list_a @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_48_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y3: a,Ys4: list_a] :
( Xs
= ( cons_a @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_49_remdups__adj_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [X2: list_a] :
( X
!= ( cons_list_a @ X2 @ nil_list_a ) )
=> ~ ! [X2: list_a,Y2: list_a,Xs2: list_list_a] :
( X
!= ( cons_list_a @ X2 @ ( cons_list_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_50_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,Y2: a,Xs2: list_a] :
( X
!= ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_51_transpose_Ocases,axiom,
! [X: list_list_list_a] :
( ( X != nil_list_list_a )
=> ( ! [Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ nil_list_a @ Xss ) )
=> ~ ! [X2: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ ( cons_list_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_52_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X2: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_53_list_Oexhaust,axiom,
! [Y: list_list_a] :
( ( Y != nil_list_a )
=> ~ ! [X212: list_a,X222: list_list_a] :
( Y
!= ( cons_list_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_54_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_55_list_OdiscI,axiom,
! [List: list_list_a,X21: list_a,X22: list_list_a] :
( ( List
= ( cons_list_a @ X21 @ X22 ) )
=> ( List != nil_list_a ) ) ).
% list.discI
thf(fact_56_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_57_list_Odistinct_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( nil_list_a
!= ( cons_list_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_58_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_59_Cons__eq__appendI,axiom,
! [X: list_a,Xs1: list_list_a,Ys: list_list_a,Xs: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_a @ Xs1 @ Zs ) )
=> ( ( cons_list_a @ X @ Xs )
= ( append_list_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_60_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_61_append__Cons,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( append_list_a @ ( cons_list_a @ X @ Xs ) @ Ys )
= ( cons_list_a @ X @ ( append_list_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_62_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_63_mem__Collect__eq,axiom,
! [A: list_list_a,P: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_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_66_Collect__mem__eq,axiom,
! [A2: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__cong,axiom,
! [P: list_list_a > $o,Q: list_list_a > $o] :
( ! [X2: list_list_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_list_list_a @ P )
= ( collect_list_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_70_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_71_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X2: list_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_72_eq__Nil__appendI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_a @ nil_list_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_73_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_74_append_Oleft__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ nil_list_a @ A )
= A ) ).
% append.left_neutral
thf(fact_75_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_76_append__Nil,axiom,
! [Ys: list_list_a] :
( ( append_list_a @ nil_list_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_77_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_78_rev__nonempty__induct,axiom,
! [Xs: list_list_a,P: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X2: list_a] : ( P @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [X2: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_list_a @ Xs2 @ ( cons_list_a @ X2 @ nil_list_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_79_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_80_append__eq__Cons__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,X: list_a,Xs: list_list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( cons_list_a @ X @ Xs ) )
= ( ( ( Ys = nil_list_a )
& ( Zs
= ( cons_list_a @ X @ Xs ) ) )
| ? [Ys3: list_list_a] :
( ( Ys
= ( cons_list_a @ X @ Ys3 ) )
& ( ( append_list_a @ Ys3 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_81_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys3: list_a] :
( ( Ys
= ( cons_a @ X @ Ys3 ) )
& ( ( append_a @ Ys3 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_82_bind__simps_I2_J,axiom,
! [X: a,Xs: list_a,F: a > list_list_a] :
( ( bind_a_list_a @ ( cons_a @ X @ Xs ) @ F )
= ( append_list_a @ ( F @ X ) @ ( bind_a_list_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_83_bind__simps_I2_J,axiom,
! [X: list_a,Xs: list_list_a,F: list_a > list_a] :
( ( bind_list_a_a @ ( cons_list_a @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_list_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_84_bind__simps_I2_J,axiom,
! [X: list_a,Xs: list_list_a,F: list_a > list_list_a] :
( ( bind_list_a_list_a @ ( cons_list_a @ X @ Xs ) @ F )
= ( append_list_a @ ( F @ X ) @ ( bind_list_a_list_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_85_bind__simps_I2_J,axiom,
! [X: a,Xs: list_a,F: a > list_a] :
( ( bind_a_a @ ( cons_a @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_86_Resid1x_Ocases,axiom,
! [X: produc8685980395799941037list_a] :
( ! [T: a] :
( X
!= ( produc6670463072477821725list_a @ T @ nil_a ) )
=> ( ! [T: a,U: a] :
( X
!= ( produc6670463072477821725list_a @ T @ ( cons_a @ U @ nil_a ) ) )
=> ~ ! [T: a,U: a,V: a,Va: list_a] :
( X
!= ( produc6670463072477821725list_a @ T @ ( cons_a @ U @ ( cons_a @ V @ Va ) ) ) ) ) ) ).
% Resid1x.cases
thf(fact_87_Residx1_Ocases,axiom,
! [X: produc2579390645249093025st_a_a] :
( ! [U: a] :
( X
!= ( produc4781227316648555537st_a_a @ nil_a @ U ) )
=> ( ! [T: a,U: a] :
( X
!= ( produc4781227316648555537st_a_a @ ( cons_a @ T @ nil_a ) @ U ) )
=> ~ ! [T: a,V: a,Va: list_a,U: a] :
( X
!= ( produc4781227316648555537st_a_a @ ( cons_a @ T @ ( cons_a @ V @ Va ) ) @ U ) ) ) ) ).
% Residx1.cases
thf(fact_88_maps__simps_I1_J,axiom,
! [F: a > list_list_a,X: a,Xs: list_a] :
( ( maps_a_list_a @ F @ ( cons_a @ X @ Xs ) )
= ( append_list_a @ ( F @ X ) @ ( maps_a_list_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_89_maps__simps_I1_J,axiom,
! [F: list_a > list_a,X: list_a,Xs: list_list_a] :
( ( maps_list_a_a @ F @ ( cons_list_a @ X @ Xs ) )
= ( append_a @ ( F @ X ) @ ( maps_list_a_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_90_maps__simps_I1_J,axiom,
! [F: list_a > list_list_a,X: list_a,Xs: list_list_a] :
( ( maps_list_a_list_a @ F @ ( cons_list_a @ X @ Xs ) )
= ( append_list_a @ ( F @ X ) @ ( maps_list_a_list_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_91_maps__simps_I1_J,axiom,
! [F: a > list_a,X: a,Xs: list_a] :
( ( maps_a_a @ F @ ( cons_a @ X @ Xs ) )
= ( append_a @ ( F @ X ) @ ( maps_a_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_92_insert__Nil,axiom,
! [X: list_a] :
( ( insert_list_a @ X @ nil_list_a )
= ( cons_list_a @ X @ nil_list_a ) ) ).
% insert_Nil
thf(fact_93_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_94_Ide,axiom,
paths_in_Ide_a @ resid @ ( cons_a @ b @ ba ) ).
% Ide
thf(fact_95_butlast__snoc,axiom,
! [Xs: list_list_a,X: list_a] :
( ( butlast_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_96_butlast__snoc,axiom,
! [Xs: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_97_list__ex1__simps_I1_J,axiom,
! [P: list_a > $o] :
~ ( list_ex1_list_a @ P @ nil_list_a ) ).
% list_ex1_simps(1)
thf(fact_98_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_99_rotate1_Osimps_I2_J,axiom,
! [X: list_a,Xs: list_list_a] :
( ( rotate1_list_a @ ( cons_list_a @ X @ Xs ) )
= ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) ) ).
% rotate1.simps(2)
thf(fact_100_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_101_last__snoc,axiom,
! [Xs: list_list_a,X: list_a] :
( ( last_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
= X ) ).
% last_snoc
thf(fact_102_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_103_map__tailrec__rev_Oelims,axiom,
! [X: a > list_a,Xa: list_a,Xb: list_list_a,Y: list_list_a] :
( ( ( map_ta1754794639992346235list_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_a )
=> ( Y != Xb ) )
=> ~ ! [A3: a,As: list_a] :
( ( Xa
= ( cons_a @ A3 @ As ) )
=> ( Y
!= ( map_ta1754794639992346235list_a @ X @ As @ ( cons_list_a @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_104_map__tailrec__rev_Oelims,axiom,
! [X: list_a > a,Xa: list_list_a,Xb: list_a,Y: list_a] :
( ( ( map_ta9088930921017855855st_a_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_list_a )
=> ( Y != Xb ) )
=> ~ ! [A3: list_a,As: list_list_a] :
( ( Xa
= ( cons_list_a @ A3 @ As ) )
=> ( Y
!= ( map_ta9088930921017855855st_a_a @ X @ As @ ( cons_a @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_105_map__tailrec__rev_Oelims,axiom,
! [X: list_a > list_a,Xa: list_list_a,Xb: list_list_a,Y: list_list_a] :
( ( ( map_ta5969768091587575669list_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_list_a )
=> ( Y != Xb ) )
=> ~ ! [A3: list_a,As: list_list_a] :
( ( Xa
= ( cons_list_a @ A3 @ As ) )
=> ( Y
!= ( map_ta5969768091587575669list_a @ X @ As @ ( cons_list_a @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_106_map__tailrec__rev_Oelims,axiom,
! [X: a > a,Xa: list_a,Xb: list_a,Y: list_a] :
( ( ( map_tailrec_rev_a_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_a )
=> ( Y != Xb ) )
=> ~ ! [A3: a,As: list_a] :
( ( Xa
= ( cons_a @ A3 @ As ) )
=> ( Y
!= ( map_tailrec_rev_a_a @ X @ As @ ( cons_a @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_107_R_Oex__un__null,axiom,
? [X2: a] :
( ! [T2: a] :
( ( ( resid @ X2 @ T2 )
= X2 )
& ( ( resid @ T2 @ X2 )
= X2 ) )
& ! [Y4: a] :
( ! [T: a] :
( ( ( resid @ Y4 @ T )
= Y4 )
& ( ( resid @ T @ Y4 )
= Y4 ) )
=> ( Y4 = X2 ) ) ) ).
% R.ex_un_null
thf(fact_108_R_Ocube,axiom,
! [V2: a,T3: a,U2: a] :
( ( resid @ ( resid @ V2 @ T3 ) @ ( resid @ U2 @ T3 ) )
= ( resid @ ( resid @ V2 @ U2 ) @ ( resid @ T3 @ U2 ) ) ) ).
% R.cube
thf(fact_109_Ide_Osimps_I1_J,axiom,
~ ( paths_in_Ide_a @ resid @ nil_a ) ).
% Ide.simps(1)
thf(fact_110_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( ( rotate1_list_a @ Xs )
= nil_list_a )
= ( Xs = nil_list_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_111_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_112_bind__simps_I1_J,axiom,
! [F: a > list_list_a] :
( ( bind_a_list_a @ nil_a @ F )
= nil_list_a ) ).
% bind_simps(1)
thf(fact_113_bind__simps_I1_J,axiom,
! [F: list_a > list_a] :
( ( bind_list_a_a @ nil_list_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_114_bind__simps_I1_J,axiom,
! [F: list_a > list_list_a] :
( ( bind_list_a_list_a @ nil_list_a @ F )
= nil_list_a ) ).
% bind_simps(1)
thf(fact_115_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_116_last__appendL,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( Ys = nil_list_a )
=> ( ( last_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( last_list_a @ Xs ) ) ) ).
% last_appendL
thf(fact_117_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_118_last__appendR,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( Ys != nil_list_a )
=> ( ( last_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( last_list_a @ Ys ) ) ) ).
% last_appendR
thf(fact_119_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_120_append__butlast__last__id,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( append_list_a @ ( butlast_list_a @ Xs ) @ ( cons_list_a @ ( last_list_a @ Xs ) @ nil_list_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_121_append__butlast__last__id,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_122_paths__in__rts_OIde_Ocong,axiom,
paths_in_Ide_list_a = paths_in_Ide_list_a ).
% paths_in_rts.Ide.cong
thf(fact_123_paths__in__rts_OIde_Ocong,axiom,
paths_in_Ide_a = paths_in_Ide_a ).
% paths_in_rts.Ide.cong
thf(fact_124_butlast_Osimps_I1_J,axiom,
( ( butlast_list_a @ nil_list_a )
= nil_list_a ) ).
% butlast.simps(1)
thf(fact_125_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_126_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_a @ nil_list_a )
= nil_list_a ) ).
% rotate1.simps(1)
thf(fact_127_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_128_snoc__eq__iff__butlast,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) )
= Ys )
= ( ( Ys != nil_list_a )
& ( ( butlast_list_a @ Ys )
= Xs )
& ( ( last_list_a @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_129_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs )
& ( ( last_a @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_130_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: list_a > a,A: list_a,As2: list_list_a,Bs: list_a] :
( ( map_ta9088930921017855855st_a_a @ F @ ( cons_list_a @ A @ As2 ) @ Bs )
= ( map_ta9088930921017855855st_a_a @ F @ As2 @ ( cons_a @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_131_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: list_a > list_a,A: list_a,As2: list_list_a,Bs: list_list_a] :
( ( map_ta5969768091587575669list_a @ F @ ( cons_list_a @ A @ As2 ) @ Bs )
= ( map_ta5969768091587575669list_a @ F @ As2 @ ( cons_list_a @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_132_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: a > a,A: a,As2: list_a,Bs: list_a] :
( ( map_tailrec_rev_a_a @ F @ ( cons_a @ A @ As2 ) @ Bs )
= ( map_tailrec_rev_a_a @ F @ As2 @ ( cons_a @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_133_last_Osimps,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_134_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_135_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_136_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_137_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs3: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Xs3 @ Ss ) )
& ( Ys
= ( append_a @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_a )
| ( Ys5 = nil_a )
| ( ( last_a @ Xs3 )
!= ( last_a @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_138_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_139_butlast__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_140_maps__simps_I2_J,axiom,
! [F: a > list_a] :
( ( maps_a_a @ F @ nil_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_141_R_Opartial__magma__axioms,axiom,
partial_magma_a @ resid ).
% R.partial_magma_axioms
thf(fact_142_Resid__cons_I1_J,axiom,
! [U3: list_a,T3: a,T4: list_a] :
( ( U3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ U3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ U3 )
= ( append_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 ) @ ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U3 @ ( cons_a @ T3 @ nil_a ) ) ) ) ) ) ) ).
% Resid_cons(1)
thf(fact_143_Resid__rec_I4_J,axiom,
! [T4: list_a,U3: list_a,T3: a,U2: a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ ( cons_a @ U2 @ U3 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ ( cons_a @ U2 @ U3 ) )
= ( append_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T3 @ U2 ) @ nil_a ) @ U3 ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ ( resid @ U2 @ T3 ) @ nil_a ) ) @ ( paths_in_Resid_a @ resid @ U3 @ ( cons_a @ ( resid @ T3 @ U2 ) @ nil_a ) ) ) ) ) ) ) ) ).
% Resid_rec(4)
thf(fact_144_seq,axiom,
seq_list_a @ ( paths_in_Resid_a @ resid ) @ ta @ ( cons_a @ b @ ba ) ).
% seq
thf(fact_145_Ide_Osimps_I2_J,axiom,
! [T3: a] :
( ( paths_in_Ide_a @ resid @ ( cons_a @ T3 @ nil_a ) )
= ( ide_a @ resid @ T3 ) ) ).
% Ide.simps(2)
thf(fact_146_Ide__imp__Ide__tl,axiom,
! [T4: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
=> ( ( ( tl_a @ T4 )
!= nil_a )
=> ( paths_in_Ide_a @ resid @ ( tl_a @ T4 ) ) ) ) ).
% Ide_imp_Ide_tl
thf(fact_147_Resid1x_Osimps_I2_J,axiom,
! [T3: a,U2: a] :
( ( paths_in_Resid1x_a @ resid @ T3 @ ( cons_a @ U2 @ nil_a ) )
= ( resid @ T3 @ U2 ) ) ).
% Resid1x.simps(2)
thf(fact_148_Resid__Ide_I1_J,axiom,
! [A2: list_a,T4: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( ( paths_in_Resid_a @ resid @ A2 @ T4 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ A2 )
= T4 ) ) ) ).
% Resid_Ide(1)
thf(fact_149_Resid__Arr__Ide__ind,axiom,
! [A2: list_a,T4: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ A2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ A2 )
= T4 ) ) ) ).
% Resid_Arr_Ide_ind
thf(fact_150_Resid__Ide__Arr__ind,axiom,
! [A2: list_a,T4: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( ( paths_in_Resid_a @ resid @ A2 @ T4 )
!= nil_a )
=> ( paths_in_Ide_a @ resid @ ( paths_in_Resid_a @ resid @ A2 @ T4 ) ) ) ) ).
% Resid_Ide_Arr_ind
thf(fact_151_cube,axiom,
! [V2: list_a,T3: list_a,U2: list_a] :
( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V2 @ T3 ) @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) ) ).
% cube
thf(fact_152_ex__un__null,axiom,
? [X2: list_a] :
( ! [T2: list_a] :
( ( ( paths_in_Resid_a @ resid @ X2 @ T2 )
= X2 )
& ( ( paths_in_Resid_a @ resid @ T2 @ X2 )
= X2 ) )
& ! [Y4: list_a] :
( ! [T: list_a] :
( ( ( paths_in_Resid_a @ resid @ Y4 @ T )
= Y4 )
& ( ( paths_in_Resid_a @ resid @ T @ Y4 )
= Y4 ) )
=> ( Y4 = X2 ) ) ) ).
% ex_un_null
thf(fact_153_R_Ocong__symmetric,axiom,
! [T3: a,U2: a] :
( ( ( ide_a @ resid @ ( resid @ T3 @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ T3 ) ) )
=> ( ( ide_a @ resid @ ( resid @ U2 @ T3 ) )
& ( ide_a @ resid @ ( resid @ T3 @ U2 ) ) ) ) ).
% R.cong_symmetric
thf(fact_154_R_Ocong__transitive,axiom,
! [T3: a,U2: a,V2: a] :
( ( ( ide_a @ resid @ ( resid @ T3 @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ T3 ) ) )
=> ( ( ( ide_a @ resid @ ( resid @ U2 @ V2 ) )
& ( ide_a @ resid @ ( resid @ V2 @ U2 ) ) )
=> ( ( ide_a @ resid @ ( resid @ T3 @ V2 ) )
& ( ide_a @ resid @ ( resid @ V2 @ T3 ) ) ) ) ) ).
% R.cong_transitive
thf(fact_155_R_Oide__backward__stable,axiom,
! [A: a,T3: a] :
( ( ide_a @ resid @ A )
=> ( ( ide_a @ resid @ ( resid @ T3 @ A ) )
=> ( ide_a @ resid @ T3 ) ) ) ).
% R.ide_backward_stable
thf(fact_156_R_Oprfx__transitive,axiom,
! [T3: a,U2: a,V2: a] :
( ( ide_a @ resid @ ( resid @ T3 @ U2 ) )
=> ( ( ide_a @ resid @ ( resid @ U2 @ V2 ) )
=> ( ide_a @ resid @ ( resid @ T3 @ V2 ) ) ) ) ).
% R.prfx_transitive
thf(fact_157_Con__sym,axiom,
! [T4: list_a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ U3 @ T4 )
!= nil_a ) ) ).
% Con_sym
thf(fact_158_Resid_Osimps_I1_J,axiom,
! [Uu: list_a] :
( ( paths_in_Resid_a @ resid @ nil_a @ Uu )
= nil_a ) ).
% Resid.simps(1)
thf(fact_159_Cube_I1_J,axiom,
! [T4: list_a,U3: list_a,V3: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) @ ( paths_in_Resid_a @ resid @ V3 @ U3 ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ V3 ) @ ( paths_in_Resid_a @ resid @ U3 @ V3 ) )
!= nil_a ) ) ).
% Cube(1)
thf(fact_160_Cube_I2_J,axiom,
! [T4: list_a,U3: list_a,V3: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) @ ( paths_in_Resid_a @ resid @ V3 @ U3 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) @ ( paths_in_Resid_a @ resid @ V3 @ U3 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ V3 ) @ ( paths_in_Resid_a @ resid @ U3 @ V3 ) ) ) ) ).
% Cube(2)
thf(fact_161_Ide__imp__Ide__last,axiom,
! [T4: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
=> ( ide_a @ resid @ ( last_a @ T4 ) ) ) ).
% Ide_imp_Ide_last
thf(fact_162_Resid1x_Osimps_I3_J,axiom,
! [T3: a,U2: a,V2: a,Va2: list_a] :
( ( paths_in_Resid1x_a @ resid @ T3 @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) )
= ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) ) ) ).
% Resid1x.simps(3)
thf(fact_163_Con__initial__right,axiom,
! [T4: list_a,U2: a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ U3 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) )
!= nil_a ) ) ).
% Con_initial_right
thf(fact_164_Con__initial__left,axiom,
! [T3: a,T4: list_a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ U3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 )
!= nil_a ) ) ).
% Con_initial_left
thf(fact_165_Resid__rec_I2_J,axiom,
! [T4: list_a,T3: a,U2: a] :
( ( T4 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ ( cons_a @ U2 @ nil_a ) )
= ( cons_a @ ( resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ ( resid @ U2 @ T3 ) @ nil_a ) ) ) ) ) ) ).
% Resid_rec(2)
thf(fact_166_Resid__rec_I3_J,axiom,
! [U3: list_a,T3: a,U2: a] :
( ( U3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ U3 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ U3 ) )
= ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T3 @ U2 ) @ nil_a ) @ U3 ) ) ) ) ).
% Resid_rec(3)
thf(fact_167_Resid__cons_I2_J,axiom,
! [U3: list_a,T4: list_a,U2: a] :
( ( U3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ U3 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ U3 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) ) @ U3 ) ) ) ) ).
% Resid_cons(2)
thf(fact_168_Resid_Osimps_I2_J,axiom,
! [V2: a,Va2: list_a] :
( ( paths_in_Resid_a @ resid @ ( cons_a @ V2 @ Va2 ) @ nil_a )
= nil_a ) ).
% Resid.simps(2)
thf(fact_169_Con__cons_I1_J,axiom,
! [T4: list_a,U3: list_a,T3: a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ U3 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U3 @ ( cons_a @ T3 @ nil_a ) ) )
!= nil_a ) ) ) ) ) ).
% Con_cons(1)
thf(fact_170_Con__cons_I2_J,axiom,
! [T4: list_a,U3: list_a,U2: a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ U3 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) ) @ U3 )
!= nil_a ) ) ) ) ) ).
% Con_cons(2)
thf(fact_171_Resid__append__ind,axiom,
! [T4: list_a,U3: list_a,V3: list_a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( V3 != nil_a )
=> ( ( ( ( paths_in_Resid_a @ resid @ ( append_a @ V3 @ T4 ) @ U3 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ V3 @ U3 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U3 @ V3 ) )
!= nil_a ) ) )
& ( ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ V3 @ U3 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T4 @ V3 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ V3 ) @ U3 )
!= nil_a ) ) )
& ( ( ( paths_in_Resid_a @ resid @ ( append_a @ V3 @ T4 ) @ U3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( append_a @ V3 @ T4 ) @ U3 )
= ( append_a @ ( paths_in_Resid_a @ resid @ V3 @ U3 ) @ ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U3 @ V3 ) ) ) ) )
& ( ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ V3 @ U3 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ V3 @ U3 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ V3 ) @ U3 ) ) ) ) ) ) ) ).
% Resid_append_ind
thf(fact_172_Con__append_I1_J,axiom,
! [T4: list_a,U3: list_a,V3: list_a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( V3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ V3 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T4 @ V3 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ U3 @ ( paths_in_Resid_a @ resid @ V3 @ T4 ) )
!= nil_a ) ) ) ) ) ) ).
% Con_append(1)
thf(fact_173_Con__append_I2_J,axiom,
! [T4: list_a,U3: list_a,V3: list_a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( V3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ U3 @ V3 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) @ V3 )
!= nil_a ) ) ) ) ) ) ).
% Con_append(2)
thf(fact_174_Resid__cons_H,axiom,
! [T4: list_a,T3: a,U3: list_a] :
( ( T4 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ U3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ U3 )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ T3 @ U3 ) @ ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U3 @ ( cons_a @ T3 @ nil_a ) ) ) ) ) ) ) ).
% Resid_cons'
thf(fact_175_Resid1x__as__Resid,axiom,
! [T3: a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ T3 @ U3 ) @ nil_a ) ) ) ).
% Resid1x_as_Resid
thf(fact_176_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_177_Resid__rec_I1_J,axiom,
! [T3: a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
= ( cons_a @ ( resid @ T3 @ U2 ) @ nil_a ) ) ) ).
% Resid_rec(1)
thf(fact_178_Con__consI_I1_J,axiom,
! [T4: list_a,U3: list_a,T3: a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U3 @ ( cons_a @ T3 @ nil_a ) ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ U3 )
!= nil_a ) ) ) ) ) ).
% Con_consI(1)
thf(fact_179_Con__consI_I2_J,axiom,
! [T4: list_a,U3: list_a,U2: a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) ) @ U3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ U3 ) )
!= nil_a ) ) ) ) ) ).
% Con_consI(2)
thf(fact_180_Resid__append2,axiom,
! [T4: list_a,U3: list_a,V3: list_a,W: list_a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( V3 != nil_a )
=> ( ( W != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ ( append_a @ V3 @ W ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ ( append_a @ V3 @ W ) )
= ( append_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ V3 ) @ W ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ U3 @ ( paths_in_Resid_a @ resid @ V3 @ T4 ) ) @ ( paths_in_Resid_a @ resid @ W @ ( paths_in_Resid_a @ resid @ T4 @ V3 ) ) ) ) ) ) ) ) ) ) ).
% Resid_append2
thf(fact_181_Resid__append_I1_J,axiom,
! [T4: list_a,U3: list_a,V3: list_a] :
( ( T4 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ V3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ V3 )
= ( append_a @ ( paths_in_Resid_a @ resid @ T4 @ V3 ) @ ( paths_in_Resid_a @ resid @ U3 @ ( paths_in_Resid_a @ resid @ V3 @ T4 ) ) ) ) ) ) ).
% Resid_append(1)
thf(fact_182_Resid__append_I2_J,axiom,
! [U3: list_a,V3: list_a,T4: list_a] :
( ( U3 != nil_a )
=> ( ( V3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ U3 @ V3 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ U3 @ V3 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) @ V3 ) ) ) ) ) ).
% Resid_append(2)
thf(fact_183_Con__appendI_I1_J,axiom,
! [T4: list_a,V3: list_a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ V3 )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ U3 @ ( paths_in_Resid_a @ resid @ V3 @ T4 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ V3 )
!= nil_a ) ) ) ).
% Con_appendI(1)
thf(fact_184_Con__appendI_I2_J,axiom,
! [T4: list_a,U3: list_a,V3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) @ V3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ U3 @ V3 ) )
!= nil_a ) ) ) ).
% Con_appendI(2)
thf(fact_185_Resid__single__ide_I2_J,axiom,
! [A: a,T4: list_a] :
( ( ide_a @ resid @ A )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ A @ nil_a ) @ T4 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ A @ nil_a ) )
= T4 ) ) ) ).
% Resid_single_ide(2)
thf(fact_186_ind,axiom,
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ ta @ ba )
=> ( ( paths_in_Ide_a @ resid @ ba )
=> ( paths_4972237869229803137Cong_a @ resid @ ( append_a @ ta @ ba ) @ ta ) ) ) ).
% ind
thf(fact_187_partial__magma_Ointro,axiom,
! [OP: a > a > a] :
( ? [X4: a] :
( ! [T: a] :
( ( ( OP @ X4 @ T )
= X4 )
& ( ( OP @ T @ X4 )
= X4 ) )
& ! [Y2: a] :
( ! [T2: a] :
( ( ( OP @ Y2 @ T2 )
= Y2 )
& ( ( OP @ T2 @ Y2 )
= Y2 ) )
=> ( Y2 = X4 ) ) )
=> ( partial_magma_a @ OP ) ) ).
% partial_magma.intro
thf(fact_188_partial__magma_Ointro,axiom,
! [OP: list_a > list_a > list_a] :
( ? [X4: list_a] :
( ! [T: list_a] :
( ( ( OP @ X4 @ T )
= X4 )
& ( ( OP @ T @ X4 )
= X4 ) )
& ! [Y2: list_a] :
( ! [T2: list_a] :
( ( ( OP @ Y2 @ T2 )
= Y2 )
& ( ( OP @ T2 @ Y2 )
= Y2 ) )
=> ( Y2 = X4 ) ) )
=> ( partial_magma_list_a @ OP ) ) ).
% partial_magma.intro
thf(fact_189_partial__magma_Oex__un__null,axiom,
! [OP: a > a > a] :
( ( partial_magma_a @ OP )
=> ? [X2: a] :
( ! [T2: a] :
( ( ( OP @ X2 @ T2 )
= X2 )
& ( ( OP @ T2 @ X2 )
= X2 ) )
& ! [Y4: a] :
( ! [T: a] :
( ( ( OP @ Y4 @ T )
= Y4 )
& ( ( OP @ T @ Y4 )
= Y4 ) )
=> ( Y4 = X2 ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_190_partial__magma_Oex__un__null,axiom,
! [OP: list_a > list_a > list_a] :
( ( partial_magma_list_a @ OP )
=> ? [X2: list_a] :
( ! [T2: list_a] :
( ( ( OP @ X2 @ T2 )
= X2 )
& ( ( OP @ T2 @ X2 )
= X2 ) )
& ! [Y4: list_a] :
( ! [T: list_a] :
( ( ( OP @ Y4 @ T )
= Y4 )
& ( ( OP @ T @ Y4 )
= Y4 ) )
=> ( Y4 = X2 ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_191_partial__magma__def,axiom,
( partial_magma_a
= ( ^ [OP2: a > a > a] :
? [X3: a] :
( ! [T5: a] :
( ( ( OP2 @ X3 @ T5 )
= X3 )
& ( ( OP2 @ T5 @ X3 )
= X3 ) )
& ! [Y3: a] :
( ! [T5: a] :
( ( ( OP2 @ Y3 @ T5 )
= Y3 )
& ( ( OP2 @ T5 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_192_partial__magma__def,axiom,
( partial_magma_list_a
= ( ^ [OP2: list_a > list_a > list_a] :
? [X3: list_a] :
( ! [T5: list_a] :
( ( ( OP2 @ X3 @ T5 )
= X3 )
& ( ( OP2 @ T5 @ X3 )
= X3 ) )
& ! [Y3: list_a] :
( ! [T5: list_a] :
( ( ( OP2 @ Y3 @ T5 )
= Y3 )
& ( ( OP2 @ T5 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_193_rts_Oseq_Ocong,axiom,
seq_list_a = seq_list_a ).
% rts.seq.cong
thf(fact_194_rts_Oseq_Ocong,axiom,
seq_a = seq_a ).
% rts.seq.cong
thf(fact_195_residuation_Oide_Ocong,axiom,
ide_a = ide_a ).
% residuation.ide.cong
thf(fact_196_residuation_Oide_Ocong,axiom,
ide_list_a = ide_list_a ).
% residuation.ide.cong
thf(fact_197_paths__in__rts_OResid_Ocong,axiom,
paths_in_Resid_a = paths_in_Resid_a ).
% paths_in_rts.Resid.cong
thf(fact_198_paths__in__rts_OResid1x_Ocong,axiom,
paths_in_Resid1x_a = paths_in_Resid1x_a ).
% paths_in_rts.Resid1x.cong
thf(fact_199_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_200_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_201_butlast__tl,axiom,
! [Xs: list_a] :
( ( butlast_a @ ( tl_a @ Xs ) )
= ( tl_a @ ( butlast_a @ Xs ) ) ) ).
% butlast_tl
thf(fact_202_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_203_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_204_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_205_last__tl,axiom,
! [Xs: list_a] :
( ( ( Xs = nil_a )
| ( ( tl_a @ Xs )
!= nil_a ) )
=> ( ( last_a @ ( tl_a @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_tl
thf(fact_206_partial__magma__axioms,axiom,
partial_magma_list_a @ ( paths_in_Resid_a @ resid ) ).
% partial_magma_axioms
thf(fact_207_rts__with__composites__axioms,axiom,
rts_wi1113109597627081724list_a @ ( paths_in_Resid_a @ resid ) ).
% rts_with_composites_axioms
thf(fact_208_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_209_PCong_Ointros_I5_J,axiom,
! [T4: list_a,B: a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ ( cons_a @ B @ nil_a ) )
=> ( ( ide_a @ resid @ B )
=> ( paths_4972237869229803137Cong_a @ resid @ ( append_a @ T4 @ ( cons_a @ B @ nil_a ) ) @ T4 ) ) ) ).
% PCong.intros(5)
thf(fact_210_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_211_composable__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 ) ) ).
% composable_iff_seq
thf(fact_212_composable__imp__seq,axiom,
! [T3: list_a,U2: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ).
% composable_imp_seq
thf(fact_213_has__composites,axiom,
! [T3: list_a,U2: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ).
% has_composites
thf(fact_214_prfx__decomp,axiom,
! [T3: a,U2: a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ ( resid @ U2 @ T3 ) @ 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 @ T3 @ nil_a ) @ ( cons_a @ ( resid @ U2 @ T3 ) @ nil_a ) ) ) ) ) ) ).
% prfx_decomp
thf(fact_215_arr__append__imp__seq,axiom,
! [T4: list_a,U3: list_a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( append_a @ T4 @ U3 ) )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 ) ) ) ) ).
% arr_append_imp_seq
thf(fact_216_R_Ocong__respects__seq,axiom,
! [T3: a,U2: a,T6: a,U4: a] :
( ( seq_a @ resid @ T3 @ U2 )
=> ( ( ( ide_a @ resid @ ( resid @ T3 @ T6 ) )
& ( ide_a @ resid @ ( resid @ T6 @ T3 ) ) )
=> ( ( ( ide_a @ resid @ ( resid @ U2 @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U2 ) ) )
=> ( seq_a @ resid @ T6 @ U4 ) ) ) ) ).
% R.cong_respects_seq
thf(fact_217_PCong_Ointros_I2_J,axiom,
! [T4: list_a,U3: list_a] :
( ( paths_4972237869229803137Cong_a @ resid @ T4 @ U3 )
=> ( paths_4972237869229803137Cong_a @ resid @ U3 @ T4 ) ) ).
% PCong.intros(2)
thf(fact_218_PCong_Ointros_I3_J,axiom,
! [T4: list_a,U3: list_a,V3: list_a] :
( ( paths_4972237869229803137Cong_a @ resid @ T4 @ U3 )
=> ( ( paths_4972237869229803137Cong_a @ resid @ U3 @ V3 )
=> ( paths_4972237869229803137Cong_a @ resid @ T4 @ V3 ) ) ) ).
% PCong.intros(3)
thf(fact_219_prfx__transitive,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ V2 ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ V2 ) ) ) ) ).
% prfx_transitive
thf(fact_220_ide__backward__stable,axiom,
! [A: list_a,T3: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ A ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ) ).
% ide_backward_stable
thf(fact_221_cong__transitive,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ V2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V2 @ U2 ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ V2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V2 @ T3 ) ) ) ) ) ).
% cong_transitive
thf(fact_222_cong__symmetric,axiom,
! [T3: list_a,U2: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) ) ) ).
% cong_symmetric
thf(fact_223_prfx__reflexive,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ T3 ) ) ) ).
% prfx_reflexive
thf(fact_224_ide__implies__arr,axiom,
! [A: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ A ) ) ).
% ide_implies_arr
thf(fact_225_cong__reflexive,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ T3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ T3 ) ) ) ) ).
% cong_reflexive
thf(fact_226_ide__char,axiom,
! [T4: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T4 )
= ( paths_in_Ide_a @ resid @ T4 ) ) ).
% ide_char
thf(fact_227_cong__respects__seq_092_060_094sub_062P,axiom,
! [T4: list_a,U3: list_a,T7: list_a,U5: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T4 @ T7 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T7 @ T4 ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U3 @ U5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U5 @ U3 ) ) )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T7 @ U5 ) ) ) ) ).
% cong_respects_seq\<^sub>P
thf(fact_228_cong__respects__seq,axiom,
! [T3: list_a,U2: list_a,T6: list_a,U4: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) ) )
=> ( ( ( 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 ) @ T6 @ U4 ) ) ) ) ).
% cong_respects_seq
thf(fact_229_composableD_I1_J,axiom,
! [T3: list_a,U2: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ).
% composableD(1)
thf(fact_230_composableD_I2_J,axiom,
! [T3: list_a,U2: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ).
% composableD(2)
thf(fact_231_PCong_Ointros_I4_J,axiom,
! [T4: list_a,U3: list_a,T7: list_a,U5: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
=> ( ( paths_4972237869229803137Cong_a @ resid @ T4 @ T7 )
=> ( ( paths_4972237869229803137Cong_a @ resid @ U3 @ U5 )
=> ( paths_4972237869229803137Cong_a @ resid @ ( append_a @ T4 @ U3 ) @ ( append_a @ T7 @ U5 ) ) ) ) ) ).
% PCong.intros(4)
thf(fact_232_cong__append__ideI_I4_J,axiom,
! [T4: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ U3 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ T4 @ U3 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ T4 ) ) ) ) ) ).
% cong_append_ideI(4)
thf(fact_233_cong__append__ideI_I3_J,axiom,
! [T4: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T4 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U3 @ ( append_a @ T4 @ U3 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ U3 ) ) ) ) ) ).
% cong_append_ideI(3)
thf(fact_234_cong__append__ideI_I2_J,axiom,
! [T4: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ U3 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ T4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ T4 @ U3 ) ) ) ) ) ) ).
% cong_append_ideI(2)
thf(fact_235_cong__append__ideI_I1_J,axiom,
! [T4: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T4 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ U3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U3 @ ( append_a @ T4 @ U3 ) ) ) ) ) ) ).
% cong_append_ideI(1)
thf(fact_236_cong__append,axiom,
! [T4: list_a,U3: list_a,T7: list_a,U5: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T4 @ T7 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T7 @ T4 ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U3 @ U5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U5 @ U3 ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ ( append_a @ T7 @ U5 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( append_a @ T7 @ U5 ) @ ( append_a @ T4 @ U3 ) ) ) ) ) ) ) ).
% cong_append
thf(fact_237_arr__append,axiom,
! [T4: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( append_a @ T4 @ U3 ) ) ) ).
% arr_append
thf(fact_238_cong__cons__ideI_I2_J,axiom,
! [T3: a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T3 @ nil_a ) @ U3 )
=> ( ( ide_a @ resid @ T3 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U3 @ ( cons_a @ T3 @ U3 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ U3 ) @ U3 ) ) ) ) ) ).
% cong_cons_ideI(2)
thf(fact_239_cong__cons__ideI_I1_J,axiom,
! [T3: a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T3 @ nil_a ) @ U3 )
=> ( ( ide_a @ resid @ T3 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ U3 ) @ U3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U3 @ ( cons_a @ T3 @ U3 ) ) ) ) ) ) ).
% cong_cons_ideI(1)
thf(fact_240_cong__cons,axiom,
! [T3: a,U3: list_a,T6: a,U5: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T3 @ nil_a ) @ U3 )
=> ( ( ( ide_a @ resid @ ( resid @ T3 @ T6 ) )
& ( ide_a @ resid @ ( resid @ T6 @ T3 ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U3 @ U5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U5 @ U3 ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ U3 ) @ ( cons_a @ T6 @ U5 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T6 @ U5 ) @ ( cons_a @ T3 @ U3 ) ) ) ) ) ) ) ).
% cong_cons
thf(fact_241__092_060open_062_092_060lbrakk_062seq_AT_A_091_093_059_AIde_A_091_093_092_060rbrakk_062_A_092_060Longrightarrow_062_APCong_A_IT_A_064_A_091_093_J_AT_092_060close_062,axiom,
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ t @ nil_a )
=> ( ( paths_in_Ide_a @ resid @ nil_a )
=> ( paths_4972237869229803137Cong_a @ resid @ ( append_a @ t @ nil_a ) @ t ) ) ) ).
% \<open>\<lbrakk>seq T []; Ide []\<rbrakk> \<Longrightarrow> PCong (T @ []) T\<close>
thf(fact_242_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_243_residuation_Oarr_Ocong,axiom,
arr_list_a = arr_list_a ).
% residuation.arr.cong
thf(fact_244_residuation_Oarr_Ocong,axiom,
arr_a = arr_a ).
% residuation.arr.cong
thf(fact_245_rts_Ocomposable_Ocong,axiom,
composable_list_a = composable_list_a ).
% rts.composable.cong
thf(fact_246_rts_Ocomposable_Ocong,axiom,
composable_a = composable_a ).
% rts.composable.cong
thf(fact_247_rts_Ocoterminal_Ocong,axiom,
coterminal_a = coterminal_a ).
% rts.coterminal.cong
thf(fact_248_rts_Ocoterminal_Ocong,axiom,
coterminal_list_a = coterminal_list_a ).
% rts.coterminal.cong
thf(fact_249_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_250_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_251_rts__with__composites_Ohas__composites,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( seq_a @ Resid @ T3 @ U2 )
=> ( composable_a @ Resid @ T3 @ U2 ) ) ) ).
% rts_with_composites.has_composites
thf(fact_252_rts__with__composites_Ohas__composites,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a] :
( ( rts_wi1113109597627081724list_a @ Resid )
=> ( ( seq_list_a @ Resid @ T3 @ U2 )
=> ( composable_list_a @ Resid @ T3 @ U2 ) ) ) ).
% rts_with_composites.has_composites
thf(fact_253_cong__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 ) ) ).
% cong_implies_coterminal
thf(fact_254_identities__form__coherent__normal__sub__rts,axiom,
cohere6429906645900029933list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( ide_list_a @ ( paths_in_Resid_a @ resid ) ) ) ).
% identities_form_coherent_normal_sub_rts
thf(fact_255_cong__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 ) ) ).
% cong_implies_coinitial
thf(fact_256_coinitial__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 ) ) ) ) ) ) ).
% coinitial_ide_are_cong
thf(fact_257_Residx1__as__Resid,axiom,
! [T4: list_a,U2: a] :
( ( paths_in_Residx1_a @ resid @ T4 @ U2 )
= ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) ) ) ).
% Residx1_as_Resid
thf(fact_258_residuation__axioms,axiom,
residuation_list_a @ ( paths_in_Resid_a @ resid ) ).
% residuation_axioms
thf(fact_259_PCong_Ointros_I6_J,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ( paths_4972237869229803137Cong_a @ resid @ ( cons_a @ T3 @ ( cons_a @ ( resid @ U2 @ T3 ) @ nil_a ) ) @ ( cons_a @ U2 @ ( cons_a @ ( resid @ T3 @ U2 ) @ nil_a ) ) ) ) ).
% PCong.intros(6)
thf(fact_260_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_261_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_262_R_OcomposableD_I2_J,axiom,
! [T3: a,U2: a] :
( ( composable_a @ resid @ T3 @ U2 )
=> ( arr_a @ resid @ U2 ) ) ).
% R.composableD(2)
thf(fact_263_R_OcomposableD_I1_J,axiom,
! [T3: a,U2: a] :
( ( composable_a @ resid @ T3 @ U2 )
=> ( arr_a @ resid @ T3 ) ) ).
% R.composableD(1)
thf(fact_264_R_Ocong__reflexive,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
=> ( ( ide_a @ resid @ ( resid @ T3 @ T3 ) )
& ( ide_a @ resid @ ( resid @ T3 @ T3 ) ) ) ) ).
% R.cong_reflexive
thf(fact_265_R_Oide__implies__arr,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( arr_a @ resid @ A ) ) ).
% R.ide_implies_arr
thf(fact_266_R_Oprfx__reflexive,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
=> ( ide_a @ resid @ ( resid @ T3 @ T3 ) ) ) ).
% R.prfx_reflexive
thf(fact_267_R_Ocon__implies__arr_I2_J,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ( arr_a @ resid @ U2 ) ) ).
% R.con_implies_arr(2)
thf(fact_268_R_Ocon__implies__arr_I1_J,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ( arr_a @ resid @ T3 ) ) ).
% R.con_implies_arr(1)
thf(fact_269_R_OarrE,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
=> ( con_a @ resid @ T3 @ T3 ) ) ).
% R.arrE
thf(fact_270_R_Oarr__def,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
= ( con_a @ resid @ T3 @ T3 ) ) ).
% R.arr_def
thf(fact_271_R_Oarr__resid,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ( arr_a @ resid @ ( resid @ T3 @ U2 ) ) ) ).
% R.arr_resid
thf(fact_272_R_Oarr__resid__iff__con,axiom,
! [T3: a,U2: a] :
( ( arr_a @ resid @ ( resid @ T3 @ U2 ) )
= ( con_a @ resid @ T3 @ U2 ) ) ).
% R.arr_resid_iff_con
thf(fact_273_R_Ocon__sym,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ( con_a @ resid @ U2 @ T3 ) ) ).
% R.con_sym
thf(fact_274_R_Oresid__reflects__con,axiom,
! [T3: a,V2: a,U2: a] :
( ( con_a @ resid @ T3 @ V2 )
=> ( ( con_a @ resid @ U2 @ V2 )
=> ( ( con_a @ resid @ ( resid @ T3 @ V2 ) @ ( resid @ U2 @ V2 ) )
=> ( con_a @ resid @ T3 @ U2 ) ) ) ) ).
% R.resid_reflects_con
thf(fact_275_R_Ocomposable__imp__seq,axiom,
! [T3: a,U2: a] :
( ( composable_a @ resid @ T3 @ U2 )
=> ( seq_a @ resid @ T3 @ U2 ) ) ).
% R.composable_imp_seq
thf(fact_276_R_Ocong__subst__left_I2_J,axiom,
! [T3: a,T6: a,U2: a] :
( ( ( ide_a @ resid @ ( resid @ T3 @ T6 ) )
& ( ide_a @ resid @ ( resid @ T6 @ T3 ) ) )
=> ( ( con_a @ resid @ T3 @ U2 )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ T3 @ U2 ) @ ( resid @ T6 @ U2 ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ T6 @ U2 ) @ ( resid @ T3 @ U2 ) ) ) ) ) ) ).
% R.cong_subst_left(2)
thf(fact_277_R_Ocong__subst__left_I1_J,axiom,
! [T3: a,T6: a,U2: a] :
( ( ( ide_a @ resid @ ( resid @ T3 @ T6 ) )
& ( ide_a @ resid @ ( resid @ T6 @ T3 ) ) )
=> ( ( con_a @ resid @ T3 @ U2 )
=> ( con_a @ resid @ T6 @ U2 ) ) ) ).
% R.cong_subst_left(1)
thf(fact_278_R_Ocong__subst__right_I2_J,axiom,
! [U2: a,U4: a,T3: a] :
( ( ( ide_a @ resid @ ( resid @ U2 @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U2 ) ) )
=> ( ( con_a @ resid @ T3 @ U2 )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ T3 @ U2 ) @ ( resid @ T3 @ U4 ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ T3 @ U4 ) @ ( resid @ T3 @ U2 ) ) ) ) ) ) ).
% R.cong_subst_right(2)
thf(fact_279_R_Ocong__subst__right_I1_J,axiom,
! [U2: a,U4: a,T3: a] :
( ( ( ide_a @ resid @ ( resid @ U2 @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U2 ) ) )
=> ( ( con_a @ resid @ T3 @ U2 )
=> ( con_a @ resid @ T3 @ U4 ) ) ) ).
% R.cong_subst_right(1)
thf(fact_280_R_Ocon__imp__coinitial__ax,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ? [A3: a] :
( ( ide_a @ resid @ A3 )
& ( con_a @ resid @ A3 @ T3 )
& ( con_a @ resid @ A3 @ U2 ) ) ) ).
% R.con_imp_coinitial_ax
thf(fact_281_R_Ocon__target,axiom,
! [T3: a,U2: a,V2: a] :
( ( ide_a @ resid @ ( resid @ T3 @ U2 ) )
=> ( ( con_a @ resid @ U2 @ V2 )
=> ( con_a @ resid @ ( resid @ T3 @ U2 ) @ ( resid @ V2 @ U2 ) ) ) ) ).
% R.con_target
thf(fact_282_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_283_R_OideE,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ~ ( ( con_a @ resid @ A @ A )
=> ( ( resid @ A @ A )
!= A ) ) ) ).
% R.ideE
thf(fact_284_R_Oide__def,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
= ( ( con_a @ resid @ A @ A )
& ( ( resid @ A @ A )
= A ) ) ) ).
% R.ide_def
thf(fact_285_R_Oide__imp__con__iff__cong,axiom,
! [T3: a,U2: a] :
( ( ide_a @ resid @ T3 )
=> ( ( ide_a @ resid @ U2 )
=> ( ( con_a @ resid @ T3 @ U2 )
= ( ( ide_a @ resid @ ( resid @ T3 @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ T3 ) ) ) ) ) ) ).
% R.ide_imp_con_iff_cong
thf(fact_286_R_Oprfx__implies__con,axiom,
! [T3: a,U2: a] :
( ( ide_a @ resid @ ( resid @ T3 @ U2 ) )
=> ( con_a @ resid @ T3 @ U2 ) ) ).
% R.prfx_implies_con
thf(fact_287_R_Oresid__arr__ide,axiom,
! [A: a,T3: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ T3 @ A )
=> ( ( resid @ T3 @ A )
= T3 ) ) ) ).
% R.resid_arr_ide
thf(fact_288_R_Oresid__ide__arr,axiom,
! [A: a,T3: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ A @ T3 )
=> ( ide_a @ resid @ ( resid @ A @ T3 ) ) ) ) ).
% R.resid_ide_arr
thf(fact_289_Residx1_Osimps_I1_J,axiom,
! [U2: a] :
( ( paths_in_Residx1_a @ resid @ nil_a @ U2 )
= nil_a ) ).
% Residx1.simps(1)
thf(fact_290_R_Ocon__imp__coinitial,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ( coinitial_a @ resid @ T3 @ U2 ) ) ).
% R.con_imp_coinitial
thf(fact_291_Resid_Osimps_I3_J,axiom,
! [T3: a,U2: a] :
( ( ( con_a @ resid @ T3 @ U2 )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
= ( cons_a @ ( resid @ T3 @ U2 ) @ nil_a ) ) )
& ( ~ ( con_a @ resid @ T3 @ U2 )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
= nil_a ) ) ) ).
% Resid.simps(3)
thf(fact_292_Con__rec_I1_J,axiom,
! [T3: a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
= ( con_a @ resid @ T3 @ U2 ) ) ).
% Con_rec(1)
thf(fact_293_Con__rec_I2_J,axiom,
! [T4: list_a,T3: a,U2: a] :
( ( T4 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
= ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ ( resid @ U2 @ T3 ) @ nil_a ) )
!= nil_a ) ) ) ) ).
% Con_rec(2)
thf(fact_294_Con__rec_I3_J,axiom,
! [U3: list_a,T3: a,U2: a] :
( ( U3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ U3 ) )
!= nil_a )
= ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T3 @ U2 ) @ nil_a ) @ U3 )
!= nil_a ) ) ) ) ).
% Con_rec(3)
thf(fact_295_Con__rec_I4_J,axiom,
! [T4: list_a,U3: list_a,T3: a,U2: a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ T4 ) @ ( cons_a @ U2 @ U3 ) )
!= nil_a )
= ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ ( resid @ U2 @ T3 ) @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T3 @ U2 ) @ nil_a ) @ U3 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ ( resid @ U2 @ T3 ) @ nil_a ) ) @ ( paths_in_Resid_a @ resid @ U3 @ ( cons_a @ ( resid @ T3 @ U2 ) @ nil_a ) ) )
!= nil_a ) ) ) ) ) ).
% Con_rec(4)
thf(fact_296_Residx1_Osimps_I2_J,axiom,
! [T3: a,U2: a] :
( ( ( con_a @ resid @ T3 @ U2 )
=> ( ( paths_in_Residx1_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U2 )
= ( cons_a @ ( resid @ T3 @ U2 ) @ nil_a ) ) )
& ( ~ ( con_a @ resid @ T3 @ U2 )
=> ( ( paths_in_Residx1_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U2 )
= nil_a ) ) ) ).
% Residx1.simps(2)
thf(fact_297_Residx1_Osimps_I3_J,axiom,
! [T3: a,U2: a,V2: a,Va2: list_a] :
( ( ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) )
!= nil_a ) )
=> ( ( paths_in_Residx1_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ U2 )
= ( cons_a @ ( resid @ T3 @ U2 ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) ) ) ) )
& ( ~ ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) )
!= nil_a ) )
=> ( ( paths_in_Residx1_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ U2 )
= nil_a ) ) ) ).
% Residx1.simps(3)
thf(fact_298_Resid_Osimps_I5_J,axiom,
! [T3: a,U2: a,V2: a,Va2: list_a] :
( ( ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ nil_a ) )
= ( cons_a @ ( resid @ T3 @ U2 ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) ) ) ) )
& ( ~ ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ nil_a ) )
= nil_a ) ) ) ).
% Resid.simps(5)
thf(fact_299_R_OarrI,axiom,
! [T3: a] :
( ( con_a @ resid @ T3 @ T3 )
=> ( arr_a @ resid @ T3 ) ) ).
% R.arrI
thf(fact_300_R_OideI,axiom,
! [A: a] :
( ( con_a @ resid @ A @ A )
=> ( ( ( resid @ A @ A )
= A )
=> ( ide_a @ resid @ A ) ) ) ).
% R.ideI
thf(fact_301_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_302_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_303_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_304_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_305_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_306_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_307_residuation_Oarr__resid__iff__con,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ ( Resid @ T3 @ U2 ) )
= ( con_a @ Resid @ T3 @ U2 ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_308_residuation_Oarr__resid__iff__con,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( arr_list_a @ Resid @ ( Resid @ T3 @ U2 ) )
= ( con_list_a @ Resid @ T3 @ U2 ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_309_residuation_Oarr__resid,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( arr_a @ Resid @ ( Resid @ T3 @ U2 ) ) ) ) ).
% residuation.arr_resid
thf(fact_310_residuation_Oarr__resid,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( arr_list_a @ Resid @ ( Resid @ T3 @ U2 ) ) ) ) ).
% residuation.arr_resid
thf(fact_311_residuation_Oarr__def,axiom,
! [Resid: a > a > a,T3: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ T3 )
= ( con_a @ Resid @ T3 @ T3 ) ) ) ).
% residuation.arr_def
thf(fact_312_residuation_Oarr__def,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( arr_list_a @ Resid @ T3 )
= ( con_list_a @ Resid @ T3 @ T3 ) ) ) ).
% residuation.arr_def
thf(fact_313_residuation_OarrI,axiom,
! [Resid: a > a > a,T3: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T3 @ T3 )
=> ( arr_a @ Resid @ T3 ) ) ) ).
% residuation.arrI
thf(fact_314_residuation_OarrI,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T3 @ T3 )
=> ( arr_list_a @ Resid @ T3 ) ) ) ).
% residuation.arrI
thf(fact_315_residuation_OarrE,axiom,
! [Resid: a > a > a,T3: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ T3 )
=> ( con_a @ Resid @ T3 @ T3 ) ) ) ).
% residuation.arrE
thf(fact_316_residuation_OarrE,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( arr_list_a @ Resid @ T3 )
=> ( con_list_a @ Resid @ T3 @ T3 ) ) ) ).
% residuation.arrE
thf(fact_317_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( arr_a @ Resid @ T3 ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_318_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( arr_list_a @ Resid @ T3 ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_319_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( arr_a @ Resid @ U2 ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_320_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( arr_list_a @ Resid @ U2 ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_321_residuation_Ocube,axiom,
! [Resid: list_a > list_a > list_a,V2: list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U2 @ T3 ) )
= ( Resid @ ( Resid @ V2 @ U2 ) @ ( Resid @ T3 @ U2 ) ) ) ) ).
% residuation.cube
thf(fact_322_residuation_Ocube,axiom,
! [Resid: a > a > a,V2: a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U2 @ T3 ) )
= ( Resid @ ( Resid @ V2 @ U2 ) @ ( Resid @ T3 @ U2 ) ) ) ) ).
% residuation.cube
thf(fact_323_rts_Ocoinitial_Ocong,axiom,
coinitial_list_a = coinitial_list_a ).
% rts.coinitial.cong
thf(fact_324_rts_Ocoinitial_Ocong,axiom,
coinitial_a = coinitial_a ).
% rts.coinitial.cong
thf(fact_325_residuation_Ocon__sym,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( con_a @ Resid @ U2 @ T3 ) ) ) ).
% residuation.con_sym
thf(fact_326_residuation_Ocon__sym,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( con_list_a @ Resid @ U2 @ T3 ) ) ) ).
% residuation.con_sym
thf(fact_327_residuation_Ocon_Ocong,axiom,
con_a = con_a ).
% residuation.con.cong
thf(fact_328_residuation_Ocon_Ocong,axiom,
con_list_a = con_list_a ).
% residuation.con.cong
thf(fact_329_paths__in__rts_OResidx1_Ocong,axiom,
paths_in_Residx1_a = paths_in_Residx1_a ).
% paths_in_rts.Residx1.cong
thf(fact_330_residuation_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( partial_magma_a @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_331_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_332_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_333_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_334_PCong_Osimps,axiom,
! [A1: list_a,A22: list_a] :
( ( paths_4972237869229803137Cong_a @ resid @ A1 @ A22 )
= ( ? [T8: list_a] :
( ( A1 = T8 )
& ( A22 = T8 )
& ( paths_in_Arr_a @ resid @ T8 ) )
| ? [T8: list_a,U6: list_a] :
( ( A1 = U6 )
& ( A22 = T8 )
& ( paths_4972237869229803137Cong_a @ resid @ T8 @ U6 ) )
| ? [T8: list_a,U6: list_a,V4: list_a] :
( ( A1 = T8 )
& ( A22 = V4 )
& ( paths_4972237869229803137Cong_a @ resid @ T8 @ U6 )
& ( paths_4972237869229803137Cong_a @ resid @ U6 @ V4 ) )
| ? [T8: list_a,U6: list_a,T9: list_a,U7: list_a] :
( ( A1
= ( append_a @ T8 @ U6 ) )
& ( A22
= ( append_a @ T9 @ U7 ) )
& ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T8 @ U6 )
& ( paths_4972237869229803137Cong_a @ resid @ T8 @ T9 )
& ( paths_4972237869229803137Cong_a @ resid @ U6 @ U7 ) )
| ? [T8: list_a,B2: a] :
( ( A1
= ( append_a @ T8 @ ( cons_a @ B2 @ nil_a ) ) )
& ( A22 = T8 )
& ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T8 @ ( cons_a @ B2 @ nil_a ) )
& ( ide_a @ resid @ B2 ) )
| ? [T5: a,U8: a] :
( ( A1
= ( cons_a @ T5 @ ( cons_a @ ( resid @ U8 @ T5 ) @ nil_a ) ) )
& ( A22
= ( cons_a @ U8 @ ( cons_a @ ( resid @ T5 @ U8 ) @ nil_a ) ) )
& ( con_a @ resid @ T5 @ U8 ) ) ) ) ).
% PCong.simps
thf(fact_335_PCong_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 )
=> ( ! [U9: list_a] :
( ( paths_4972237869229803137Cong_a @ resid @ A1 @ U9 )
=> ~ ( paths_4972237869229803137Cong_a @ resid @ U9 @ A22 ) )
=> ( ! [T10: list_a,U9: list_a] :
( ( A1
= ( append_a @ T10 @ U9 ) )
=> ! [T11: list_a,U10: list_a] :
( ( A22
= ( append_a @ T11 @ U10 ) )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T10 @ U9 )
=> ( ( paths_4972237869229803137Cong_a @ resid @ T10 @ T11 )
=> ~ ( paths_4972237869229803137Cong_a @ resid @ U9 @ U10 ) ) ) ) )
=> ( ! [T10: list_a,B3: a] :
( ( A1
= ( append_a @ T10 @ ( cons_a @ B3 @ nil_a ) ) )
=> ( ( A22 = T10 )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T10 @ ( cons_a @ B3 @ nil_a ) )
=> ~ ( ide_a @ resid @ B3 ) ) ) )
=> ~ ! [T: a,U: a] :
( ( A1
= ( cons_a @ T @ ( cons_a @ ( resid @ U @ T ) @ nil_a ) ) )
=> ( ( A22
= ( cons_a @ U @ ( cons_a @ ( resid @ T @ U ) @ nil_a ) ) )
=> ~ ( con_a @ resid @ T @ U ) ) ) ) ) ) ) ) ) ).
% PCong.cases
thf(fact_336_joinable__implies__coinitial,axiom,
! [T3: list_a,U2: list_a] :
( ( joinable_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ).
% joinable_implies_coinitial
thf(fact_337_Resid_Osimps_I6_J,axiom,
! [T3: a,U2: a,Vb: a,Vc: list_a,V2: a,Va2: list_a] :
( ( ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ Vb @ Vc ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ ( cons_a @ Vb @ Vc ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ Vb @ Vc ) ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ T3 @ U2 ) ) ) ) ) )
& ( ~ ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ Vb @ Vc ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ ( cons_a @ Vb @ Vc ) ) )
= nil_a ) ) ) ).
% Resid.simps(6)
thf(fact_338_Resid_Osimps_I7_J,axiom,
! [T3: a,U2: a,V2: a,Va2: list_a,Vb: a,Vc: list_a] :
( ( ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ T3 @ U2 ) ) ) ) ) )
& ( ~ ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) )
= nil_a ) ) ) ).
% Resid.simps(7)
thf(fact_339_R_Ojoinable__implies__coinitial,axiom,
! [T3: a,U2: a] :
( ( joinable_a @ resid @ T3 @ U2 )
=> ( coinitial_a @ resid @ T3 @ U2 ) ) ).
% R.joinable_implies_coinitial
thf(fact_340_Resid_Osimps_I4_J,axiom,
! [T3: a,U2: a,V2: a,Va2: list_a] :
( ( ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) )
!= ( partial_null_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) ) @ nil_a ) ) )
& ( ~ ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) )
!= ( partial_null_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) )
= nil_a ) ) ) ).
% Resid.simps(4)
thf(fact_341_R_Oresiduation__axioms,axiom,
residuation_a @ resid ).
% R.residuation_axioms
thf(fact_342_resid__reflects__con,axiom,
! [T3: list_a,V2: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ V2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ V2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ V2 ) @ ( paths_in_Resid_a @ resid @ U2 @ V2 ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ) ) ).
% resid_reflects_con
thf(fact_343_con__sym,axiom,
! [T3: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 ) ) ).
% con_sym
thf(fact_344_R_Onull__eqI,axiom,
! [N: a] :
( ! [T: a] :
( ( ( resid @ N @ T )
= N )
& ( ( resid @ T @ N )
= N ) )
=> ( N
= ( partial_null_a @ resid ) ) ) ).
% R.null_eqI
thf(fact_345_R_Ocube__ax,axiom,
! [V2: a,T3: a,U2: a] :
( ( ( resid @ ( resid @ V2 @ T3 ) @ ( resid @ U2 @ T3 ) )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ ( resid @ V2 @ T3 ) @ ( resid @ U2 @ T3 ) )
= ( resid @ ( resid @ V2 @ U2 ) @ ( resid @ T3 @ U2 ) ) ) ) ).
% R.cube_ax
thf(fact_346_R_Ocon__sym__ax,axiom,
! [T3: a,U2: a] :
( ( ( resid @ T3 @ U2 )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ U2 @ T3 )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_sym_ax
thf(fact_347_R_Ocon__imp__arr__resid,axiom,
! [T3: a,U2: a] :
( ( ( resid @ T3 @ U2 )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ ( resid @ T3 @ U2 ) @ ( resid @ T3 @ U2 ) )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_imp_arr_resid
thf(fact_348_con__char,axiom,
! [T4: list_a,U3: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
= ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a ) ) ).
% con_char
thf(fact_349_resid__ide__arr,axiom,
! [A: list_a,T3: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ T3 )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A @ T3 ) ) ) ) ).
% resid_ide_arr
thf(fact_350_resid__arr__ide,axiom,
! [A: list_a,T3: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ A )
=> ( ( paths_in_Resid_a @ resid @ T3 @ A )
= T3 ) ) ) ).
% resid_arr_ide
thf(fact_351_prfx__implies__con,axiom,
! [T3: list_a,U2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ).
% prfx_implies_con
thf(fact_352_ide__imp__con__iff__cong,axiom,
! [T3: list_a,U2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
= ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) ) ) ) ) ) ).
% ide_imp_con_iff_cong
thf(fact_353_ide__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 ) ) ) ).
% ide_def
thf(fact_354_ideE,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 ) ) ) ).
% ideE
thf(fact_355_con__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 ) ) ) ) ) ) ).
% con_transitive_on_ide
thf(fact_356_con__target,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ V2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ V2 @ U2 ) ) ) ) ).
% con_target
thf(fact_357_con__imp__coinitial__ax,axiom,
! [T3: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ? [A3: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A3 )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A3 @ T3 )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A3 @ U2 ) ) ) ).
% con_imp_coinitial_ax
thf(fact_358_cong__subst__right_I1_J,axiom,
! [U2: list_a,U4: list_a,T3: 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 ) @ T3 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U4 ) ) ) ).
% cong_subst_right(1)
thf(fact_359_cong__subst__right_I2_J,axiom,
! [U2: list_a,U4: list_a,T3: 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 ) @ T3 @ U2 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T3 @ U4 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U4 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_360_cong__subst__left_I1_J,axiom,
! [T3: list_a,T6: list_a,U2: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T6 @ U2 ) ) ) ).
% cong_subst_left(1)
thf(fact_361_cong__subst__left_I2_J,axiom,
! [T3: list_a,T6: list_a,U2: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T6 @ U2 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T6 @ U2 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_362_arr__resid__iff__con,axiom,
! [T3: list_a,U2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ).
% arr_resid_iff_con
thf(fact_363_arr__resid,axiom,
! [T3: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) ) ).
% arr_resid
thf(fact_364_arr__def,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ T3 ) ) ).
% arr_def
thf(fact_365_arrE,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ T3 ) ) ).
% arrE
thf(fact_366_con__implies__arr_I1_J,axiom,
! [T3: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ).
% con_implies_arr(1)
thf(fact_367_con__implies__arr_I2_J,axiom,
! [T3: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ).
% con_implies_arr(2)
thf(fact_368_Arr_Osimps_I1_J,axiom,
~ ( paths_in_Arr_a @ resid @ nil_a ) ).
% Arr.simps(1)
thf(fact_369_Ide__implies__Arr,axiom,
! [T4: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
=> ( paths_in_Arr_a @ resid @ T4 ) ) ).
% Ide_implies_Arr
thf(fact_370_R_Ocon__def,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
= ( ( resid @ T3 @ U2 )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_def
thf(fact_371_R_OconE,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ( ( resid @ T3 @ U2 )
!= ( partial_null_a @ resid ) ) ) ).
% R.conE
thf(fact_372_PCong_Ointros_I1_J,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( paths_4972237869229803137Cong_a @ resid @ T4 @ T4 ) ) ).
% PCong.intros(1)
thf(fact_373_R_Onot__arr__null,axiom,
~ ( arr_a @ resid @ ( partial_null_a @ resid ) ) ).
% R.not_arr_null
thf(fact_374_con__imp__coinitial,axiom,
! [T3: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ).
% con_imp_coinitial
thf(fact_375_Resid1x__null,axiom,
! [T4: list_a] :
( ( paths_in_Resid1x_a @ resid @ ( partial_null_a @ resid ) @ T4 )
= ( partial_null_a @ resid ) ) ).
% Resid1x_null
thf(fact_376_joinable__implies__con,axiom,
! [T3: list_a,U2: list_a] :
( ( joinable_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ).
% joinable_implies_con
thf(fact_377_R_Ojoinable__implies__con,axiom,
! [T3: a,U2: a] :
( ( joinable_a @ resid @ T3 @ U2 )
=> ( con_a @ resid @ T3 @ U2 ) ) ).
% R.joinable_implies_con
thf(fact_378_Con__implies__Arr_I2_J,axiom,
! [T4: list_a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ U3 ) ) ).
% Con_implies_Arr(2)
thf(fact_379_Con__implies__Arr_I1_J,axiom,
! [T4: list_a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ T4 ) ) ).
% Con_implies_Arr(1)
thf(fact_380_Arr__iff__Con__self,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
= ( ( paths_in_Resid_a @ resid @ T4 @ T4 )
!= nil_a ) ) ).
% Arr_iff_Con_self
thf(fact_381_Con__Arr__self,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Resid_a @ resid @ T4 @ T4 )
!= nil_a ) ) ).
% Con_Arr_self
thf(fact_382_Con__imp__Arr__Resid,axiom,
! [T4: list_a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) ) ) ).
% Con_imp_Arr_Resid
thf(fact_383_arr__char,axiom,
! [T4: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T4 )
= ( paths_in_Arr_a @ resid @ T4 ) ) ).
% arr_char
thf(fact_384_Resid__Arr__self,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( paths_in_Ide_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ T4 ) ) ) ).
% Resid_Arr_self
thf(fact_385_Resid1x_Osimps_I1_J,axiom,
! [T3: a] :
( ( paths_in_Resid1x_a @ resid @ T3 @ nil_a )
= ( partial_null_a @ resid ) ) ).
% Resid1x.simps(1)
thf(fact_386_Resid1x__ide,axiom,
! [A: a,T4: list_a] :
( ( ide_a @ resid @ A )
=> ( ( ( paths_in_Resid1x_a @ resid @ A @ T4 )
!= ( partial_null_a @ resid ) )
=> ( ide_a @ resid @ ( paths_in_Resid1x_a @ resid @ A @ T4 ) ) ) ) ).
% Resid1x_ide
thf(fact_387_Arr__imp__Arr__tl,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( ( tl_a @ T4 )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ ( tl_a @ T4 ) ) ) ) ).
% Arr_imp_Arr_tl
thf(fact_388_Arr__imp__arr__last,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( arr_a @ resid @ ( last_a @ T4 ) ) ) ).
% Arr_imp_arr_last
thf(fact_389_Arr__Resid__single,axiom,
! [T4: list_a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) ) ) ) ).
% Arr_Resid_single
thf(fact_390_Arr_Osimps_I2_J,axiom,
! [T3: a] :
( ( paths_in_Arr_a @ resid @ ( cons_a @ T3 @ nil_a ) )
= ( arr_a @ resid @ T3 ) ) ).
% Arr.simps(2)
thf(fact_391_Resid1x_Oelims,axiom,
! [X: a,Xa: list_a,Y: a] :
( ( ( paths_in_Resid1x_a @ resid @ X @ Xa )
= Y )
=> ( ( ( Xa = nil_a )
=> ( Y
!= ( partial_null_a @ resid ) ) )
=> ( ! [U: a] :
( ( Xa
= ( cons_a @ U @ nil_a ) )
=> ( Y
!= ( resid @ X @ U ) ) )
=> ~ ! [U: a,V: a,Va: list_a] :
( ( Xa
= ( cons_a @ U @ ( cons_a @ V @ Va ) ) )
=> ( Y
!= ( paths_in_Resid1x_a @ resid @ ( resid @ X @ U ) @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ).
% Resid1x.elims
thf(fact_392_Con__sym1,axiom,
! [T4: list_a,U2: a] :
( ( ( paths_in_Residx1_a @ resid @ T4 @ U2 )
!= nil_a )
= ( ( paths_in_Resid1x_a @ resid @ U2 @ T4 )
!= ( partial_null_a @ resid ) ) ) ).
% Con_sym1
thf(fact_393_R_Onull__is__zero_I1_J,axiom,
! [T3: a] :
( ( resid @ ( partial_null_a @ resid ) @ T3 )
= ( partial_null_a @ resid ) ) ).
% R.null_is_zero(1)
thf(fact_394_R_Onull__is__zero_I2_J,axiom,
! [T3: a] :
( ( resid @ T3 @ ( partial_null_a @ resid ) )
= ( partial_null_a @ resid ) ) ).
% R.null_is_zero(2)
thf(fact_395_conI_092_060_094sub_062P,axiom,
! [T4: list_a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 ) ) ).
% conI\<^sub>P
thf(fact_396_ideI,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 ) ) ) ).
% ideI
thf(fact_397_arrI,axiom,
! [T3: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ T3 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ).
% arrI
thf(fact_398_R_OconI,axiom,
! [T3: a,U2: a] :
( ( ( resid @ T3 @ U2 )
!= ( partial_null_a @ resid ) )
=> ( con_a @ resid @ T3 @ U2 ) ) ).
% R.conI
thf(fact_399_arrI_092_060_094sub_062P,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T4 ) ) ).
% arrI\<^sub>P
thf(fact_400_partial__magma_Onull_Ocong,axiom,
partial_null_a = partial_null_a ).
% partial_magma.null.cong
thf(fact_401_partial__magma_Onull_Ocong,axiom,
partial_null_list_a = partial_null_list_a ).
% partial_magma.null.cong
thf(fact_402_paths__in__rts_OArr_Ocong,axiom,
paths_in_Arr_a = paths_in_Arr_a ).
% paths_in_rts.Arr.cong
thf(fact_403_rts_Ojoinable_Ocong,axiom,
joinable_list_a = joinable_list_a ).
% rts.joinable.cong
thf(fact_404_rts_Ojoinable_Ocong,axiom,
joinable_a = joinable_a ).
% rts.joinable.cong
thf(fact_405_residuation_Ocon__imp__arr__resid,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ T3 @ U2 )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T3 @ U2 ) )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.con_imp_arr_resid
thf(fact_406_residuation_Ocon__imp__arr__resid,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( ( Resid @ T3 @ U2 )
!= ( partial_null_list_a @ Resid ) )
=> ( ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T3 @ U2 ) )
!= ( partial_null_list_a @ Resid ) ) ) ) ).
% residuation.con_imp_arr_resid
thf(fact_407_residuation_Ocon__sym__ax,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ T3 @ U2 )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ U2 @ T3 )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.con_sym_ax
thf(fact_408_residuation_Ocon__sym__ax,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( ( Resid @ T3 @ U2 )
!= ( partial_null_list_a @ Resid ) )
=> ( ( Resid @ U2 @ T3 )
!= ( partial_null_list_a @ Resid ) ) ) ) ).
% residuation.con_sym_ax
thf(fact_409_residuation_Ocube__ax,axiom,
! [Resid: a > a > a,V2: a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U2 @ T3 ) )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U2 @ T3 ) )
= ( Resid @ ( Resid @ V2 @ U2 ) @ ( Resid @ T3 @ U2 ) ) ) ) ) ).
% residuation.cube_ax
thf(fact_410_residuation_Ocube__ax,axiom,
! [Resid: list_a > list_a > list_a,V2: list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U2 @ T3 ) )
!= ( partial_null_list_a @ Resid ) )
=> ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U2 @ T3 ) )
= ( Resid @ ( Resid @ V2 @ U2 ) @ ( Resid @ T3 @ U2 ) ) ) ) ) ).
% residuation.cube_ax
thf(fact_411_partial__magma_Onull__eqI,axiom,
! [OP: a > a > a,N: a] :
( ( partial_magma_a @ OP )
=> ( ! [T: a] :
( ( ( OP @ N @ T )
= N )
& ( ( OP @ T @ N )
= N ) )
=> ( N
= ( partial_null_a @ OP ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_412_partial__magma_Onull__eqI,axiom,
! [OP: list_a > list_a > list_a,N: list_a] :
( ( partial_magma_list_a @ OP )
=> ( ! [T: list_a] :
( ( ( OP @ N @ T )
= N )
& ( ( OP @ T @ N )
= N ) )
=> ( N
= ( partial_null_list_a @ OP ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_413_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP: a > a > a,T3: a] :
( ( partial_magma_a @ OP )
=> ( ( OP @ ( partial_null_a @ OP ) @ T3 )
= ( partial_null_a @ OP ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_414_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP: list_a > list_a > list_a,T3: list_a] :
( ( partial_magma_list_a @ OP )
=> ( ( OP @ ( partial_null_list_a @ OP ) @ T3 )
= ( partial_null_list_a @ OP ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_415_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP: a > a > a,T3: a] :
( ( partial_magma_a @ OP )
=> ( ( OP @ T3 @ ( partial_null_a @ OP ) )
= ( partial_null_a @ OP ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_416_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP: list_a > list_a > list_a,T3: list_a] :
( ( partial_magma_list_a @ OP )
=> ( ( OP @ T3 @ ( partial_null_list_a @ OP ) )
= ( partial_null_list_a @ OP ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_417_residuation_Ocon__def,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T3 @ U2 )
= ( ( Resid @ T3 @ U2 )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.con_def
thf(fact_418_residuation_Ocon__def,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
= ( ( Resid @ T3 @ U2 )
!= ( partial_null_list_a @ Resid ) ) ) ) ).
% residuation.con_def
thf(fact_419_residuation_OconI,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ T3 @ U2 )
!= ( partial_null_a @ Resid ) )
=> ( con_a @ Resid @ T3 @ U2 ) ) ) ).
% residuation.conI
thf(fact_420_residuation_OconI,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( ( Resid @ T3 @ U2 )
!= ( partial_null_list_a @ Resid ) )
=> ( con_list_a @ Resid @ T3 @ U2 ) ) ) ).
% residuation.conI
thf(fact_421_residuation_OconE,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( ( Resid @ T3 @ U2 )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.conE
thf(fact_422_residuation_OconE,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( ( Resid @ T3 @ U2 )
!= ( partial_null_list_a @ Resid ) ) ) ) ).
% residuation.conE
thf(fact_423_residuation_Onot__arr__null,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ~ ( arr_a @ Resid @ ( partial_null_a @ Resid ) ) ) ).
% residuation.not_arr_null
thf(fact_424_residuation_Onot__arr__null,axiom,
! [Resid: list_a > list_a > list_a] :
( ( residuation_list_a @ Resid )
=> ~ ( arr_list_a @ Resid @ ( partial_null_list_a @ Resid ) ) ) ).
% residuation.not_arr_null
thf(fact_425_Resid1x__as__Resid_H,axiom,
! [T3: a,U3: list_a] :
( ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 )
!= nil_a )
=> ( ( paths_in_Resid1x_a @ resid @ T3 @ U3 )
= ( hd_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 ) ) ) )
& ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 )
= nil_a )
=> ( ( paths_in_Resid1x_a @ resid @ T3 @ U3 )
= ( partial_null_a @ resid ) ) ) ) ).
% Resid1x_as_Resid'
thf(fact_426_reflects__con,axiom,
! [T3: a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ ( if_list_a @ ( arr_a @ resid @ T3 ) @ ( cons_a @ T3 @ 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 @ T3 @ U2 ) ) ).
% reflects_con
thf(fact_427_Con__single__ide__ind,axiom,
! [A: a,T4: list_a] :
( ( ide_a @ resid @ A )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ A @ nil_a ) @ T4 )
!= nil_a )
= ( ( paths_in_Arr_a @ resid @ T4 )
& ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T4 ) ) ) ) ) ).
% Con_single_ide_ind
thf(fact_428_paths__in__rts__with__coherent__normal_OPCong_Osimps,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,A1: list_list_a,A22: list_list_a] :
( ( paths_9180082184051307706list_a @ Resid @ NN )
=> ( ( paths_9002432012371690887list_a @ Resid @ A1 @ A22 )
= ( ? [T8: list_list_a] :
( ( A1 = T8 )
& ( A22 = T8 )
& ( paths_in_Arr_list_a @ Resid @ T8 ) )
| ? [T8: list_list_a,U6: list_list_a] :
( ( A1 = U6 )
& ( A22 = T8 )
& ( paths_9002432012371690887list_a @ Resid @ T8 @ U6 ) )
| ? [T8: list_list_a,U6: list_list_a,V4: list_list_a] :
( ( A1 = T8 )
& ( A22 = V4 )
& ( paths_9002432012371690887list_a @ Resid @ T8 @ U6 )
& ( paths_9002432012371690887list_a @ Resid @ U6 @ V4 ) )
| ? [T8: list_list_a,U6: list_list_a,T9: list_list_a,U7: list_list_a] :
( ( A1
= ( append_list_a @ T8 @ U6 ) )
& ( A22
= ( append_list_a @ T9 @ U7 ) )
& ( seq_list_list_a @ ( paths_8620460302779588466list_a @ Resid ) @ T8 @ U6 )
& ( paths_9002432012371690887list_a @ Resid @ T8 @ T9 )
& ( paths_9002432012371690887list_a @ Resid @ U6 @ U7 ) )
| ? [T8: list_list_a,B2: list_a] :
( ( A1
= ( append_list_a @ T8 @ ( cons_list_a @ B2 @ nil_list_a ) ) )
& ( A22 = T8 )
& ( seq_list_list_a @ ( paths_8620460302779588466list_a @ Resid ) @ T8 @ ( cons_list_a @ B2 @ nil_list_a ) )
& ( ide_list_a @ Resid @ B2 ) )
| ? [T5: list_a,U8: list_a] :
( ( A1
= ( cons_list_a @ T5 @ ( cons_list_a @ ( Resid @ U8 @ T5 ) @ nil_list_a ) ) )
& ( A22
= ( cons_list_a @ U8 @ ( cons_list_a @ ( Resid @ T5 @ U8 ) @ nil_list_a ) ) )
& ( con_list_a @ Resid @ T5 @ U8 ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.simps
thf(fact_429_paths__in__rts__with__coherent__normal_OPCong_Osimps,axiom,
! [Resid: a > a > a,NN: set_a,A1: list_a,A22: list_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( paths_4972237869229803137Cong_a @ Resid @ A1 @ A22 )
= ( ? [T8: list_a] :
( ( A1 = T8 )
& ( A22 = T8 )
& ( paths_in_Arr_a @ Resid @ T8 ) )
| ? [T8: list_a,U6: list_a] :
( ( A1 = U6 )
& ( A22 = T8 )
& ( paths_4972237869229803137Cong_a @ Resid @ T8 @ U6 ) )
| ? [T8: list_a,U6: list_a,V4: list_a] :
( ( A1 = T8 )
& ( A22 = V4 )
& ( paths_4972237869229803137Cong_a @ Resid @ T8 @ U6 )
& ( paths_4972237869229803137Cong_a @ Resid @ U6 @ V4 ) )
| ? [T8: list_a,U6: list_a,T9: list_a,U7: list_a] :
( ( A1
= ( append_a @ T8 @ U6 ) )
& ( A22
= ( append_a @ T9 @ U7 ) )
& ( seq_list_a @ ( paths_in_Resid_a @ Resid ) @ T8 @ U6 )
& ( paths_4972237869229803137Cong_a @ Resid @ T8 @ T9 )
& ( paths_4972237869229803137Cong_a @ Resid @ U6 @ U7 ) )
| ? [T8: list_a,B2: a] :
( ( A1
= ( append_a @ T8 @ ( cons_a @ B2 @ nil_a ) ) )
& ( A22 = T8 )
& ( seq_list_a @ ( paths_in_Resid_a @ Resid ) @ T8 @ ( cons_a @ B2 @ nil_a ) )
& ( ide_a @ Resid @ B2 ) )
| ? [T5: a,U8: a] :
( ( A1
= ( cons_a @ T5 @ ( cons_a @ ( Resid @ U8 @ T5 ) @ nil_a ) ) )
& ( A22
= ( cons_a @ U8 @ ( cons_a @ ( Resid @ T5 @ U8 ) @ nil_a ) ) )
& ( con_a @ Resid @ T5 @ U8 ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.simps
thf(fact_430_paths__in__rts__with__coherent__normal_OPCong_Ocases,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,A1: list_list_a,A22: list_list_a] :
( ( paths_9180082184051307706list_a @ Resid @ NN )
=> ( ( paths_9002432012371690887list_a @ Resid @ A1 @ A22 )
=> ( ( ( A22 = A1 )
=> ~ ( paths_in_Arr_list_a @ Resid @ A1 ) )
=> ( ~ ( paths_9002432012371690887list_a @ Resid @ A22 @ A1 )
=> ( ! [U9: list_list_a] :
( ( paths_9002432012371690887list_a @ Resid @ A1 @ U9 )
=> ~ ( paths_9002432012371690887list_a @ Resid @ U9 @ A22 ) )
=> ( ! [T10: list_list_a,U9: list_list_a] :
( ( A1
= ( append_list_a @ T10 @ U9 ) )
=> ! [T11: list_list_a,U10: list_list_a] :
( ( A22
= ( append_list_a @ T11 @ U10 ) )
=> ( ( seq_list_list_a @ ( paths_8620460302779588466list_a @ Resid ) @ T10 @ U9 )
=> ( ( paths_9002432012371690887list_a @ Resid @ T10 @ T11 )
=> ~ ( paths_9002432012371690887list_a @ Resid @ U9 @ U10 ) ) ) ) )
=> ( ! [T10: list_list_a,B3: list_a] :
( ( A1
= ( append_list_a @ T10 @ ( cons_list_a @ B3 @ nil_list_a ) ) )
=> ( ( A22 = T10 )
=> ( ( seq_list_list_a @ ( paths_8620460302779588466list_a @ Resid ) @ T10 @ ( cons_list_a @ B3 @ nil_list_a ) )
=> ~ ( ide_list_a @ Resid @ B3 ) ) ) )
=> ~ ! [T: list_a,U: list_a] :
( ( A1
= ( cons_list_a @ T @ ( cons_list_a @ ( Resid @ U @ T ) @ nil_list_a ) ) )
=> ( ( A22
= ( cons_list_a @ U @ ( cons_list_a @ ( Resid @ T @ U ) @ nil_list_a ) ) )
=> ~ ( con_list_a @ Resid @ T @ U ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.cases
thf(fact_431_paths__in__rts__with__coherent__normal_OPCong_Ocases,axiom,
! [Resid: a > a > a,NN: set_a,A1: list_a,A22: list_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( paths_4972237869229803137Cong_a @ Resid @ A1 @ A22 )
=> ( ( ( A22 = A1 )
=> ~ ( paths_in_Arr_a @ Resid @ A1 ) )
=> ( ~ ( paths_4972237869229803137Cong_a @ Resid @ A22 @ A1 )
=> ( ! [U9: list_a] :
( ( paths_4972237869229803137Cong_a @ Resid @ A1 @ U9 )
=> ~ ( paths_4972237869229803137Cong_a @ Resid @ U9 @ A22 ) )
=> ( ! [T10: list_a,U9: list_a] :
( ( A1
= ( append_a @ T10 @ U9 ) )
=> ! [T11: list_a,U10: list_a] :
( ( A22
= ( append_a @ T11 @ U10 ) )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ Resid ) @ T10 @ U9 )
=> ( ( paths_4972237869229803137Cong_a @ Resid @ T10 @ T11 )
=> ~ ( paths_4972237869229803137Cong_a @ Resid @ U9 @ U10 ) ) ) ) )
=> ( ! [T10: list_a,B3: a] :
( ( A1
= ( append_a @ T10 @ ( cons_a @ B3 @ nil_a ) ) )
=> ( ( A22 = T10 )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ Resid ) @ T10 @ ( cons_a @ B3 @ nil_a ) )
=> ~ ( ide_a @ Resid @ B3 ) ) ) )
=> ~ ! [T: a,U: a] :
( ( A1
= ( cons_a @ T @ ( cons_a @ ( Resid @ U @ T ) @ nil_a ) ) )
=> ( ( A22
= ( cons_a @ U @ ( cons_a @ ( Resid @ T @ U ) @ nil_a ) ) )
=> ~ ( con_a @ Resid @ T @ U ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.cases
thf(fact_432_Con__IdeI_I2_J,axiom,
! [A2: list_a,T4: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( ( paths_in_Srcs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ A2 ) )
=> ( ( paths_in_Resid_a @ resid @ T4 @ A2 )
!= nil_a ) ) ) ) ).
% Con_IdeI(2)
thf(fact_433_Srcs__are__con,axiom,
! [A: a,T4: list_a,A4: a] :
( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( ( member_a @ A4 @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( con_a @ resid @ A @ A4 ) ) ) ).
% Srcs_are_con
thf(fact_434_con__imp__arr__resid,axiom,
! [T3: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T3 @ U2 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% con_imp_arr_resid
thf(fact_435_con__sym__ax,axiom,
! [T3: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T3 @ U2 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ U2 @ T3 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% con_sym_ax
thf(fact_436_cube__ax,axiom,
! [V2: list_a,T3: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V2 @ T3 ) @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V2 @ T3 ) @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) ) ) ).
% cube_ax
thf(fact_437_null__eqI,axiom,
! [N: list_a] :
( ! [T: list_a] :
( ( ( paths_in_Resid_a @ resid @ N @ T )
= N )
& ( ( paths_in_Resid_a @ resid @ T @ N )
= N ) )
=> ( N
= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% null_eqI
thf(fact_438_Con__imp__eq__Srcs,axiom,
! [T4: list_a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ U3 ) ) ) ).
% Con_imp_eq_Srcs
thf(fact_439_null__char,axiom,
( ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) )
= nil_a ) ).
% null_char
thf(fact_440_Srcs__con__closed,axiom,
! [A: a,T4: list_a,A4: a] :
( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( ( ide_a @ resid @ A4 )
=> ( ( con_a @ resid @ A @ A4 )
=> ( member_a @ A4 @ ( paths_in_Srcs_a @ resid @ T4 ) ) ) ) ) ).
% Srcs_con_closed
thf(fact_441_conE,axiom,
! [T3: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( paths_in_Resid_a @ resid @ T3 @ U2 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% conE
thf(fact_442_con__def,axiom,
! [T3: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
= ( ( paths_in_Resid_a @ resid @ T3 @ U2 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% con_def
thf(fact_443_not__arr__null,axiom,
~ ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ).
% not_arr_null
thf(fact_444_Arr__imp__arr__hd,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( arr_a @ resid @ ( hd_a @ T4 ) ) ) ).
% Arr_imp_arr_hd
thf(fact_445_Ide__imp__Ide__hd,axiom,
! [T4: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
=> ( ide_a @ resid @ ( hd_a @ T4 ) ) ) ).
% Ide_imp_Ide_hd
thf(fact_446_Resid__Arr__Src,axiom,
! [T4: list_a,A: a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ A @ nil_a ) )
= T4 ) ) ) ).
% Resid_Arr_Src
thf(fact_447_Con__Ide__iff,axiom,
! [A2: list_a,T4: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( ( paths_in_Resid_a @ resid @ A2 @ T4 )
!= nil_a )
= ( ( paths_in_Arr_a @ resid @ T4 )
& ( ( paths_in_Srcs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ A2 ) ) ) ) ) ).
% Con_Ide_iff
thf(fact_448_Con__IdeI_I1_J,axiom,
! [A2: list_a,T4: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( ( paths_in_Srcs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ A2 ) )
=> ( ( paths_in_Resid_a @ resid @ A2 @ T4 )
!= nil_a ) ) ) ) ).
% Con_IdeI(1)
thf(fact_449_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_450_null__is__zero_I2_J,axiom,
! [T3: list_a] :
( ( paths_in_Resid_a @ resid @ T3 @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ).
% null_is_zero(2)
thf(fact_451_null__is__zero_I1_J,axiom,
! [T3: list_a] :
( ( paths_in_Resid_a @ resid @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) @ T3 )
= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ).
% null_is_zero(1)
thf(fact_452_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_453_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_454_Srcs__append,axiom,
! [T4: list_a,U3: list_a] :
( ( T4 != nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( append_a @ T4 @ U3 ) )
= ( paths_in_Srcs_a @ resid @ T4 ) ) ) ).
% Srcs_append
thf(fact_455_conI,axiom,
! [T3: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T3 @ U2 )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ).
% conI
thf(fact_456_Con__single__ideI_I1_J,axiom,
! [A: a,T4: list_a] :
( ( ide_a @ resid @ A )
=> ( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ A @ nil_a ) @ T4 )
!= nil_a ) ) ) ) ).
% Con_single_ideI(1)
thf(fact_457_Con__single__ideI_I2_J,axiom,
! [A: a,T4: list_a] :
( ( ide_a @ resid @ A )
=> ( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ A @ nil_a ) )
!= nil_a ) ) ) ) ).
% Con_single_ideI(2)
thf(fact_458_paths__in__rts_OSrcs_Ocong,axiom,
paths_in_Srcs_a = paths_in_Srcs_a ).
% paths_in_rts.Srcs.cong
thf(fact_459_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_460_paths__in__rts__with__coherent__normal_OPCong_Ointros_I3_J,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,U3: list_a,V3: list_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( paths_4972237869229803137Cong_a @ Resid @ T4 @ U3 )
=> ( ( paths_4972237869229803137Cong_a @ Resid @ U3 @ V3 )
=> ( paths_4972237869229803137Cong_a @ Resid @ T4 @ V3 ) ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.intros(3)
thf(fact_461_paths__in__rts__with__coherent__normal_OPCong_Ointros_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,U3: list_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( paths_4972237869229803137Cong_a @ Resid @ T4 @ U3 )
=> ( paths_4972237869229803137Cong_a @ Resid @ U3 @ T4 ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.intros(2)
thf(fact_462_paths__in__rts__with__coherent__normal_Oaxioms_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a] :
( ( paths_9180082184051307706list_a @ Resid @ NN )
=> ( cohere6429906645900029933list_a @ Resid @ NN ) ) ).
% paths_in_rts_with_coherent_normal.axioms(2)
thf(fact_463_paths__in__rts__with__coherent__normal_Oaxioms_I2_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( cohere6072184133013167079_rts_a @ Resid @ NN ) ) ).
% paths_in_rts_with_coherent_normal.axioms(2)
thf(fact_464_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs3: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_a @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_a )
| ( Ys5 = nil_a )
| ( ( hd_a @ Xs3 )
!= ( hd_a @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_465_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_466_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_467_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_468_paths__in__rts__with__coherent__normal_OPCong_Ointros_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( paths_4972237869229803137Cong_a @ Resid @ T4 @ T4 ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.intros(1)
thf(fact_469_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_470_rotate1__hd__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( rotate1_a @ Xs )
= ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_471_paths__in__rts__with__coherent__normal_OPCong_Ointros_I6_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a] :
( ( paths_9180082184051307706list_a @ Resid @ NN )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( paths_9002432012371690887list_a @ Resid @ ( cons_list_a @ T3 @ ( cons_list_a @ ( Resid @ U2 @ T3 ) @ nil_list_a ) ) @ ( cons_list_a @ U2 @ ( cons_list_a @ ( Resid @ T3 @ U2 ) @ nil_list_a ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.intros(6)
thf(fact_472_paths__in__rts__with__coherent__normal_OPCong_Ointros_I6_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( paths_4972237869229803137Cong_a @ Resid @ ( cons_a @ T3 @ ( cons_a @ ( Resid @ U2 @ T3 ) @ nil_a ) ) @ ( cons_a @ U2 @ ( cons_a @ ( Resid @ T3 @ U2 ) @ nil_a ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.intros(6)
thf(fact_473_paths__in__rts__with__coherent__normal_OPCong_Ointros_I4_J,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,U3: list_a,T7: list_a,U5: list_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ U3 )
=> ( ( paths_4972237869229803137Cong_a @ Resid @ T4 @ T7 )
=> ( ( paths_4972237869229803137Cong_a @ Resid @ U3 @ U5 )
=> ( paths_4972237869229803137Cong_a @ Resid @ ( append_a @ T4 @ U3 ) @ ( append_a @ T7 @ U5 ) ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.intros(4)
thf(fact_474_paths__in__rts__with__coherent__normal_OPCong_Ointros_I5_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T4: list_list_a,B: list_a] :
( ( paths_9180082184051307706list_a @ Resid @ NN )
=> ( ( seq_list_list_a @ ( paths_8620460302779588466list_a @ Resid ) @ T4 @ ( cons_list_a @ B @ nil_list_a ) )
=> ( ( ide_list_a @ Resid @ B )
=> ( paths_9002432012371690887list_a @ Resid @ ( append_list_a @ T4 @ ( cons_list_a @ B @ nil_list_a ) ) @ T4 ) ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.intros(5)
thf(fact_475_paths__in__rts__with__coherent__normal_OPCong_Ointros_I5_J,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,B: a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ ( cons_a @ B @ nil_a ) )
=> ( ( ide_a @ Resid @ B )
=> ( paths_4972237869229803137Cong_a @ Resid @ ( append_a @ T4 @ ( cons_a @ B @ nil_a ) ) @ T4 ) ) ) ) ).
% paths_in_rts_with_coherent_normal.PCong.intros(5)
thf(fact_476_Arr__has__Src,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Srcs_a @ resid @ T4 )
!= bot_bot_set_a ) ) ).
% Arr_has_Src
thf(fact_477_Srcs_Osimps_I1_J,axiom,
( ( paths_in_Srcs_a @ resid @ nil_a )
= bot_bot_set_a ) ).
% Srcs.simps(1)
thf(fact_478_sources__cons,axiom,
! [T3: a,T4: list_a] :
( ( paths_in_Arr_a @ resid @ ( cons_a @ T3 @ T4 ) )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T3 @ T4 ) )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T3 @ nil_a ) ) ) ) ).
% sources_cons
thf(fact_479_joinable__def,axiom,
! [T3: list_a,U2: list_a] :
( ( joinable_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
= ( ? [X5: list_a] : ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ X5 ) ) ) ).
% joinable_def
thf(fact_480_Resid1x_Opelims,axiom,
! [X: a,Xa: list_a,Y: a] :
( ( ( paths_in_Resid1x_a @ resid @ X @ Xa )
= Y )
=> ( ( accp_P3213725926765619766list_a @ ( paths_6492648068886854876_rel_a @ resid ) @ ( produc6670463072477821725list_a @ X @ Xa ) )
=> ( ( ( Xa = nil_a )
=> ( ( Y
= ( partial_null_a @ resid ) )
=> ~ ( accp_P3213725926765619766list_a @ ( paths_6492648068886854876_rel_a @ resid ) @ ( produc6670463072477821725list_a @ X @ nil_a ) ) ) )
=> ( ! [U: a] :
( ( Xa
= ( cons_a @ U @ nil_a ) )
=> ( ( Y
= ( resid @ X @ U ) )
=> ~ ( accp_P3213725926765619766list_a @ ( paths_6492648068886854876_rel_a @ resid ) @ ( produc6670463072477821725list_a @ X @ ( cons_a @ U @ nil_a ) ) ) ) )
=> ~ ! [U: a,V: a,Va: list_a] :
( ( Xa
= ( cons_a @ U @ ( cons_a @ V @ Va ) ) )
=> ( ( Y
= ( paths_in_Resid1x_a @ resid @ ( resid @ X @ U ) @ ( cons_a @ V @ Va ) ) )
=> ~ ( accp_P3213725926765619766list_a @ ( paths_6492648068886854876_rel_a @ resid ) @ ( produc6670463072477821725list_a @ X @ ( cons_a @ U @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ) ) ).
% Resid1x.pelims
thf(fact_481_Srcs__Resid__Arr__single,axiom,
! [T4: list_a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) ) )
= ( targets_a @ resid @ U2 ) ) ) ).
% Srcs_Resid_Arr_single
thf(fact_482_R_Otargets__cong__closed,axiom,
! [B: a,T3: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T3 ) )
=> ( ( ( ide_a @ resid @ ( resid @ B @ B4 ) )
& ( ide_a @ resid @ ( resid @ B4 @ B ) ) )
=> ( member_a @ B4 @ ( targets_a @ resid @ T3 ) ) ) ) ).
% R.targets_cong_closed
thf(fact_483_R_Otargets__are__cong,axiom,
! [B: a,T3: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T3 ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T3 ) )
=> ( ( ide_a @ resid @ ( resid @ B @ B4 ) )
& ( ide_a @ resid @ ( resid @ B4 @ B ) ) ) ) ) ).
% R.targets_are_cong
thf(fact_484_R_Otarget__is__ide,axiom,
! [A: a,T3: a] :
( ( member_a @ A @ ( targets_a @ resid @ T3 ) )
=> ( ide_a @ resid @ A ) ) ).
% R.target_is_ide
thf(fact_485_R_Otargets__resid__sym,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ( ( targets_a @ resid @ ( resid @ T3 @ U2 ) )
= ( targets_a @ resid @ ( resid @ U2 @ T3 ) ) ) ) ).
% R.targets_resid_sym
thf(fact_486_R_Otargets__are__con,axiom,
! [B: a,T3: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T3 ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T3 ) )
=> ( con_a @ resid @ B @ B4 ) ) ) ).
% R.targets_are_con
thf(fact_487_join__of__symmetric,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ V2 ) ) ).
% join_of_symmetric
thf(fact_488_sources__cong__closed,axiom,
! [A: list_a,T3: list_a,A4: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( ( 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 ) @ T3 ) ) ) ) ).
% sources_cong_closed
thf(fact_489_sources__are__cong,axiom,
! [A: list_a,T3: list_a,A4: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( member_list_a @ A4 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( 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 ) ) ) ) ) ).
% sources_are_cong
thf(fact_490_source__is__ide,axiom,
! [A: list_a,T3: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A ) ) ).
% source_is_ide
thf(fact_491_sources__are__con,axiom,
! [A: list_a,T3: list_a,A4: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( member_list_a @ A4 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ A4 ) ) ) ).
% sources_are_con
thf(fact_492_R_Otargets__con__closed,axiom,
! [B: a,T3: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T3 ) )
=> ( ( ide_a @ resid @ B4 )
=> ( ( con_a @ resid @ B @ B4 )
=> ( member_a @ B4 @ ( targets_a @ resid @ T3 ) ) ) ) ) ).
% R.targets_con_closed
thf(fact_493_join__of__un__upto__cong,axiom,
! [T3: list_a,U2: list_a,V2: list_a,V5: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V5 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V2 @ V5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V5 @ V2 ) ) ) ) ) ).
% join_of_un_upto_cong
thf(fact_494_join__of__resid,axiom,
! [T3: list_a,U2: list_a,W2: list_a,V2: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ W2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ V2 @ W2 )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ V2 ) @ ( paths_in_Resid_a @ resid @ U2 @ V2 ) @ ( paths_in_Resid_a @ resid @ W2 @ V2 ) ) ) ) ).
% join_of_resid
thf(fact_495_con__with__join__of__iff_I1_J,axiom,
! [T3: list_a,U2: list_a,W2: list_a,V2: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ W2 )
=> ( ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ V2 )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V2 @ U2 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W2 @ V2 ) ) ) ).
% con_with_join_of_iff(1)
thf(fact_496_con__with__join__of__iff_I2_J,axiom,
! [T3: list_a,U2: list_a,W2: list_a,V2: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ W2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W2 @ V2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ V2 )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V2 @ T3 ) @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) ) ) ) ) ).
% con_with_join_of_iff(2)
thf(fact_497_sources__join__of_I1_J,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V2 ) ) ) ).
% sources_join_of(1)
thf(fact_498_sources__join__of_I2_J,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V2 ) ) ) ).
% sources_join_of(2)
thf(fact_499_join__of__arr__self,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ T3 @ T3 ) ) ).
% join_of_arr_self
thf(fact_500_R_Oarr__iff__has__target,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
= ( ( targets_a @ resid @ T3 )
!= bot_bot_set_a ) ) ).
% R.arr_iff_has_target
thf(fact_501_R_Ocoterminal__iff,axiom,
! [T3: a,T6: a] :
( ( coterminal_a @ resid @ T3 @ T6 )
= ( ( arr_a @ resid @ T3 )
& ( arr_a @ resid @ T6 )
& ( ( targets_a @ resid @ T3 )
= ( targets_a @ resid @ T6 ) ) ) ) ).
% R.coterminal_iff
thf(fact_502_R_OcoterminalE,axiom,
! [T3: a,U2: a] :
( ( coterminal_a @ resid @ T3 @ U2 )
=> ~ ( ( arr_a @ resid @ T3 )
=> ( ( arr_a @ resid @ U2 )
=> ( ( targets_a @ resid @ T3 )
!= ( targets_a @ resid @ U2 ) ) ) ) ) ).
% R.coterminalE
thf(fact_503_in__sourcesE,axiom,
! [A: list_a,T3: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ~ ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ~ ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ A ) ) ) ).
% in_sourcesE
thf(fact_504_sources__con__closed,axiom,
! [A: list_a,T3: list_a,A4: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( 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 ) @ T3 ) ) ) ) ) ).
% sources_con_closed
thf(fact_505_join__of__arr__src_I2_J,axiom,
! [T3: list_a,A: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ A @ T3 ) ) ) ).
% join_of_arr_src(2)
thf(fact_506_join__of__arr__src_I1_J,axiom,
! [T3: list_a,A: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ A @ T3 @ T3 ) ) ) ).
% join_of_arr_src(1)
thf(fact_507_coinitialE,axiom,
! [T3: list_a,U2: list_a] :
( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ~ ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
!= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ) ) ).
% coinitialE
thf(fact_508_coinitial__iff,axiom,
! [T3: list_a,T6: list_a] :
( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ T6 )
= ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
& ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T6 )
& ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) ) ) ) ).
% coinitial_iff
thf(fact_509_R_OcoterminalI,axiom,
! [T3: a,U2: a] :
( ( arr_a @ resid @ T3 )
=> ( ( ( targets_a @ resid @ T3 )
= ( targets_a @ resid @ U2 ) )
=> ( coterminal_a @ resid @ T3 @ U2 ) ) ) ).
% R.coterminalI
thf(fact_510_in__sourcesI,axiom,
! [A: list_a,T3: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ A )
=> ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ) ) ).
% in_sourcesI
thf(fact_511_sources__append,axiom,
! [T4: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ ( append_a @ T4 @ U3 ) )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T4 ) ) ) ).
% sources_append
thf(fact_512_coinitialI,axiom,
! [T3: list_a,U2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ) ).
% coinitialI
thf(fact_513_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T3 )
=> ( ( 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 @ T3 )
= ( sources_a @ Resid @ U2 ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T3 @ U4 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_514_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,U4: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( arr_list_a @ Resid @ T3 )
=> ( ( 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 @ T3 )
= ( sources_list_a @ Resid @ U2 ) )
=> ( ( member_list_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T3 @ U4 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_515_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: a > a > a,NN: set_a,V2: a,V5: a,W2: a,W3: a,T3: a,T6: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( member_a @ V2 @ NN )
=> ( ( member_a @ V5 @ NN )
=> ( ( member_a @ W2 @ NN )
=> ( ( member_a @ W3 @ NN )
=> ( ( ( sources_a @ Resid @ V2 )
= ( sources_a @ Resid @ W2 ) )
=> ( ( ( sources_a @ Resid @ V5 )
= ( sources_a @ Resid @ W3 ) )
=> ( ( ( targets_a @ Resid @ W2 )
= ( targets_a @ Resid @ W3 ) )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T3 @ V2 ) @ ( Resid @ T6 @ V5 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ V5 ) @ ( Resid @ T3 @ V2 ) ) @ NN ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T3 @ W2 ) @ ( Resid @ T6 @ W3 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ W3 ) @ ( Resid @ T3 @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_516_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,V2: list_a,V5: list_a,W2: list_a,W3: list_a,T3: list_a,T6: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( member_list_a @ V2 @ NN )
=> ( ( member_list_a @ V5 @ NN )
=> ( ( member_list_a @ W2 @ NN )
=> ( ( member_list_a @ W3 @ NN )
=> ( ( ( sources_list_a @ Resid @ V2 )
= ( sources_list_a @ Resid @ W2 ) )
=> ( ( ( sources_list_a @ Resid @ V5 )
= ( sources_list_a @ Resid @ W3 ) )
=> ( ( ( targets_list_a @ Resid @ W2 )
= ( targets_list_a @ Resid @ W3 ) )
=> ( ( ( member_list_a @ ( Resid @ ( Resid @ T3 @ V2 ) @ ( Resid @ T6 @ V5 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ V5 ) @ ( Resid @ T3 @ V2 ) ) @ NN ) )
=> ( ( member_list_a @ ( Resid @ ( Resid @ T3 @ W2 ) @ ( Resid @ T6 @ W3 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ W3 ) @ ( Resid @ T3 @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_517_rts_Otargets_Ocong,axiom,
targets_a = targets_a ).
% rts.targets.cong
thf(fact_518_rts_Otargets_Ocong,axiom,
targets_list_a = targets_list_a ).
% rts.targets.cong
thf(fact_519_rts_Osources_Ocong,axiom,
sources_list_a = sources_list_a ).
% rts.sources.cong
thf(fact_520_rts_Osources_Ocong,axiom,
sources_a = sources_a ).
% rts.sources.cong
thf(fact_521_rts_Ojoin__of_Ocong,axiom,
join_of_list_a = join_of_list_a ).
% rts.join_of.cong
thf(fact_522_rts_Ojoin__of_Ocong,axiom,
join_of_a = join_of_a ).
% rts.join_of.cong
thf(fact_523_Ide_Oelims_I3_J,axiom,
! [X: list_a] :
( ~ ( paths_in_Ide_a @ resid @ X )
=> ( ( X != nil_a )
=> ( ! [T: a] :
( ( X
= ( cons_a @ T @ nil_a ) )
=> ( ide_a @ resid @ T ) )
=> ~ ! [T: a,V: a,Va: list_a] :
( ( X
= ( cons_a @ T @ ( cons_a @ V @ Va ) ) )
=> ( ( ide_a @ resid @ T )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ).
% Ide.elims(3)
thf(fact_524_Ide_Oelims_I2_J,axiom,
! [X: list_a] :
( ( paths_in_Ide_a @ resid @ X )
=> ( ! [T: a] :
( ( X
= ( cons_a @ T @ nil_a ) )
=> ~ ( ide_a @ resid @ T ) )
=> ~ ! [T: a,V: a,Va: list_a] :
( ( X
= ( cons_a @ T @ ( cons_a @ V @ Va ) ) )
=> ~ ( ( ide_a @ resid @ T )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ).
% Ide.elims(2)
thf(fact_525_Ide_Oelims_I1_J,axiom,
! [X: list_a,Y: $o] :
( ( ( paths_in_Ide_a @ resid @ X )
= Y )
=> ( ( ( X = nil_a )
=> Y )
=> ( ! [T: a] :
( ( X
= ( cons_a @ T @ nil_a ) )
=> ( Y
= ( ~ ( ide_a @ resid @ T ) ) ) )
=> ~ ! [T: a,V: a,Va: list_a] :
( ( X
= ( cons_a @ T @ ( cons_a @ V @ Va ) ) )
=> ( Y
= ( ~ ( ( ide_a @ resid @ T )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ) ) ).
% Ide.elims(1)
thf(fact_526_Arr_Oelims_I1_J,axiom,
! [X: list_a,Y: $o] :
( ( ( paths_in_Arr_a @ resid @ X )
= Y )
=> ( ( ( X = nil_a )
=> Y )
=> ( ! [T: a] :
( ( X
= ( cons_a @ T @ nil_a ) )
=> ( Y
= ( ~ ( arr_a @ resid @ T ) ) ) )
=> ~ ! [T: a,V: a,Va: list_a] :
( ( X
= ( cons_a @ T @ ( cons_a @ V @ Va ) ) )
=> ( Y
= ( ~ ( ( arr_a @ resid @ T )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ) ) ).
% Arr.elims(1)
thf(fact_527_Arr_Oelims_I2_J,axiom,
! [X: list_a] :
( ( paths_in_Arr_a @ resid @ X )
=> ( ! [T: a] :
( ( X
= ( cons_a @ T @ nil_a ) )
=> ~ ( arr_a @ resid @ T ) )
=> ~ ! [T: a,V: a,Va: list_a] :
( ( X
= ( cons_a @ T @ ( cons_a @ V @ Va ) ) )
=> ~ ( ( arr_a @ resid @ T )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ).
% Arr.elims(2)
thf(fact_528_R_Osources__join__of_I2_J,axiom,
! [T3: a,U2: a,V2: a] :
( ( join_of_a @ resid @ T3 @ U2 @ V2 )
=> ( ( sources_a @ resid @ U2 )
= ( sources_a @ resid @ V2 ) ) ) ).
% R.sources_join_of(2)
thf(fact_529_R_Osources__join__of_I1_J,axiom,
! [T3: a,U2: a,V2: a] :
( ( join_of_a @ resid @ T3 @ U2 @ V2 )
=> ( ( sources_a @ resid @ T3 )
= ( sources_a @ resid @ V2 ) ) ) ).
% R.sources_join_of(1)
thf(fact_530_R_Ojoin__of__symmetric,axiom,
! [T3: a,U2: a,V2: a] :
( ( join_of_a @ resid @ T3 @ U2 @ V2 )
=> ( join_of_a @ resid @ U2 @ T3 @ V2 ) ) ).
% R.join_of_symmetric
thf(fact_531_R_Osource__is__ide,axiom,
! [A: a,T3: a] :
( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ( ide_a @ resid @ A ) ) ).
% R.source_is_ide
thf(fact_532_R_Osources__are__cong,axiom,
! [A: a,T3: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T3 ) )
=> ( ( ide_a @ resid @ ( resid @ A @ A4 ) )
& ( ide_a @ resid @ ( resid @ A4 @ A ) ) ) ) ) ).
% R.sources_are_cong
thf(fact_533_R_Osources__cong__closed,axiom,
! [A: a,T3: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ( ( ( ide_a @ resid @ ( resid @ A @ A4 ) )
& ( ide_a @ resid @ ( resid @ A4 @ A ) ) )
=> ( member_a @ A4 @ ( sources_a @ resid @ T3 ) ) ) ) ).
% R.sources_cong_closed
thf(fact_534_R_Ojoin__of__un__upto__cong,axiom,
! [T3: a,U2: a,V2: a,V5: a] :
( ( join_of_a @ resid @ T3 @ U2 @ V2 )
=> ( ( join_of_a @ resid @ T3 @ U2 @ V5 )
=> ( ( ide_a @ resid @ ( resid @ V2 @ V5 ) )
& ( ide_a @ resid @ ( resid @ V5 @ V2 ) ) ) ) ) ).
% R.join_of_un_upto_cong
thf(fact_535_R_Osources__are__con,axiom,
! [A: a,T3: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T3 ) )
=> ( con_a @ resid @ A @ A4 ) ) ) ).
% R.sources_are_con
thf(fact_536_R_Ocon__with__join__of__iff_I2_J,axiom,
! [T3: a,U2: a,W2: a,V2: a] :
( ( join_of_a @ resid @ T3 @ U2 @ W2 )
=> ( ( con_a @ resid @ W2 @ V2 )
=> ( ( con_a @ resid @ T3 @ V2 )
& ( con_a @ resid @ ( resid @ V2 @ T3 ) @ ( resid @ U2 @ T3 ) ) ) ) ) ).
% R.con_with_join_of_iff(2)
thf(fact_537_R_Ocon__with__join__of__iff_I1_J,axiom,
! [T3: a,U2: a,W2: a,V2: a] :
( ( join_of_a @ resid @ T3 @ U2 @ W2 )
=> ( ( ( con_a @ resid @ U2 @ V2 )
& ( con_a @ resid @ ( resid @ V2 @ U2 ) @ ( resid @ T3 @ U2 ) ) )
=> ( con_a @ resid @ W2 @ V2 ) ) ) ).
% R.con_with_join_of_iff(1)
thf(fact_538_R_Ojoin__of__resid,axiom,
! [T3: a,U2: a,W2: a,V2: a] :
( ( join_of_a @ resid @ T3 @ U2 @ W2 )
=> ( ( con_a @ resid @ V2 @ W2 )
=> ( join_of_a @ resid @ ( resid @ T3 @ V2 ) @ ( resid @ U2 @ V2 ) @ ( resid @ W2 @ V2 ) ) ) ) ).
% R.join_of_resid
thf(fact_539_R_Ojoin__of__arr__src_I2_J,axiom,
! [T3: a,A: a] :
( ( arr_a @ resid @ T3 )
=> ( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ( join_of_a @ resid @ T3 @ A @ T3 ) ) ) ).
% R.join_of_arr_src(2)
thf(fact_540_R_Ojoin__of__arr__src_I1_J,axiom,
! [T3: a,A: a] :
( ( arr_a @ resid @ T3 )
=> ( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ( join_of_a @ resid @ A @ T3 @ T3 ) ) ) ).
% R.join_of_arr_src(1)
thf(fact_541_R_Ojoin__of__arr__self,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
=> ( join_of_a @ resid @ T3 @ T3 @ T3 ) ) ).
% R.join_of_arr_self
thf(fact_542_R_Oresid__source__in__targets,axiom,
! [A: a,T3: a] :
( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ( member_a @ ( resid @ A @ T3 ) @ ( targets_a @ resid @ T3 ) ) ) ).
% R.resid_source_in_targets
thf(fact_543_R_Otargets__join__of_I1_J,axiom,
! [T3: a,U2: a,V2: a] :
( ( join_of_a @ resid @ T3 @ U2 @ V2 )
=> ( ( targets_a @ resid @ ( resid @ T3 @ U2 ) )
= ( targets_a @ resid @ V2 ) ) ) ).
% R.targets_join_of(1)
thf(fact_544_R_Otargets__join__of_I2_J,axiom,
! [T3: a,U2: a,V2: a] :
( ( join_of_a @ resid @ T3 @ U2 @ V2 )
=> ( ( targets_a @ resid @ ( resid @ U2 @ T3 ) )
= ( targets_a @ resid @ V2 ) ) ) ).
% R.targets_join_of(2)
thf(fact_545_R_Ojoinable__def,axiom,
! [T3: a,U2: a] :
( ( joinable_a @ resid @ T3 @ U2 )
= ( ? [X5: a] : ( join_of_a @ resid @ T3 @ U2 @ X5 ) ) ) ).
% R.joinable_def
thf(fact_546_target__is__ide,axiom,
! [A: list_a,T3: list_a] :
( ( member_list_a @ A @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A ) ) ).
% target_is_ide
thf(fact_547_targets__are__cong,axiom,
! [B: list_a,T3: list_a,B4: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( member_list_a @ B4 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B @ B4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B4 @ B ) ) ) ) ) ).
% targets_are_cong
thf(fact_548_targets__cong__closed,axiom,
! [B: list_a,T3: list_a,B4: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B @ B4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B4 @ B ) ) )
=> ( member_list_a @ B4 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ) ) ).
% targets_cong_closed
thf(fact_549_targets__are__con,axiom,
! [B: list_a,T3: list_a,B4: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( member_list_a @ B4 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ B @ B4 ) ) ) ).
% targets_are_con
thf(fact_550_targets__resid__sym,axiom,
! [T3: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) ) ) ) ).
% targets_resid_sym
thf(fact_551_resid__source__in__targets,axiom,
! [A: list_a,T3: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( member_list_a @ ( paths_in_Resid_a @ resid @ A @ T3 ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ) ).
% resid_source_in_targets
thf(fact_552_Srcs_Osimps_I3_J,axiom,
! [T3: a,V2: a,Va2: list_a] :
( ( paths_in_Srcs_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) )
= ( sources_a @ resid @ T3 ) ) ).
% Srcs.simps(3)
thf(fact_553_R_Oin__sourcesE,axiom,
! [A: a,T3: a] :
( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ~ ( ( ide_a @ resid @ A )
=> ~ ( con_a @ resid @ T3 @ A ) ) ) ).
% R.in_sourcesE
thf(fact_554_R_Osources__con__closed,axiom,
! [A: a,T3: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ( ( ide_a @ resid @ A4 )
=> ( ( con_a @ resid @ A @ A4 )
=> ( member_a @ A4 @ ( sources_a @ resid @ T3 ) ) ) ) ) ).
% R.sources_con_closed
thf(fact_555_targets__join__of_I2_J,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ V2 ) ) ) ).
% targets_join_of(2)
thf(fact_556_targets__join__of_I1_J,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ V2 ) ) ) ).
% targets_join_of(1)
thf(fact_557_R_Oarr__iff__has__source,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
= ( ( sources_a @ resid @ T3 )
!= bot_bot_set_a ) ) ).
% R.arr_iff_has_source
thf(fact_558_R_OcoinitialE,axiom,
! [T3: a,U2: a] :
( ( coinitial_a @ resid @ T3 @ U2 )
=> ~ ( ( arr_a @ resid @ T3 )
=> ( ( arr_a @ resid @ U2 )
=> ( ( sources_a @ resid @ T3 )
!= ( sources_a @ resid @ U2 ) ) ) ) ) ).
% R.coinitialE
thf(fact_559_R_Ocoinitial__iff,axiom,
! [T3: a,T6: a] :
( ( coinitial_a @ resid @ T3 @ T6 )
= ( ( arr_a @ resid @ T3 )
& ( arr_a @ resid @ T6 )
& ( ( sources_a @ resid @ T3 )
= ( sources_a @ resid @ T6 ) ) ) ) ).
% R.coinitial_iff
thf(fact_560_R_OcomposableD_I3_J,axiom,
! [T3: a,U2: a] :
( ( composable_a @ resid @ T3 @ U2 )
=> ( ( targets_a @ resid @ T3 )
= ( sources_a @ resid @ U2 ) ) ) ).
% R.composableD(3)
thf(fact_561_Srcs_Osimps_I2_J,axiom,
! [T3: a] :
( ( paths_in_Srcs_a @ resid @ ( cons_a @ T3 @ nil_a ) )
= ( sources_a @ resid @ T3 ) ) ).
% Srcs.simps(2)
thf(fact_562_targets__con__closed,axiom,
! [B: list_a,T3: list_a,B4: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ B4 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ B @ B4 )
=> ( member_list_a @ B4 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ) ) ) ).
% targets_con_closed
thf(fact_563_Ide__imp__sources__eq__targets,axiom,
! [T4: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T4 )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T4 ) ) ) ).
% Ide_imp_sources_eq_targets
thf(fact_564_Srcs__are__ide,axiom,
! [T4: list_a] : ( ord_less_eq_set_a @ ( paths_in_Srcs_a @ resid @ T4 ) @ ( collect_a @ ( ide_a @ resid ) ) ) ).
% Srcs_are_ide
thf(fact_565_composableD_I3_J,axiom,
! [T3: list_a,U2: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ).
% composableD(3)
thf(fact_566_coterminal__iff,axiom,
! [T3: list_a,T6: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ T6 )
= ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
& ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T6 )
& ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) ) ) ) ).
% coterminal_iff
thf(fact_567_coterminalE,axiom,
! [T3: list_a,U2: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ~ ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
!= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ) ) ).
% coterminalE
thf(fact_568_R_Oseq__def,axiom,
! [T3: a,U2: a] :
( ( seq_a @ resid @ T3 @ U2 )
= ( ( arr_a @ resid @ T3 )
& ( arr_a @ resid @ U2 )
& ( ( targets_a @ resid @ T3 )
= ( sources_a @ resid @ U2 ) ) ) ) ).
% R.seq_def
thf(fact_569_R_OseqE,axiom,
! [T3: a,U2: a] :
( ( seq_a @ resid @ T3 @ U2 )
=> ~ ( ( arr_a @ resid @ T3 )
=> ( ( arr_a @ resid @ U2 )
=> ( ( targets_a @ resid @ T3 )
!= ( sources_a @ resid @ U2 ) ) ) ) ) ).
% R.seqE
thf(fact_570_Srcs__simp_092_060_094sub_062P,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Srcs_a @ resid @ T4 )
= ( sources_a @ resid @ ( hd_a @ T4 ) ) ) ) ).
% Srcs_simp\<^sub>P
thf(fact_571_Srcs_Oelims,axiom,
! [X: list_a,Y: set_a] :
( ( ( paths_in_Srcs_a @ resid @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != bot_bot_set_a ) )
=> ( ! [T: a] :
( ( X
= ( cons_a @ T @ nil_a ) )
=> ( Y
!= ( sources_a @ resid @ T ) ) )
=> ~ ! [T: a] :
( ? [V: a,Va: list_a] :
( X
= ( cons_a @ T @ ( cons_a @ V @ Va ) ) )
=> ( Y
!= ( sources_a @ resid @ T ) ) ) ) ) ) ).
% Srcs.elims
thf(fact_572_seq__def,axiom,
! [T3: list_a,U2: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
= ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
& ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
& ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ) ).
% seq_def
thf(fact_573_seqE,axiom,
! [T3: list_a,U2: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ~ ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
!= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ) ) ).
% seqE
thf(fact_574_Arr_Osimps_I3_J,axiom,
! [T3: a,V2: a,Va2: list_a] :
( ( paths_in_Arr_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) )
= ( ( arr_a @ resid @ T3 )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V2 @ Va2 ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va2 ) ) ) ) ) ).
% Arr.simps(3)
thf(fact_575_Ide_Osimps_I3_J,axiom,
! [T3: a,V2: a,Va2: list_a] :
( ( paths_in_Ide_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) )
= ( ( ide_a @ resid @ T3 )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V2 @ Va2 ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va2 ) ) ) ) ) ).
% Ide.simps(3)
thf(fact_576_Arr_Oelims_I3_J,axiom,
! [X: list_a] :
( ~ ( paths_in_Arr_a @ resid @ X )
=> ( ( X != nil_a )
=> ( ! [T: a] :
( ( X
= ( cons_a @ T @ nil_a ) )
=> ( arr_a @ resid @ T ) )
=> ~ ! [T: a,V: a,Va: list_a] :
( ( X
= ( cons_a @ T @ ( cons_a @ V @ Va ) ) )
=> ( ( arr_a @ resid @ T )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ).
% Arr.elims(3)
thf(fact_577_R_Oin__sourcesI,axiom,
! [A: a,T3: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ T3 @ A )
=> ( member_a @ A @ ( sources_a @ resid @ T3 ) ) ) ) ).
% R.in_sourcesI
thf(fact_578_R_Osources__resid,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ( ( sources_a @ resid @ ( resid @ T3 @ U2 ) )
= ( targets_a @ resid @ U2 ) ) ) ).
% R.sources_resid
thf(fact_579_R_OcoinitialI,axiom,
! [T3: a,U2: a] :
( ( arr_a @ resid @ T3 )
=> ( ( ( sources_a @ resid @ T3 )
= ( sources_a @ resid @ U2 ) )
=> ( coinitial_a @ resid @ T3 @ U2 ) ) ) ).
% R.coinitialI
thf(fact_580_sources__resid,axiom,
! [T3: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ).
% sources_resid
thf(fact_581_targets__append,axiom,
! [T4: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( append_a @ T4 @ U3 ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U3 ) ) ) ).
% targets_append
thf(fact_582_coterminalI,axiom,
! [T3: list_a,U2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ) ).
% coterminalI
thf(fact_583_R_OseqI,axiom,
! [T3: a,U2: a] :
( ( arr_a @ resid @ T3 )
=> ( ( arr_a @ resid @ U2 )
=> ( ( ( targets_a @ resid @ T3 )
= ( sources_a @ resid @ U2 ) )
=> ( seq_a @ resid @ T3 @ U2 ) ) ) ) ).
% R.seqI
thf(fact_584_seqI,axiom,
! [T3: list_a,U2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U2 )
=> ( ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 ) ) ) ) ).
% seqI
thf(fact_585_Arr__consI_092_060_094sub_062P,axiom,
! [T3: a,U3: list_a] :
( ( arr_a @ resid @ T3 )
=> ( ( paths_in_Arr_a @ resid @ U3 )
=> ( ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( paths_in_Arr_a @ resid @ ( cons_a @ T3 @ U3 ) ) ) ) ) ).
% Arr_consI\<^sub>P
thf(fact_586_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U3: list_list_a] :
( ( paths_2227358642439567705list_a @ Resid )
=> ( ! [T: list_a,U: list_a] :
( ( coinitial_list_a @ Resid @ T @ U )
=> ( con_list_a @ Resid @ T @ U ) )
=> ( ( arr_list_a @ Resid @ T3 )
=> ( ( paths_in_Arr_list_a @ Resid @ U3 )
=> ( ( ( sources_list_a @ Resid @ T3 )
= ( paths_in_Srcs_list_a @ Resid @ U3 ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T3 @ nil_list_a ) @ U3 )
!= nil_list_a ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_587_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: a > a > a,T3: a,U3: list_a] :
( ( paths_8083246678931295059_rts_a @ Resid )
=> ( ! [T: a,U: a] :
( ( coinitial_a @ Resid @ T @ U )
=> ( con_a @ Resid @ T @ U ) )
=> ( ( arr_a @ Resid @ T3 )
=> ( ( paths_in_Arr_a @ Resid @ U3 )
=> ( ( ( sources_a @ Resid @ T3 )
= ( paths_in_Srcs_a @ Resid @ U3 ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T3 @ nil_a ) @ U3 )
!= nil_a ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_588_Ide__char,axiom,
! [T4: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
= ( ( paths_in_Arr_a @ resid @ T4 )
& ( ord_less_eq_set_a @ ( set_a2 @ T4 ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ) ).
% Ide_char
thf(fact_589_arr__iff__has__source,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
!= bot_bot_set_list_a ) ) ).
% arr_iff_has_source
thf(fact_590_R_Ocoterminal__def,axiom,
! [T3: a,U2: a] :
( ( coterminal_a @ resid @ T3 @ U2 )
= ( ( inf_inf_set_a @ ( targets_a @ resid @ T3 ) @ ( targets_a @ resid @ U2 ) )
!= bot_bot_set_a ) ) ).
% R.coterminal_def
thf(fact_591_Ide__append__iff_092_060_094sub_062P,axiom,
! [T4: list_a,U3: list_a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( paths_in_Ide_a @ resid @ ( append_a @ T4 @ U3 ) )
= ( ( paths_in_Ide_a @ resid @ T4 )
& ( paths_in_Ide_a @ resid @ U3 )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U3 ) ) ) ) ) ) ).
% Ide_append_iff\<^sub>P
thf(fact_592_Trgs_Osimps_I3_J,axiom,
! [T3: a,V2: a,Va2: list_a] :
( ( paths_in_Trgs_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) )
= ( paths_in_Trgs_a @ resid @ ( cons_a @ V2 @ Va2 ) ) ) ).
% Trgs.simps(3)
thf(fact_593_Trgs__are__con,axiom,
! [B: a,T4: list_a,B4: a] :
( ( member_a @ B @ ( paths_in_Trgs_a @ resid @ T4 ) )
=> ( ( member_a @ B4 @ ( paths_in_Trgs_a @ resid @ T4 ) )
=> ( con_a @ resid @ B @ B4 ) ) ) ).
% Trgs_are_con
thf(fact_594_Trgs__con__closed,axiom,
! [B: a,T4: list_a,B4: a] :
( ( member_a @ B @ ( paths_in_Trgs_a @ resid @ T4 ) )
=> ( ( ide_a @ resid @ B4 )
=> ( ( con_a @ resid @ B @ B4 )
=> ( member_a @ B4 @ ( paths_in_Trgs_a @ resid @ T4 ) ) ) ) ) ).
% Trgs_con_closed
thf(fact_595_Trgs_Osimps_I1_J,axiom,
( ( paths_in_Trgs_a @ resid @ nil_a )
= bot_bot_set_a ) ).
% Trgs.simps(1)
thf(fact_596_Trgs__are__ide,axiom,
! [T4: list_a] : ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( collect_a @ ( ide_a @ resid ) ) ) ).
% Trgs_are_ide
thf(fact_597_Arr__has__Trg,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Trgs_a @ resid @ T4 )
!= bot_bot_set_a ) ) ).
% Arr_has_Trg
thf(fact_598_R_Osources__eqI,axiom,
! [T3: a,T6: a] :
( ( ( inf_inf_set_a @ ( sources_a @ resid @ T3 ) @ ( sources_a @ resid @ T6 ) )
!= bot_bot_set_a )
=> ( ( sources_a @ resid @ T3 )
= ( sources_a @ resid @ T6 ) ) ) ).
% R.sources_eqI
thf(fact_599_Srcs__eqI,axiom,
! [T4: list_a,T7: list_a] :
( ( ( inf_inf_set_a @ ( paths_in_Srcs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ T7 ) )
!= bot_bot_set_a )
=> ( ( paths_in_Srcs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ T7 ) ) ) ).
% Srcs_eqI
thf(fact_600_R_Otargets__eqI,axiom,
! [T3: a,T6: a] :
( ( ( inf_inf_set_a @ ( targets_a @ resid @ T3 ) @ ( targets_a @ resid @ T6 ) )
!= bot_bot_set_a )
=> ( ( targets_a @ resid @ T3 )
= ( targets_a @ resid @ T6 ) ) ) ).
% R.targets_eqI
thf(fact_601_Trgs__eqI,axiom,
! [T4: list_a,T7: list_a] :
( ( ( inf_inf_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Trgs_a @ resid @ T7 ) )
!= bot_bot_set_a )
=> ( ( paths_in_Trgs_a @ resid @ T4 )
= ( paths_in_Trgs_a @ resid @ T7 ) ) ) ).
% Trgs_eqI
thf(fact_602_Trgs__Resid__sym__Arr__single,axiom,
! [T4: list_a,U2: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) ) )
= ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ U2 @ nil_a ) @ T4 ) ) ) ) ).
% Trgs_Resid_sym_Arr_single
thf(fact_603_Trgs_Osimps_I2_J,axiom,
! [T3: a] :
( ( paths_in_Trgs_a @ resid @ ( cons_a @ T3 @ nil_a ) )
= ( targets_a @ resid @ T3 ) ) ).
% Trgs.simps(2)
thf(fact_604_seq__implies__Trgs__eq__Srcs,axiom,
! [T4: list_a,U3: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Arr_a @ resid @ U3 )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( ( paths_in_Trgs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ U3 ) ) ) ) ) ).
% seq_implies_Trgs_eq_Srcs
thf(fact_605_arr__iff__has__target,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
!= bot_bot_set_list_a ) ) ).
% arr_iff_has_target
thf(fact_606_R_Ocon__imp__common__source,axiom,
! [T3: a,U2: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ( ( inf_inf_set_a @ ( sources_a @ resid @ T3 ) @ ( sources_a @ resid @ U2 ) )
!= bot_bot_set_a ) ) ).
% R.con_imp_common_source
thf(fact_607_set__Arr__subset__arr,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ord_less_eq_set_a @ ( set_a2 @ T4 ) @ ( collect_a @ ( arr_a @ resid ) ) ) ) ).
% set_Arr_subset_arr
thf(fact_608_set__Ide__subset__ide,axiom,
! [T4: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
=> ( ord_less_eq_set_a @ ( set_a2 @ T4 ) @ ( collect_a @ ( ide_a @ resid ) ) ) ) ).
% set_Ide_subset_ide
thf(fact_609_R_Ocoinitial__def,axiom,
! [T3: a,U2: a] :
( ( coinitial_a @ resid @ T3 @ U2 )
= ( ( inf_inf_set_a @ ( sources_a @ resid @ T3 ) @ ( sources_a @ resid @ U2 ) )
!= bot_bot_set_a ) ) ).
% R.coinitial_def
thf(fact_610_Trgs__simp_092_060_094sub_062P,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Trgs_a @ resid @ T4 )
= ( targets_a @ resid @ ( last_a @ T4 ) ) ) ) ).
% Trgs_simp\<^sub>P
thf(fact_611_Srcs__Resid__single__Arr,axiom,
! [U2: a,T4: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ U2 @ nil_a ) @ T4 )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ U2 @ nil_a ) @ T4 ) )
= ( paths_in_Trgs_a @ resid @ T4 ) ) ) ).
% Srcs_Resid_single_Arr
thf(fact_612_Arr__appendE_092_060_094sub_062P,axiom,
! [T4: list_a,U3: list_a] :
( ( paths_in_Arr_a @ resid @ ( append_a @ T4 @ U3 ) )
=> ( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ~ ( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Arr_a @ resid @ U3 )
=> ( ( paths_in_Trgs_a @ resid @ T4 )
!= ( paths_in_Srcs_a @ resid @ U3 ) ) ) ) ) ) ) ).
% Arr_appendE\<^sub>P
thf(fact_613_seq__char,axiom,
! [T4: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
= ( ( paths_in_Arr_a @ resid @ T4 )
& ( paths_in_Arr_a @ resid @ U3 )
& ( ( paths_in_Trgs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ U3 ) ) ) ) ).
% seq_char
thf(fact_614_Trgs_Oelims,axiom,
! [X: list_a,Y: set_a] :
( ( ( paths_in_Trgs_a @ resid @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != bot_bot_set_a ) )
=> ( ! [T: a] :
( ( X
= ( cons_a @ T @ nil_a ) )
=> ( Y
!= ( targets_a @ resid @ T ) ) )
=> ~ ! [T: a,V: a,Va: list_a] :
( ( X
= ( cons_a @ T @ ( cons_a @ V @ Va ) ) )
=> ( Y
!= ( paths_in_Trgs_a @ resid @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ).
% Trgs.elims
thf(fact_615_Arr__append__iff_092_060_094sub_062P,axiom,
! [T4: list_a,U3: list_a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( paths_in_Arr_a @ resid @ ( append_a @ T4 @ U3 ) )
= ( ( paths_in_Arr_a @ resid @ T4 )
& ( paths_in_Arr_a @ resid @ U3 )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U3 ) ) ) ) ) ) ).
% Arr_append_iff\<^sub>P
thf(fact_616_set__rotate1,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rotate1_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_617_seq__char_H,axiom,
! [T4: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
= ( ( paths_in_Arr_a @ resid @ T4 )
& ( paths_in_Arr_a @ resid @ U3 )
& ( ( inf_inf_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U3 ) )
!= bot_bot_set_a ) ) ) ).
% seq_char'
thf(fact_618_in__set__insert,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ( insert_list_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_619_in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_620_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_621_set__empty,axiom,
! [Xs: list_list_a] :
( ( ( set_list_a2 @ Xs )
= bot_bot_set_list_a )
= ( Xs = nil_list_a ) ) ).
% set_empty
thf(fact_622_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_623_set__empty2,axiom,
! [Xs: list_list_a] :
( ( bot_bot_set_list_a
= ( set_list_a2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% set_empty2
thf(fact_624_not__in__set__insert,axiom,
! [X: list_a,Xs: list_list_a] :
( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ( insert_list_a @ X @ Xs )
= ( cons_list_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_625_not__in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= ( cons_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_626_Trgs__Resid__sym,axiom,
! [T4: list_a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
=> ( ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) )
= ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ U3 @ T4 ) ) ) ) ).
% Trgs_Resid_sym
thf(fact_627_Trgs__append,axiom,
! [U3: list_a,T4: list_a] :
( ( U3 != nil_a )
=> ( ( paths_in_Trgs_a @ resid @ ( append_a @ T4 @ U3 ) )
= ( paths_in_Trgs_a @ resid @ U3 ) ) ) ).
% Trgs_append
thf(fact_628_Srcs__Resid,axiom,
! [T4: list_a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) )
= ( paths_in_Trgs_a @ resid @ U3 ) ) ) ).
% Srcs_Resid
thf(fact_629_Arr__appendI_092_060_094sub_062P,axiom,
! [T4: list_a,U3: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Arr_a @ resid @ U3 )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( paths_in_Arr_a @ resid @ ( append_a @ T4 @ U3 ) ) ) ) ) ).
% Arr_appendI\<^sub>P
thf(fact_630_Ide__appendI_092_060_094sub_062P,axiom,
! [T4: list_a,U3: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
=> ( ( paths_in_Ide_a @ resid @ U3 )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( paths_in_Ide_a @ resid @ ( append_a @ T4 @ U3 ) ) ) ) ) ).
% Ide_appendI\<^sub>P
thf(fact_631_IdeI,axiom,
! [T4: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ T4 ) @ ( collect_a @ ( ide_a @ resid ) ) )
=> ( paths_in_Ide_a @ resid @ T4 ) ) ) ).
% IdeI
thf(fact_632_seqI_092_060_094sub_062P,axiom,
! [T4: list_a,U3: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Arr_a @ resid @ U3 )
=> ( ( ( inf_inf_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U3 ) )
!= bot_bot_set_a )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 ) ) ) ) ).
% seqI\<^sub>P
thf(fact_633_subset__code_I1_J,axiom,
! [Xs: list_list_a,B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B5 )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X3 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_634_subset__code_I1_J,axiom,
! [Xs: list_a,B5: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B5 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_635_paths__in__rts_OTrgs_Ocong,axiom,
paths_in_Trgs_a = paths_in_Trgs_a ).
% paths_in_rts.Trgs.cong
thf(fact_636_set__ConsD,axiom,
! [Y: list_a,X: list_a,Xs: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_list_a @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_637_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_638_list_Oset__cases,axiom,
! [E: list_a,A: list_list_a] :
( ( member_list_a @ E @ ( set_list_a2 @ A ) )
=> ( ! [Z2: list_list_a] :
( A
!= ( cons_list_a @ E @ Z2 ) )
=> ~ ! [Z1: list_a,Z2: list_list_a] :
( ( A
= ( cons_list_a @ Z1 @ Z2 ) )
=> ~ ( member_list_a @ E @ ( set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_639_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z2: list_a] :
( A
!= ( cons_a @ E @ Z2 ) )
=> ~ ! [Z1: a,Z2: list_a] :
( ( A
= ( cons_a @ Z1 @ Z2 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_640_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_641_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_642_list_Oset__intros_I2_J,axiom,
! [Y: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ X22 ) )
=> ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_643_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_644_in__set__butlastD,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ Xs ) ) )
=> ( member_list_a @ X @ ( set_list_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_645_in__set__butlastD,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ ( butlast_a @ Xs ) ) )
=> ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_646_list__ex1__iff,axiom,
( list_ex1_list_a
= ( ^ [P2: list_a > $o,Xs4: list_list_a] :
? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs4 ) )
& ( P2 @ X3 )
& ! [Y3: list_a] :
( ( ( member_list_a @ Y3 @ ( set_list_a2 @ Xs4 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_647_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P2: a > $o,Xs4: list_a] :
? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs4 ) )
& ( P2 @ X3 )
& ! [Y3: a] :
( ( ( member_a @ Y3 @ ( set_a2 @ Xs4 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_648_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_649_empty__set,axiom,
( bot_bot_set_list_a
= ( set_list_a2 @ nil_list_a ) ) ).
% empty_set
thf(fact_650_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_651_split__list,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_652_split__list,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_653_split__list__last,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_654_split__list__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_655_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_656_split__list__first,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_657_split__list__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_658_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_659_append__Cons__eq__iff,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,Xs5: list_list_a,Ys6: list_list_a] :
( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ~ ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
=> ( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ Ys ) )
= ( append_list_a @ Xs5 @ ( cons_list_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_660_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys: list_a,Xs5: list_a,Ys6: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
= ( append_a @ Xs5 @ ( cons_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_661_in__set__conv__decomp,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_a,Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_662_in__set__conv__decomp,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_663_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_a,X2: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_664_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_665_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_a,X2: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_666_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_667_in__set__conv__decomp__last,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_668_in__set__conv__decomp__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_669_in__set__conv__decomp__first,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_670_in__set__conv__decomp__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_671_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys4: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_672_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys4: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Ys4 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_673_list_Oset__sel_I1_J,axiom,
! [A: list_list_a] :
( ( A != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ A ) @ ( set_list_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_674_list_Oset__sel_I1_J,axiom,
! [A: list_a] :
( ( A != nil_a )
=> ( member_a @ ( hd_a @ A ) @ ( set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_675_hd__in__set,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ Xs ) @ ( set_list_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_676_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_677_list_Oset__sel_I2_J,axiom,
! [A: list_list_a,X: list_a] :
( ( A != nil_list_a )
=> ( ( member_list_a @ X @ ( set_list_a2 @ ( tl_list_a @ A ) ) )
=> ( member_list_a @ X @ ( set_list_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_678_list_Oset__sel_I2_J,axiom,
! [A: list_a,X: a] :
( ( A != nil_a )
=> ( ( member_a @ X @ ( set_a2 @ ( tl_a @ A ) ) )
=> ( member_a @ X @ ( set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_679_last__in__set,axiom,
! [As2: list_list_a] :
( ( As2 != nil_list_a )
=> ( member_list_a @ ( last_list_a @ As2 ) @ ( set_list_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_680_last__in__set,axiom,
! [As2: list_a] :
( ( As2 != nil_a )
=> ( member_a @ ( last_a @ As2 ) @ ( set_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_681_in__set__butlast__appendI,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ Xs ) ) )
| ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ Ys ) ) ) )
=> ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ ( append_list_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_682_in__set__butlast__appendI,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( member_a @ X @ ( set_a2 @ ( butlast_a @ Xs ) ) )
| ( member_a @ X @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
=> ( member_a @ X @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_683_List_Oinsert__def,axiom,
( insert_list_a
= ( ^ [X3: list_a,Xs4: list_list_a] : ( if_list_list_a @ ( member_list_a @ X3 @ ( set_list_a2 @ Xs4 ) ) @ Xs4 @ ( cons_list_a @ X3 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_684_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X3: a,Xs4: list_a] : ( if_list_a @ ( member_a @ X3 @ ( set_a2 @ Xs4 ) ) @ Xs4 @ ( cons_a @ X3 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_685_paths__in__confluent__rts_Oconfluence_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a] :
( ( paths_8083246678931295059_rts_a @ Resid )
=> ( ( coinitial_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ U3 )
=> ( con_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ U3 ) ) ) ).
% paths_in_confluent_rts.confluence\<^sub>P
thf(fact_686_paths__in__confluent__rts_Oconfluence__ind,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a] :
( ( paths_8083246678931295059_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( ( paths_in_Arr_a @ Resid @ U3 )
=> ( ( ( paths_in_Srcs_a @ Resid @ T4 )
= ( paths_in_Srcs_a @ Resid @ U3 ) )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ U3 )
!= nil_a ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_ind
thf(fact_687_const__ide__is__Ide,axiom,
! [T4: list_a] :
( ( T4 != nil_a )
=> ( ( ide_a @ resid @ ( hd_a @ T4 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ T4 ) @ ( insert_a2 @ ( hd_a @ T4 ) @ bot_bot_set_a ) )
=> ( paths_in_Ide_a @ resid @ T4 ) ) ) ) ).
% const_ide_is_Ide
thf(fact_688_coterminal__def,axiom,
! [T3: list_a,U2: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
= ( ( inf_inf_set_list_a @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
!= bot_bot_set_list_a ) ) ).
% coterminal_def
thf(fact_689_coinitial__def,axiom,
! [T3: list_a,U2: list_a] :
( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
= ( ( inf_inf_set_list_a @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
!= bot_bot_set_list_a ) ) ).
% coinitial_def
thf(fact_690_con__imp__common__source,axiom,
! [T3: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( inf_inf_set_list_a @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
!= bot_bot_set_list_a ) ) ).
% con_imp_common_source
thf(fact_691_sources__eqI,axiom,
! [T3: list_a,T6: list_a] :
( ( ( inf_inf_set_list_a @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) )
!= bot_bot_set_list_a )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) ) ) ).
% sources_eqI
thf(fact_692_targets__eqI,axiom,
! [T3: list_a,T6: list_a] :
( ( ( inf_inf_set_list_a @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) )
!= bot_bot_set_list_a )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) ) ) ).
% targets_eqI
thf(fact_693_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a2 @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_694_set__insert,axiom,
! [X: a,Xs: list_a] :
( ( set_a2 @ ( insert_a @ X @ Xs ) )
= ( insert_a2 @ X @ ( set_a2 @ Xs ) ) ) ).
% set_insert
thf(fact_695_in__targetsE,axiom,
! [B: list_a,T3: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ~ ( ( 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 ) @ T3 ) @ B ) ) ) ).
% in_targetsE
thf(fact_696_can__select__set__list__ex1,axiom,
! [P: a > $o,A2: list_a] :
( ( can_select_a @ P @ ( set_a2 @ A2 ) )
= ( list_ex1_a @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_697_coterminal__iff__con__trg,axiom,
! [T3: list_a,U2: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ).
% coterminal_iff_con_trg
thf(fact_698_R_Oin__targetsE,axiom,
! [B: a,T3: a] :
( ( member_a @ B @ ( targets_a @ resid @ T3 ) )
=> ~ ( ( ide_a @ resid @ B )
=> ~ ( con_a @ resid @ ( trg_a @ resid @ T3 ) @ B ) ) ) ).
% R.in_targetsE
thf(fact_699_R_Otrg__def,axiom,
! [T3: a] :
( ( trg_a @ resid @ T3 )
= ( resid @ T3 @ T3 ) ) ).
% R.trg_def
thf(fact_700_trg__def,axiom,
! [T3: list_a] :
( ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( paths_in_Resid_a @ resid @ T3 @ T3 ) ) ).
% trg_def
thf(fact_701_R_Oide__trg,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
=> ( ide_a @ resid @ ( trg_a @ resid @ T3 ) ) ) ).
% R.ide_trg
thf(fact_702_R_Otrg__in__targets,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
=> ( member_a @ ( trg_a @ resid @ T3 ) @ ( targets_a @ resid @ T3 ) ) ) ).
% R.trg_in_targets
thf(fact_703_R_Ocoterminal__iff__con__trg,axiom,
! [T3: a,U2: a] :
( ( coterminal_a @ resid @ T3 @ U2 )
= ( con_a @ resid @ ( trg_a @ resid @ T3 ) @ ( trg_a @ resid @ U2 ) ) ) ).
% R.coterminal_iff_con_trg
thf(fact_704_ide__trg,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ) ).
% ide_trg
thf(fact_705_trg__in__targets,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( member_list_a @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ) ).
% trg_in_targets
thf(fact_706_R_Oin__targetsI,axiom,
! [B: a,T3: a] :
( ( ide_a @ resid @ B )
=> ( ( con_a @ resid @ ( trg_a @ resid @ T3 ) @ B )
=> ( member_a @ B @ ( targets_a @ resid @ T3 ) ) ) ) ).
% R.in_targetsI
thf(fact_707_in__targetsI,axiom,
! [B: list_a,T3: 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 ) @ T3 ) @ B )
=> ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ) ) ).
% in_targetsI
thf(fact_708_can__select__def,axiom,
( can_select_a
= ( ^ [P2: a > $o,A5: set_a] :
? [X3: a] :
( ( member_a @ X3 @ A5 )
& ( P2 @ X3 )
& ! [Y3: a] :
( ( ( member_a @ Y3 @ A5 )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_709_can__select__def,axiom,
( can_select_list_a
= ( ^ [P2: list_a > $o,A5: set_list_a] :
? [X3: list_a] :
( ( member_list_a @ X3 @ A5 )
& ( P2 @ X3 )
& ! [Y3: list_a] :
( ( ( member_list_a @ Y3 @ A5 )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_710_residuation_Otrg__def,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( trg_list_a @ Resid @ T3 )
= ( Resid @ T3 @ T3 ) ) ) ).
% residuation.trg_def
thf(fact_711_residuation_Otrg__def,axiom,
! [Resid: a > a > a,T3: a] :
( ( residuation_a @ Resid )
=> ( ( trg_a @ Resid @ T3 )
= ( Resid @ T3 @ T3 ) ) ) ).
% residuation.trg_def
thf(fact_712_residuation_Oresid__arr__self,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( Resid @ T3 @ T3 )
= ( trg_list_a @ Resid @ T3 ) ) ) ).
% residuation.resid_arr_self
thf(fact_713_residuation_Oresid__arr__self,axiom,
! [Resid: a > a > a,T3: a] :
( ( residuation_a @ Resid )
=> ( ( Resid @ T3 @ T3 )
= ( trg_a @ Resid @ T3 ) ) ) ).
% residuation.resid_arr_self
thf(fact_714_residuation_Otrg_Ocong,axiom,
trg_list_a = trg_list_a ).
% residuation.trg.cong
thf(fact_715_residuation_Otrg_Ocong,axiom,
trg_a = trg_a ).
% residuation.trg.cong
thf(fact_716_rts__axioms_Ointro,axiom,
! [Resid: list_a > list_a > list_a] :
( ! [T: list_a] :
( ( arr_list_a @ Resid @ T )
=> ( ide_list_a @ Resid @ ( trg_list_a @ Resid @ T ) ) )
=> ( ! [A3: list_a,T: list_a] :
( ( ide_list_a @ Resid @ A3 )
=> ( ( con_list_a @ Resid @ T @ A3 )
=> ( ( Resid @ T @ A3 )
= T ) ) )
=> ( ! [A3: list_a,T: list_a] :
( ( ide_list_a @ Resid @ A3 )
=> ( ( con_list_a @ Resid @ A3 @ T )
=> ( ide_list_a @ Resid @ ( Resid @ A3 @ T ) ) ) )
=> ( ! [T: list_a,U: list_a] :
( ( con_list_a @ Resid @ T @ U )
=> ? [A6: list_a] :
( ( ide_list_a @ Resid @ A6 )
& ( con_list_a @ Resid @ A6 @ T )
& ( con_list_a @ Resid @ A6 @ U ) ) )
=> ( ! [T: list_a,U: list_a,V: list_a] :
( ( ide_list_a @ Resid @ ( Resid @ T @ U ) )
=> ( ( con_list_a @ Resid @ U @ V )
=> ( con_list_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ V @ U ) ) ) )
=> ( rts_axioms_list_a @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_717_rts__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T: a] :
( ( arr_a @ Resid @ T )
=> ( ide_a @ Resid @ ( trg_a @ Resid @ T ) ) )
=> ( ! [A3: a,T: a] :
( ( ide_a @ Resid @ A3 )
=> ( ( con_a @ Resid @ T @ A3 )
=> ( ( Resid @ T @ A3 )
= T ) ) )
=> ( ! [A3: a,T: a] :
( ( ide_a @ Resid @ A3 )
=> ( ( con_a @ Resid @ A3 @ T )
=> ( ide_a @ Resid @ ( Resid @ A3 @ T ) ) ) )
=> ( ! [T: a,U: a] :
( ( con_a @ Resid @ T @ U )
=> ? [A6: a] :
( ( ide_a @ Resid @ A6 )
& ( con_a @ Resid @ A6 @ T )
& ( con_a @ Resid @ A6 @ U ) ) )
=> ( ! [T: a,U: a,V: a] :
( ( ide_a @ Resid @ ( Resid @ T @ U ) )
=> ( ( con_a @ Resid @ U @ V )
=> ( con_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ V @ U ) ) ) )
=> ( rts_axioms_a @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_718_rts__axioms__def,axiom,
( rts_axioms_list_a
= ( ^ [Resid2: list_a > list_a > list_a] :
( ! [T5: list_a] :
( ( arr_list_a @ Resid2 @ T5 )
=> ( ide_list_a @ Resid2 @ ( trg_list_a @ Resid2 @ T5 ) ) )
& ! [A7: list_a,T5: list_a] :
( ( ide_list_a @ Resid2 @ A7 )
=> ( ( con_list_a @ Resid2 @ T5 @ A7 )
=> ( ( Resid2 @ T5 @ A7 )
= T5 ) ) )
& ! [A7: list_a,T5: list_a] :
( ( ide_list_a @ Resid2 @ A7 )
=> ( ( con_list_a @ Resid2 @ A7 @ T5 )
=> ( ide_list_a @ Resid2 @ ( Resid2 @ A7 @ T5 ) ) ) )
& ! [T5: list_a,U8: list_a] :
( ( con_list_a @ Resid2 @ T5 @ U8 )
=> ? [A7: list_a] :
( ( ide_list_a @ Resid2 @ A7 )
& ( con_list_a @ Resid2 @ A7 @ T5 )
& ( con_list_a @ Resid2 @ A7 @ U8 ) ) )
& ! [T5: list_a,U8: list_a,V6: list_a] :
( ( ide_list_a @ Resid2 @ ( Resid2 @ T5 @ U8 ) )
=> ( ( con_list_a @ Resid2 @ U8 @ V6 )
=> ( con_list_a @ Resid2 @ ( Resid2 @ T5 @ U8 ) @ ( Resid2 @ V6 @ U8 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_719_rts__axioms__def,axiom,
( rts_axioms_a
= ( ^ [Resid2: a > a > a] :
( ! [T5: a] :
( ( arr_a @ Resid2 @ T5 )
=> ( ide_a @ Resid2 @ ( trg_a @ Resid2 @ T5 ) ) )
& ! [A7: a,T5: a] :
( ( ide_a @ Resid2 @ A7 )
=> ( ( con_a @ Resid2 @ T5 @ A7 )
=> ( ( Resid2 @ T5 @ A7 )
= T5 ) ) )
& ! [A7: a,T5: a] :
( ( ide_a @ Resid2 @ A7 )
=> ( ( con_a @ Resid2 @ A7 @ T5 )
=> ( ide_a @ Resid2 @ ( Resid2 @ A7 @ T5 ) ) ) )
& ! [T5: a,U8: a] :
( ( con_a @ Resid2 @ T5 @ U8 )
=> ? [A7: a] :
( ( ide_a @ Resid2 @ A7 )
& ( con_a @ Resid2 @ A7 @ T5 )
& ( con_a @ Resid2 @ A7 @ U8 ) ) )
& ! [T5: a,U8: a,V6: a] :
( ( ide_a @ Resid2 @ ( Resid2 @ T5 @ U8 ) )
=> ( ( con_a @ Resid2 @ U8 @ V6 )
=> ( con_a @ Resid2 @ ( Resid2 @ T5 @ U8 ) @ ( Resid2 @ V6 @ U8 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_720_paths__in__rts__with__coherent__normal_ONPath__folding,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U3: list_list_a] :
( ( paths_9180082184051307706list_a @ Resid @ NN )
=> ( ( paths_8553768783572469479list_a @ Resid @ NN @ U3 )
=> ? [U: list_a] :
( ( member_list_a @ U @ NN )
& ( ( sources_list_a @ Resid @ U )
= ( paths_in_Srcs_list_a @ Resid @ U3 ) )
& ( ( targets_list_a @ Resid @ U )
= ( paths_in_Trgs_list_a @ Resid @ U3 ) )
& ! [T2: list_a] :
( ( con_list_list_a @ ( paths_8620460302779588466list_a @ Resid ) @ ( cons_list_a @ T2 @ nil_list_a ) @ U3 )
=> ( ( member_list_list_a @ ( paths_8620460302779588466list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T2 @ nil_list_a ) @ U3 ) @ ( cons_list_a @ ( Resid @ T2 @ U ) @ nil_list_a ) ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ NN ) ) )
& ( member_list_list_a @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ ( Resid @ T2 @ U ) @ nil_list_a ) @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T2 @ nil_list_a ) @ U3 ) ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ NN ) ) ) ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.NPath_folding
thf(fact_721_paths__in__rts__with__coherent__normal_ONPath__folding,axiom,
! [Resid: a > a > a,NN: set_a,U3: list_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U3 )
=> ? [U: a] :
( ( member_a @ U @ NN )
& ( ( sources_a @ Resid @ U )
= ( paths_in_Srcs_a @ Resid @ U3 ) )
& ( ( targets_a @ Resid @ U )
= ( paths_in_Trgs_a @ Resid @ U3 ) )
& ! [T2: a] :
( ( con_list_a @ ( paths_in_Resid_a @ Resid ) @ ( cons_a @ T2 @ nil_a ) @ U3 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T2 @ nil_a ) @ U3 ) @ ( cons_a @ ( Resid @ T2 @ U ) @ nil_a ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( cons_a @ ( Resid @ T2 @ U ) @ nil_a ) @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T2 @ nil_a ) @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.NPath_folding
thf(fact_722_append__is__composite__of,axiom,
! [T4: list_a,U3: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 )
=> ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U3 @ ( append_a @ T4 @ U3 ) ) ) ).
% append_is_composite_of
thf(fact_723_composite__ofE,axiom,
! [U2: list_a,T3: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ V2 )
=> ~ ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ V2 ) )
=> ~ ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V2 @ U2 ) @ T3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ ( paths_in_Resid_a @ resid @ V2 @ U2 ) ) ) ) ) ) ).
% composite_ofE
thf(fact_724_composite__of__cancel__left,axiom,
! [T3: list_a,U2: list_a,V2: list_a,U4: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U4 @ V2 )
=> ( ( 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 ) ) ) ) ) ).
% composite_of_cancel_left
thf(fact_725_composite__of__def,axiom,
! [U2: list_a,T3: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ V2 )
= ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ V2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V2 @ U2 ) @ T3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ ( paths_in_Resid_a @ resid @ V2 @ U2 ) ) ) ) ) ).
% composite_of_def
thf(fact_726_composite__of__ide__self,axiom,
! [A: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ A @ A @ A ) ) ).
% composite_of_ide_self
thf(fact_727_composite__of__unq__upto__cong,axiom,
! [U2: list_a,T3: list_a,V2: list_a,V5: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ V2 )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ V5 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V2 @ V5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V5 @ V2 ) ) ) ) ) ).
% composite_of_unq_upto_cong
thf(fact_728_diamond__commutes__upto__cong,axiom,
! [T3: list_a,U2: list_a,V2: list_a,V5: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) @ V2 )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ V5 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V2 @ V5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V5 @ V2 ) ) ) ) ) ).
% diamond_commutes_upto_cong
thf(fact_729_con__prfx__composite__of_I2_J,axiom,
! [T3: list_a,U2: list_a,W2: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ W2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W2 @ V2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ V2 ) ) ) ).
% con_prfx_composite_of(2)
thf(fact_730_con__prfx__composite__of_I1_J,axiom,
! [T3: list_a,U2: list_a,W2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ W2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ W2 ) ) ).
% con_prfx_composite_of(1)
thf(fact_731_resid__composite__of_I4_J,axiom,
! [T3: list_a,U2: list_a,W2: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ W2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W2 @ V2 )
=> ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ V2 ) @ ( paths_in_Resid_a @ resid @ U2 @ ( paths_in_Resid_a @ resid @ V2 @ T3 ) ) @ ( paths_in_Resid_a @ resid @ W2 @ V2 ) ) ) ) ).
% resid_composite_of(4)
thf(fact_732_resid__composite__of_I2_J,axiom,
! [T3: list_a,U2: list_a,W2: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ W2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W2 @ V2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V2 @ T3 ) @ U2 ) ) ) ).
% resid_composite_of(2)
thf(fact_733_resid__composite__of_I1_J,axiom,
! [T3: list_a,U2: list_a,W2: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ W2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W2 @ V2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V2 @ T3 ) @ ( paths_in_Resid_a @ resid @ W2 @ T3 ) ) ) ) ).
% resid_composite_of(1)
thf(fact_734_bounded__imp__con,axiom,
! [T3: list_a,U2: list_a,V2: list_a,T6: list_a,U4: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T6 @ U4 @ V2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ T6 ) ) ) ).
% bounded_imp_con
thf(fact_735_con__composite__of__iff,axiom,
! [T3: list_a,U2: list_a,V2: list_a,W2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W2 @ V2 )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ W2 @ T3 ) @ U2 ) ) ) ).
% con_composite_of_iff
thf(fact_736_sources__composite__of,axiom,
! [U2: list_a,T3: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ V2 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) ) ) ).
% sources_composite_of
thf(fact_737_arr__composite__of,axiom,
! [U2: list_a,T3: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ V2 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ V2 ) ) ).
% arr_composite_of
thf(fact_738_targets__composite__of,axiom,
! [U2: list_a,T3: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ V2 )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ V2 )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ) ).
% targets_composite_of
thf(fact_739_obtains__composite__of,axiom,
! [G: list_a,F: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ G @ F )
=> ~ ! [H: list_a] :
~ ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ G @ F @ H ) ) ).
% obtains_composite_of
thf(fact_740_join__ofE,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ~ ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) @ V2 )
=> ~ ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ V2 ) ) ) ).
% join_ofE
thf(fact_741_join__of__def,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
= ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) @ V2 )
& ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ V2 ) ) ) ).
% join_of_def
thf(fact_742_composable__def,axiom,
! [T3: list_a,U2: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
= ( ? [X5: list_a] : ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ X5 ) ) ) ).
% composable_def
thf(fact_743_composite__of__ide__arr,axiom,
! [A: list_a,T3: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ A @ T3 @ T3 )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ A ) ) ) ).
% composite_of_ide_arr
thf(fact_744_composite__of__arr__ide,axiom,
! [B: list_a,T3: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ B )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ B @ T3 )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ T3 ) @ B ) ) ) ).
% composite_of_arr_ide
thf(fact_745_resid__composite__of_I3_J,axiom,
! [T3: list_a,U2: list_a,W2: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ W2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W2 @ V2 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V2 @ W2 ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V2 @ T3 ) @ U2 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V2 @ T3 ) @ U2 ) @ ( paths_in_Resid_a @ resid @ V2 @ W2 ) ) ) ) ) ) ).
% resid_composite_of(3)
thf(fact_746_composite__of__source__arr,axiom,
! [T3: list_a,A: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ A @ T3 @ T3 ) ) ) ).
% composite_of_source_arr
thf(fact_747_composite__of__arr__target,axiom,
! [T3: list_a,B: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ B @ T3 ) ) ) ).
% composite_of_arr_target
thf(fact_748_composite__ofI,axiom,
! [U2: list_a,V2: list_a,T3: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ V2 ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V2 @ U2 ) @ T3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ ( paths_in_Resid_a @ resid @ V2 @ U2 ) ) ) )
=> ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ V2 ) ) ) ).
% composite_ofI
thf(fact_749_join__ofI,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) @ V2 )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ V2 )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 ) ) ) ).
% join_ofI
thf(fact_750_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,T6: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T3 @ U2 @ T6 )
=> ( ( member_a @ U2 @ NN )
=> ( ( member_a @ ( Resid @ T6 @ T3 ) @ NN )
& ( member_a @ ( Resid @ T3 @ T6 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_751_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,T6: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( composite_of_list_a @ Resid @ T3 @ U2 @ T6 )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN )
& ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_752_rts_Ocomposite__of_Ocong,axiom,
composite_of_list_a = composite_of_list_a ).
% rts.composite_of.cong
thf(fact_753_rts_Ocomposite__of_Ocong,axiom,
composite_of_a = composite_of_a ).
% rts.composite_of.cong
thf(fact_754_paths__in__rts__with__normal_ONPath_Ocong,axiom,
paths_3208254328100665313Path_a = paths_3208254328100665313Path_a ).
% paths_in_rts_with_normal.NPath.cong
thf(fact_755_rts__with__composites_Odiamond__commutes__upto__cong,axiom,
! [Resid: a > a > a,T3: a,U2: a,V2: a,V5: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( composite_of_a @ Resid @ T3 @ ( Resid @ U2 @ T3 ) @ V2 )
=> ( ( composite_of_a @ Resid @ U2 @ ( Resid @ T3 @ U2 ) @ V5 )
=> ( ( ide_a @ Resid @ ( Resid @ V2 @ V5 ) )
& ( ide_a @ Resid @ ( Resid @ V5 @ V2 ) ) ) ) ) ) ).
% rts_with_composites.diamond_commutes_upto_cong
thf(fact_756_rts__with__composites_Odiamond__commutes__upto__cong,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,U2: list_a,V2: list_a,V5: list_a] :
( ( rts_wi1113109597627081724list_a @ Resid )
=> ( ( composite_of_list_a @ Resid @ T3 @ ( Resid @ U2 @ T3 ) @ V2 )
=> ( ( composite_of_list_a @ Resid @ U2 @ ( Resid @ T3 @ U2 ) @ V5 )
=> ( ( ide_list_a @ Resid @ ( Resid @ V2 @ V5 ) )
& ( ide_list_a @ Resid @ ( Resid @ V5 @ V2 ) ) ) ) ) ) ).
% rts_with_composites.diamond_commutes_upto_cong
thf(fact_757_rts__with__composites_Oobtains__composite__of,axiom,
! [Resid: a > a > a,G: a,F: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( seq_a @ Resid @ G @ F )
=> ~ ! [H: a] :
~ ( composite_of_a @ Resid @ G @ F @ H ) ) ) ).
% rts_with_composites.obtains_composite_of
thf(fact_758_rts__with__composites_Oobtains__composite__of,axiom,
! [Resid: list_a > list_a > list_a,G: list_a,F: list_a] :
( ( rts_wi1113109597627081724list_a @ Resid )
=> ( ( seq_list_a @ Resid @ G @ F )
=> ~ ! [H: list_a] :
~ ( composite_of_list_a @ Resid @ G @ F @ H ) ) ) ).
% rts_with_composites.obtains_composite_of
thf(fact_759_paths__in__rts__with__coherent__normal_Ocoherent__normal__extends__to__paths,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( cohere6429906645900029933list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) ) ).
% paths_in_rts_with_coherent_normal.coherent_normal_extends_to_paths
thf(fact_760_paths__in__rts__with__coherent__normal_OCong_092_060_094sub_0620__append__Arr__NPath,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,U3: list_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( T4 != nil_a )
=> ( ( paths_in_Arr_a @ Resid @ ( append_a @ T4 @ U3 ) )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U3 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( append_a @ T4 @ U3 ) @ T4 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ Resid @ T4 @ ( append_a @ T4 @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.Cong\<^sub>0_append_Arr_NPath
thf(fact_761_paths__in__rts__with__coherent__normal_OCoherent,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,U3: list_a,U5: list_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U3 )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U5 )
=> ( ( ( paths_in_Srcs_a @ Resid @ T4 )
= ( paths_in_Srcs_a @ Resid @ U3 ) )
=> ( ( ( paths_in_Srcs_a @ Resid @ U3 )
= ( paths_in_Srcs_a @ Resid @ U5 ) )
=> ( ( ( paths_in_Trgs_a @ Resid @ U3 )
= ( paths_in_Trgs_a @ Resid @ U5 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) @ ( paths_in_Resid_a @ Resid @ T4 @ U5 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U5 ) @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.Coherent
thf(fact_762_paths__in__rts__with__coherent__normal_OCoherent__single,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U3: list_list_a,U5: list_list_a] :
( ( paths_9180082184051307706list_a @ Resid @ NN )
=> ( ( arr_list_a @ Resid @ T3 )
=> ( ( paths_8553768783572469479list_a @ Resid @ NN @ U3 )
=> ( ( paths_8553768783572469479list_a @ Resid @ NN @ U5 )
=> ( ( ( sources_list_a @ Resid @ T3 )
= ( paths_in_Srcs_list_a @ Resid @ U3 ) )
=> ( ( ( paths_in_Srcs_list_a @ Resid @ U3 )
= ( paths_in_Srcs_list_a @ Resid @ U5 ) )
=> ( ( ( paths_in_Trgs_list_a @ Resid @ U3 )
= ( paths_in_Trgs_list_a @ Resid @ U5 ) )
=> ( ( member_list_list_a @ ( paths_8620460302779588466list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T3 @ nil_list_a ) @ U3 ) @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T3 @ nil_list_a ) @ U5 ) ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ NN ) ) )
& ( member_list_list_a @ ( paths_8620460302779588466list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T3 @ nil_list_a ) @ U5 ) @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T3 @ nil_list_a ) @ U3 ) ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.Coherent_single
thf(fact_763_paths__in__rts__with__coherent__normal_OCoherent__single,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U3: list_a,U5: list_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T3 )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U3 )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U5 )
=> ( ( ( sources_a @ Resid @ T3 )
= ( paths_in_Srcs_a @ Resid @ U3 ) )
=> ( ( ( paths_in_Srcs_a @ Resid @ U3 )
= ( paths_in_Srcs_a @ Resid @ U5 ) )
=> ( ( ( paths_in_Trgs_a @ Resid @ U3 )
= ( paths_in_Trgs_a @ Resid @ U5 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T3 @ nil_a ) @ U3 ) @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T3 @ nil_a ) @ U5 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T3 @ nil_a ) @ U5 ) @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T3 @ nil_a ) @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.Coherent_single
thf(fact_764_composite__of__single__single,axiom,
! [T3: a,U2: a,V2: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ V2 )
=> ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) @ ( append_a @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) ) ) ) ).
% composite_of_single_single
thf(fact_765_normal__sub__rts__axioms__def,axiom,
( normal2939518615156061708list_a
= ( ^ [Resid2: list_a > list_a > list_a,NN2: set_list_a] :
( ! [T5: list_a] :
( ( member_list_a @ T5 @ NN2 )
=> ( arr_list_a @ Resid2 @ T5 ) )
& ! [A7: list_a] :
( ( ide_list_a @ Resid2 @ A7 )
=> ( member_list_a @ A7 @ NN2 ) )
& ! [U8: list_a,T5: list_a] :
( ( member_list_a @ U8 @ NN2 )
=> ( ( coinitial_list_a @ Resid2 @ T5 @ U8 )
=> ( member_list_a @ ( Resid2 @ U8 @ T5 ) @ NN2 ) ) )
& ! [U8: list_a,T5: list_a] :
( ( member_list_a @ U8 @ NN2 )
=> ( ( member_list_a @ ( Resid2 @ T5 @ U8 ) @ NN2 )
=> ( member_list_a @ T5 @ NN2 ) ) )
& ! [U8: list_a,T5: list_a] :
( ( member_list_a @ U8 @ NN2 )
=> ( ( seq_list_a @ Resid2 @ U8 @ T5 )
=> ? [X5: list_a] : ( composite_of_list_a @ Resid2 @ U8 @ T5 @ X5 ) ) )
& ! [U8: list_a,T5: list_a] :
( ( member_list_a @ U8 @ NN2 )
=> ( ( seq_list_a @ Resid2 @ T5 @ U8 )
=> ? [X5: list_a] : ( composite_of_list_a @ Resid2 @ T5 @ U8 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_766_normal__sub__rts__axioms__def,axiom,
( normal7698203753654205830ioms_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
( ! [T5: a] :
( ( member_a @ T5 @ NN2 )
=> ( arr_a @ Resid2 @ T5 ) )
& ! [A7: a] :
( ( ide_a @ Resid2 @ A7 )
=> ( member_a @ A7 @ NN2 ) )
& ! [U8: a,T5: a] :
( ( member_a @ U8 @ NN2 )
=> ( ( coinitial_a @ Resid2 @ T5 @ U8 )
=> ( member_a @ ( Resid2 @ U8 @ T5 ) @ NN2 ) ) )
& ! [U8: a,T5: a] :
( ( member_a @ U8 @ NN2 )
=> ( ( member_a @ ( Resid2 @ T5 @ U8 ) @ NN2 )
=> ( member_a @ T5 @ NN2 ) ) )
& ! [U8: a,T5: a] :
( ( member_a @ U8 @ NN2 )
=> ( ( seq_a @ Resid2 @ U8 @ T5 )
=> ? [X5: a] : ( composite_of_a @ Resid2 @ U8 @ T5 @ X5 ) ) )
& ! [U8: a,T5: a] :
( ( member_a @ U8 @ NN2 )
=> ( ( seq_a @ Resid2 @ T5 @ U8 )
=> ? [X5: a] : ( composite_of_a @ Resid2 @ T5 @ U8 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_767_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_list_a,Resid: list_a > list_a > list_a] :
( ! [T: list_a] :
( ( member_list_a @ T @ NN )
=> ( arr_list_a @ Resid @ T ) )
=> ( ! [A3: list_a] :
( ( ide_list_a @ Resid @ A3 )
=> ( member_list_a @ A3 @ NN ) )
=> ( ! [U: list_a,T: list_a] :
( ( member_list_a @ U @ NN )
=> ( ( coinitial_list_a @ Resid @ T @ U )
=> ( member_list_a @ ( Resid @ U @ T ) @ NN ) ) )
=> ( ! [U: list_a,T: list_a] :
( ( member_list_a @ U @ NN )
=> ( ( member_list_a @ ( Resid @ T @ U ) @ NN )
=> ( member_list_a @ T @ NN ) ) )
=> ( ! [U: list_a,T: list_a] :
( ( member_list_a @ U @ NN )
=> ( ( seq_list_a @ Resid @ U @ T )
=> ? [X_1: list_a] : ( composite_of_list_a @ Resid @ U @ T @ X_1 ) ) )
=> ( ! [U: list_a,T: list_a] :
( ( member_list_a @ U @ NN )
=> ( ( seq_list_a @ Resid @ T @ U )
=> ? [X_1: list_a] : ( composite_of_list_a @ Resid @ T @ U @ X_1 ) ) )
=> ( normal2939518615156061708list_a @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_768_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_a,Resid: a > a > a] :
( ! [T: a] :
( ( member_a @ T @ NN )
=> ( arr_a @ Resid @ T ) )
=> ( ! [A3: a] :
( ( ide_a @ Resid @ A3 )
=> ( member_a @ A3 @ NN ) )
=> ( ! [U: a,T: a] :
( ( member_a @ U @ NN )
=> ( ( coinitial_a @ Resid @ T @ U )
=> ( member_a @ ( Resid @ U @ T ) @ NN ) ) )
=> ( ! [U: a,T: a] :
( ( member_a @ U @ NN )
=> ( ( member_a @ ( Resid @ T @ U ) @ NN )
=> ( member_a @ T @ NN ) ) )
=> ( ! [U: a,T: a] :
( ( member_a @ U @ NN )
=> ( ( seq_a @ Resid @ U @ T )
=> ? [X_1: a] : ( composite_of_a @ Resid @ U @ T @ X_1 ) ) )
=> ( ! [U: a,T: a] :
( ( member_a @ U @ NN )
=> ( ( seq_a @ Resid @ T @ U )
=> ? [X_1: a] : ( composite_of_a @ Resid @ T @ U @ X_1 ) ) )
=> ( normal7698203753654205830ioms_a @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_769_R_Ocomposite__of__unq__upto__cong,axiom,
! [U2: a,T3: a,V2: a,V5: a] :
( ( composite_of_a @ resid @ U2 @ T3 @ V2 )
=> ( ( composite_of_a @ resid @ U2 @ T3 @ V5 )
=> ( ( ide_a @ resid @ ( resid @ V2 @ V5 ) )
& ( ide_a @ resid @ ( resid @ V5 @ V2 ) ) ) ) ) ).
% R.composite_of_unq_upto_cong
thf(fact_770_R_Ocomposite__of__ide__self,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( composite_of_a @ resid @ A @ A @ A ) ) ).
% R.composite_of_ide_self
thf(fact_771_R_Ocomposite__of__def,axiom,
! [U2: a,T3: a,V2: a] :
( ( composite_of_a @ resid @ U2 @ T3 @ V2 )
= ( ( ide_a @ resid @ ( resid @ U2 @ V2 ) )
& ( ide_a @ resid @ ( resid @ ( resid @ V2 @ U2 ) @ T3 ) )
& ( ide_a @ resid @ ( resid @ T3 @ ( resid @ V2 @ U2 ) ) ) ) ) ).
% R.composite_of_def
thf(fact_772_R_Ocomposite__of__cancel__left,axiom,
! [T3: a,U2: a,V2: a,U4: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ V2 )
=> ( ( composite_of_a @ resid @ T3 @ U4 @ V2 )
=> ( ( ide_a @ resid @ ( resid @ U2 @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U2 ) ) ) ) ) ).
% R.composite_of_cancel_left
thf(fact_773_R_Ocomposite__ofE,axiom,
! [U2: a,T3: a,V2: a] :
( ( composite_of_a @ resid @ U2 @ T3 @ V2 )
=> ~ ( ( ide_a @ resid @ ( resid @ U2 @ V2 ) )
=> ~ ( ( ide_a @ resid @ ( resid @ ( resid @ V2 @ U2 ) @ T3 ) )
& ( ide_a @ resid @ ( resid @ T3 @ ( resid @ V2 @ U2 ) ) ) ) ) ) ).
% R.composite_ofE
thf(fact_774_R_Ocon__composite__of__iff,axiom,
! [T3: a,U2: a,V2: a,W2: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ V2 )
=> ( ( con_a @ resid @ W2 @ V2 )
= ( con_a @ resid @ ( resid @ W2 @ T3 ) @ U2 ) ) ) ).
% R.con_composite_of_iff
thf(fact_775_R_Obounded__imp__con,axiom,
! [T3: a,U2: a,V2: a,T6: a,U4: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ V2 )
=> ( ( composite_of_a @ resid @ T6 @ U4 @ V2 )
=> ( con_a @ resid @ T3 @ T6 ) ) ) ).
% R.bounded_imp_con
thf(fact_776_R_Oresid__composite__of_I1_J,axiom,
! [T3: a,U2: a,W2: a,V2: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ W2 )
=> ( ( con_a @ resid @ W2 @ V2 )
=> ( con_a @ resid @ ( resid @ V2 @ T3 ) @ ( resid @ W2 @ T3 ) ) ) ) ).
% R.resid_composite_of(1)
thf(fact_777_R_Oresid__composite__of_I2_J,axiom,
! [T3: a,U2: a,W2: a,V2: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ W2 )
=> ( ( con_a @ resid @ W2 @ V2 )
=> ( con_a @ resid @ ( resid @ V2 @ T3 ) @ U2 ) ) ) ).
% R.resid_composite_of(2)
thf(fact_778_R_Oresid__composite__of_I4_J,axiom,
! [T3: a,U2: a,W2: a,V2: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ W2 )
=> ( ( con_a @ resid @ W2 @ V2 )
=> ( composite_of_a @ resid @ ( resid @ T3 @ V2 ) @ ( resid @ U2 @ ( resid @ V2 @ T3 ) ) @ ( resid @ W2 @ V2 ) ) ) ) ).
% R.resid_composite_of(4)
thf(fact_779_R_Ocon__prfx__composite__of_I1_J,axiom,
! [T3: a,U2: a,W2: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ W2 )
=> ( con_a @ resid @ T3 @ W2 ) ) ).
% R.con_prfx_composite_of(1)
thf(fact_780_R_Ocon__prfx__composite__of_I2_J,axiom,
! [T3: a,U2: a,W2: a,V2: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ W2 )
=> ( ( con_a @ resid @ W2 @ V2 )
=> ( con_a @ resid @ T3 @ V2 ) ) ) ).
% R.con_prfx_composite_of(2)
thf(fact_781_R_Osources__composite__of,axiom,
! [U2: a,T3: a,V2: a] :
( ( composite_of_a @ resid @ U2 @ T3 @ V2 )
=> ( ( sources_a @ resid @ V2 )
= ( sources_a @ resid @ U2 ) ) ) ).
% R.sources_composite_of
thf(fact_782_R_Oarr__composite__of,axiom,
! [U2: a,T3: a,V2: a] :
( ( composite_of_a @ resid @ U2 @ T3 @ V2 )
=> ( arr_a @ resid @ V2 ) ) ).
% R.arr_composite_of
thf(fact_783_R_Otargets__composite__of,axiom,
! [U2: a,T3: a,V2: a] :
( ( composite_of_a @ resid @ U2 @ T3 @ V2 )
=> ( ( targets_a @ resid @ V2 )
= ( targets_a @ resid @ T3 ) ) ) ).
% R.targets_composite_of
thf(fact_784_R_Ojoin__of__def,axiom,
! [T3: a,U2: a,V2: a] :
( ( join_of_a @ resid @ T3 @ U2 @ V2 )
= ( ( composite_of_a @ resid @ T3 @ ( resid @ U2 @ T3 ) @ V2 )
& ( composite_of_a @ resid @ U2 @ ( resid @ T3 @ U2 ) @ V2 ) ) ) ).
% R.join_of_def
thf(fact_785_R_Ojoin__ofE,axiom,
! [T3: a,U2: a,V2: a] :
( ( join_of_a @ resid @ T3 @ U2 @ V2 )
=> ~ ( ( composite_of_a @ resid @ T3 @ ( resid @ U2 @ T3 ) @ V2 )
=> ~ ( composite_of_a @ resid @ U2 @ ( resid @ T3 @ U2 ) @ V2 ) ) ) ).
% R.join_ofE
thf(fact_786_R_Ocomposable__def,axiom,
! [T3: a,U2: a] :
( ( composable_a @ resid @ T3 @ U2 )
= ( ? [X5: a] : ( composite_of_a @ resid @ T3 @ U2 @ X5 ) ) ) ).
% R.composable_def
thf(fact_787_R_Ocomposite__of__ide__arr,axiom,
! [A: a,T3: a] :
( ( ide_a @ resid @ A )
=> ( ( composite_of_a @ resid @ A @ T3 @ T3 )
= ( con_a @ resid @ T3 @ A ) ) ) ).
% R.composite_of_ide_arr
thf(fact_788_R_Ocomposite__of__arr__ide,axiom,
! [B: a,T3: a] :
( ( ide_a @ resid @ B )
=> ( ( composite_of_a @ resid @ T3 @ B @ T3 )
= ( con_a @ resid @ ( resid @ T3 @ T3 ) @ B ) ) ) ).
% R.composite_of_arr_ide
thf(fact_789_R_Oresid__composite__of_I3_J,axiom,
! [T3: a,U2: a,W2: a,V2: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ W2 )
=> ( ( con_a @ resid @ W2 @ V2 )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ V2 @ W2 ) @ ( resid @ ( resid @ V2 @ T3 ) @ U2 ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ ( resid @ V2 @ T3 ) @ U2 ) @ ( resid @ V2 @ W2 ) ) ) ) ) ) ).
% R.resid_composite_of(3)
thf(fact_790_R_Ocomposite__of__source__arr,axiom,
! [T3: a,A: a] :
( ( arr_a @ resid @ T3 )
=> ( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ( composite_of_a @ resid @ A @ T3 @ T3 ) ) ) ).
% R.composite_of_source_arr
thf(fact_791_R_Ocomposite__of__arr__target,axiom,
! [T3: a,B: a] :
( ( arr_a @ resid @ T3 )
=> ( ( member_a @ B @ ( targets_a @ resid @ T3 ) )
=> ( composite_of_a @ resid @ T3 @ B @ T3 ) ) ) ).
% R.composite_of_arr_target
thf(fact_792_R_Ocomposite__ofI,axiom,
! [U2: a,V2: a,T3: a] :
( ( ide_a @ resid @ ( resid @ U2 @ V2 ) )
=> ( ( ( ide_a @ resid @ ( resid @ ( resid @ V2 @ U2 ) @ T3 ) )
& ( ide_a @ resid @ ( resid @ T3 @ ( resid @ V2 @ U2 ) ) ) )
=> ( composite_of_a @ resid @ U2 @ T3 @ V2 ) ) ) ).
% R.composite_ofI
thf(fact_793_R_Ojoin__ofI,axiom,
! [T3: a,U2: a,V2: a] :
( ( composite_of_a @ resid @ T3 @ ( resid @ U2 @ T3 ) @ V2 )
=> ( ( composite_of_a @ resid @ U2 @ ( resid @ T3 @ U2 ) @ V2 )
=> ( join_of_a @ resid @ T3 @ U2 @ V2 ) ) ) ).
% R.join_ofI
thf(fact_794_NPath__folding,axiom,
! [U3: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ nn @ U3 )
=> ? [U: a] :
( ( member_a @ U @ nn )
& ( ( sources_a @ resid @ U )
= ( paths_in_Srcs_a @ resid @ U3 ) )
& ( ( targets_a @ resid @ U )
= ( paths_in_Trgs_a @ resid @ U3 ) )
& ! [T2: a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T2 @ nil_a ) @ U3 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U3 ) @ ( cons_a @ ( resid @ T2 @ U ) @ nil_a ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T2 @ U ) @ nil_a ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ).
% NPath_folding
thf(fact_795_Coherent__single,axiom,
! [T3: a,U3: list_a,U5: list_a] :
( ( arr_a @ resid @ T3 )
=> ( ( paths_3208254328100665313Path_a @ resid @ nn @ U3 )
=> ( ( paths_3208254328100665313Path_a @ resid @ nn @ U5 )
=> ( ( ( sources_a @ resid @ T3 )
= ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( ( ( paths_in_Srcs_a @ resid @ U3 )
= ( paths_in_Srcs_a @ resid @ U5 ) )
=> ( ( ( paths_in_Trgs_a @ resid @ U3 )
= ( paths_in_Trgs_a @ resid @ U5 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U5 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U5 ) @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ) ) ) ).
% Coherent_single
thf(fact_796_paths__in__rts__with__normal_ONPath__Resid__single__Arr,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U3: list_list_a] :
( ( paths_8755422415569132860list_a @ Resid @ NN )
=> ( ( member_list_a @ T3 @ NN )
=> ( ( paths_in_Arr_list_a @ Resid @ U3 )
=> ( ( ( sources_list_a @ Resid @ T3 )
= ( paths_in_Srcs_list_a @ Resid @ U3 ) )
=> ( paths_8553768783572469479list_a @ Resid @ NN @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T3 @ nil_list_a ) @ U3 ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid_single_Arr
thf(fact_797_paths__in__rts__with__normal_ONPath__Resid__single__Arr,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U3: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( member_a @ T3 @ NN )
=> ( ( paths_in_Arr_a @ Resid @ U3 )
=> ( ( ( sources_a @ Resid @ T3 )
= ( paths_in_Srcs_a @ Resid @ U3 ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T3 @ nil_a ) @ U3 ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid_single_Arr
thf(fact_798_paths__in__rts__with__coherent__normal__axioms,axiom,
paths_6252547876539050804rmal_a @ resid @ nn ).
% paths_in_rts_with_coherent_normal_axioms
thf(fact_799_paths__in__rts__with__normal__axioms,axiom,
paths_3682460814909619382rmal_a @ resid @ nn ).
% paths_in_rts_with_normal_axioms
thf(fact_800_N_OCong_092_060_094sub_0620__symmetric,axiom,
! [T3: a,T6: a] :
( ( ( member_a @ ( resid @ T3 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T3 ) @ nn ) )
=> ( ( member_a @ ( resid @ T6 @ T3 ) @ nn )
& ( member_a @ ( resid @ T3 @ T6 ) @ nn ) ) ) ).
% N.Cong\<^sub>0_symmetric
thf(fact_801_N_OCong_092_060_094sub_0620__transitive,axiom,
! [T3: a,T6: a,T12: a] :
( ( ( member_a @ ( resid @ T3 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T3 ) @ nn ) )
=> ( ( ( member_a @ ( resid @ T6 @ T12 ) @ nn )
& ( member_a @ ( resid @ T12 @ T6 ) @ nn ) )
=> ( ( member_a @ ( resid @ T3 @ T12 ) @ nn )
& ( member_a @ ( resid @ T12 @ T3 ) @ nn ) ) ) ) ).
% N.Cong\<^sub>0_transitive
thf(fact_802_N_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [T3: a,U2: a,T6: a] :
( ( ( member_a @ ( resid @ ( resid @ T3 @ U2 ) @ ( resid @ T6 @ U2 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T6 @ U2 ) @ ( resid @ T3 @ U2 ) ) @ nn ) )
=> ( ( member_a @ U2 @ nn )
=> ( ( member_a @ ( resid @ T3 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T3 ) @ nn ) ) ) ) ).
% N.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_803_N_Obackward__stable,axiom,
! [U2: a,T3: a] :
( ( member_a @ U2 @ nn )
=> ( ( member_a @ ( resid @ T3 @ U2 ) @ nn )
=> ( member_a @ T3 @ nn ) ) ) ).
% N.backward_stable
thf(fact_804_N_Oide__closed,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( member_a @ A @ nn ) ) ).
% N.ide_closed
thf(fact_805_N_Oprfx__closed,axiom,
! [U2: a,T3: a] :
( ( member_a @ U2 @ nn )
=> ( ( ide_a @ resid @ ( resid @ T3 @ U2 ) )
=> ( member_a @ T3 @ nn ) ) ) ).
% N.prfx_closed
thf(fact_806_N_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [T3: a,T6: a,U2: a] :
( ( ( member_a @ ( resid @ T3 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T3 ) @ nn ) )
=> ( ( con_a @ resid @ T3 @ U2 )
=> ( ( member_a @ ( resid @ ( resid @ T3 @ U2 ) @ ( resid @ T6 @ U2 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T6 @ U2 ) @ ( resid @ T3 @ U2 ) ) @ nn ) ) ) ) ).
% N.Cong\<^sub>0_subst_left(2)
thf(fact_807_N_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [T3: a,T6: a,U2: a] :
( ( ( member_a @ ( resid @ T3 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T3 ) @ nn ) )
=> ( ( con_a @ resid @ T3 @ U2 )
=> ( con_a @ resid @ T6 @ U2 ) ) ) ).
% N.Cong\<^sub>0_subst_left(1)
thf(fact_808_N_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [U2: a,U4: a,T3: a] :
( ( ( member_a @ ( resid @ U2 @ U4 ) @ nn )
& ( member_a @ ( resid @ U4 @ U2 ) @ nn ) )
=> ( ( con_a @ resid @ T3 @ U2 )
=> ( ( member_a @ ( resid @ ( resid @ ( resid @ T3 @ U2 ) @ ( resid @ U4 @ U2 ) ) @ ( resid @ ( resid @ T3 @ U4 ) @ ( resid @ U2 @ U4 ) ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ ( resid @ T3 @ U4 ) @ ( resid @ U2 @ U4 ) ) @ ( resid @ ( resid @ T3 @ U2 ) @ ( resid @ U4 @ U2 ) ) ) @ nn ) ) ) ) ).
% N.Cong\<^sub>0_subst_right(2)
thf(fact_809_N_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [U2: a,U4: a,T3: a] :
( ( ( member_a @ ( resid @ U2 @ U4 ) @ nn )
& ( member_a @ ( resid @ U4 @ U2 ) @ nn ) )
=> ( ( con_a @ resid @ T3 @ U2 )
=> ( con_a @ resid @ T3 @ U4 ) ) ) ).
% N.Cong\<^sub>0_subst_right(1)
thf(fact_810_N_OCong_092_060_094sub_0620__imp__con,axiom,
! [T3: a,T6: a] :
( ( ( member_a @ ( resid @ T3 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T3 ) @ nn ) )
=> ( con_a @ resid @ T3 @ T6 ) ) ).
% N.Cong\<^sub>0_imp_con
thf(fact_811_N_OCong_092_060_094sub_0620__subst__Con,axiom,
! [T3: a,T6: a,U2: a,U4: a] :
( ( ( member_a @ ( resid @ T3 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T3 ) @ nn ) )
=> ( ( ( member_a @ ( resid @ U2 @ U4 ) @ nn )
& ( member_a @ ( resid @ U4 @ U2 ) @ nn ) )
=> ( ( con_a @ resid @ T3 @ U2 )
= ( con_a @ resid @ T6 @ U4 ) ) ) ) ).
% N.Cong\<^sub>0_subst_Con
thf(fact_812_N_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [T3: a,T6: a] :
( ( ( member_a @ ( resid @ T3 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T3 ) @ nn ) )
=> ( ( sources_a @ resid @ T3 )
= ( sources_a @ resid @ T6 ) ) ) ).
% N.Cong\<^sub>0_imp_coinitial
thf(fact_813_N_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [T3: a,T6: a,U2: a] :
( ( ( member_a @ ( resid @ T3 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T3 ) @ nn ) )
=> ( ( member_a @ U2 @ nn )
=> ( ( ( sources_a @ resid @ T3 )
= ( sources_a @ resid @ U2 ) )
=> ( ( member_a @ ( resid @ ( resid @ T3 @ U2 ) @ ( resid @ T6 @ U2 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T6 @ U2 ) @ ( resid @ T3 @ U2 ) ) @ nn ) ) ) ) ) ).
% N.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_814_N_OCong_092_060_094sub_0620__reflexive,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
=> ( ( member_a @ ( resid @ T3 @ T3 ) @ nn )
& ( member_a @ ( resid @ T3 @ T3 ) @ nn ) ) ) ).
% N.Cong\<^sub>0_reflexive
thf(fact_815_N_Oelements__are__arr,axiom,
! [T3: a] :
( ( member_a @ T3 @ nn )
=> ( arr_a @ resid @ T3 ) ) ).
% N.elements_are_arr
thf(fact_816_N_Ofactor__closed_I2_J,axiom,
! [T3: a,U2: a,V2: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ V2 )
=> ( ( member_a @ V2 @ nn )
=> ( member_a @ U2 @ nn ) ) ) ).
% N.factor_closed(2)
thf(fact_817_N_Ofactor__closed_I1_J,axiom,
! [T3: a,U2: a,V2: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ V2 )
=> ( ( member_a @ V2 @ nn )
=> ( member_a @ T3 @ nn ) ) ) ).
% N.factor_closed(1)
thf(fact_818_N_OCong_092_060_094sub_0620__cancel__left,axiom,
! [T3: a,U2: a,V2: a,U4: a,V5: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ V2 )
=> ( ( composite_of_a @ resid @ T3 @ U4 @ V5 )
=> ( ( ( member_a @ ( resid @ V2 @ V5 ) @ nn )
& ( member_a @ ( resid @ V5 @ V2 ) @ nn ) )
=> ( ( member_a @ ( resid @ U2 @ U4 ) @ nn )
& ( member_a @ ( resid @ U4 @ U2 ) @ nn ) ) ) ) ) ).
% N.Cong\<^sub>0_cancel_left
thf(fact_819_N_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [T3: a,U2: a,T6: a] :
( ( composite_of_a @ resid @ T3 @ U2 @ T6 )
=> ( ( member_a @ U2 @ nn )
=> ( ( member_a @ ( resid @ T6 @ T3 ) @ nn )
& ( member_a @ ( resid @ T3 @ T6 ) @ nn ) ) ) ) ).
% N.Cong\<^sub>0_composite_of_arr_normal
thf(fact_820_N_OCong_092_060_094sub_0620__iff,axiom,
! [T3: a,T6: a] :
( ( ( member_a @ ( resid @ T3 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T3 ) @ nn ) )
= ( ? [U8: a,U11: a,V6: a,V7: a] :
( ( member_a @ U8 @ nn )
& ( member_a @ U11 @ nn )
& ( member_a @ ( resid @ V6 @ V7 ) @ nn )
& ( member_a @ ( resid @ V7 @ V6 ) @ nn )
& ( composite_of_a @ resid @ T3 @ U8 @ V6 )
& ( composite_of_a @ resid @ T6 @ U11 @ V7 ) ) ) ) ).
% N.Cong\<^sub>0_iff
thf(fact_821_N_Ocomposite__closed,axiom,
! [T3: a,U2: a,V2: a] :
( ( member_a @ T3 @ nn )
=> ( ( member_a @ U2 @ nn )
=> ( ( composite_of_a @ resid @ T3 @ U2 @ V2 )
=> ( member_a @ V2 @ nn ) ) ) ) ).
% N.composite_closed
thf(fact_822_N_Ocomposite__of__arr__normal,axiom,
! [Arr: a > $o,T3: a,U2: a,T6: a] :
( ( Arr @ T3 )
=> ( ( member_a @ U2 @ nn )
=> ( ( composite_of_a @ resid @ T3 @ U2 @ T6 )
=> ( ( member_a @ ( resid @ T6 @ T3 ) @ nn )
& ( member_a @ ( resid @ T3 @ T6 ) @ nn ) ) ) ) ) ).
% N.composite_of_arr_normal
thf(fact_823_N_Oforward__stable,axiom,
! [U2: a,T3: a] :
( ( member_a @ U2 @ nn )
=> ( ( coinitial_a @ resid @ T3 @ U2 )
=> ( member_a @ ( resid @ U2 @ T3 ) @ nn ) ) ) ).
% N.forward_stable
thf(fact_824_Backward__stable,axiom,
! [U3: list_a,T4: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ nn @ U3 )
=> ( ( paths_3208254328100665313Path_a @ resid @ nn @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) )
=> ( paths_3208254328100665313Path_a @ resid @ nn @ T4 ) ) ) ).
% Backward_stable
thf(fact_825_Cong_092_060_094sub_0620__symmetric,axiom,
! [T3: list_a,T6: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ).
% Cong\<^sub>0_symmetric
thf(fact_826_Cong_092_060_094sub_0620__transitive,axiom,
! [T3: list_a,T6: list_a,T12: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T12 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T12 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T12 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T12 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ).
% Cong\<^sub>0_transitive
thf(fact_827_Resid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [T3: list_a,U2: list_a,T6: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T6 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T6 @ U2 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ).
% Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_828_backward__stable,axiom,
! [U2: list_a,T3: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( member_list_a @ T3 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ).
% backward_stable
thf(fact_829_NPath__implies__Arr,axiom,
! [T4: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ nn @ T4 )
=> ( paths_in_Arr_a @ resid @ T4 ) ) ).
% NPath_implies_Arr
thf(fact_830_N_Ocoherent__normal__sub__rts__axioms,axiom,
cohere6072184133013167079_rts_a @ resid @ nn ).
% N.coherent_normal_sub_rts_axioms
thf(fact_831_N_OResid__along__normal__preserves__reflects__con,axiom,
! [U2: a,T3: a,T6: a] :
( ( member_a @ U2 @ nn )
=> ( ( ( sources_a @ resid @ T3 )
= ( sources_a @ resid @ U2 ) )
=> ( ( con_a @ resid @ ( resid @ T3 @ U2 ) @ ( resid @ T6 @ U2 ) )
= ( con_a @ resid @ T3 @ T6 ) ) ) ) ).
% N.Resid_along_normal_preserves_reflects_con
thf(fact_832_Ide__implies__NPath,axiom,
! [T4: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
=> ( paths_3208254328100665313Path_a @ resid @ nn @ T4 ) ) ).
% Ide_implies_NPath
thf(fact_833_N_Ocoherent_H,axiom,
! [V2: a,V5: a,W2: a,W3: a,T3: a,T6: a] :
( ( member_a @ V2 @ nn )
=> ( ( member_a @ V5 @ nn )
=> ( ( member_a @ W2 @ nn )
=> ( ( member_a @ W3 @ nn )
=> ( ( ( sources_a @ resid @ V2 )
= ( sources_a @ resid @ W2 ) )
=> ( ( ( sources_a @ resid @ V5 )
= ( sources_a @ resid @ W3 ) )
=> ( ( ( targets_a @ resid @ W2 )
= ( targets_a @ resid @ W3 ) )
=> ( ( ( member_a @ ( resid @ ( resid @ T3 @ V2 ) @ ( resid @ T6 @ V5 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T6 @ V5 ) @ ( resid @ T3 @ V2 ) ) @ nn ) )
=> ( ( member_a @ ( resid @ ( resid @ T3 @ W2 ) @ ( resid @ T6 @ W3 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T6 @ W3 ) @ ( resid @ T3 @ W2 ) ) @ nn ) ) ) ) ) ) ) ) ) ) ).
% N.coherent'
thf(fact_834_N_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [T3: a,U2: a,V2: a,V5: a] :
( ( con_a @ resid @ T3 @ U2 )
=> ( ( composite_of_a @ resid @ T3 @ ( resid @ U2 @ T3 ) @ V2 )
=> ( ( composite_of_a @ resid @ U2 @ ( resid @ T3 @ U2 ) @ V5 )
=> ( ( member_a @ ( resid @ V2 @ V5 ) @ nn )
& ( member_a @ ( resid @ V5 @ V2 ) @ nn ) ) ) ) ) ).
% N.diamond_commutes_upto_Cong\<^sub>0
thf(fact_835_N_Oresid__along__elem__preserves__con,axiom,
! [T3: a,T6: a,U2: a] :
( ( con_a @ resid @ T3 @ T6 )
=> ( ( coinitial_a @ resid @ T3 @ U2 )
=> ( ( member_a @ U2 @ nn )
=> ( con_a @ resid @ ( resid @ T3 @ U2 ) @ ( resid @ T6 @ U2 ) ) ) ) ) ).
% N.resid_along_elem_preserves_con
thf(fact_836_N_Ocomposite__closed__left,axiom,
! [U2: a,T3: a] :
( ( member_a @ U2 @ nn )
=> ( ( seq_a @ resid @ U2 @ T3 )
=> ? [X_12: a] : ( composite_of_a @ resid @ U2 @ T3 @ X_12 ) ) ) ).
% N.composite_closed_left
thf(fact_837_N_Ocomposite__closed__right,axiom,
! [U2: a,T3: a] :
( ( member_a @ U2 @ nn )
=> ( ( seq_a @ resid @ T3 @ U2 )
=> ? [X_12: a] : ( composite_of_a @ resid @ T3 @ U2 @ X_12 ) ) ) ).
% N.composite_closed_right
thf(fact_838_prfx__closed,axiom,
! [U2: list_a,T3: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
=> ( member_list_a @ T3 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ).
% prfx_closed
thf(fact_839_ide__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 @ nn ) ) ) ) ).
% ide_closed
thf(fact_840_Cong_092_060_094sub_0620__subst__Con,axiom,
! [T3: list_a,T6: list_a,U2: list_a,U4: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T6 @ U4 ) ) ) ) ).
% Cong\<^sub>0_subst_Con
thf(fact_841_Cong_092_060_094sub_0620__imp__con,axiom,
! [T3: list_a,T6: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ T6 ) ) ).
% Cong\<^sub>0_imp_con
thf(fact_842_Cong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [U2: list_a,U4: list_a,T3: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U4 ) ) ) ).
% Cong\<^sub>0_subst_right(1)
thf(fact_843_Cong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [U2: list_a,U4: list_a,T3: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U4 ) @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U4 ) @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ).
% Cong\<^sub>0_subst_right(2)
thf(fact_844_Cong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [T3: list_a,T6: list_a,U2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T6 @ U2 ) ) ) ).
% Cong\<^sub>0_subst_left(1)
thf(fact_845_Cong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [T3: list_a,T6: list_a,U2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T6 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T6 @ U2 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ).
% Cong\<^sub>0_subst_left(2)
thf(fact_846_Cong_092_060_094sub_0620__imp__coinitial,axiom,
! [T3: list_a,T6: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) ) ) ).
% Cong\<^sub>0_imp_coinitial
thf(fact_847_Resid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [T3: list_a,T6: list_a,U2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T6 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T6 @ U2 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ).
% Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_848_Cong_092_060_094sub_0620__append__resid__NPath,axiom,
! [T4: list_a,U3: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ nn @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ ( paths_in_Resid_a @ resid @ U3 @ T4 ) ) @ U3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U3 @ ( append_a @ T4 @ ( paths_in_Resid_a @ resid @ U3 @ T4 ) ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ).
% Cong\<^sub>0_append_resid_NPath
thf(fact_849_elements__are__arr,axiom,
! [T3: list_a] :
( ( member_list_a @ T3 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ).
% elements_are_arr
thf(fact_850_Cong_092_060_094sub_0620__reflexive,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ).
% Cong\<^sub>0_reflexive
thf(fact_851_factor__closed_I2_J,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( ( member_list_a @ V2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ).
% factor_closed(2)
thf(fact_852_factor__closed_I1_J,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( ( member_list_a @ V2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( member_list_a @ T3 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ).
% factor_closed(1)
thf(fact_853_Cong_092_060_094sub_0620__cancel__left,axiom,
! [T3: list_a,U2: list_a,V2: list_a,U4: list_a,V5: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U4 @ V5 )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ V2 @ V5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ V5 @ V2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ U4 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U4 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ).
% Cong\<^sub>0_cancel_left
thf(fact_854_Cong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [T3: list_a,U2: list_a,T6: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ T6 )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ).
% Cong\<^sub>0_composite_of_arr_normal
thf(fact_855_Cong_092_060_094sub_0620__iff,axiom,
! [T3: list_a,T6: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
= ( ? [U8: list_a,U11: list_a,V6: list_a,V7: list_a] :
( ( member_list_a @ U8 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ U11 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ V6 @ V7 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ V7 @ V6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U8 @ V6 )
& ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T6 @ U11 @ V7 ) ) ) ) ).
% Cong\<^sub>0_iff
thf(fact_856_composite__closed,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( member_list_a @ T3 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ V2 )
=> ( member_list_a @ V2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ).
% composite_closed
thf(fact_857_composite__of__arr__normal,axiom,
! [Arr: list_a > $o,T3: list_a,U2: list_a,T6: list_a] :
( ( Arr @ T3 )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ T6 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ).
% composite_of_arr_normal
thf(fact_858_N_Ocoherent,axiom,
! [T3: a,U2: a,U4: a] :
( ( arr_a @ resid @ T3 )
=> ( ( 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 @ T3 )
= ( sources_a @ resid @ U2 ) )
=> ( ( member_a @ ( resid @ ( resid @ T3 @ U2 ) @ ( resid @ T3 @ U4 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T3 @ U4 ) @ ( resid @ T3 @ U2 ) ) @ nn ) ) ) ) ) ) ) ) ).
% N.coherent
thf(fact_859_forward__stable,axiom,
! [U2: list_a,T3: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ).
% forward_stable
thf(fact_860_Backward__stable__single,axiom,
! [U3: list_a,T3: a] :
( ( paths_3208254328100665313Path_a @ resid @ nn @ U3 )
=> ( ( paths_3208254328100665313Path_a @ resid @ nn @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 ) )
=> ( paths_3208254328100665313Path_a @ resid @ nn @ ( cons_a @ T3 @ nil_a ) ) ) ) ).
% Backward_stable_single
thf(fact_861_Resid__NPath__preserves__reflects__Con,axiom,
! [U3: list_a,T4: list_a,T7: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ nn @ U3 )
=> ( ( ( paths_in_Srcs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) @ ( paths_in_Resid_a @ resid @ T7 @ U3 ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ T4 @ T7 )
!= nil_a ) ) ) ) ).
% Resid_NPath_preserves_reflects_Con
thf(fact_862_Cong_092_060_094sub_0620__cancel__left_092_060_094sub_062C_092_060_094sub_062S,axiom,
! [T4: list_a,U3: list_a,U5: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ ( append_a @ T4 @ U5 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U5 ) @ ( append_a @ T4 @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( U5 != nil_a )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ U3 @ U5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U5 @ U3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ) ).
% Cong\<^sub>0_cancel_left\<^sub>C\<^sub>S
thf(fact_863_coherent__normal__sub__rts__axioms,axiom,
cohere6429906645900029933list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ).
% coherent_normal_sub_rts_axioms
thf(fact_864_Resid__along__normal__preserves__reflects__con,axiom,
! [U2: list_a,T3: list_a,T6: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T6 @ U2 ) )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ T6 ) ) ) ) ).
% Resid_along_normal_preserves_reflects_con
thf(fact_865_NPath__Resid,axiom,
! [T4: list_a,U3: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ nn @ T4 )
=> ( ( paths_in_Arr_a @ resid @ U3 )
=> ( ( ( paths_in_Srcs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( paths_3208254328100665313Path_a @ resid @ nn @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) ) ) ) ) ).
% NPath_Resid
thf(fact_866_diamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [T3: list_a,U2: list_a,V2: list_a,V5: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) @ V2 )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ V5 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ V2 @ V5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ V5 @ V2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ).
% diamond_commutes_upto_Cong\<^sub>0
thf(fact_867_coherent_H,axiom,
! [V2: list_a,V5: list_a,W2: list_a,W3: list_a,T3: list_a,T6: list_a] :
( ( member_list_a @ V2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( member_list_a @ V5 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( member_list_a @ W2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( member_list_a @ W3 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V2 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ W2 ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V5 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ W3 ) )
=> ( ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ W2 )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ W3 ) )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ V2 ) @ ( paths_in_Resid_a @ resid @ T6 @ V5 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T6 @ V5 ) @ ( paths_in_Resid_a @ resid @ T3 @ V2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ W2 ) @ ( paths_in_Resid_a @ resid @ T6 @ W3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T6 @ W3 ) @ ( paths_in_Resid_a @ resid @ T3 @ W2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ) ) ) ) ) ).
% coherent'
thf(fact_868_composite__closed__left,axiom,
! [U2: list_a,T3: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 )
=> ? [X_12: list_a] : ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ X_12 ) ) ) ).
% composite_closed_left
thf(fact_869_composite__closed__right,axiom,
! [U2: list_a,T3: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ? [X_12: list_a] : ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 @ X_12 ) ) ) ).
% composite_closed_right
thf(fact_870_resid__along__elem__preserves__con,axiom,
! [T3: list_a,T6: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ T6 )
=> ( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T6 @ U2 ) ) ) ) ) ).
% resid_along_elem_preserves_con
thf(fact_871_NPath__def,axiom,
! [T4: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ nn @ T4 )
= ( ( paths_in_Arr_a @ resid @ T4 )
& ( ord_less_eq_set_a @ ( set_a2 @ T4 ) @ nn ) ) ) ).
% NPath_def
thf(fact_872_Cong_092_060_094sub_0620__append__Arr__NPath,axiom,
! [T4: list_a,U3: list_a] :
( ( T4 != nil_a )
=> ( ( paths_in_Arr_a @ resid @ ( append_a @ T4 @ U3 ) )
=> ( ( paths_3208254328100665313Path_a @ resid @ nn @ U3 )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( append_a @ T4 @ U3 ) @ T4 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ T4 @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ).
% Cong\<^sub>0_append_Arr_NPath
thf(fact_873_Coherent,axiom,
! [T4: list_a,U3: list_a,U5: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_3208254328100665313Path_a @ resid @ nn @ U3 )
=> ( ( paths_3208254328100665313Path_a @ resid @ nn @ U5 )
=> ( ( ( paths_in_Srcs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( ( ( paths_in_Srcs_a @ resid @ U3 )
= ( paths_in_Srcs_a @ resid @ U5 ) )
=> ( ( ( paths_in_Trgs_a @ resid @ U3 )
= ( paths_in_Trgs_a @ resid @ U5 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) @ ( paths_in_Resid_a @ resid @ T4 @ U5 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U5 ) @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ) ) ) ).
% Coherent
thf(fact_874_coherent,axiom,
! [T3: list_a,U2: list_a,U4: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( member_list_a @ U4 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( ( 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 ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T3 @ U4 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U4 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ) ) ) ).
% coherent
thf(fact_875_NPath__append,axiom,
! [T4: list_a,U3: list_a] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( paths_3208254328100665313Path_a @ resid @ nn @ ( append_a @ T4 @ U3 ) )
= ( ( paths_3208254328100665313Path_a @ resid @ nn @ T4 )
& ( paths_3208254328100665313Path_a @ resid @ nn @ U3 )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U3 ) ) ) ) ) ) ).
% NPath_append
thf(fact_876_NPath__Resid__Arr__single,axiom,
! [T4: list_a,U2: a] :
( ( paths_3208254328100665313Path_a @ resid @ nn @ T4 )
=> ( ( arr_a @ resid @ U2 )
=> ( ( ( paths_in_Srcs_a @ resid @ T4 )
= ( sources_a @ resid @ U2 ) )
=> ( paths_3208254328100665313Path_a @ resid @ nn @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U2 @ nil_a ) ) ) ) ) ) ).
% NPath_Resid_Arr_single
thf(fact_877_NPath__Resid__single__Arr,axiom,
! [T3: a,U3: list_a] :
( ( member_a @ T3 @ nn )
=> ( ( paths_in_Arr_a @ resid @ U3 )
=> ( ( ( sources_a @ resid @ T3 )
= ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( paths_3208254328100665313Path_a @ resid @ nn @ ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ U3 ) ) ) ) ) ).
% NPath_Resid_single_Arr
thf(fact_878_NPath__appendI,axiom,
! [T4: list_a,U3: list_a] :
( ( paths_3208254328100665313Path_a @ resid @ nn @ T4 )
=> ( ( paths_3208254328100665313Path_a @ resid @ nn @ U3 )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U3 ) )
=> ( paths_3208254328100665313Path_a @ resid @ nn @ ( append_a @ T4 @ U3 ) ) ) ) ) ).
% NPath_appendI
thf(fact_879_paths__in__rts__with__normal_OBackward__stable,axiom,
! [Resid: a > a > a,NN: set_a,U3: list_a,T4: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U3 )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ T4 ) ) ) ) ).
% paths_in_rts_with_normal.Backward_stable
thf(fact_880_paths__in__rts__with__normal_ONPath__implies__Arr,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ T4 )
=> ( paths_in_Arr_a @ Resid @ T4 ) ) ) ).
% paths_in_rts_with_normal.NPath_implies_Arr
thf(fact_881_paths__in__rts__with__normal_OIde__implies__NPath,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_in_Ide_a @ Resid @ T4 )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ T4 ) ) ) ).
% paths_in_rts_with_normal.Ide_implies_NPath
thf(fact_882_paths__in__rts__with__normal_OCong_092_060_094sub_0620__append__resid__NPath,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,U3: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) )
=> ( ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( append_a @ T4 @ ( paths_in_Resid_a @ Resid @ U3 @ T4 ) ) @ U3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ Resid @ U3 @ ( append_a @ T4 @ ( paths_in_Resid_a @ Resid @ U3 @ T4 ) ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Cong\<^sub>0_append_resid_NPath
thf(fact_883_paths__in__rts__with__normal_OBackward__stable__single,axiom,
! [Resid: a > a > a,NN: set_a,U3: list_a,T3: a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U3 )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T3 @ nil_a ) @ U3 ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ ( cons_a @ T3 @ nil_a ) ) ) ) ) ).
% paths_in_rts_with_normal.Backward_stable_single
thf(fact_884_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,T4: list_a,U3: list_a,U5: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( append_a @ T4 @ U3 ) @ ( append_a @ T4 @ U5 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ Resid @ ( append_a @ T4 @ U5 ) @ ( append_a @ T4 @ U3 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) )
=> ( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( U5 != nil_a )
=> ( ( member_list_a @ ( paths_in_Resid_a @ Resid @ U3 @ U5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ Resid @ U5 @ U3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Cong\<^sub>0_cancel_left\<^sub>C\<^sub>S
thf(fact_885_paths__in__rts__with__normal_OResid__NPath__preserves__reflects__Con,axiom,
! [Resid: a > a > a,NN: set_a,U3: list_a,T4: list_a,T7: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U3 )
=> ( ( ( paths_in_Srcs_a @ Resid @ T4 )
= ( paths_in_Srcs_a @ Resid @ U3 ) )
=> ( ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) @ ( paths_in_Resid_a @ Resid @ T7 @ U3 ) )
!= nil_a )
= ( ( paths_in_Resid_a @ Resid @ T4 @ T7 )
!= nil_a ) ) ) ) ) ).
% paths_in_rts_with_normal.Resid_NPath_preserves_reflects_Con
thf(fact_886_paths__in__rts__with__normal_ONPath__def,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ T4 )
= ( ( paths_in_Arr_a @ Resid @ T4 )
& ( ord_less_eq_set_a @ ( set_a2 @ T4 ) @ NN ) ) ) ) ).
% paths_in_rts_with_normal.NPath_def
thf(fact_887_paths__in__rts__with__normal_ONPath__Resid,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,U3: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ T4 )
=> ( ( paths_in_Arr_a @ Resid @ U3 )
=> ( ( ( paths_in_Srcs_a @ Resid @ T4 )
= ( paths_in_Srcs_a @ Resid @ U3 ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid
thf(fact_888_paths__in__rts__with__normal_ONPath__appendI,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,U3: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ T4 )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U3 )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ Resid @ T4 ) @ ( paths_in_Srcs_a @ Resid @ U3 ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ ( append_a @ T4 @ U3 ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_appendI
thf(fact_889_paths__in__rts__with__normal_ONPath__append,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,U3: list_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ ( append_a @ T4 @ U3 ) )
= ( ( paths_3208254328100665313Path_a @ Resid @ NN @ T4 )
& ( paths_3208254328100665313Path_a @ Resid @ NN @ U3 )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ Resid @ T4 ) @ ( paths_in_Srcs_a @ Resid @ U3 ) ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_append
thf(fact_890_paths__in__rts__with__normal_ONPath__Resid__Arr__single,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T4: list_list_a,U2: list_a] :
( ( paths_8755422415569132860list_a @ Resid @ NN )
=> ( ( paths_8553768783572469479list_a @ Resid @ NN @ T4 )
=> ( ( arr_list_a @ Resid @ U2 )
=> ( ( ( paths_in_Srcs_list_a @ Resid @ T4 )
= ( sources_list_a @ Resid @ U2 ) )
=> ( paths_8553768783572469479list_a @ Resid @ NN @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U2 @ nil_list_a ) ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid_Arr_single
thf(fact_891_paths__in__rts__with__normal_ONPath__Resid__Arr__single,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,U2: a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ T4 )
=> ( ( arr_a @ Resid @ U2 )
=> ( ( ( paths_in_Srcs_a @ Resid @ T4 )
= ( sources_a @ Resid @ U2 ) )
=> ( paths_3208254328100665313Path_a @ Resid @ NN @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U2 @ nil_a ) ) ) ) ) ) ) ).
% paths_in_rts_with_normal.NPath_Resid_Arr_single
thf(fact_892_arr__in__Cong__class,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( member_list_a @ T3 @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 ) ) ) ).
% arr_in_Cong_class
thf(fact_893_N_Oarr__in__Cong__class,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
=> ( member_a @ T3 @ ( normal7408713899360725774lass_a @ resid @ nn @ T3 ) ) ) ).
% N.arr_in_Cong_class
thf(fact_894_Cong__class__eqI_H,axiom,
! [T13: set_list_a,U12: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T13 )
=> ( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ U12 )
=> ( ( ( inf_inf_set_list_a @ T13 @ U12 )
!= bot_bot_set_list_a )
=> ( T13 = U12 ) ) ) ) ).
% Cong_class_eqI'
thf(fact_895_Cong__class__memb__is__arr,axiom,
! [T13: set_list_a,T3: list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T13 )
=> ( ( member_list_a @ T3 @ T13 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ) ).
% Cong_class_memb_is_arr
thf(fact_896_Cong__class__is__nonempty,axiom,
! [T13: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T13 )
=> ( T13 != bot_bot_set_list_a ) ) ).
% Cong_class_is_nonempty
thf(fact_897_is__Cong__class__def,axiom,
! [T13: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T13 )
= ( ? [T5: list_a] :
( ( member_list_a @ T5 @ T13 )
& ( T13
= ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T5 ) ) ) ) ) ).
% is_Cong_class_def
thf(fact_898_is__Cong__classI,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 ) ) ) ).
% is_Cong_classI
thf(fact_899_rep__in__Cong__class,axiom,
! [T13: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T13 )
=> ( member_list_a @ ( normal5825812266234710965list_a @ T13 ) @ T13 ) ) ).
% rep_in_Cong_class
thf(fact_900_normal__sub__rts_Ois__Cong__class_Ocong,axiom,
normal4947954984078133662list_a = normal4947954984078133662list_a ).
% normal_sub_rts.is_Cong_class.cong
thf(fact_901_normal__sub__rts_Ois__Cong__class_Ocong,axiom,
normal8595587647932138008lass_a = normal8595587647932138008lass_a ).
% normal_sub_rts.is_Cong_class.cong
thf(fact_902_normal__sub__rts_OCong__class_Ocong,axiom,
normal8738828432243163028list_a = normal8738828432243163028list_a ).
% normal_sub_rts.Cong_class.cong
thf(fact_903_normal__sub__rts_OCong__class_Ocong,axiom,
normal7408713899360725774lass_a = normal7408713899360725774lass_a ).
% normal_sub_rts.Cong_class.cong
thf(fact_904_Cong__class__rep,axiom,
! [T13: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T13 )
=> ( ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ ( normal5825812266234710965list_a @ T13 ) )
= T13 ) ) ).
% Cong_class_rep
thf(fact_905_Cong_H_Osimps,axiom,
! [A1: list_a,A22: list_a] :
( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ A1 @ A22 )
= ( ? [T5: list_a,U8: list_a] :
( ( A1 = U8 )
& ( A22 = T5 )
& ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T5 @ U8 ) )
| ? [T5: list_a,U8: list_a,V6: list_a] :
( ( A1 = T5 )
& ( A22 = V6 )
& ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T5 @ U8 )
& ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ U8 @ V6 ) )
| ? [T5: list_a,U8: list_a] :
( ( A1 = T5 )
& ( A22 = U8 )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T5 @ U8 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U8 @ T5 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
| ? [T5: list_a,U8: list_a] :
( ( A1 = T5 )
& ( A22
= ( paths_in_Resid_a @ resid @ T5 @ U8 ) )
& ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T5 )
& ( member_list_a @ U8 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T5 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U8 ) ) ) ) ) ).
% Cong'.simps
thf(fact_906_N_OCong__class__memb__is__arr,axiom,
! [T13: set_a,T3: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T13 )
=> ( ( member_a @ T3 @ T13 )
=> ( arr_a @ resid @ T3 ) ) ) ).
% N.Cong_class_memb_is_arr
thf(fact_907_N_OCong__class__is__nonempty,axiom,
! [T13: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T13 )
=> ( T13 != bot_bot_set_a ) ) ).
% N.Cong_class_is_nonempty
thf(fact_908_N_Ois__Cong__class__def,axiom,
! [T13: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T13 )
= ( ? [T5: a] :
( ( member_a @ T5 @ T13 )
& ( T13
= ( normal7408713899360725774lass_a @ resid @ nn @ T5 ) ) ) ) ) ).
% N.is_Cong_class_def
thf(fact_909_N_Ois__Cong__classI,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
=> ( normal8595587647932138008lass_a @ resid @ nn @ ( normal7408713899360725774lass_a @ resid @ nn @ T3 ) ) ) ).
% N.is_Cong_classI
thf(fact_910_N_OCong__class__eqI_H,axiom,
! [T13: set_a,U12: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T13 )
=> ( ( normal8595587647932138008lass_a @ resid @ nn @ U12 )
=> ( ( ( inf_inf_set_a @ T13 @ U12 )
!= bot_bot_set_a )
=> ( T13 = U12 ) ) ) ) ).
% N.Cong_class_eqI'
thf(fact_911_Cong_H__if,axiom,
! [U2: list_a,U4: list_a,T3: list_a,T6: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( member_list_a @ U4 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T6 @ U4 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T6 @ U4 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 ) ) ) ) ).
% Cong'_if
thf(fact_912_Cong_H_Ointros_I1_J,axiom,
! [T3: list_a,U2: list_a] :
( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ U2 )
=> ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ U2 @ T3 ) ) ).
% Cong'.intros(1)
thf(fact_913_Cong_H_Ointros_I2_J,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ U2 )
=> ( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ U2 @ V2 )
=> ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ V2 ) ) ) ).
% Cong'.intros(2)
thf(fact_914_Cong_H_Ointros_I3_J,axiom,
! [T3: list_a,U2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ U2 ) ) ).
% Cong'.intros(3)
thf(fact_915_Cong_H_Ointros_I4_J,axiom,
! [T3: list_a,U2: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) ) ) ) ).
% Cong'.intros(4)
thf(fact_916_Cong_H_Ocases,axiom,
! [A1: list_a,A22: list_a] :
( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ A1 @ A22 )
=> ( ~ ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ A22 @ A1 )
=> ( ! [U: list_a] :
( ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ A1 @ U )
=> ~ ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ U @ A22 ) )
=> ( ~ ( ( member_list_a @ ( paths_in_Resid_a @ resid @ A1 @ A22 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ A22 @ A1 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ~ ! [U: list_a] :
( ( A22
= ( paths_in_Resid_a @ resid @ A1 @ U ) )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ A1 )
=> ( ( member_list_a @ U @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ A1 )
!= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ) ) ) ) ) ) ) ).
% Cong'.cases
thf(fact_917_N_Orep__in__Cong__class,axiom,
! [T13: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T13 )
=> ( member_a @ ( normal3259722184653208495_rep_a @ T13 ) @ T13 ) ) ).
% N.rep_in_Cong_class
thf(fact_918_normal__sub__rts_OCong_H_Ocong,axiom,
normal4820288888589421745list_a = normal4820288888589421745list_a ).
% normal_sub_rts.Cong'.cong
thf(fact_919_normal__sub__rts_OCong_H_Ocong,axiom,
normal_sub_Cong_a2 = normal_sub_Cong_a2 ).
% normal_sub_rts.Cong'.cong
thf(fact_920_N_OCong_H_Osimps,axiom,
! [A1: a,A22: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ A1 @ A22 )
= ( ? [T5: a,U8: a] :
( ( A1 = U8 )
& ( A22 = T5 )
& ( normal_sub_Cong_a2 @ resid @ nn @ T5 @ U8 ) )
| ? [T5: a,U8: a,V6: a] :
( ( A1 = T5 )
& ( A22 = V6 )
& ( normal_sub_Cong_a2 @ resid @ nn @ T5 @ U8 )
& ( normal_sub_Cong_a2 @ resid @ nn @ U8 @ V6 ) )
| ? [T5: a,U8: a] :
( ( A1 = T5 )
& ( A22 = U8 )
& ( member_a @ ( resid @ T5 @ U8 ) @ nn )
& ( member_a @ ( resid @ U8 @ T5 ) @ nn ) )
| ? [T5: a,U8: a] :
( ( A1 = T5 )
& ( A22
= ( resid @ T5 @ U8 ) )
& ( arr_a @ resid @ T5 )
& ( member_a @ U8 @ nn )
& ( ( sources_a @ resid @ T5 )
= ( sources_a @ resid @ U8 ) ) ) ) ) ).
% N.Cong'.simps
thf(fact_921_N_OCong_H_Ointros_I3_J,axiom,
! [T3: a,U2: a] :
( ( ( member_a @ ( resid @ T3 @ U2 ) @ nn )
& ( member_a @ ( resid @ U2 @ T3 ) @ nn ) )
=> ( normal_sub_Cong_a2 @ resid @ nn @ T3 @ U2 ) ) ).
% N.Cong'.intros(3)
thf(fact_922_N_OCong_H_Ointros_I2_J,axiom,
! [T3: a,U2: a,V2: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ T3 @ U2 )
=> ( ( normal_sub_Cong_a2 @ resid @ nn @ U2 @ V2 )
=> ( normal_sub_Cong_a2 @ resid @ nn @ T3 @ V2 ) ) ) ).
% N.Cong'.intros(2)
thf(fact_923_N_OCong_H_Ointros_I1_J,axiom,
! [T3: a,U2: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ T3 @ U2 )
=> ( normal_sub_Cong_a2 @ resid @ nn @ U2 @ T3 ) ) ).
% N.Cong'.intros(1)
thf(fact_924_N_OCong_H__if,axiom,
! [U2: a,U4: a,T3: a,T6: a] :
( ( member_a @ U2 @ nn )
=> ( ( member_a @ U4 @ nn )
=> ( ( ( member_a @ ( resid @ ( resid @ T3 @ U2 ) @ ( resid @ T6 @ U4 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T6 @ U4 ) @ ( resid @ T3 @ U2 ) ) @ nn ) )
=> ( normal_sub_Cong_a2 @ resid @ nn @ T3 @ T6 ) ) ) ) ).
% N.Cong'_if
thf(fact_925_N_OCong_H_Ointros_I4_J,axiom,
! [T3: a,U2: a] :
( ( arr_a @ resid @ T3 )
=> ( ( member_a @ U2 @ nn )
=> ( ( ( sources_a @ resid @ T3 )
= ( sources_a @ resid @ U2 ) )
=> ( normal_sub_Cong_a2 @ resid @ nn @ T3 @ ( resid @ T3 @ U2 ) ) ) ) ) ).
% N.Cong'.intros(4)
thf(fact_926_N_OCong_H_Ocases,axiom,
! [A1: a,A22: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ A1 @ A22 )
=> ( ~ ( normal_sub_Cong_a2 @ resid @ nn @ A22 @ A1 )
=> ( ! [U: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ A1 @ U )
=> ~ ( normal_sub_Cong_a2 @ resid @ nn @ U @ A22 ) )
=> ( ~ ( ( member_a @ ( resid @ A1 @ A22 ) @ nn )
& ( member_a @ ( resid @ A22 @ A1 ) @ nn ) )
=> ~ ! [U: a] :
( ( A22
= ( resid @ A1 @ U ) )
=> ( ( arr_a @ resid @ A1 )
=> ( ( member_a @ U @ nn )
=> ( ( sources_a @ resid @ A1 )
!= ( sources_a @ resid @ U ) ) ) ) ) ) ) ) ) ).
% N.Cong'.cases
thf(fact_927_N_OCong__class__rep,axiom,
! [T13: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T13 )
=> ( ( normal7408713899360725774lass_a @ resid @ nn @ ( normal3259722184653208495_rep_a @ T13 ) )
= T13 ) ) ).
% N.Cong_class_rep
thf(fact_928_Cong__append__NPath__Arr,axiom,
! [T4: list_a,U3: list_a] :
( ( T4 != nil_a )
=> ( ( paths_in_Arr_a @ resid @ ( append_a @ U3 @ T4 ) )
=> ( ( paths_3208254328100665313Path_a @ resid @ nn @ U3 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ ( append_a @ U3 @ T4 ) @ T4 ) ) ) ) ).
% Cong_append_NPath_Arr
thf(fact_929_Cong__class__memb__Cong__rep,axiom,
! [T13: set_list_a,T3: list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T13 )
=> ( ( member_list_a @ T3 @ T13 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ ( normal5825812266234710965list_a @ T13 ) ) ) ) ).
% Cong_class_memb_Cong_rep
thf(fact_930_Cong__iff__cong,axiom,
! [T3: list_a,U2: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( ide_list_a @ ( paths_in_Resid_a @ resid ) ) ) @ T3 @ U2 )
= ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) ) ) ) ).
% Cong_iff_cong
thf(fact_931_normal__is__Cong__closed,axiom,
! [T3: list_a,T6: list_a] :
( ( member_list_a @ T3 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
=> ( member_list_a @ T6 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ).
% normal_is_Cong_closed
thf(fact_932_Cong__transitive,axiom,
! [T3: list_a,T12: list_a,T6: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T12 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T12 @ T6 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 ) ) ) ).
% Cong_transitive
thf(fact_933_Cong__symmetric,axiom,
! [T3: list_a,T6: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T6 @ T3 ) ) ).
% Cong_symmetric
thf(fact_934_Cong__def,axiom,
! [T3: list_a,T6: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
= ( ? [U8: list_a,U11: list_a] :
( ( member_list_a @ U8 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ U11 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U8 ) @ ( paths_in_Resid_a @ resid @ T6 @ U11 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T6 @ U11 ) @ ( paths_in_Resid_a @ resid @ T3 @ U8 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ).
% Cong_def
thf(fact_935_Cong_092_060_094sub_0620__implies__Cong,axiom,
! [T3: list_a,T6: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ T6 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ T6 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 ) ) ).
% Cong\<^sub>0_implies_Cong
thf(fact_936_CongE,axiom,
! [T3: list_a,T6: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
=> ~ ! [U: list_a] :
( ( member_list_a @ U @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ! [U13: list_a] :
( ( member_list_a @ U13 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ~ ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U ) @ ( paths_in_Resid_a @ resid @ T6 @ U13 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T6 @ U13 ) @ ( paths_in_Resid_a @ resid @ T3 @ U ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) ) ) ) ) ).
% CongE
thf(fact_937_Cong__closure__props_I1_J,axiom,
! [T3: list_a,U2: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ U2 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ U2 @ T3 ) ) ).
% Cong_closure_props(1)
thf(fact_938_Cong__closure__props_I2_J,axiom,
! [T3: list_a,U2: list_a,V2: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ U2 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ U2 @ V2 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ V2 ) ) ) ).
% Cong_closure_props(2)
thf(fact_939_Cong__closure__props_I3_J,axiom,
! [T3: list_a,U2: list_a] :
( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ U2 @ T3 ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ U2 ) ) ).
% Cong_closure_props(3)
thf(fact_940_sources__are__Cong,axiom,
! [A: list_a,T3: list_a,A4: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( member_list_a @ A4 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ A @ A4 ) ) ) ).
% sources_are_Cong
thf(fact_941_in__sources__respects__Cong,axiom,
! [T3: list_a,T6: list_a,A: list_a,A4: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
=> ( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( member_list_a @ A4 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ A @ A4 ) ) ) ) ).
% in_sources_respects_Cong
thf(fact_942_Cong__closure__props_I4_J,axiom,
! [U2: list_a,T3: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) ) ) ).
% Cong_closure_props(4)
thf(fact_943_Cong__imp__arr_I2_J,axiom,
! [T3: list_a,T6: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) ) ).
% Cong_imp_arr(2)
thf(fact_944_Cong__imp__arr_I1_J,axiom,
! [T3: list_a,T6: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) ) ).
% Cong_imp_arr(1)
thf(fact_945_Cong__reflexive,axiom,
! [T3: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T3 ) ) ).
% Cong_reflexive
thf(fact_946_targets__are__Cong,axiom,
! [B: list_a,T3: list_a,B4: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( member_list_a @ B4 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ B @ B4 ) ) ) ).
% targets_are_Cong
thf(fact_947_in__targets__respects__Cong,axiom,
! [T3: list_a,T6: list_a,B: list_a,B4: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
=> ( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T3 ) )
=> ( ( member_list_a @ B4 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ B @ B4 ) ) ) ) ).
% in_targets_respects_Cong
thf(fact_948_Cong__composite__of__normal__arr,axiom,
! [U2: list_a,T3: list_a,T6: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ T6 )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T6 @ T3 ) ) ) ).
% Cong_composite_of_normal_arr
thf(fact_949_Cong__char,axiom,
! [T3: list_a,T6: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
= ( normal4820288888589421745list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 ) ) ).
% Cong_char
thf(fact_950_Cong__class__membs__are__Cong,axiom,
! [T13: set_list_a,T3: list_a,T6: list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T13 )
=> ( ( member_list_a @ T3 @ T13 )
=> ( ( member_list_a @ T6 @ T13 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 ) ) ) ) ).
% Cong_class_membs_are_Cong
thf(fact_951_Cong__class__eqI,axiom,
! [T3: list_a,T6: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
=> ( ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 )
= ( normal8738828432243163028list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T6 ) ) ) ).
% Cong_class_eqI
thf(fact_952_Cong__subst_I2_J,axiom,
! [T3: list_a,T6: list_a,U2: list_a,U4: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ U2 @ U4 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U4 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T6 @ U4 ) ) ) ) ) ) ).
% Cong_subst(2)
thf(fact_953_Cong__subst_I1_J,axiom,
! [T3: list_a,T6: list_a,U2: list_a,U4: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ U2 @ U4 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U4 ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T6 @ U4 ) ) ) ) ) ).
% Cong_subst(1)
thf(fact_954_Cong__subst__con,axiom,
! [T3: list_a,U2: list_a,T6: list_a,U4: list_a] :
( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U2 ) )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U4 ) )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ U2 @ U4 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T3 @ U2 )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T6 @ U4 ) ) ) ) ) ) ).
% Cong_subst_con
thf(fact_955_composite__of__normal__arr,axiom,
! [T3: list_a,U2: list_a,T6: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T3 )
=> ( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U2 @ T3 @ T6 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T6 @ T3 ) ) ) ) ).
% composite_of_normal_arr
thf(fact_956_is__Cong__classE,axiom,
! [T13: set_list_a] :
( ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T13 )
=> ~ ( ( T13 != bot_bot_set_list_a )
=> ( ! [T2: list_a] :
( ( member_list_a @ T2 @ T13 )
=> ! [T14: list_a] :
( ( member_list_a @ T14 @ T13 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T2 @ T14 ) ) )
=> ~ ! [T2: list_a] :
( ( member_list_a @ T2 @ T13 )
=> ! [T14: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T14 @ T2 )
=> ( member_list_a @ T14 @ T13 ) ) ) ) ) ) ).
% is_Cong_classE
thf(fact_957_Srcs__respects__Cong,axiom,
! [T4: list_a,T7: list_a,A: a,A4: a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T4 @ T7 )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( ( member_a @ A4 @ ( paths_in_Srcs_a @ resid @ T7 ) )
=> ( 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 ) ) ) ) ) ).
% Srcs_respects_Cong
thf(fact_958_Trgs__respects__Cong,axiom,
! [T4: list_a,T7: list_a,B: a,B4: a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T4 @ T7 )
=> ( ( member_a @ B @ ( paths_in_Trgs_a @ resid @ T4 ) )
=> ( ( member_a @ B4 @ ( paths_in_Trgs_a @ resid @ T7 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ ( cons_a @ B @ nil_a ) @ ( cons_a @ B4 @ nil_a ) ) ) ) ) ).
% Trgs_respects_Cong
thf(fact_959_CongI,axiom,
! [U2: list_a,U4: list_a,T3: list_a,T6: list_a] :
( ( member_list_a @ U2 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( member_list_a @ U4 @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
=> ( ( ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T6 @ U4 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) )
& ( member_list_a @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T6 @ U4 ) @ ( paths_in_Resid_a @ resid @ T3 @ U2 ) ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T3 @ T6 ) ) ) ) ).
% CongI
thf(fact_960_is__Cong__classI_H,axiom,
! [T13: set_list_a] :
( ( T13 != bot_bot_set_list_a )
=> ( ! [T: list_a,T15: list_a] :
( ( member_list_a @ T @ T13 )
=> ( ( member_list_a @ T15 @ T13 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T @ T15 ) ) )
=> ( ! [T: list_a,T15: list_a] :
( ( member_list_a @ T @ T13 )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T15 @ T )
=> ( member_list_a @ T15 @ T13 ) ) )
=> ( normal4947954984078133662list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) @ T13 ) ) ) ) ).
% is_Cong_classI'
thf(fact_961_normal__sub__rts_OCong_Ocong,axiom,
normal4889798360446511898list_a = normal4889798360446511898list_a ).
% normal_sub_rts.Cong.cong
thf(fact_962_normal__sub__rts_OCong_Ocong,axiom,
normal_sub_Cong_a = normal_sub_Cong_a ).
% normal_sub_rts.Cong.cong
thf(fact_963_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,T3: a,T6: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ U2 @ T3 @ T6 )
=> ( ( member_a @ U2 @ NN )
=> ( normal_sub_Cong_a @ Resid @ NN @ T6 @ T3 ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_964_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,T3: list_a,T6: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( composite_of_list_a @ Resid @ U2 @ T3 @ T6 )
=> ( ( member_list_a @ U2 @ NN )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T6 @ T3 ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_965_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,T6: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( ( sources_a @ Resid @ T3 )
= ( sources_a @ Resid @ U2 ) )
=> ( ( ( sources_a @ Resid @ T6 )
= ( sources_a @ Resid @ U4 ) )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_a @ Resid @ T3 @ U2 )
= ( con_a @ Resid @ T6 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_966_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,T6: list_a,U4: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( ( sources_list_a @ Resid @ T3 )
= ( sources_list_a @ Resid @ U2 ) )
=> ( ( ( sources_list_a @ Resid @ T6 )
= ( sources_list_a @ Resid @ U4 ) )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
= ( con_list_a @ Resid @ T6 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_967_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a,U2: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( ( ( sources_a @ Resid @ T6 )
= ( sources_a @ Resid @ U4 ) )
=> ( con_a @ Resid @ T6 @ U4 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_968_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a,U2: list_a,U4: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( ( ( sources_list_a @ Resid @ T6 )
= ( sources_list_a @ Resid @ U4 ) )
=> ( con_list_a @ Resid @ T6 @ U4 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_969_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a,U2: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( ( ( sources_a @ Resid @ T6 )
= ( sources_a @ Resid @ U4 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_970_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a,U2: list_a,U4: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ U2 @ U4 )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( ( ( sources_list_a @ Resid @ T6 )
= ( sources_list_a @ Resid @ U4 ) )
=> ( normal4889798360446511898list_a @ Resid @ NN @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_971_paths__in__rts__with__normal_OSrcs__respects__Cong,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T4: list_list_a,T7: 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 ) ) @ T4 @ T7 )
=> ( ( member_list_a @ A @ ( paths_in_Srcs_list_a @ Resid @ T4 ) )
=> ( ( member_list_a @ A4 @ ( paths_in_Srcs_list_a @ Resid @ T7 ) )
=> ( 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_972_paths__in__rts__with__normal_OSrcs__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,T7: 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 ) ) @ T4 @ T7 )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ Resid @ T4 ) )
=> ( ( member_a @ A4 @ ( paths_in_Srcs_a @ Resid @ T7 ) )
=> ( 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_973_paths__in__rts__with__normal_OTrgs__respects__Cong,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T4: list_list_a,T7: list_list_a,B: list_a,B4: list_a] :
( ( paths_8755422415569132860list_a @ Resid @ NN )
=> ( ( normal2844150749108697120list_a @ ( paths_8620460302779588466list_a @ Resid ) @ ( collect_list_list_a @ ( paths_8553768783572469479list_a @ Resid @ NN ) ) @ T4 @ T7 )
=> ( ( member_list_a @ B @ ( paths_in_Trgs_list_a @ Resid @ T4 ) )
=> ( ( member_list_a @ B4 @ ( paths_in_Trgs_list_a @ Resid @ T7 ) )
=> ( 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 @ B4 @ nil_list_a ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Trgs_respects_Cong
thf(fact_974_paths__in__rts__with__normal_OTrgs__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,T7: list_a,B: a,B4: a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) @ T4 @ T7 )
=> ( ( member_a @ B @ ( paths_in_Trgs_a @ Resid @ T4 ) )
=> ( ( member_a @ B4 @ ( paths_in_Trgs_a @ Resid @ T7 ) )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) @ ( cons_a @ B @ nil_a ) @ ( cons_a @ B4 @ nil_a ) ) ) ) ) ) ).
% paths_in_rts_with_normal.Trgs_respects_Cong
thf(fact_975_paths__in__rts__with__coherent__normal_OCong__append__NPath__Arr,axiom,
! [Resid: a > a > a,NN: set_a,T4: list_a,U3: list_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( ( T4 != nil_a )
=> ( ( paths_in_Arr_a @ Resid @ ( append_a @ U3 @ T4 ) )
=> ( ( paths_3208254328100665313Path_a @ Resid @ NN @ U3 )
=> ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) @ ( append_a @ U3 @ T4 ) @ T4 ) ) ) ) ) ).
% paths_in_rts_with_coherent_normal.Cong_append_NPath_Arr
thf(fact_976_N_OCong__class__memb__Cong__rep,axiom,
! [T13: set_a,T3: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T13 )
=> ( ( member_a @ T3 @ T13 )
=> ( normal_sub_Cong_a @ resid @ nn @ T3 @ ( normal3259722184653208495_rep_a @ T13 ) ) ) ) ).
% N.Cong_class_memb_Cong_rep
thf(fact_977_N_Ois__Cong__classE,axiom,
! [T13: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T13 )
=> ~ ( ( T13 != bot_bot_set_a )
=> ( ! [T2: a] :
( ( member_a @ T2 @ T13 )
=> ! [T14: a] :
( ( member_a @ T14 @ T13 )
=> ( normal_sub_Cong_a @ resid @ nn @ T2 @ T14 ) ) )
=> ~ ! [T2: a] :
( ( member_a @ T2 @ T13 )
=> ! [T14: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T14 @ T2 )
=> ( member_a @ T14 @ T13 ) ) ) ) ) ) ).
% N.is_Cong_classE
thf(fact_978_R_OCong__iff__cong,axiom,
! [T3: a,U2: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T3 @ U2 )
= ( ( ide_a @ resid @ ( resid @ T3 @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ T3 ) ) ) ) ).
% R.Cong_iff_cong
thf(fact_979_N_OCong__closure__props_I3_J,axiom,
! [T3: a,U2: a] :
( ( ( member_a @ ( resid @ T3 @ U2 ) @ nn )
& ( member_a @ ( resid @ U2 @ T3 ) @ nn ) )
=> ( normal_sub_Cong_a @ resid @ nn @ T3 @ U2 ) ) ).
% N.Cong_closure_props(3)
thf(fact_980_N_OCong__closure__props_I2_J,axiom,
! [T3: a,U2: a,V2: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ U2 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ U2 @ V2 )
=> ( normal_sub_Cong_a @ resid @ nn @ T3 @ V2 ) ) ) ).
% N.Cong_closure_props(2)
thf(fact_981_N_OCong__closure__props_I1_J,axiom,
! [T3: a,U2: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ U2 )
=> ( normal_sub_Cong_a @ resid @ nn @ U2 @ T3 ) ) ).
% N.Cong_closure_props(1)
thf(fact_982_N_OCongE,axiom,
! [T3: a,T6: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
=> ~ ! [U: a] :
( ( member_a @ U @ nn )
=> ! [U13: a] :
( ( member_a @ U13 @ nn )
=> ~ ( ( member_a @ ( resid @ ( resid @ T3 @ U ) @ ( resid @ T6 @ U13 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T6 @ U13 ) @ ( resid @ T3 @ U ) ) @ nn ) ) ) ) ) ).
% N.CongE
thf(fact_983_N_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [T3: a,T6: a] :
( ( ( member_a @ ( resid @ T3 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T3 ) @ nn ) )
=> ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 ) ) ).
% N.Cong\<^sub>0_implies_Cong
thf(fact_984_N_OCong__def,axiom,
! [T3: a,T6: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
= ( ? [U8: a,U11: a] :
( ( member_a @ U8 @ nn )
& ( member_a @ U11 @ nn )
& ( member_a @ ( resid @ ( resid @ T3 @ U8 ) @ ( resid @ T6 @ U11 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T6 @ U11 ) @ ( resid @ T3 @ U8 ) ) @ nn ) ) ) ) ).
% N.Cong_def
thf(fact_985_N_OCong__symmetric,axiom,
! [T3: a,T6: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
=> ( normal_sub_Cong_a @ resid @ nn @ T6 @ T3 ) ) ).
% N.Cong_symmetric
thf(fact_986_N_OCong__transitive,axiom,
! [T3: a,T12: a,T6: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T12 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T12 @ T6 )
=> ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 ) ) ) ).
% N.Cong_transitive
thf(fact_987_N_Onormal__is__Cong__closed,axiom,
! [T3: a,T6: a] :
( ( member_a @ T3 @ nn )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
=> ( member_a @ T6 @ nn ) ) ) ).
% N.normal_is_Cong_closed
thf(fact_988_N_Osources__are__Cong,axiom,
! [A: a,T3: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T3 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ A @ A4 ) ) ) ).
% N.sources_are_Cong
thf(fact_989_N_Oin__sources__respects__Cong,axiom,
! [T3: a,T6: a,A: a,A4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
=> ( ( member_a @ A @ ( sources_a @ resid @ T3 ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T6 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ A @ A4 ) ) ) ) ).
% N.in_sources_respects_Cong
thf(fact_990_N_OCong__closure__props_I4_J,axiom,
! [U2: a,T3: a] :
( ( member_a @ U2 @ nn )
=> ( ( ( sources_a @ resid @ T3 )
= ( sources_a @ resid @ U2 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ T3 @ ( resid @ T3 @ U2 ) ) ) ) ).
% N.Cong_closure_props(4)
thf(fact_991_N_OCong__reflexive,axiom,
! [T3: a] :
( ( arr_a @ resid @ T3 )
=> ( normal_sub_Cong_a @ resid @ nn @ T3 @ T3 ) ) ).
% N.Cong_reflexive
thf(fact_992_N_OCong__imp__arr_I1_J,axiom,
! [T3: a,T6: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
=> ( arr_a @ resid @ T3 ) ) ).
% N.Cong_imp_arr(1)
thf(fact_993_N_OCong__imp__arr_I2_J,axiom,
! [T3: a,T6: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
=> ( arr_a @ resid @ T6 ) ) ).
% N.Cong_imp_arr(2)
thf(fact_994_N_Otargets__are__Cong,axiom,
! [B: a,T3: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T3 ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T3 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ B @ B4 ) ) ) ).
% N.targets_are_Cong
thf(fact_995_N_Oin__targets__respects__Cong,axiom,
! [T3: a,T6: a,B: a,B4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
=> ( ( member_a @ B @ ( targets_a @ resid @ T3 ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T6 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ B @ B4 ) ) ) ) ).
% N.in_targets_respects_Cong
thf(fact_996_N_OCong__composite__of__normal__arr,axiom,
! [U2: a,T3: a,T6: a] :
( ( composite_of_a @ resid @ U2 @ T3 @ T6 )
=> ( ( member_a @ U2 @ nn )
=> ( normal_sub_Cong_a @ resid @ nn @ T6 @ T3 ) ) ) ).
% N.Cong_composite_of_normal_arr
thf(fact_997_N_OCong__char,axiom,
! [T3: a,T6: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
= ( normal_sub_Cong_a2 @ resid @ nn @ T3 @ T6 ) ) ).
% N.Cong_char
thf(fact_998_N_OCong__class__membs__are__Cong,axiom,
! [T13: set_a,T3: a,T6: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T13 )
=> ( ( member_a @ T3 @ T13 )
=> ( ( member_a @ T6 @ T13 )
=> ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 ) ) ) ) ).
% N.Cong_class_membs_are_Cong
thf(fact_999_N_OCong__class__eqI,axiom,
! [T3: a,T6: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
=> ( ( normal7408713899360725774lass_a @ resid @ nn @ T3 )
= ( normal7408713899360725774lass_a @ resid @ nn @ T6 ) ) ) ).
% N.Cong_class_eqI
thf(fact_1000_N_OCong__subst_I2_J,axiom,
! [T3: a,T6: a,U2: a,U4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ U2 @ U4 )
=> ( ( con_a @ resid @ T3 @ U2 )
=> ( ( ( sources_a @ resid @ T6 )
= ( sources_a @ resid @ U4 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ ( resid @ T3 @ U2 ) @ ( resid @ T6 @ U4 ) ) ) ) ) ) ).
% N.Cong_subst(2)
thf(fact_1001_N_OCong__subst_I1_J,axiom,
! [T3: a,T6: a,U2: a,U4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ U2 @ U4 )
=> ( ( con_a @ resid @ T3 @ U2 )
=> ( ( ( sources_a @ resid @ T6 )
= ( sources_a @ resid @ U4 ) )
=> ( con_a @ resid @ T6 @ U4 ) ) ) ) ) ).
% N.Cong_subst(1)
thf(fact_1002_N_OCong__subst__con,axiom,
! [T3: a,U2: a,T6: a,U4: a] :
( ( ( sources_a @ resid @ T3 )
= ( sources_a @ resid @ U2 ) )
=> ( ( ( sources_a @ resid @ T6 )
= ( sources_a @ resid @ U4 ) )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ U2 @ U4 )
=> ( ( con_a @ resid @ T3 @ U2 )
= ( con_a @ resid @ T6 @ U4 ) ) ) ) ) ) ).
% N.Cong_subst_con
thf(fact_1003_N_Ocomposite__of__normal__arr,axiom,
! [T3: a,U2: a,T6: a] :
( ( arr_a @ resid @ T3 )
=> ( ( member_a @ U2 @ nn )
=> ( ( composite_of_a @ resid @ U2 @ T3 @ T6 )
=> ( normal_sub_Cong_a @ resid @ nn @ T6 @ T3 ) ) ) ) ).
% N.composite_of_normal_arr
thf(fact_1004_N_OCongI,axiom,
! [U2: a,U4: a,T3: a,T6: a] :
( ( member_a @ U2 @ nn )
=> ( ( member_a @ U4 @ nn )
=> ( ( ( member_a @ ( resid @ ( resid @ T3 @ U2 ) @ ( resid @ T6 @ U4 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T6 @ U4 ) @ ( resid @ T3 @ U2 ) ) @ nn ) )
=> ( normal_sub_Cong_a @ resid @ nn @ T3 @ T6 ) ) ) ) ).
% N.CongI
thf(fact_1005_N_Ois__Cong__classI_H,axiom,
! [T13: set_a] :
( ( T13 != bot_bot_set_a )
=> ( ! [T: a,T15: a] :
( ( member_a @ T @ T13 )
=> ( ( member_a @ T15 @ T13 )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T15 ) ) )
=> ( ! [T: a,T15: a] :
( ( member_a @ T @ T13 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T15 @ T )
=> ( member_a @ T15 @ T13 ) ) )
=> ( normal8595587647932138008lass_a @ resid @ nn @ T13 ) ) ) ) ).
% N.is_Cong_classI'
thf(fact_1006_normal__sub__rts__axioms,axiom,
normal6589580540804570479list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ resid @ nn ) ) ).
% normal_sub_rts_axioms
thf(fact_1007_length__Resid,axiom,
! [T4: list_a,U3: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
=> ( ( size_size_list_a @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) )
= ( size_size_list_a @ T4 ) ) ) ).
% length_Resid
thf(fact_1008_identities__form__normal__sub__rts,axiom,
normal6589580540804570479list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( ide_list_a @ ( paths_in_Resid_a @ resid ) ) ) ).
% identities_form_normal_sub_rts
thf(fact_1009_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us2: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us2 )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us2 )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_1010_length__rotate1,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate1
thf(fact_1011_normal__sub__rts_OCong__class__membs__are__Cong,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T13: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ T13 )
=> ( ( member_list_a @ T3 @ T13 )
=> ( ( member_list_a @ T6 @ T13 )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_membs_are_Cong
thf(fact_1012_normal__sub__rts_OCong__class__membs__are__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T13: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T13 )
=> ( ( member_a @ T3 @ T13 )
=> ( ( member_a @ T6 @ T13 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_membs_are_Cong
thf(fact_1013_normal__sub__rts_OCong__char,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
= ( normal4820288888589421745list_a @ Resid @ NN @ T3 @ T6 ) ) ) ).
% normal_sub_rts.Cong_char
thf(fact_1014_normal__sub__rts_OCong__char,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
= ( normal_sub_Cong_a2 @ Resid @ NN @ T3 @ T6 ) ) ) ).
% normal_sub_rts.Cong_char
thf(fact_1015_normal__sub__rts_Oin__targets__respects__Cong,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a,B: list_a,B4: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
=> ( ( member_list_a @ B @ ( targets_list_a @ Resid @ T3 ) )
=> ( ( member_list_a @ B4 @ ( targets_list_a @ Resid @ T6 ) )
=> ( normal4889798360446511898list_a @ Resid @ NN @ B @ B4 ) ) ) ) ) ).
% normal_sub_rts.in_targets_respects_Cong
thf(fact_1016_normal__sub__rts_Oin__targets__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a,B: a,B4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T3 ) )
=> ( ( member_a @ B4 @ ( targets_a @ Resid @ T6 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ B @ B4 ) ) ) ) ) ).
% normal_sub_rts.in_targets_respects_Cong
thf(fact_1017_normal__sub__rts_Oin__sources__respects__Cong,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a,A: list_a,A4: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
=> ( ( member_list_a @ A @ ( sources_list_a @ Resid @ T3 ) )
=> ( ( member_list_a @ A4 @ ( sources_list_a @ Resid @ T6 ) )
=> ( normal4889798360446511898list_a @ Resid @ NN @ A @ A4 ) ) ) ) ) ).
% normal_sub_rts.in_sources_respects_Cong
thf(fact_1018_normal__sub__rts_Oin__sources__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a,A: a,A4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T3 ) )
=> ( ( member_a @ A4 @ ( sources_a @ Resid @ T6 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ A @ A4 ) ) ) ) ) ).
% normal_sub_rts.in_sources_respects_Cong
thf(fact_1019_normal__sub__rts_Otargets__are__Cong,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,B: list_a,T3: list_a,B4: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ B @ ( targets_list_a @ Resid @ T3 ) )
=> ( ( member_list_a @ B4 @ ( targets_list_a @ Resid @ T3 ) )
=> ( normal4889798360446511898list_a @ Resid @ NN @ B @ B4 ) ) ) ) ).
% normal_sub_rts.targets_are_Cong
thf(fact_1020_normal__sub__rts_Otargets__are__Cong,axiom,
! [Resid: a > a > a,NN: set_a,B: a,T3: a,B4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T3 ) )
=> ( ( member_a @ B4 @ ( targets_a @ Resid @ T3 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ B @ B4 ) ) ) ) ).
% normal_sub_rts.targets_are_Cong
thf(fact_1021_normal__sub__rts_Osources__are__Cong,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,A: list_a,T3: list_a,A4: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ A @ ( sources_list_a @ Resid @ T3 ) )
=> ( ( member_list_a @ A4 @ ( sources_list_a @ Resid @ T3 ) )
=> ( normal4889798360446511898list_a @ Resid @ NN @ A @ A4 ) ) ) ) ).
% normal_sub_rts.sources_are_Cong
thf(fact_1022_normal__sub__rts_Osources__are__Cong,axiom,
! [Resid: a > a > a,NN: set_a,A: a,T3: a,A4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T3 ) )
=> ( ( member_a @ A4 @ ( sources_a @ Resid @ T3 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ A @ A4 ) ) ) ) ).
% normal_sub_rts.sources_are_Cong
thf(fact_1023_normal__sub__rts_OCong__closure__props_I4_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( ( sources_list_a @ Resid @ T3 )
= ( sources_list_a @ Resid @ U2 ) )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ ( Resid @ T3 @ U2 ) ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(4)
thf(fact_1024_normal__sub__rts_OCong__closure__props_I4_J,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( ( sources_a @ Resid @ T3 )
= ( sources_a @ Resid @ U2 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T3 @ ( Resid @ T3 @ U2 ) ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(4)
thf(fact_1025_normal__sub__rts_OCong__reflexive,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( arr_list_a @ Resid @ T3 )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T3 ) ) ) ).
% normal_sub_rts.Cong_reflexive
thf(fact_1026_normal__sub__rts_OCong__reflexive,axiom,
! [Resid: a > a > a,NN: set_a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T3 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T3 ) ) ) ).
% normal_sub_rts.Cong_reflexive
thf(fact_1027_normal__sub__rts_OCong__imp__arr_I1_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
=> ( arr_list_a @ Resid @ T3 ) ) ) ).
% normal_sub_rts.Cong_imp_arr(1)
thf(fact_1028_normal__sub__rts_OCong__imp__arr_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
=> ( arr_a @ Resid @ T3 ) ) ) ).
% normal_sub_rts.Cong_imp_arr(1)
thf(fact_1029_normal__sub__rts_OCong__imp__arr_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
=> ( arr_list_a @ Resid @ T6 ) ) ) ).
% normal_sub_rts.Cong_imp_arr(2)
thf(fact_1030_normal__sub__rts_OCong__imp__arr_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
=> ( arr_a @ Resid @ T6 ) ) ) ).
% normal_sub_rts.Cong_imp_arr(2)
thf(fact_1031_normal__sub__rts_OCong__closure__props_I3_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ U2 ) @ NN )
& ( member_list_a @ ( Resid @ U2 @ T3 ) @ NN ) )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ U2 ) ) ) ).
% normal_sub_rts.Cong_closure_props(3)
thf(fact_1032_normal__sub__rts_OCong__closure__props_I3_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ U2 ) @ NN )
& ( member_a @ ( Resid @ U2 @ T3 ) @ NN ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T3 @ U2 ) ) ) ).
% normal_sub_rts.Cong_closure_props(3)
thf(fact_1033_normal__sub__rts_OCong__closure__props_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,V2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ U2 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ U2 @ V2 )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ V2 ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(2)
thf(fact_1034_normal__sub__rts_OCong__closure__props_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,V2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ U2 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U2 @ V2 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T3 @ V2 ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(2)
thf(fact_1035_normal__sub__rts_OCong__closure__props_I1_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ U2 )
=> ( normal4889798360446511898list_a @ Resid @ NN @ U2 @ T3 ) ) ) ).
% normal_sub_rts.Cong_closure_props(1)
thf(fact_1036_normal__sub__rts_OCong__closure__props_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ U2 )
=> ( normal_sub_Cong_a @ Resid @ NN @ U2 @ T3 ) ) ) ).
% normal_sub_rts.Cong_closure_props(1)
thf(fact_1037_normal__sub__rts_OCongE,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
=> ~ ! [U: list_a] :
( ( member_list_a @ U @ NN )
=> ! [U13: list_a] :
( ( member_list_a @ U13 @ NN )
=> ~ ( ( member_list_a @ ( Resid @ ( Resid @ T3 @ U ) @ ( Resid @ T6 @ U13 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ U13 ) @ ( Resid @ T3 @ U ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.CongE
thf(fact_1038_normal__sub__rts_OCongE,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
=> ~ ! [U: a] :
( ( member_a @ U @ NN )
=> ! [U13: a] :
( ( member_a @ U13 @ NN )
=> ~ ( ( member_a @ ( Resid @ ( Resid @ T3 @ U ) @ ( Resid @ T6 @ U13 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U13 ) @ ( Resid @ T3 @ U ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.CongE
thf(fact_1039_normal__sub__rts_OCongI,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,U4: list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( member_list_a @ U4 @ NN )
=> ( ( ( member_list_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U4 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ U4 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 ) ) ) ) ) ).
% normal_sub_rts.CongI
thf(fact_1040_normal__sub__rts_OCongI,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,U4: a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( member_a @ U4 @ NN )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U4 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U4 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 ) ) ) ) ) ).
% normal_sub_rts.CongI
thf(fact_1041_normal__sub__rts_OCong__def,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
= ( ? [U8: list_a,U11: list_a] :
( ( member_list_a @ U8 @ NN )
& ( member_list_a @ U11 @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T3 @ U8 ) @ ( Resid @ T6 @ U11 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ U11 ) @ ( Resid @ T3 @ U8 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong_def
thf(fact_1042_normal__sub__rts_OCong__def,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
= ( ? [U8: a,U11: a] :
( ( member_a @ U8 @ NN )
& ( member_a @ U11 @ NN )
& ( member_a @ ( Resid @ ( Resid @ T3 @ U8 ) @ ( Resid @ T6 @ U11 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U11 ) @ ( Resid @ T3 @ U8 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong_def
thf(fact_1043_normal__sub__rts_OCong__symmetric,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T6 @ T3 ) ) ) ).
% normal_sub_rts.Cong_symmetric
thf(fact_1044_normal__sub__rts_OCong__symmetric,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T6 @ T3 ) ) ) ).
% normal_sub_rts.Cong_symmetric
thf(fact_1045_normal__sub__rts_OCong__transitive,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T12: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T12 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T12 @ T6 )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 ) ) ) ) ).
% normal_sub_rts.Cong_transitive
thf(fact_1046_normal__sub__rts_OCong__transitive,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T12: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T12 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T12 @ T6 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 ) ) ) ) ).
% normal_sub_rts.Cong_transitive
thf(fact_1047_normal__sub__rts_Onormal__is__Cong__closed,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ T3 @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
=> ( member_list_a @ T6 @ NN ) ) ) ) ).
% normal_sub_rts.normal_is_Cong_closed
thf(fact_1048_normal__sub__rts_Onormal__is__Cong__closed,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ T3 @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
=> ( member_a @ T6 @ NN ) ) ) ) ).
% normal_sub_rts.normal_is_Cong_closed
thf(fact_1049_normal__sub__rts_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_implies_Cong
thf(fact_1050_normal__sub__rts_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_implies_Cong
thf(fact_1051_normal__sub__rts_OCong__class__eqI,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ T6 )
=> ( ( normal8738828432243163028list_a @ Resid @ NN @ T3 )
= ( normal8738828432243163028list_a @ Resid @ NN @ T6 ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI
thf(fact_1052_normal__sub__rts_OCong__class__eqI,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T6 )
=> ( ( normal7408713899360725774lass_a @ Resid @ NN @ T3 )
= ( normal7408713899360725774lass_a @ Resid @ NN @ T6 ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI
thf(fact_1053_normal__sub__rts_OCong__class__is__nonempty,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T13: set_list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ T13 )
=> ( T13 != bot_bot_set_list_a ) ) ) ).
% normal_sub_rts.Cong_class_is_nonempty
thf(fact_1054_normal__sub__rts_OCong__class__is__nonempty,axiom,
! [Resid: a > a > a,NN: set_a,T13: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T13 )
=> ( T13 != bot_bot_set_a ) ) ) ).
% normal_sub_rts.Cong_class_is_nonempty
thf(fact_1055_normal__sub__rts_Ois__Cong__class__def,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T13: set_list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ T13 )
= ( ? [T5: list_a] :
( ( member_list_a @ T5 @ T13 )
& ( T13
= ( normal8738828432243163028list_a @ Resid @ NN @ T5 ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_class_def
thf(fact_1056_normal__sub__rts_Ois__Cong__class__def,axiom,
! [Resid: a > a > a,NN: set_a,T13: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T13 )
= ( ? [T5: a] :
( ( member_a @ T5 @ T13 )
& ( T13
= ( normal7408713899360725774lass_a @ Resid @ NN @ T5 ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_class_def
thf(fact_1057_normal__sub__rts_OCong__class__memb__is__arr,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T13: set_list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ T13 )
=> ( ( member_list_a @ T3 @ T13 )
=> ( arr_list_a @ Resid @ T3 ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_is_arr
thf(fact_1058_normal__sub__rts_OCong__class__memb__is__arr,axiom,
! [Resid: a > a > a,NN: set_a,T13: set_a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T13 )
=> ( ( member_a @ T3 @ T13 )
=> ( arr_a @ Resid @ T3 ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_is_arr
thf(fact_1059_normal__sub__rts_OCong_H_Ointros_I3_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ U2 ) @ NN )
& ( member_list_a @ ( Resid @ U2 @ T3 ) @ NN ) )
=> ( normal4820288888589421745list_a @ Resid @ NN @ T3 @ U2 ) ) ) ).
% normal_sub_rts.Cong'.intros(3)
thf(fact_1060_normal__sub__rts_OCong_H_Ointros_I3_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ U2 ) @ NN )
& ( member_a @ ( Resid @ U2 @ T3 ) @ NN ) )
=> ( normal_sub_Cong_a2 @ Resid @ NN @ T3 @ U2 ) ) ) ).
% normal_sub_rts.Cong'.intros(3)
thf(fact_1061_normal__sub__rts_OCong_H_Ointros_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,V2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4820288888589421745list_a @ Resid @ NN @ T3 @ U2 )
=> ( ( normal4820288888589421745list_a @ Resid @ NN @ U2 @ V2 )
=> ( normal4820288888589421745list_a @ Resid @ NN @ T3 @ V2 ) ) ) ) ).
% normal_sub_rts.Cong'.intros(2)
thf(fact_1062_normal__sub__rts_OCong_H_Ointros_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,V2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a2 @ Resid @ NN @ T3 @ U2 )
=> ( ( normal_sub_Cong_a2 @ Resid @ NN @ U2 @ V2 )
=> ( normal_sub_Cong_a2 @ Resid @ NN @ T3 @ V2 ) ) ) ) ).
% normal_sub_rts.Cong'.intros(2)
thf(fact_1063_normal__sub__rts_OCong_H_Ointros_I1_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4820288888589421745list_a @ Resid @ NN @ T3 @ U2 )
=> ( normal4820288888589421745list_a @ Resid @ NN @ U2 @ T3 ) ) ) ).
% normal_sub_rts.Cong'.intros(1)
thf(fact_1064_normal__sub__rts_OCong_H_Ointros_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a2 @ Resid @ NN @ T3 @ U2 )
=> ( normal_sub_Cong_a2 @ Resid @ NN @ U2 @ T3 ) ) ) ).
% normal_sub_rts.Cong'.intros(1)
thf(fact_1065_normal__sub__rts_OCong_H__if,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,U4: list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( member_list_a @ U4 @ NN )
=> ( ( ( member_list_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U4 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ U4 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) )
=> ( normal4820288888589421745list_a @ Resid @ NN @ T3 @ T6 ) ) ) ) ) ).
% normal_sub_rts.Cong'_if
thf(fact_1066_normal__sub__rts_OCong_H__if,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,U4: a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( member_a @ U4 @ NN )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U4 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U4 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) )
=> ( normal_sub_Cong_a2 @ Resid @ NN @ T3 @ T6 ) ) ) ) ) ).
% normal_sub_rts.Cong'_if
thf(fact_1067_normal__sub__rts_Orep__in__Cong__class,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T13: set_list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ T13 )
=> ( member_list_a @ ( normal5825812266234710965list_a @ T13 ) @ T13 ) ) ) ).
% normal_sub_rts.rep_in_Cong_class
thf(fact_1068_normal__sub__rts_Orep__in__Cong__class,axiom,
! [Resid: a > a > a,NN: set_a,T13: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T13 )
=> ( member_a @ ( normal3259722184653208495_rep_a @ T13 ) @ T13 ) ) ) ).
% normal_sub_rts.rep_in_Cong_class
thf(fact_1069_paths__in__rts__with__normal_Oaxioms_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a] :
( ( paths_8755422415569132860list_a @ Resid @ NN )
=> ( normal6589580540804570479list_a @ Resid @ NN ) ) ).
% paths_in_rts_with_normal.axioms(2)
thf(fact_1070_paths__in__rts__with__normal_Oaxioms_I2_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( normal_sub_rts_a @ Resid @ NN ) ) ).
% paths_in_rts_with_normal.axioms(2)
thf(fact_1071_normal__sub__rts_OResid__along__normal__preserves__reflects__con,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( ( sources_a @ Resid @ T3 )
= ( sources_a @ Resid @ U2 ) )
=> ( ( con_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U2 ) )
= ( con_a @ Resid @ T3 @ T6 ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_reflects_con
thf(fact_1072_normal__sub__rts_OResid__along__normal__preserves__reflects__con,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( ( sources_list_a @ Resid @ T3 )
= ( sources_list_a @ Resid @ U2 ) )
=> ( ( con_list_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U2 ) )
= ( con_list_a @ Resid @ T3 @ T6 ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_reflects_con
thf(fact_1073_normal__sub__rts_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,V2: list_a,V5: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( ( composite_of_list_a @ Resid @ T3 @ ( Resid @ U2 @ T3 ) @ V2 )
=> ( ( composite_of_list_a @ Resid @ U2 @ ( Resid @ T3 @ U2 ) @ V5 )
=> ( ( member_list_a @ ( Resid @ V2 @ V5 ) @ NN )
& ( member_list_a @ ( Resid @ V5 @ V2 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.diamond_commutes_upto_Cong\<^sub>0
thf(fact_1074_normal__sub__rts_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,V2: a,V5: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( ( composite_of_a @ Resid @ T3 @ ( Resid @ U2 @ T3 ) @ V2 )
=> ( ( composite_of_a @ Resid @ U2 @ ( Resid @ T3 @ U2 ) @ V5 )
=> ( ( member_a @ ( Resid @ V2 @ V5 ) @ NN )
& ( member_a @ ( Resid @ V5 @ V2 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.diamond_commutes_upto_Cong\<^sub>0
thf(fact_1075_normal__sub__rts_Ocomposite__closed__right,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( seq_list_a @ Resid @ T3 @ U2 )
=> ? [X_12: list_a] : ( composite_of_list_a @ Resid @ T3 @ U2 @ X_12 ) ) ) ) ).
% normal_sub_rts.composite_closed_right
thf(fact_1076_normal__sub__rts_Ocomposite__closed__right,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( seq_a @ Resid @ T3 @ U2 )
=> ? [X_12: a] : ( composite_of_a @ Resid @ T3 @ U2 @ X_12 ) ) ) ) ).
% normal_sub_rts.composite_closed_right
thf(fact_1077_normal__sub__rts_Ocomposite__closed__left,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( seq_list_a @ Resid @ U2 @ T3 )
=> ? [X_12: list_a] : ( composite_of_list_a @ Resid @ U2 @ T3 @ X_12 ) ) ) ) ).
% normal_sub_rts.composite_closed_left
thf(fact_1078_normal__sub__rts_Ocomposite__closed__left,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( seq_a @ Resid @ U2 @ T3 )
=> ? [X_12: a] : ( composite_of_a @ Resid @ U2 @ T3 @ X_12 ) ) ) ) ).
% normal_sub_rts.composite_closed_left
thf(fact_1079_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1080_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1081_normal__sub__rts_Oresid__along__elem__preserves__con,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a,U2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( con_a @ Resid @ T3 @ T6 )
=> ( ( coinitial_a @ Resid @ T3 @ U2 )
=> ( ( member_a @ U2 @ NN )
=> ( con_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U2 ) ) ) ) ) ) ).
% normal_sub_rts.resid_along_elem_preserves_con
thf(fact_1082_normal__sub__rts_Oresid__along__elem__preserves__con,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a,U2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( con_list_a @ Resid @ T3 @ T6 )
=> ( ( coinitial_list_a @ Resid @ T3 @ U2 )
=> ( ( member_list_a @ U2 @ NN )
=> ( con_list_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U2 ) ) ) ) ) ) ).
% normal_sub_rts.resid_along_elem_preserves_con
thf(fact_1083_normal__sub__rts_Ofactor__closed_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,V2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( composite_of_list_a @ Resid @ T3 @ U2 @ V2 )
=> ( ( member_list_a @ V2 @ NN )
=> ( member_list_a @ U2 @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(2)
thf(fact_1084_normal__sub__rts_Ofactor__closed_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,V2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T3 @ U2 @ V2 )
=> ( ( member_a @ V2 @ NN )
=> ( member_a @ U2 @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(2)
thf(fact_1085_normal__sub__rts_Ofactor__closed_I1_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,V2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( composite_of_list_a @ Resid @ T3 @ U2 @ V2 )
=> ( ( member_list_a @ V2 @ NN )
=> ( member_list_a @ T3 @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(1)
thf(fact_1086_normal__sub__rts_Ofactor__closed_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,V2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T3 @ U2 @ V2 )
=> ( ( member_a @ V2 @ NN )
=> ( member_a @ T3 @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(1)
thf(fact_1087_normal__sub__rts_OCong_092_060_094sub_0620__iff,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN ) )
= ( ? [U8: list_a,U11: list_a,V6: list_a,V7: list_a] :
( ( member_list_a @ U8 @ NN )
& ( member_list_a @ U11 @ NN )
& ( member_list_a @ ( Resid @ V6 @ V7 ) @ NN )
& ( member_list_a @ ( Resid @ V7 @ V6 ) @ NN )
& ( composite_of_list_a @ Resid @ T3 @ U8 @ V6 )
& ( composite_of_list_a @ Resid @ T6 @ U11 @ V7 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_iff
thf(fact_1088_normal__sub__rts_OCong_092_060_094sub_0620__iff,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T3 ) @ NN ) )
= ( ? [U8: a,U11: a,V6: a,V7: a] :
( ( member_a @ U8 @ NN )
& ( member_a @ U11 @ NN )
& ( member_a @ ( Resid @ V6 @ V7 ) @ NN )
& ( member_a @ ( Resid @ V7 @ V6 ) @ NN )
& ( composite_of_a @ Resid @ T3 @ U8 @ V6 )
& ( composite_of_a @ Resid @ T6 @ U11 @ V7 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_iff
thf(fact_1089_normal__sub__rts_Ocomposite__closed,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,V2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ T3 @ NN )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( composite_of_list_a @ Resid @ T3 @ U2 @ V2 )
=> ( member_list_a @ V2 @ NN ) ) ) ) ) ).
% normal_sub_rts.composite_closed
thf(fact_1090_normal__sub__rts_Ocomposite__closed,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,V2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ T3 @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( composite_of_a @ Resid @ T3 @ U2 @ V2 )
=> ( member_a @ V2 @ NN ) ) ) ) ) ).
% normal_sub_rts.composite_closed
thf(fact_1091_normal__sub__rts_Ocomposite__of__arr__normal,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,Arr: list_a > $o,T3: list_a,U2: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( Arr @ T3 )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( composite_of_list_a @ Resid @ T3 @ U2 @ T6 )
=> ( ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN )
& ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.composite_of_arr_normal
thf(fact_1092_normal__sub__rts_Ocomposite__of__arr__normal,axiom,
! [Resid: a > a > a,NN: set_a,Arr: a > $o,T3: a,U2: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( Arr @ T3 )
=> ( ( member_a @ U2 @ NN )
=> ( ( composite_of_a @ Resid @ T3 @ U2 @ T6 )
=> ( ( member_a @ ( Resid @ T6 @ T3 ) @ NN )
& ( member_a @ ( Resid @ T3 @ T6 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.composite_of_arr_normal
thf(fact_1093_normal__sub__rts_OCong_092_060_094sub_0620__cancel__left,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,V2: list_a,U4: list_a,V5: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( composite_of_list_a @ Resid @ T3 @ U2 @ V2 )
=> ( ( composite_of_list_a @ Resid @ T3 @ U4 @ V5 )
=> ( ( ( member_list_a @ ( Resid @ V2 @ V5 ) @ NN )
& ( member_list_a @ ( Resid @ V5 @ V2 ) @ NN ) )
=> ( ( member_list_a @ ( Resid @ U2 @ U4 ) @ NN )
& ( member_list_a @ ( Resid @ U4 @ U2 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_cancel_left
thf(fact_1094_normal__sub__rts_OCong_092_060_094sub_0620__cancel__left,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,V2: a,U4: a,V5: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T3 @ U2 @ V2 )
=> ( ( composite_of_a @ Resid @ T3 @ U4 @ V5 )
=> ( ( ( member_a @ ( Resid @ V2 @ V5 ) @ NN )
& ( member_a @ ( Resid @ V5 @ V2 ) @ NN ) )
=> ( ( member_a @ ( Resid @ U2 @ U4 ) @ NN )
& ( member_a @ ( Resid @ U4 @ U2 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_cancel_left
thf(fact_1095_normal__sub__rts_Oaxioms_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( normal2939518615156061708list_a @ Resid @ NN ) ) ).
% normal_sub_rts.axioms(2)
thf(fact_1096_normal__sub__rts_Oaxioms_I2_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( normal7698203753654205830ioms_a @ Resid @ NN ) ) ).
% normal_sub_rts.axioms(2)
thf(fact_1097_coherent__normal__sub__rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( normal_sub_rts_a @ Resid @ NN ) ) ).
% coherent_normal_sub_rts.axioms(1)
thf(fact_1098_coherent__normal__sub__rts_Oaxioms_I1_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( normal6589580540804570479list_a @ Resid @ NN ) ) ).
% coherent_normal_sub_rts.axioms(1)
thf(fact_1099_normal__sub__rts_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a,U2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( ( sources_list_a @ Resid @ T3 )
= ( sources_list_a @ Resid @ U2 ) )
=> ( ( member_list_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U2 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ U2 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_1100_normal__sub__rts_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a,U2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( member_a @ U2 @ NN )
=> ( ( ( sources_a @ Resid @ T3 )
= ( sources_a @ Resid @ U2 ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U2 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U2 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_1101_normal__sub__rts_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( sources_list_a @ Resid @ T3 )
= ( sources_list_a @ Resid @ T6 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_coinitial
thf(fact_1102_normal__sub__rts_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( sources_a @ Resid @ T3 )
= ( sources_a @ Resid @ T6 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_coinitial
thf(fact_1103_normal__sub__rts_Oprfx__closed,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( ide_a @ Resid @ ( Resid @ T3 @ U2 ) )
=> ( member_a @ T3 @ NN ) ) ) ) ).
% normal_sub_rts.prfx_closed
thf(fact_1104_normal__sub__rts_Oprfx__closed,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( ide_list_a @ Resid @ ( Resid @ T3 @ U2 ) )
=> ( member_list_a @ T3 @ NN ) ) ) ) ).
% normal_sub_rts.prfx_closed
thf(fact_1105_normal__sub__rts_Oide__closed,axiom,
! [Resid: a > a > a,NN: set_a,A: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ide_a @ Resid @ A )
=> ( member_a @ A @ NN ) ) ) ).
% normal_sub_rts.ide_closed
thf(fact_1106_normal__sub__rts_Oide__closed,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,A: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ide_list_a @ Resid @ A )
=> ( member_list_a @ A @ NN ) ) ) ).
% normal_sub_rts.ide_closed
thf(fact_1107_normal__sub__rts_OCong_092_060_094sub_0620__reflexive,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( arr_list_a @ Resid @ T3 )
=> ( ( member_list_a @ ( Resid @ T3 @ T3 ) @ NN )
& ( member_list_a @ ( Resid @ T3 @ T3 ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_reflexive
thf(fact_1108_normal__sub__rts_OCong_092_060_094sub_0620__reflexive,axiom,
! [Resid: a > a > a,NN: set_a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T3 )
=> ( ( member_a @ ( Resid @ T3 @ T3 ) @ NN )
& ( member_a @ ( Resid @ T3 @ T3 ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_reflexive
thf(fact_1109_normal__sub__rts_Oelements__are__arr,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ T3 @ NN )
=> ( arr_list_a @ Resid @ T3 ) ) ) ).
% normal_sub_rts.elements_are_arr
thf(fact_1110_normal__sub__rts_Oelements__are__arr,axiom,
! [Resid: a > a > a,NN: set_a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ T3 @ NN )
=> ( arr_a @ Resid @ T3 ) ) ) ).
% normal_sub_rts.elements_are_arr
thf(fact_1111_normal__sub__rts_Obackward__stable,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( member_list_a @ ( Resid @ T3 @ U2 ) @ NN )
=> ( member_list_a @ T3 @ NN ) ) ) ) ).
% normal_sub_rts.backward_stable
thf(fact_1112_normal__sub__rts_Obackward__stable,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( member_a @ ( Resid @ T3 @ U2 ) @ NN )
=> ( member_a @ T3 @ NN ) ) ) ) ).
% normal_sub_rts.backward_stable
thf(fact_1113_normal__sub__rts_OCong_092_060_094sub_0620__symmetric,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN )
& ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_symmetric
thf(fact_1114_normal__sub__rts_OCong_092_060_094sub_0620__symmetric,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( member_a @ ( Resid @ T6 @ T3 ) @ NN )
& ( member_a @ ( Resid @ T3 @ T6 ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_symmetric
thf(fact_1115_normal__sub__rts_OCong_092_060_094sub_0620__transitive,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a,T12: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( ( member_list_a @ ( Resid @ T6 @ T12 ) @ NN )
& ( member_list_a @ ( Resid @ T12 @ T6 ) @ NN ) )
=> ( ( member_list_a @ ( Resid @ T3 @ T12 ) @ NN )
& ( member_list_a @ ( Resid @ T12 @ T3 ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_transitive
thf(fact_1116_normal__sub__rts_OCong_092_060_094sub_0620__transitive,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a,T12: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( ( member_a @ ( Resid @ T6 @ T12 ) @ NN )
& ( member_a @ ( Resid @ T12 @ T6 ) @ NN ) )
=> ( ( member_a @ ( Resid @ T3 @ T12 ) @ NN )
& ( member_a @ ( Resid @ T12 @ T3 ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_transitive
thf(fact_1117_normal__sub__rts_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U2 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ U2 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_1118_normal__sub__rts_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U2 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U2 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) )
=> ( ( member_a @ U2 @ NN )
=> ( ( member_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T3 ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_1119_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1120_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1121_normal__sub__rts_Oforward__stable,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( coinitial_list_a @ Resid @ T3 @ U2 )
=> ( member_list_a @ ( Resid @ U2 @ T3 ) @ NN ) ) ) ) ).
% normal_sub_rts.forward_stable
thf(fact_1122_normal__sub__rts_Oforward__stable,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( coinitial_a @ Resid @ T3 @ U2 )
=> ( member_a @ ( Resid @ U2 @ T3 ) @ NN ) ) ) ) ).
% normal_sub_rts.forward_stable
thf(fact_1123_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a,U2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( ( member_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U2 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U2 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(2)
thf(fact_1124_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a,U2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( ( member_list_a @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ T6 @ U2 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ U2 ) @ ( Resid @ T3 @ U2 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(2)
thf(fact_1125_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a,U2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( con_a @ Resid @ T6 @ U2 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(1)
thf(fact_1126_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a,U2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( con_list_a @ Resid @ T6 @ U2 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(1)
thf(fact_1127_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,U4: a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ U2 @ U4 ) @ NN )
& ( member_a @ ( Resid @ U4 @ U2 ) @ NN ) )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( ( member_a @ ( Resid @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ U4 @ U2 ) ) @ ( Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ U2 @ U4 ) ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ U2 @ U4 ) ) @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ U4 @ U2 ) ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(2)
thf(fact_1128_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,U4: list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ U2 @ U4 ) @ NN )
& ( member_list_a @ ( Resid @ U4 @ U2 ) @ NN ) )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( ( member_list_a @ ( Resid @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ U4 @ U2 ) ) @ ( Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ U2 @ U4 ) ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ U2 @ U4 ) ) @ ( Resid @ ( Resid @ T3 @ U2 ) @ ( Resid @ U4 @ U2 ) ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(2)
thf(fact_1129_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,U2: a,U4: a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ U2 @ U4 ) @ NN )
& ( member_a @ ( Resid @ U4 @ U2 ) @ NN ) )
=> ( ( con_a @ Resid @ T3 @ U2 )
=> ( con_a @ Resid @ T3 @ U4 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(1)
thf(fact_1130_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U2: list_a,U4: list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ U2 @ U4 ) @ NN )
& ( member_list_a @ ( Resid @ U4 @ U2 ) @ NN ) )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
=> ( con_list_a @ Resid @ T3 @ U4 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(1)
thf(fact_1131_normal__sub__rts_OCong_092_060_094sub_0620__imp__con,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( con_a @ Resid @ T3 @ T6 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_con
thf(fact_1132_normal__sub__rts_OCong_092_060_094sub_0620__imp__con,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( con_list_a @ Resid @ T3 @ T6 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_con
thf(fact_1133_normal__sub__rts_OCong_092_060_094sub_0620__subst__Con,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,T6: a,U2: a,U4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( ( member_a @ ( Resid @ U2 @ U4 ) @ NN )
& ( member_a @ ( Resid @ U4 @ U2 ) @ NN ) )
=> ( ( con_a @ Resid @ T3 @ U2 )
= ( con_a @ Resid @ T6 @ U4 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_Con
thf(fact_1134_normal__sub__rts_OCong_092_060_094sub_0620__subst__Con,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,T6: list_a,U2: list_a,U4: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( ( member_list_a @ ( Resid @ T3 @ T6 ) @ NN )
& ( member_list_a @ ( Resid @ T6 @ T3 ) @ NN ) )
=> ( ( ( member_list_a @ ( Resid @ U2 @ U4 ) @ NN )
& ( member_list_a @ ( Resid @ U4 @ U2 ) @ NN ) )
=> ( ( con_list_a @ Resid @ T3 @ U2 )
= ( con_list_a @ Resid @ T6 @ U4 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_Con
thf(fact_1135_normal__sub__rts_Oarr__in__Cong__class,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( arr_list_a @ Resid @ T3 )
=> ( member_list_a @ T3 @ ( normal8738828432243163028list_a @ Resid @ NN @ T3 ) ) ) ) ).
% normal_sub_rts.arr_in_Cong_class
thf(fact_1136_normal__sub__rts_Oarr__in__Cong__class,axiom,
! [Resid: a > a > a,NN: set_a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T3 )
=> ( member_a @ T3 @ ( normal7408713899360725774lass_a @ Resid @ NN @ T3 ) ) ) ) ).
% normal_sub_rts.arr_in_Cong_class
thf(fact_1137_normal__sub__rts_OCong__class__eqI_H,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T13: set_list_a,U12: set_list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ T13 )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ U12 )
=> ( ( ( inf_inf_set_list_a @ T13 @ U12 )
!= bot_bot_set_list_a )
=> ( T13 = U12 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI'
thf(fact_1138_normal__sub__rts_OCong__class__eqI_H,axiom,
! [Resid: a > a > a,NN: set_a,T13: set_a,U12: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T13 )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ U12 )
=> ( ( ( inf_inf_set_a @ T13 @ U12 )
!= bot_bot_set_a )
=> ( T13 = U12 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI'
thf(fact_1139_normal__sub__rts_Ocomposite__of__normal__arr,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a,T6: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( arr_list_a @ Resid @ T3 )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( composite_of_list_a @ Resid @ U2 @ T3 @ T6 )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T6 @ T3 ) ) ) ) ) ).
% normal_sub_rts.composite_of_normal_arr
thf(fact_1140_normal__sub__rts_Ocomposite__of__normal__arr,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T3 )
=> ( ( member_a @ U2 @ NN )
=> ( ( composite_of_a @ Resid @ U2 @ T3 @ T6 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T6 @ T3 ) ) ) ) ) ).
% normal_sub_rts.composite_of_normal_arr
thf(fact_1141_normal__sub__rts_OCong_H_Ointros_I4_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a,U2: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( arr_list_a @ Resid @ T3 )
=> ( ( member_list_a @ U2 @ NN )
=> ( ( ( sources_list_a @ Resid @ T3 )
= ( sources_list_a @ Resid @ U2 ) )
=> ( normal4820288888589421745list_a @ Resid @ NN @ T3 @ ( Resid @ T3 @ U2 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.intros(4)
thf(fact_1142_normal__sub__rts_OCong_H_Ointros_I4_J,axiom,
! [Resid: a > a > a,NN: set_a,T3: a,U2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T3 )
=> ( ( member_a @ U2 @ NN )
=> ( ( ( sources_a @ Resid @ T3 )
= ( sources_a @ Resid @ U2 ) )
=> ( normal_sub_Cong_a2 @ Resid @ NN @ T3 @ ( Resid @ T3 @ U2 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.intros(4)
thf(fact_1143_normal__sub__rts_OCong_H_Ocases,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,A1: list_a,A22: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4820288888589421745list_a @ Resid @ NN @ A1 @ A22 )
=> ( ~ ( normal4820288888589421745list_a @ Resid @ NN @ A22 @ A1 )
=> ( ! [U: list_a] :
( ( normal4820288888589421745list_a @ Resid @ NN @ A1 @ U )
=> ~ ( normal4820288888589421745list_a @ Resid @ NN @ U @ A22 ) )
=> ( ~ ( ( member_list_a @ ( Resid @ A1 @ A22 ) @ NN )
& ( member_list_a @ ( Resid @ A22 @ A1 ) @ NN ) )
=> ~ ! [U: list_a] :
( ( A22
= ( Resid @ A1 @ U ) )
=> ( ( arr_list_a @ Resid @ A1 )
=> ( ( member_list_a @ U @ NN )
=> ( ( sources_list_a @ Resid @ A1 )
!= ( sources_list_a @ Resid @ U ) ) ) ) ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.cases
thf(fact_1144_normal__sub__rts_OCong_H_Ocases,axiom,
! [Resid: a > a > a,NN: set_a,A1: a,A22: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a2 @ Resid @ NN @ A1 @ A22 )
=> ( ~ ( normal_sub_Cong_a2 @ Resid @ NN @ A22 @ A1 )
=> ( ! [U: a] :
( ( normal_sub_Cong_a2 @ Resid @ NN @ A1 @ U )
=> ~ ( normal_sub_Cong_a2 @ Resid @ NN @ U @ A22 ) )
=> ( ~ ( ( member_a @ ( Resid @ A1 @ A22 ) @ NN )
& ( member_a @ ( Resid @ A22 @ A1 ) @ NN ) )
=> ~ ! [U: a] :
( ( A22
= ( Resid @ A1 @ U ) )
=> ( ( arr_a @ Resid @ A1 )
=> ( ( member_a @ U @ NN )
=> ( ( sources_a @ Resid @ A1 )
!= ( sources_a @ Resid @ U ) ) ) ) ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.cases
thf(fact_1145_normal__sub__rts_OCong_H_Osimps,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,A1: list_a,A22: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4820288888589421745list_a @ Resid @ NN @ A1 @ A22 )
= ( ? [T5: list_a,U8: list_a] :
( ( A1 = U8 )
& ( A22 = T5 )
& ( normal4820288888589421745list_a @ Resid @ NN @ T5 @ U8 ) )
| ? [T5: list_a,U8: list_a,V6: list_a] :
( ( A1 = T5 )
& ( A22 = V6 )
& ( normal4820288888589421745list_a @ Resid @ NN @ T5 @ U8 )
& ( normal4820288888589421745list_a @ Resid @ NN @ U8 @ V6 ) )
| ? [T5: list_a,U8: list_a] :
( ( A1 = T5 )
& ( A22 = U8 )
& ( member_list_a @ ( Resid @ T5 @ U8 ) @ NN )
& ( member_list_a @ ( Resid @ U8 @ T5 ) @ NN ) )
| ? [T5: list_a,U8: list_a] :
( ( A1 = T5 )
& ( A22
= ( Resid @ T5 @ U8 ) )
& ( arr_list_a @ Resid @ T5 )
& ( member_list_a @ U8 @ NN )
& ( ( sources_list_a @ Resid @ T5 )
= ( sources_list_a @ Resid @ U8 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.simps
thf(fact_1146_normal__sub__rts_OCong_H_Osimps,axiom,
! [Resid: a > a > a,NN: set_a,A1: a,A22: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a2 @ Resid @ NN @ A1 @ A22 )
= ( ? [T5: a,U8: a] :
( ( A1 = U8 )
& ( A22 = T5 )
& ( normal_sub_Cong_a2 @ Resid @ NN @ T5 @ U8 ) )
| ? [T5: a,U8: a,V6: a] :
( ( A1 = T5 )
& ( A22 = V6 )
& ( normal_sub_Cong_a2 @ Resid @ NN @ T5 @ U8 )
& ( normal_sub_Cong_a2 @ Resid @ NN @ U8 @ V6 ) )
| ? [T5: a,U8: a] :
( ( A1 = T5 )
& ( A22 = U8 )
& ( member_a @ ( Resid @ T5 @ U8 ) @ NN )
& ( member_a @ ( Resid @ U8 @ T5 ) @ NN ) )
| ? [T5: a,U8: a] :
( ( A1 = T5 )
& ( A22
= ( Resid @ T5 @ U8 ) )
& ( arr_a @ Resid @ T5 )
& ( member_a @ U8 @ NN )
& ( ( sources_a @ Resid @ T5 )
= ( sources_a @ Resid @ U8 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.simps
thf(fact_1147_normal__sub__rts_Ois__Cong__classE,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T13: set_list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ T13 )
=> ~ ( ( T13 != bot_bot_set_list_a )
=> ( ! [T2: list_a] :
( ( member_list_a @ T2 @ T13 )
=> ! [T14: list_a] :
( ( member_list_a @ T14 @ T13 )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T2 @ T14 ) ) )
=> ~ ! [T2: list_a] :
( ( member_list_a @ T2 @ T13 )
=> ! [T14: list_a] :
( ( normal4889798360446511898list_a @ Resid @ NN @ T14 @ T2 )
=> ( member_list_a @ T14 @ T13 ) ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classE
thf(fact_1148_normal__sub__rts_Ois__Cong__classE,axiom,
! [Resid: a > a > a,NN: set_a,T13: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T13 )
=> ~ ( ( T13 != bot_bot_set_a )
=> ( ! [T2: a] :
( ( member_a @ T2 @ T13 )
=> ! [T14: a] :
( ( member_a @ T14 @ T13 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T2 @ T14 ) ) )
=> ~ ! [T2: a] :
( ( member_a @ T2 @ T13 )
=> ! [T14: a] :
( ( normal_sub_Cong_a @ Resid @ NN @ T14 @ T2 )
=> ( member_a @ T14 @ T13 ) ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classE
thf(fact_1149_normal__sub__rts_Ois__Cong__classI_H,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T13: set_list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( T13 != bot_bot_set_list_a )
=> ( ! [T: list_a,T15: list_a] :
( ( member_list_a @ T @ T13 )
=> ( ( member_list_a @ T15 @ T13 )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T @ T15 ) ) )
=> ( ! [T: list_a,T15: list_a] :
( ( member_list_a @ T @ T13 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T15 @ T )
=> ( member_list_a @ T15 @ T13 ) ) )
=> ( normal4947954984078133662list_a @ Resid @ NN @ T13 ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classI'
thf(fact_1150_normal__sub__rts_Ois__Cong__classI_H,axiom,
! [Resid: a > a > a,NN: set_a,T13: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( T13 != bot_bot_set_a )
=> ( ! [T: a,T15: a] :
( ( member_a @ T @ T13 )
=> ( ( member_a @ T15 @ T13 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T15 ) ) )
=> ( ! [T: a,T15: a] :
( ( member_a @ T @ T13 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T15 @ T )
=> ( member_a @ T15 @ T13 ) ) )
=> ( normal8595587647932138008lass_a @ Resid @ NN @ T13 ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classI'
thf(fact_1151_paths__in__rts__with__normal_Onormal__extends__to__paths,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( normal6589580540804570479list_a @ ( paths_in_Resid_a @ Resid ) @ ( collect_list_a @ ( paths_3208254328100665313Path_a @ Resid @ NN ) ) ) ) ).
% paths_in_rts_with_normal.normal_extends_to_paths
thf(fact_1152_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X2: a,Xs3: list_a,Y2: a,Ys5: list_a] :
( ( X2 != Y2 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X2 @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1153_normal__sub__rts_Ois__Cong__classI,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( arr_list_a @ Resid @ T3 )
=> ( normal4947954984078133662list_a @ Resid @ NN @ ( normal8738828432243163028list_a @ Resid @ NN @ T3 ) ) ) ) ).
% normal_sub_rts.is_Cong_classI
thf(fact_1154_normal__sub__rts_Ois__Cong__classI,axiom,
! [Resid: a > a > a,NN: set_a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T3 )
=> ( normal8595587647932138008lass_a @ Resid @ NN @ ( normal7408713899360725774lass_a @ Resid @ NN @ T3 ) ) ) ) ).
% normal_sub_rts.is_Cong_classI
thf(fact_1155_normal__sub__rts_OCong__class__memb__Cong__rep,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T13: set_list_a,T3: list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ T13 )
=> ( ( member_list_a @ T3 @ T13 )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T3 @ ( normal5825812266234710965list_a @ T13 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_Cong_rep
thf(fact_1156_normal__sub__rts_OCong__class__memb__Cong__rep,axiom,
! [Resid: a > a > a,NN: set_a,T13: set_a,T3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T13 )
=> ( ( member_a @ T3 @ T13 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T3 @ ( normal3259722184653208495_rep_a @ T13 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_Cong_rep
thf(fact_1157_normal__sub__rts_OCong__class__rep,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T13: set_list_a] :
( ( normal6589580540804570479list_a @ Resid @ NN )
=> ( ( normal4947954984078133662list_a @ Resid @ NN @ T13 )
=> ( ( normal8738828432243163028list_a @ Resid @ NN @ ( normal5825812266234710965list_a @ T13 ) )
= T13 ) ) ) ).
% normal_sub_rts.Cong_class_rep
thf(fact_1158_normal__sub__rts_OCong__class__rep,axiom,
! [Resid: a > a > a,NN: set_a,T13: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T13 )
=> ( ( normal7408713899360725774lass_a @ Resid @ NN @ ( normal3259722184653208495_rep_a @ T13 ) )
= T13 ) ) ) ).
% normal_sub_rts.Cong_class_rep
thf(fact_1159_length__Residx1,axiom,
! [T4: list_a,U2: a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( paths_in_Residx1_a @ resid @ T4 @ U2 ) ) @ ( size_size_list_a @ T4 ) ) ).
% length_Residx1
thf(fact_1160_prod_Oinject,axiom,
! [X1: a,X23: list_a,Y1: a,Y23: list_a] :
( ( ( produc6670463072477821725list_a @ X1 @ X23 )
= ( produc6670463072477821725list_a @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_1161_prod_Oinject,axiom,
! [X1: list_a,X23: a,Y1: list_a,Y23: a] :
( ( ( produc4781227316648555537st_a_a @ X1 @ X23 )
= ( produc4781227316648555537st_a_a @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_1162_prod_Oinject,axiom,
! [X1: list_a,X23: list_a,Y1: list_a,Y23: list_a] :
( ( ( produc6837034575241423639list_a @ X1 @ X23 )
= ( produc6837034575241423639list_a @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_1163_R_Oidentities__form__normal__sub__rts,axiom,
normal_sub_rts_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ).
% R.identities_form_normal_sub_rts
thf(fact_1164_N_Onormal__sub__rts__axioms,axiom,
normal_sub_rts_a @ resid @ nn ).
% N.normal_sub_rts_axioms
thf(fact_1165_old_Oprod_Oinject,axiom,
! [A: a,B: list_a,A4: a,B4: list_a] :
( ( ( produc6670463072477821725list_a @ A @ B )
= ( produc6670463072477821725list_a @ A4 @ B4 ) )
= ( ( A = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_1166_old_Oprod_Oinject,axiom,
! [A: list_a,B: a,A4: list_a,B4: a] :
( ( ( produc4781227316648555537st_a_a @ A @ B )
= ( produc4781227316648555537st_a_a @ A4 @ B4 ) )
= ( ( A = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_1167_old_Oprod_Oinject,axiom,
! [A: list_a,B: list_a,A4: list_a,B4: list_a] :
( ( ( produc6837034575241423639list_a @ A @ B )
= ( produc6837034575241423639list_a @ A4 @ B4 ) )
= ( ( A = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_1168_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1169_map__tailrec__rev_Ocases,axiom,
! [X: produc1473018763691903991list_a] :
( ! [F2: a > a,Bs2: list_a] :
( X
!= ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ nil_a @ Bs2 ) ) )
=> ~ ! [F2: a > a,A3: a,As: list_a,Bs2: list_a] :
( X
!= ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ ( cons_a @ A3 @ As ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_1170_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1171_splice_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ~ ! [X2: a,Xs2: list_a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_1172_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ( ! [P3: a > a > $o,X2: a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X2 @ nil_a ) ) )
=> ~ ! [P3: a > a > $o,X2: a,Y2: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1173_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ~ ! [P3: a > a > $o,X2: a,Ys2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X2 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1174_Pair__inject,axiom,
! [A: a,B: list_a,A4: a,B4: list_a] :
( ( ( produc6670463072477821725list_a @ A @ B )
= ( produc6670463072477821725list_a @ A4 @ B4 ) )
=> ~ ( ( A = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_1175_Pair__inject,axiom,
! [A: list_a,B: a,A4: list_a,B4: a] :
( ( ( produc4781227316648555537st_a_a @ A @ B )
= ( produc4781227316648555537st_a_a @ A4 @ B4 ) )
=> ~ ( ( A = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_1176_Pair__inject,axiom,
! [A: list_a,B: list_a,A4: list_a,B4: list_a] :
( ( ( produc6837034575241423639list_a @ A @ B )
= ( produc6837034575241423639list_a @ A4 @ B4 ) )
=> ~ ( ( A = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_1177_prod__cases,axiom,
! [P: produc8685980395799941037list_a > $o,P4: produc8685980395799941037list_a] :
( ! [A3: a,B3: list_a] : ( P @ ( produc6670463072477821725list_a @ A3 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_1178_prod__cases,axiom,
! [P: produc2579390645249093025st_a_a > $o,P4: produc2579390645249093025st_a_a] :
( ! [A3: list_a,B3: a] : ( P @ ( produc4781227316648555537st_a_a @ A3 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_1179_prod__cases,axiom,
! [P: produc9164743771328383783list_a > $o,P4: produc9164743771328383783list_a] :
( ! [A3: list_a,B3: list_a] : ( P @ ( produc6837034575241423639list_a @ A3 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_1180_surj__pair,axiom,
! [P4: produc8685980395799941037list_a] :
? [X2: a,Y2: list_a] :
( P4
= ( produc6670463072477821725list_a @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_1181_surj__pair,axiom,
! [P4: produc2579390645249093025st_a_a] :
? [X2: list_a,Y2: a] :
( P4
= ( produc4781227316648555537st_a_a @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_1182_surj__pair,axiom,
! [P4: produc9164743771328383783list_a] :
? [X2: list_a,Y2: list_a] :
( P4
= ( produc6837034575241423639list_a @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_1183_old_Oprod_Oexhaust,axiom,
! [Y: produc8685980395799941037list_a] :
~ ! [A3: a,B3: list_a] :
( Y
!= ( produc6670463072477821725list_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_1184_old_Oprod_Oexhaust,axiom,
! [Y: produc2579390645249093025st_a_a] :
~ ! [A3: list_a,B3: a] :
( Y
!= ( produc4781227316648555537st_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_1185_old_Oprod_Oexhaust,axiom,
! [Y: produc9164743771328383783list_a] :
~ ! [A3: list_a,B3: list_a] :
( Y
!= ( produc6837034575241423639list_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_1186_Resid__cons__ind,axiom,
! [T4: list_a,U3: list_a,N: nat] :
( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_size_list_a @ T4 ) @ ( size_size_list_a @ U3 ) ) @ N )
=> ( ! [T2: a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T4 ) @ U3 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U3 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U3 @ ( cons_a @ T2 @ nil_a ) ) )
!= nil_a ) ) )
& ! [U14: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U14 @ U3 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U14 @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U14 @ nil_a ) ) @ U3 )
!= nil_a ) ) )
& ! [T2: a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T4 ) @ U3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T4 ) @ U3 )
= ( append_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U3 ) @ ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U3 @ ( cons_a @ T2 @ nil_a ) ) ) ) ) )
& ! [U14: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U14 @ U3 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U14 @ U3 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U14 @ nil_a ) ) @ U3 ) ) ) ) ) ) ) ).
% Resid_cons_ind
thf(fact_1187_Con__sym__ind,axiom,
! [T4: list_a,U3: list_a,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_size_list_a @ T4 ) @ ( size_size_list_a @ U3 ) ) @ N )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ U3 @ T4 )
!= nil_a ) ) ) ).
% Con_sym_ind
thf(fact_1188_length__Resid__ind,axiom,
! [T4: list_a,U3: list_a,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_size_list_a @ T4 ) @ ( size_size_list_a @ U3 ) ) @ N )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
=> ( ( size_size_list_a @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) )
= ( size_size_list_a @ T4 ) ) ) ) ).
% length_Resid_ind
thf(fact_1189_Cube__ind,axiom,
! [T4: list_a,U3: list_a,V3: list_a,N: nat] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U3 )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ V3 @ T4 )
!= nil_a )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_list_a @ T4 ) @ ( size_size_list_a @ U3 ) ) @ ( size_size_list_a @ V3 ) ) @ N )
=> ( ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V3 @ T4 ) @ ( paths_in_Resid_a @ resid @ U3 @ T4 ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V3 @ U3 ) @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) )
!= nil_a ) )
& ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V3 @ T4 ) @ ( paths_in_Resid_a @ resid @ U3 @ T4 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V3 @ T4 ) @ ( paths_in_Resid_a @ resid @ U3 @ T4 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V3 @ U3 ) @ ( paths_in_Resid_a @ resid @ T4 @ U3 ) ) ) ) ) ) ) ) ).
% Cube_ind
thf(fact_1190_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_1191_Resid_Opsimps_I7_J,axiom,
! [T3: a,Vb: a,Vc: list_a,U2: a,V2: a,Va2: list_a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T3 @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) ) )
=> ( ( ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ T3 @ U2 ) ) ) ) ) )
& ( ~ ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) )
= nil_a ) ) ) ) ).
% Resid.psimps(7)
thf(fact_1192_Resid_Opsimps_I1_J,axiom,
! [Uu: list_a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ nil_a @ Uu ) )
=> ( ( paths_in_Resid_a @ resid @ nil_a @ Uu )
= nil_a ) ) ).
% Resid.psimps(1)
thf(fact_1193_Resid_Opsimps_I2_J,axiom,
! [V2: a,Va2: list_a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ V2 @ Va2 ) @ nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ V2 @ Va2 ) @ nil_a )
= nil_a ) ) ).
% Resid.psimps(2)
thf(fact_1194_Resid_Opsimps_I3_J,axiom,
! [T3: a,U2: a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) ) )
=> ( ( ( con_a @ resid @ T3 @ U2 )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
= ( cons_a @ ( resid @ T3 @ U2 ) @ nil_a ) ) )
& ( ~ ( con_a @ resid @ T3 @ U2 )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ nil_a ) )
= nil_a ) ) ) ) ).
% Resid.psimps(3)
thf(fact_1195_Resid_Opsimps_I5_J,axiom,
! [T3: a,V2: a,Va2: list_a,U2: a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ nil_a ) ) )
=> ( ( ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ nil_a ) )
= ( cons_a @ ( resid @ T3 @ U2 ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) ) ) ) )
& ( ~ ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ nil_a ) )
= nil_a ) ) ) ) ).
% Resid.psimps(5)
thf(fact_1196_Resid_Opsimps_I4_J,axiom,
! [T3: a,U2: a,V2: a,Va2: list_a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) ) )
=> ( ( ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) )
!= ( partial_null_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) ) @ nil_a ) ) )
& ( ~ ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) )
!= ( partial_null_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ nil_a ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) )
= nil_a ) ) ) ) ).
% Resid.psimps(4)
thf(fact_1197_Resid_Opsimps_I6_J,axiom,
! [T3: a,V2: a,Va2: list_a,U2: a,Vb: a,Vc: list_a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ ( cons_a @ Vb @ Vc ) ) ) )
=> ( ( ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ Vb @ Vc ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ ( cons_a @ Vb @ Vc ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ Vb @ Vc ) ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ T3 @ U2 ) ) ) ) ) )
& ( ~ ( ( con_a @ resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T3 @ U2 ) @ ( cons_a @ Vb @ Vc ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V2 @ Va2 ) @ ( resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ ( cons_a @ Vb @ Vc ) ) )
= nil_a ) ) ) ) ).
% Resid.psimps(6)
thf(fact_1198_paths__in__rts_OResid_Opsimps_I7_J,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,Vb: list_a,Vc: list_list_a,U2: list_a,V2: list_a,Va2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( accp_P8911511002227036208list_a @ ( paths_3409276745531958875list_a @ Resid ) @ ( produc8696003437204565271list_a @ ( cons_list_a @ T3 @ ( cons_list_a @ Vb @ Vc ) ) @ ( cons_list_a @ U2 @ ( cons_list_a @ V2 @ Va2 ) ) ) )
=> ( ( ( ( con_list_a @ Resid @ T3 @ U2 )
& ( ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_list_a @ V2 @ Va2 ) )
!= ( partial_null_list_a @ Resid ) )
& ( ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V2 @ Va2 ) @ ( Resid @ T3 @ U2 ) ) )
!= nil_list_a ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T3 @ ( cons_list_a @ Vb @ Vc ) ) @ ( cons_list_a @ U2 @ ( cons_list_a @ V2 @ Va2 ) ) )
= ( cons_list_a @ ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_list_a @ V2 @ Va2 ) ) @ ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V2 @ Va2 ) @ ( Resid @ T3 @ U2 ) ) ) ) ) )
& ( ~ ( ( con_list_a @ Resid @ T3 @ U2 )
& ( ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_list_a @ V2 @ Va2 ) )
!= ( partial_null_list_a @ Resid ) )
& ( ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V2 @ Va2 ) @ ( Resid @ T3 @ U2 ) ) )
!= nil_list_a ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T3 @ ( cons_list_a @ Vb @ Vc ) ) @ ( cons_list_a @ U2 @ ( cons_list_a @ V2 @ Va2 ) ) )
= nil_list_a ) ) ) ) ) ).
% paths_in_rts.Resid.psimps(7)
thf(fact_1199_paths__in__rts_OResid_Opsimps_I7_J,axiom,
! [Resid: a > a > a,T3: a,Vb: a,Vc: list_a,U2: a,V2: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ Resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T3 @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) ) )
=> ( ( ( ( con_a @ Resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) )
!= ( partial_null_a @ Resid ) )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V2 @ Va2 ) @ ( Resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T3 @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) ) @ ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V2 @ Va2 ) @ ( Resid @ T3 @ U2 ) ) ) ) ) )
& ( ~ ( ( con_a @ Resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_a @ V2 @ Va2 ) )
!= ( partial_null_a @ Resid ) )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V2 @ Va2 ) @ ( Resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T3 @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U2 @ ( cons_a @ V2 @ Va2 ) ) )
= nil_a ) ) ) ) ) ).
% paths_in_rts.Resid.psimps(7)
thf(fact_1200_paths__in__rts_OResid_Opsimps_I6_J,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,V2: list_a,Va2: list_list_a,U2: list_a,Vb: list_a,Vc: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( accp_P8911511002227036208list_a @ ( paths_3409276745531958875list_a @ Resid ) @ ( produc8696003437204565271list_a @ ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va2 ) ) @ ( cons_list_a @ U2 @ ( cons_list_a @ Vb @ Vc ) ) ) )
=> ( ( ( ( con_list_a @ Resid @ T3 @ U2 )
& ( ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_list_a @ Vb @ Vc ) )
!= ( partial_null_list_a @ Resid ) )
& ( ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V2 @ Va2 ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ T3 @ U2 ) ) )
!= nil_list_a ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va2 ) ) @ ( cons_list_a @ U2 @ ( cons_list_a @ Vb @ Vc ) ) )
= ( cons_list_a @ ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_list_a @ Vb @ Vc ) ) @ ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V2 @ Va2 ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ T3 @ U2 ) ) ) ) ) )
& ( ~ ( ( con_list_a @ Resid @ T3 @ U2 )
& ( ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_list_a @ Vb @ Vc ) )
!= ( partial_null_list_a @ Resid ) )
& ( ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V2 @ Va2 ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ T3 @ U2 ) ) )
!= nil_list_a ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va2 ) ) @ ( cons_list_a @ U2 @ ( cons_list_a @ Vb @ Vc ) ) )
= nil_list_a ) ) ) ) ) ).
% paths_in_rts.Resid.psimps(6)
thf(fact_1201_paths__in__rts_OResid_Opsimps_I6_J,axiom,
! [Resid: a > a > a,T3: a,V2: a,Va2: list_a,U2: a,Vb: a,Vc: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ Resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ ( cons_a @ Vb @ Vc ) ) ) )
=> ( ( ( ( con_a @ Resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_a @ Vb @ Vc ) )
!= ( partial_null_a @ Resid ) )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V2 @ Va2 ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ ( cons_a @ Vb @ Vc ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_a @ Vb @ Vc ) ) @ ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V2 @ Va2 ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ T3 @ U2 ) ) ) ) ) )
& ( ~ ( ( con_a @ Resid @ T3 @ U2 )
& ( ( paths_in_Resid1x_a @ Resid @ ( Resid @ T3 @ U2 ) @ ( cons_a @ Vb @ Vc ) )
!= ( partial_null_a @ Resid ) )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V2 @ Va2 ) @ ( Resid @ U2 @ T3 ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ T3 @ U2 ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) @ ( cons_a @ U2 @ ( cons_a @ Vb @ Vc ) ) )
= nil_a ) ) ) ) ) ).
% paths_in_rts.Resid.psimps(6)
thf(fact_1202_paths__in__rts__axioms,axiom,
paths_in_rts_a @ resid ).
% paths_in_rts_axioms
thf(fact_1203_paths__in__rts_OResid1x_Ocases,axiom,
! [Resid: a > a > a,X: produc8685980395799941037list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ! [T: a] :
( X
!= ( produc6670463072477821725list_a @ T @ nil_a ) )
=> ( ! [T: a,U: a] :
( X
!= ( produc6670463072477821725list_a @ T @ ( cons_a @ U @ nil_a ) ) )
=> ~ ! [T: a,U: a,V: a,Va: list_a] :
( X
!= ( produc6670463072477821725list_a @ T @ ( cons_a @ U @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ).
% paths_in_rts.Resid1x.cases
thf(fact_1204_paths__in__rts_OResidx1_Ocases,axiom,
! [Resid: a > a > a,X: produc2579390645249093025st_a_a] :
( ( paths_in_rts_a @ Resid )
=> ( ! [U: a] :
( X
!= ( produc4781227316648555537st_a_a @ nil_a @ U ) )
=> ( ! [T: a,U: a] :
( X
!= ( produc4781227316648555537st_a_a @ ( cons_a @ T @ nil_a ) @ U ) )
=> ~ ! [T: a,V: a,Va: list_a,U: a] :
( X
!= ( produc4781227316648555537st_a_a @ ( cons_a @ T @ ( cons_a @ V @ Va ) ) @ U ) ) ) ) ) ).
% paths_in_rts.Residx1.cases
thf(fact_1205_paths__in__rts_Olength__Residx1,axiom,
! [Resid: a > a > a,T4: list_a,U2: a] :
( ( paths_in_rts_a @ Resid )
=> ( ord_less_eq_nat @ ( size_size_list_a @ ( paths_in_Residx1_a @ Resid @ T4 @ U2 ) ) @ ( size_size_list_a @ T4 ) ) ) ).
% paths_in_rts.length_Residx1
thf(fact_1206_paths__in__rts_Olength__Resid,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U3 )
!= nil_a )
=> ( ( size_size_list_a @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) )
= ( size_size_list_a @ T4 ) ) ) ) ).
% paths_in_rts.length_Resid
thf(fact_1207_paths__in__rts_OCon__append_I2_J,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a,V3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( V3 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( append_a @ U3 @ V3 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ Resid @ T4 @ U3 )
!= nil_a )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) @ V3 )
!= nil_a ) ) ) ) ) ) ) ).
% paths_in_rts.Con_append(2)
thf(fact_1208_paths__in__rts_OCon__append_I1_J,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a,V3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( V3 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( append_a @ T4 @ U3 ) @ V3 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ Resid @ T4 @ V3 )
!= nil_a )
& ( ( paths_in_Resid_a @ Resid @ U3 @ ( paths_in_Resid_a @ Resid @ V3 @ T4 ) )
!= nil_a ) ) ) ) ) ) ) ).
% paths_in_rts.Con_append(1)
thf(fact_1209_paths__in__rts_OCon__appendI_I2_J,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a,V3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U3 )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) @ V3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ ( append_a @ U3 @ V3 ) )
!= nil_a ) ) ) ) ).
% paths_in_rts.Con_appendI(2)
thf(fact_1210_paths__in__rts_OCon__appendI_I1_J,axiom,
! [Resid: a > a > a,T4: list_a,V3: list_a,U3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ V3 )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ U3 @ ( paths_in_Resid_a @ Resid @ V3 @ T4 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( append_a @ T4 @ U3 ) @ V3 )
!= nil_a ) ) ) ) ).
% paths_in_rts.Con_appendI(1)
thf(fact_1211_paths__in__rts_OResid__append_I2_J,axiom,
! [Resid: a > a > a,U3: list_a,V3: list_a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( U3 != nil_a )
=> ( ( V3 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( append_a @ U3 @ V3 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ ( append_a @ U3 @ V3 ) )
= ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) @ V3 ) ) ) ) ) ) ).
% paths_in_rts.Resid_append(2)
thf(fact_1212_paths__in__rts_OResid__append_I1_J,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a,V3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( append_a @ T4 @ U3 ) @ V3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( append_a @ T4 @ U3 ) @ V3 )
= ( append_a @ ( paths_in_Resid_a @ Resid @ T4 @ V3 ) @ ( paths_in_Resid_a @ Resid @ U3 @ ( paths_in_Resid_a @ Resid @ V3 @ T4 ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_append(1)
thf(fact_1213_paths__in__rts_OResid__append2,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a,V3: list_a,W: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( V3 != nil_a )
=> ( ( W != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( append_a @ T4 @ U3 ) @ ( append_a @ V3 @ W ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( append_a @ T4 @ U3 ) @ ( append_a @ V3 @ W ) )
= ( append_a @ ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ V3 ) @ W ) @ ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ U3 @ ( paths_in_Resid_a @ Resid @ V3 @ T4 ) ) @ ( paths_in_Resid_a @ Resid @ W @ ( paths_in_Resid_a @ Resid @ T4 @ V3 ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_append2
thf(fact_1214_paths__in__rts_OResid__append__ind,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a,V3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U3 != nil_a )
=> ( ( V3 != nil_a )
=> ( ( ( ( paths_in_Resid_a @ Resid @ ( append_a @ V3 @ T4 ) @ U3 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ Resid @ V3 @ U3 )
!= nil_a )
& ( ( paths_in_Resid_a @ Resid @ T4 @ ( paths_in_Resid_a @ Resid @ U3 @ V3 ) )
!= nil_a ) ) )
& ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( append_a @ V3 @ U3 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ Resid @ T4 @ V3 )
!= nil_a )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ V3 ) @ U3 )
!= nil_a ) ) )
& ( ( ( paths_in_Resid_a @ Resid @ ( append_a @ V3 @ T4 ) @ U3 )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( append_a @ V3 @ T4 ) @ U3 )
= ( append_a @ ( paths_in_Resid_a @ Resid @ V3 @ U3 ) @ ( paths_in_Resid_a @ Resid @ T4 @ ( paths_in_Resid_a @ Resid @ U3 @ V3 ) ) ) ) )
& ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( append_a @ V3 @ U3 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ ( append_a @ V3 @ U3 ) )
= ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ V3 ) @ U3 ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_append_ind
thf(fact_1215_paths__in__rts_OCon__imp__Arr__Resid,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U3 )
!= nil_a )
=> ( paths_in_Arr_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) ) ) ) ).
% paths_in_rts.Con_imp_Arr_Resid
thf(fact_1216_paths__in__rts_OArr__iff__Con__self,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
= ( ( paths_in_Resid_a @ Resid @ T4 @ T4 )
!= nil_a ) ) ) ).
% paths_in_rts.Arr_iff_Con_self
thf(fact_1217_paths__in__rts_OCon__Arr__self,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ T4 )
!= nil_a ) ) ) ).
% paths_in_rts.Con_Arr_self
thf(fact_1218_paths__in__rts_OCon__implies__Arr_I1_J,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U3 )
!= nil_a )
=> ( paths_in_Arr_a @ Resid @ T4 ) ) ) ).
% paths_in_rts.Con_implies_Arr(1)
thf(fact_1219_paths__in__rts_OCon__implies__Arr_I2_J,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U3 )
!= nil_a )
=> ( paths_in_Arr_a @ Resid @ U3 ) ) ) ).
% paths_in_rts.Con_implies_Arr(2)
thf(fact_1220_paths__in__rts_OCon__imp__eq__Srcs,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U3 )
!= nil_a )
=> ( ( paths_in_Srcs_a @ Resid @ T4 )
= ( paths_in_Srcs_a @ Resid @ U3 ) ) ) ) ).
% paths_in_rts.Con_imp_eq_Srcs
thf(fact_1221_paths__in__rts_OSrcs_Osimps_I1_J,axiom,
! [Resid: list_a > list_a > list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Srcs_list_a @ Resid @ nil_list_a )
= bot_bot_set_list_a ) ) ).
% paths_in_rts.Srcs.simps(1)
thf(fact_1222_paths__in__rts_OSrcs_Osimps_I1_J,axiom,
! [Resid: a > a > a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Srcs_a @ Resid @ nil_a )
= bot_bot_set_a ) ) ).
% paths_in_rts.Srcs.simps(1)
thf(fact_1223_paths__in__rts_OResid__Ide_I1_J,axiom,
! [Resid: a > a > a,A2: list_a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ A2 )
=> ( ( ( paths_in_Resid_a @ Resid @ A2 @ T4 )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ A2 )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_Ide(1)
thf(fact_1224_paths__in__rts_OResid__Arr__Ide__ind,axiom,
! [Resid: a > a > a,A2: list_a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ A2 )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ A2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ A2 )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_Arr_Ide_ind
thf(fact_1225_paths__in__rts_OResid__Ide__Arr__ind,axiom,
! [Resid: a > a > a,A2: list_a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ A2 )
=> ( ( ( paths_in_Resid_a @ Resid @ A2 @ T4 )
!= nil_a )
=> ( paths_in_Ide_a @ Resid @ ( paths_in_Resid_a @ Resid @ A2 @ T4 ) ) ) ) ) ).
% paths_in_rts.Resid_Ide_Arr_ind
thf(fact_1226_paths__in__rts_OTrgs__Resid__sym,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U3 )
!= nil_a )
=> ( ( paths_in_Trgs_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U3 ) )
= ( paths_in_Trgs_a @ Resid @ ( paths_in_Resid_a @ Resid @ U3 @ T4 ) ) ) ) ) ).
% paths_in_rts.Trgs_Resid_sym
thf(fact_1227_paths__in__rts_OconI_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U3 )
!= nil_a )
=> ( con_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ U3 ) ) ) ).
% paths_in_rts.conI\<^sub>P
thf(fact_1228_paths__in__rts_Ocon__char,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( con_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ U3 )
= ( ( paths_in_Resid_a @ Resid @ T4 @ U3 )
!= nil_a ) ) ) ).
% paths_in_rts.con_char
thf(fact_1229_paths__in__rts_OSrcs__append,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( paths_in_Srcs_a @ Resid @ ( append_a @ T4 @ U3 ) )
= ( paths_in_Srcs_a @ Resid @ T4 ) ) ) ) ).
% paths_in_rts.Srcs_append
thf(fact_1230_paths__in__rts_OTrgs_Osimps_I1_J,axiom,
! [Resid: list_a > list_a > list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Trgs_list_a @ Resid @ nil_list_a )
= bot_bot_set_list_a ) ) ).
% paths_in_rts.Trgs.simps(1)
thf(fact_1231_paths__in__rts_OTrgs_Osimps_I1_J,axiom,
! [Resid: a > a > a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Trgs_a @ Resid @ nil_a )
= bot_bot_set_a ) ) ).
% paths_in_rts.Trgs.simps(1)
thf(fact_1232_paths__in__rts_OSrcs__eqI,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,T7: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( inf_inf_set_list_a @ ( paths_in_Srcs_list_a @ Resid @ T4 ) @ ( paths_in_Srcs_list_a @ Resid @ T7 ) )
!= bot_bot_set_list_a )
=> ( ( paths_in_Srcs_list_a @ Resid @ T4 )
= ( paths_in_Srcs_list_a @ Resid @ T7 ) ) ) ) ).
% paths_in_rts.Srcs_eqI
thf(fact_1233_paths__in__rts_OSrcs__eqI,axiom,
! [Resid: a > a > a,T4: list_a,T7: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( inf_inf_set_a @ ( paths_in_Srcs_a @ Resid @ T4 ) @ ( paths_in_Srcs_a @ Resid @ T7 ) )
!= bot_bot_set_a )
=> ( ( paths_in_Srcs_a @ Resid @ T4 )
= ( paths_in_Srcs_a @ Resid @ T7 ) ) ) ) ).
% paths_in_rts.Srcs_eqI
thf(fact_1234_paths__in__rts_OSrcs__are__ide,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ord_le8861187494160871172list_a @ ( paths_in_Srcs_list_a @ Resid @ T4 ) @ ( collect_list_a @ ( ide_list_a @ Resid ) ) ) ) ).
% paths_in_rts.Srcs_are_ide
thf(fact_1235_paths__in__rts_OSrcs__are__ide,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ord_less_eq_set_a @ ( paths_in_Srcs_a @ Resid @ T4 ) @ ( collect_a @ ( ide_a @ Resid ) ) ) ) ).
% paths_in_rts.Srcs_are_ide
thf(fact_1236_paths__in__rts_OSrcs_Osimps_I3_J,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a,V2: list_a,Va2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va2 ) ) )
= ( sources_list_a @ Resid @ T3 ) ) ) ).
% paths_in_rts.Srcs.simps(3)
thf(fact_1237_paths__in__rts_OSrcs_Osimps_I3_J,axiom,
! [Resid: a > a > a,T3: a,V2: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Srcs_a @ Resid @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va2 ) ) )
= ( sources_a @ Resid @ T3 ) ) ) ).
% paths_in_rts.Srcs.simps(3)
thf(fact_1238_paths__in__rts_OTrgs__append,axiom,
! [Resid: a > a > a,U3: list_a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( U3 != nil_a )
=> ( ( paths_in_Trgs_a @ Resid @ ( append_a @ T4 @ U3 ) )
= ( paths_in_Trgs_a @ Resid @ U3 ) ) ) ) ).
% paths_in_rts.Trgs_append
thf(fact_1239_paths__in__rts_OTrgs__eqI,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,T7: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( inf_inf_set_list_a @ ( paths_in_Trgs_list_a @ Resid @ T4 ) @ ( paths_in_Trgs_list_a @ Resid @ T7 ) )
!= bot_bot_set_list_a )
=> ( ( paths_in_Trgs_list_a @ Resid @ T4 )
= ( paths_in_Trgs_list_a @ Resid @ T7 ) ) ) ) ).
% paths_in_rts.Trgs_eqI
thf(fact_1240_paths__in__rts_OTrgs__eqI,axiom,
! [Resid: a > a > a,T4: list_a,T7: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( inf_inf_set_a @ ( paths_in_Trgs_a @ Resid @ T4 ) @ ( paths_in_Trgs_a @ Resid @ T7 ) )
!= bot_bot_set_a )
=> ( ( paths_in_Trgs_a @ Resid @ T4 )
= ( paths_in_Trgs_a @ Resid @ T7 ) ) ) ) ).
% paths_in_rts.Trgs_eqI
thf(fact_1241_paths__in__rts_OResid1x_Osimps_I2_J,axiom,
! [Resid: a > a > a,T3: a,U2: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Resid1x_a @ Resid @ T3 @ ( cons_a @ U2 @ nil_a ) )
= ( Resid @ T3 @ U2 ) ) ) ).
% paths_in_rts.Resid1x.simps(2)
thf(fact_1242_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: list_a > list_a > list_a,A: list_a,T4: list_list_a,A4: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( member_list_a @ A @ ( paths_in_Srcs_list_a @ Resid @ T4 ) )
=> ( ( ide_list_a @ Resid @ A4 )
=> ( ( con_list_a @ Resid @ A @ A4 )
=> ( member_list_a @ A4 @ ( paths_in_Srcs_list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_1243_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: a > a > a,A: a,T4: list_a,A4: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ Resid @ T4 ) )
=> ( ( ide_a @ Resid @ A4 )
=> ( ( con_a @ Resid @ A @ A4 )
=> ( member_a @ A4 @ ( paths_in_Srcs_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_1244_paths__in__rts_OTrgs__are__ide,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ord_le8861187494160871172list_a @ ( paths_in_Trgs_list_a @ Resid @ T4 ) @ ( collect_list_a @ ( ide_list_a @ Resid ) ) ) ) ).
% paths_in_rts.Trgs_are_ide
thf(fact_1245_paths__in__rts_OTrgs__are__ide,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ Resid @ T4 ) @ ( collect_a @ ( ide_a @ Resid ) ) ) ) ).
% paths_in_rts.Trgs_are_ide
thf(fact_1246_paths__in__rts_OArr__has__Src,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Arr_list_a @ Resid @ T4 )
=> ( ( paths_in_Srcs_list_a @ Resid @ T4 )
!= bot_bot_set_list_a ) ) ) ).
% paths_in_rts.Arr_has_Src
thf(fact_1247_paths__in__rts_OArr__has__Src,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( ( paths_in_Srcs_a @ Resid @ T4 )
!= bot_bot_set_a ) ) ) ).
% paths_in_rts.Arr_has_Src
thf(fact_1248_paths__in__rts_OTrgs__con__closed,axiom,
! [Resid: list_a > list_a > list_a,B: list_a,T4: list_list_a,B4: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( member_list_a @ B @ ( paths_in_Trgs_list_a @ Resid @ T4 ) )
=> ( ( ide_list_a @ Resid @ B4 )
=> ( ( con_list_a @ Resid @ B @ B4 )
=> ( member_list_a @ B4 @ ( paths_in_Trgs_list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Trgs_con_closed
thf(fact_1249_paths__in__rts_OTrgs__con__closed,axiom,
! [Resid: a > a > a,B: a,T4: list_a,B4: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( member_a @ B @ ( paths_in_Trgs_a @ Resid @ T4 ) )
=> ( ( ide_a @ Resid @ B4 )
=> ( ( con_a @ Resid @ B @ B4 )
=> ( member_a @ B4 @ ( paths_in_Trgs_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Trgs_con_closed
thf(fact_1250_paths__in__rts_OResid__Arr__self,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( paths_in_Ide_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ T4 ) ) ) ) ).
% paths_in_rts.Resid_Arr_self
thf(fact_1251_paths__in__rts_Oide__char,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 )
= ( paths_in_Ide_a @ Resid @ T4 ) ) ) ).
% paths_in_rts.ide_char
thf(fact_1252_paths__in__rts_Ois__rts__with__composites,axiom,
! [Resid: a > a > a] :
( ( paths_in_rts_a @ Resid )
=> ( rts_wi1113109597627081724list_a @ ( paths_in_Resid_a @ Resid ) ) ) ).
% paths_in_rts.is_rts_with_composites
thf(fact_1253_paths__in__rts_Ois__partial__magma,axiom,
! [Resid: a > a > a] :
( ( paths_in_rts_a @ Resid )
=> ( partial_magma_list_a @ ( paths_in_Resid_a @ Resid ) ) ) ).
% paths_in_rts.is_partial_magma
thf(fact_1254_paths__in__rts_Ois__residuation,axiom,
! [Resid: a > a > a] :
( ( paths_in_rts_a @ Resid )
=> ( residuation_list_a @ ( paths_in_Resid_a @ Resid ) ) ) ).
% paths_in_rts.is_residuation
thf(fact_1255_paths__in__rts_OArr__has__Trg,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Arr_list_a @ Resid @ T4 )
=> ( ( paths_in_Trgs_list_a @ Resid @ T4 )
!= bot_bot_set_list_a ) ) ) ).
% paths_in_rts.Arr_has_Trg
thf(fact_1256_paths__in__rts_OArr__has__Trg,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( ( paths_in_Trgs_a @ Resid @ T4 )
!= bot_bot_set_a ) ) ) ).
% paths_in_rts.Arr_has_Trg
thf(fact_1257_paths__in__rts_Ocong__respects__seq_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T4: list_a,U3: list_a,T7: list_a,U5: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ U3 )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ Resid ) @ ( paths_in_Resid_a @ Resid @ T4 @ T7 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ Resid ) @ ( paths_in_Resid_a @ Resid @ T7 @ T4 ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ Resid ) @ ( paths_in_Resid_a @ Resid @ U3 @ U5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ Resid ) @ ( paths_in_Resid_a @ Resid @ U5 @ U3 ) ) )
=> ( seq_list_a @ ( paths_in_Resid_a @ Resid ) @ T7 @ U5 ) ) ) ) ) ).
% paths_in_rts.cong_respects_seq\<^sub>P
thf(fact_1258_paths__in__rts_OArr__imp__Arr__tl,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( ( ( tl_a @ T4 )
!= nil_a )
=> ( paths_in_Arr_a @ Resid @ ( tl_a @ T4 ) ) ) ) ) ).
% paths_in_rts.Arr_imp_Arr_tl
thf(fact_1259_paths__in__rts_Oarr__char,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 )
= ( paths_in_Arr_a @ Resid @ T4 ) ) ) ).
% paths_in_rts.arr_char
thf(fact_1260_paths__in__rts_OarrI_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( arr_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 ) ) ) ).
% paths_in_rts.arrI\<^sub>P
thf(fact_1261_paths__in__rts_Onull__char,axiom,
! [Resid: a > a > a] :
( ( paths_in_rts_a @ Resid )
=> ( ( partial_null_list_a @ ( paths_in_Resid_a @ Resid ) )
= nil_a ) ) ).
% paths_in_rts.null_char
thf(fact_1262_paths__in__rts_OIde__imp__Ide__tl,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ T4 )
=> ( ( ( tl_a @ T4 )
!= nil_a )
=> ( paths_in_Ide_a @ Resid @ ( tl_a @ T4 ) ) ) ) ) ).
% paths_in_rts.Ide_imp_Ide_tl
thf(fact_1263_paths__in__rts_OArr__imp__arr__hd,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Arr_list_a @ Resid @ T4 )
=> ( arr_list_a @ Resid @ ( hd_list_a @ T4 ) ) ) ) ).
% paths_in_rts.Arr_imp_arr_hd
thf(fact_1264_paths__in__rts_OArr__imp__arr__hd,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( arr_a @ Resid @ ( hd_a @ T4 ) ) ) ) ).
% paths_in_rts.Arr_imp_arr_hd
thf(fact_1265_paths__in__rts_OResid1x_Osimps_I1_J,axiom,
! [Resid: list_a > list_a > list_a,T3: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_1777230443808135851list_a @ Resid @ T3 @ nil_list_a )
= ( partial_null_list_a @ Resid ) ) ) ).
% paths_in_rts.Resid1x.simps(1)
thf(fact_1266_paths__in__rts_OResid1x_Osimps_I1_J,axiom,
! [Resid: a > a > a,T3: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Resid1x_a @ Resid @ T3 @ nil_a )
= ( partial_null_a @ Resid ) ) ) ).
% paths_in_rts.Resid1x.simps(1)
thf(fact_1267_paths__in__rts_OIde__imp__Ide__hd,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Ide_list_a @ Resid @ T4 )
=> ( ide_list_a @ Resid @ ( hd_list_a @ T4 ) ) ) ) ).
% paths_in_rts.Ide_imp_Ide_hd
thf(fact_1268_paths__in__rts_OIde__imp__Ide__hd,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ T4 )
=> ( ide_a @ Resid @ ( hd_a @ T4 ) ) ) ) ).
% paths_in_rts.Ide_imp_Ide_hd
thf(fact_1269_paths__in__rts_OResid1x__ide,axiom,
! [Resid: list_a > list_a > list_a,A: list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ide_list_a @ Resid @ A )
=> ( ( ( paths_1777230443808135851list_a @ Resid @ A @ T4 )
!= ( partial_null_list_a @ Resid ) )
=> ( ide_list_a @ Resid @ ( paths_1777230443808135851list_a @ Resid @ A @ T4 ) ) ) ) ) ).
% paths_in_rts.Resid1x_ide
thf(fact_1270_paths__in__rts_OResid1x__ide,axiom,
! [Resid: a > a > a,A: a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( ( paths_in_Resid1x_a @ Resid @ A @ T4 )
!= ( partial_null_a @ Resid ) )
=> ( ide_a @ Resid @ ( paths_in_Resid1x_a @ Resid @ A @ T4 ) ) ) ) ) ).
% paths_in_rts.Resid1x_ide
thf(fact_1271_paths__in__rts_OArr__imp__arr__last,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Arr_list_a @ Resid @ T4 )
=> ( arr_list_a @ Resid @ ( last_list_a @ T4 ) ) ) ) ).
% paths_in_rts.Arr_imp_arr_last
thf(fact_1272_paths__in__rts_OArr__imp__arr__last,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( arr_a @ Resid @ ( last_a @ T4 ) ) ) ) ).
% paths_in_rts.Arr_imp_arr_last
thf(fact_1273_paths__in__rts_OIde__imp__Ide__last,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Ide_list_a @ Resid @ T4 )
=> ( ide_list_a @ Resid @ ( last_list_a @ T4 ) ) ) ) ).
% paths_in_rts.Ide_imp_Ide_last
thf(fact_1274_paths__in__rts_OIde__imp__Ide__last,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ T4 )
=> ( ide_a @ Resid @ ( last_a @ T4 ) ) ) ) ).
% paths_in_rts.Ide_imp_Ide_last
thf(fact_1275_paths__in__rts__with__normal_Oaxioms_I3_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( paths_3682460814909619382rmal_a @ Resid @ NN )
=> ( paths_in_rts_a @ Resid ) ) ).
% paths_in_rts_with_normal.axioms(3)
thf(fact_1276_paths__in__rts__with__coherent__normal_Oaxioms_I3_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( paths_6252547876539050804rmal_a @ Resid @ NN )
=> ( paths_in_rts_a @ Resid ) ) ).
% paths_in_rts_with_coherent_normal.axioms(3)
% Helper facts (5)
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
! [X: list_list_a,Y: list_list_a] :
( ( if_list_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
! [X: list_list_a,Y: list_list_a] :
( ( if_list_list_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( append_a @ ta @ ( cons_a @ b @ ba ) )
= ( append_a @ ( append_a @ ta @ ( cons_a @ b @ nil_a ) ) @ ba ) ) ).
%------------------------------------------------------------------------------