TPTP Problem File: SLH0086^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : ResiduatedTransitionSystem/0000_ResiduatedTransitionSystem/prob_05846_238489__14253128_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1403 ( 397 unt; 118 typ; 0 def)
% Number of atoms : 4408 (1759 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 17066 ( 807 ~; 24 |; 461 &;13759 @)
% ( 0 <=>;2015 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 984 ( 984 >; 0 *; 0 +; 0 <<)
% Number of symbols : 115 ( 112 usr; 11 con; 0-4 aty)
% Number of variables : 3531 ( 98 ^;3321 !; 112 ?;3531 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:48:19.742
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
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 (112)
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__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
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_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_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_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_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_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
bot_bot_list_a_o: list_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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_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_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__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_It__List__Olist_Itf__a_J_J,type,
partia5616633150074602416list_a: ( list_list_a > list_list_a > list_list_a ) > list_list_a ).
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__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_OArr__rel_001t__List__Olist_Itf__a_J,type,
paths_7232963753683178177list_a: list_list_a > list_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OArr__rel_001tf__a,type,
paths_in_Arr_rel_a: list_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_OIde__rel_001t__List__Olist_Itf__a_J,type,
paths_4942548060537684810list_a: list_list_a > list_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OIde__rel_001tf__a,type,
paths_in_Ide_rel_a: list_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_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_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_OSrcs__rel_001tf__a,type,
paths_in_Srcs_rel_a: list_a > list_a > $o ).
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_OTrgs__rel_001tf__a,type,
paths_in_Trgs_rel_a: list_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_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_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_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_Itf__a_J,type,
is_singleton_list_a: set_list_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_Itf__a_J,type,
the_elem_list_a: set_list_a > list_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
accp_list_list_a: ( list_list_a > list_list_a > $o ) > list_list_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_Itf__a_J,type,
accp_list_a: ( list_a > list_a > $o ) > 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_Ta____,type,
ta: list_a ).
thf(sy_v_Ua____,type,
ua: list_a ).
thf(sy_v_Va____,type,
va: list_a ).
thf(sy_v_resid,type,
resid: a > a > a ).
thf(sy_v_v____,type,
v: a ).
% Relevant facts (1278)
thf(fact_0_U,axiom,
ua != nil_a ).
% U
thf(fact_1_T,axiom,
ta != nil_a ).
% T
thf(fact_2_R_Ocube,axiom,
! [V: a,T: a,U: a] :
( ( resid @ ( resid @ V @ T ) @ ( resid @ U @ T ) )
= ( resid @ ( resid @ V @ U ) @ ( resid @ T @ U ) ) ) ).
% R.cube
thf(fact_3_R_Oex__un__null,axiom,
? [X: a] :
( ! [T2: a] :
( ( ( resid @ X @ T2 )
= X )
& ( ( resid @ T2 @ X )
= X ) )
& ! [Y: a] :
( ! [T3: a] :
( ( ( resid @ Y @ T3 )
= Y )
& ( ( resid @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% R.ex_un_null
thf(fact_4_Trgs_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [T3: a] :
( X2
!= ( cons_a @ T3 @ nil_a ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( X2
!= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ).
% Trgs.cases
thf(fact_5_cube,axiom,
! [V: list_a,T: list_a,U: list_a] :
( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ T ) @ ( paths_in_Resid_a @ resid @ U @ T ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ U ) @ ( paths_in_Resid_a @ resid @ T @ U ) ) ) ).
% cube
thf(fact_6_ex__un__null,axiom,
? [X: list_a] :
( ! [T2: list_a] :
( ( ( paths_in_Resid_a @ resid @ X @ T2 )
= X )
& ( ( paths_in_Resid_a @ resid @ T2 @ X )
= X ) )
& ! [Y: list_a] :
( ! [T3: list_a] :
( ( ( paths_in_Resid_a @ resid @ Y @ T3 )
= Y )
& ( ( paths_in_Resid_a @ resid @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% ex_un_null
thf(fact_7_Cube_I2_J,axiom,
! [T4: list_a,U2: list_a,V3: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) @ ( paths_in_Resid_a @ resid @ V3 @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) @ ( paths_in_Resid_a @ resid @ V3 @ U2 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ V3 ) @ ( paths_in_Resid_a @ resid @ U2 @ V3 ) ) ) ) ).
% Cube(2)
thf(fact_8_Cube_I1_J,axiom,
! [T4: list_a,U2: list_a,V3: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) @ ( paths_in_Resid_a @ resid @ V3 @ U2 ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ V3 ) @ ( paths_in_Resid_a @ resid @ U2 @ V3 ) )
!= nil_a ) ) ).
% Cube(1)
thf(fact_9_Resid_Osimps_I1_J,axiom,
! [Uu: list_a] :
( ( paths_in_Resid_a @ resid @ nil_a @ Uu )
= nil_a ) ).
% Resid.simps(1)
thf(fact_10_Con__sym,axiom,
! [T4: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ U2 @ T4 )
!= nil_a ) ) ).
% Con_sym
thf(fact_11_Con__cons_I2_J,axiom,
! [T4: list_a,U2: list_a,U: a] :
( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ U2 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) ) @ U2 )
!= nil_a ) ) ) ) ) ).
% Con_cons(2)
thf(fact_12_Con__cons_I1_J,axiom,
! [T4: list_a,U2: list_a,T: a] :
( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ U2 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ U2 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U2 @ ( cons_a @ T @ nil_a ) ) )
!= nil_a ) ) ) ) ) ).
% Con_cons(1)
thf(fact_13_Resid_Osimps_I2_J,axiom,
! [V: a,Va2: list_a] :
( ( paths_in_Resid_a @ resid @ ( cons_a @ V @ Va2 ) @ nil_a )
= nil_a ) ).
% Resid.simps(2)
thf(fact_14_Resid__cons_I2_J,axiom,
! [U2: list_a,T4: list_a,U: a] :
( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ U2 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) ) @ U2 ) ) ) ) ).
% Resid_cons(2)
thf(fact_15_Resid__rec_I3_J,axiom,
! [U2: list_a,T: a,U: a] :
( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ U2 ) )
= ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T @ U ) @ nil_a ) @ U2 ) ) ) ) ).
% Resid_rec(3)
thf(fact_16_Resid__rec_I2_J,axiom,
! [T4: list_a,T: a,U: a] :
( ( T4 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ nil_a ) )
= ( cons_a @ ( resid @ T @ U ) @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ ( resid @ U @ T ) @ nil_a ) ) ) ) ) ) ).
% Resid_rec(2)
thf(fact_17_Con__initial__left,axiom,
! [T: a,T4: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ U2 )
!= nil_a ) ) ).
% Con_initial_left
thf(fact_18_Con__initial__right,axiom,
! [T4: list_a,U: a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a ) ) ).
% Con_initial_right
thf(fact_19__092_060open_062_092_060And_062U_AT_O_A_092_060lbrakk_062T_A_092_060noteq_062_A_091_093_059_AU_A_092_060noteq_062_A_091_093_059_A_091_093_A_092_060noteq_062_A_091_093_092_060rbrakk_062_A_092_060Longrightarrow_062_A_I_I_091_093_A_064_AT_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_A_092_060noteq_062_A_091_093_J_A_061_A_I_091_093_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_A_092_060noteq_062_A_091_093_A_092_060and_062_AT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IU_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091_093_J_A_092_060noteq_062_A_091_093_J_A_092_060and_062_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_I_091_093_A_064_AU_J_A_092_060noteq_062_A_091_093_J_A_061_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091_093_A_092_060noteq_062_A_091_093_A_092_060and_062_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091_093_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_A_092_060noteq_062_A_091_093_J_A_092_060and_062_A_I_I_091_093_A_064_AT_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_A_092_060noteq_062_A_091_093_A_092_060longrightarrow_062_A_I_091_093_A_064_AT_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_A_061_A_091_093_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_A_064_AT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IU_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091_093_J_J_A_092_060and_062_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_I_091_093_A_064_AU_J_A_092_060noteq_062_A_091_093_A_092_060longrightarrow_062_AT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_I_091_093_A_064_AU_J_A_061_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091_093_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_092_060close_062,axiom,
! [T4: list_a,U2: list_a] :
( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( nil_a != nil_a )
=> ( ( ( ( paths_in_Resid_a @ resid @ ( append_a @ nil_a @ T4 ) @ U2 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ nil_a @ U2 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U2 @ nil_a ) )
!= nil_a ) ) )
& ( ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ nil_a @ U2 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T4 @ nil_a )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ nil_a ) @ U2 )
!= nil_a ) ) )
& ( ( ( paths_in_Resid_a @ resid @ ( append_a @ nil_a @ T4 ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( append_a @ nil_a @ T4 ) @ U2 )
= ( append_a @ ( paths_in_Resid_a @ resid @ nil_a @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U2 @ nil_a ) ) ) ) )
& ( ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ nil_a @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ nil_a @ U2 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ nil_a ) @ U2 ) ) ) ) ) ) ) ).
% \<open>\<And>U T. \<lbrakk>T \<noteq> []; U \<noteq> []; [] \<noteq> []\<rbrakk> \<Longrightarrow> (([] @ T) \<^sup>*\\<^sup>* U \<noteq> []) = ([] \<^sup>*\\<^sup>* U \<noteq> [] \<and> T \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* []) \<noteq> []) \<and> (T \<^sup>*\\<^sup>* ([] @ U) \<noteq> []) = (T \<^sup>*\\<^sup>* [] \<noteq> [] \<and> (T \<^sup>*\\<^sup>* []) \<^sup>*\\<^sup>* U \<noteq> []) \<and> (([] @ T) \<^sup>*\\<^sup>* U \<noteq> [] \<longrightarrow> ([] @ T) \<^sup>*\\<^sup>* U = [] \<^sup>*\\<^sup>* U @ T \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* [])) \<and> (T \<^sup>*\\<^sup>* ([] @ U) \<noteq> [] \<longrightarrow> T \<^sup>*\\<^sup>* ([] @ U) = (T \<^sup>*\\<^sup>* []) \<^sup>*\\<^sup>* U)\<close>
thf(fact_20_Resid__cons_I1_J,axiom,
! [U2: list_a,T: a,T4: list_a] :
( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ U2 )
= ( append_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U2 @ ( cons_a @ T @ nil_a ) ) ) ) ) ) ) ).
% Resid_cons(1)
thf(fact_21_Resid__rec_I4_J,axiom,
! [T4: list_a,U2: list_a,T: a,U: a] :
( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ U2 ) )
= ( append_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T @ U ) @ nil_a ) @ U2 ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ ( resid @ U @ T ) @ nil_a ) ) @ ( paths_in_Resid_a @ resid @ U2 @ ( cons_a @ ( resid @ T @ U ) @ nil_a ) ) ) ) ) ) ) ) ).
% Resid_rec(4)
thf(fact_22_ind,axiom,
! [T4: list_a,U2: list_a] :
( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( va != nil_a )
=> ( ( ( ( paths_in_Resid_a @ resid @ ( append_a @ va @ T4 ) @ U2 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ va @ U2 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U2 @ va ) )
!= nil_a ) ) )
& ( ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ va @ U2 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T4 @ va )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ va ) @ U2 )
!= nil_a ) ) )
& ( ( ( paths_in_Resid_a @ resid @ ( append_a @ va @ T4 ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( append_a @ va @ T4 ) @ U2 )
= ( append_a @ ( paths_in_Resid_a @ resid @ va @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U2 @ va ) ) ) ) )
& ( ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ va @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( append_a @ va @ U2 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ va ) @ U2 ) ) ) ) ) ) ) ).
% ind
thf(fact_23__092_060open_062T_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_I_Iv_A_D_AV_J_A_064_AU_J_A_092_060noteq_062_A_091_093_A_092_060Longrightarrow_062_AT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_Iv_A_D_AV_J_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
( ( ( paths_in_Resid_a @ resid @ ta @ ( append_a @ ( cons_a @ v @ va ) @ ua ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ta @ ( cons_a @ v @ va ) )
!= nil_a ) ) ).
% \<open>T \<^sup>*\\<^sup>* ((v # V) @ U) \<noteq> [] \<Longrightarrow> T \<^sup>*\\<^sup>* (v # V) \<noteq> []\<close>
thf(fact_24__092_060open_062_I_Iv_A_D_AV_J_A_064_AT_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_A_092_060noteq_062_A_091_093_A_092_060Longrightarrow_062_A_Iv_A_D_AV_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
( ( ( paths_in_Resid_a @ resid @ ( append_a @ ( cons_a @ v @ va ) @ ta ) @ ua )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ v @ va ) @ ua )
!= nil_a ) ) ).
% \<open>((v # V) @ T) \<^sup>*\\<^sup>* U \<noteq> [] \<Longrightarrow> (v # V) \<^sup>*\\<^sup>* U \<noteq> []\<close>
thf(fact_25__092_060open_062_I_Iv_A_D_AV_J_A_064_AT_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_A_092_060noteq_062_A_091_093_A_092_060Longrightarrow_062_AT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IU_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_Iv_A_D_AV_J_J_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
( ( ( paths_in_Resid_a @ resid @ ( append_a @ ( cons_a @ v @ va ) @ ta ) @ ua )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ta @ ( paths_in_Resid_a @ resid @ ua @ ( cons_a @ v @ va ) ) )
!= nil_a ) ) ).
% \<open>((v # V) @ T) \<^sup>*\\<^sup>* U \<noteq> [] \<Longrightarrow> T \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* (v # V)) \<noteq> []\<close>
thf(fact_26__C1_C,axiom,
( ( ( paths_in_Resid_a @ resid @ ( append_a @ ( cons_a @ v @ va ) @ ta ) @ ua )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( append_a @ ( cons_a @ v @ va ) @ ta ) @ ua )
= ( append_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ v @ va ) @ ua ) @ ( paths_in_Resid_a @ resid @ ta @ ( paths_in_Resid_a @ resid @ ua @ ( cons_a @ v @ va ) ) ) ) ) ) ).
% "1"
thf(fact_27__C2_C,axiom,
( ( ( paths_in_Resid_a @ resid @ ta @ ( append_a @ ( cons_a @ v @ va ) @ ua ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ta @ ( append_a @ ( cons_a @ v @ va ) @ ua ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ta @ ( cons_a @ v @ va ) ) @ ua ) ) ) ).
% "2"
thf(fact_28_Con__consI_I2_J,axiom,
! [T4: list_a,U2: list_a,U: a] :
( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ U2 ) )
!= nil_a ) ) ) ) ) ).
% Con_consI(2)
thf(fact_29_Con__consI_I1_J,axiom,
! [T4: list_a,U2: list_a,T: a] :
( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ U2 )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U2 @ ( cons_a @ T @ nil_a ) ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ U2 )
!= nil_a ) ) ) ) ) ).
% Con_consI(1)
thf(fact_30_Resid__rec_I1_J,axiom,
! [T: a,U: a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_a ) )
= ( cons_a @ ( resid @ T @ U ) @ nil_a ) ) ) ).
% Resid_rec(1)
thf(fact_31_paths__in__rts__axioms,axiom,
paths_in_rts_a @ resid ).
% paths_in_rts_axioms
thf(fact_32_paths__in__rts_OResid_Ocong,axiom,
paths_in_Resid_a = paths_in_Resid_a ).
% paths_in_rts.Resid.cong
thf(fact_33_R_Opartial__magma__axioms,axiom,
partial_magma_a @ resid ).
% R.partial_magma_axioms
thf(fact_34_partial__magma__axioms,axiom,
partial_magma_list_a @ ( paths_in_Resid_a @ resid ) ).
% partial_magma_axioms
thf(fact_35_append1__eq__conv,axiom,
! [Xs: list_a,X2: a,Ys: list_a,Y2: a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y2 @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_36_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_37_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_38_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_39_self__append__conv,axiom,
! [Y2: list_a,Ys: list_a] :
( ( Y2
= ( append_a @ Y2 @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_40_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_41_self__append__conv2,axiom,
! [Y2: list_a,Xs: list_a] :
( ( Y2
= ( append_a @ Xs @ Y2 ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_42_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_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_44_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_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_46_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_47_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X: list_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_48_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_49_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_50_Resid1x__as__Resid,axiom,
! [T: a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ U2 )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ T @ U2 ) @ nil_a ) ) ) ).
% Resid1x_as_Resid
thf(fact_51_Resid1x_Osimps_I3_J,axiom,
! [T: a,U: a,V: a,Va2: list_a] :
( ( paths_in_Resid1x_a @ resid @ T @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) )
= ( paths_in_Resid1x_a @ resid @ ( resid @ T @ U ) @ ( cons_a @ V @ Va2 ) ) ) ).
% Resid1x.simps(3)
thf(fact_52_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_53_Resid1x_Osimps_I2_J,axiom,
! [T: a,U: a] :
( ( paths_in_Resid1x_a @ resid @ T @ ( cons_a @ U @ nil_a ) )
= ( resid @ T @ U ) ) ).
% Resid1x.simps(2)
thf(fact_54_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_55_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_56_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_57_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_58_Resid__cons_H,axiom,
! [T4: list_a,T: a,U2: list_a] :
( ( T4 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ U2 )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ T @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U2 @ ( cons_a @ T @ nil_a ) ) ) ) ) ) ) ).
% Resid_cons'
thf(fact_59_paths__in__rts_OResid1x_Ocong,axiom,
paths_in_Resid1x_a = paths_in_Resid1x_a ).
% paths_in_rts.Resid1x.cong
thf(fact_60_partial__magma__def,axiom,
( partial_magma_a
= ( ^ [OP: a > a > a] :
? [X3: a] :
( ! [T5: a] :
( ( ( OP @ X3 @ T5 )
= X3 )
& ( ( OP @ T5 @ X3 )
= X3 ) )
& ! [Y3: a] :
( ! [T5: a] :
( ( ( OP @ Y3 @ T5 )
= Y3 )
& ( ( OP @ T5 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_61_partial__magma__def,axiom,
( partial_magma_list_a
= ( ^ [OP: list_a > list_a > list_a] :
? [X3: list_a] :
( ! [T5: list_a] :
( ( ( OP @ X3 @ T5 )
= X3 )
& ( ( OP @ T5 @ X3 )
= X3 ) )
& ! [Y3: list_a] :
( ! [T5: list_a] :
( ( ( OP @ Y3 @ T5 )
= Y3 )
& ( ( OP @ T5 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_62_partial__magma_Oex__un__null,axiom,
! [OP2: a > a > a] :
( ( partial_magma_a @ OP2 )
=> ? [X: a] :
( ! [T2: a] :
( ( ( OP2 @ X @ T2 )
= X )
& ( ( OP2 @ T2 @ X )
= X ) )
& ! [Y: a] :
( ! [T3: a] :
( ( ( OP2 @ Y @ T3 )
= Y )
& ( ( OP2 @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_63_partial__magma_Oex__un__null,axiom,
! [OP2: list_a > list_a > list_a] :
( ( partial_magma_list_a @ OP2 )
=> ? [X: list_a] :
( ! [T2: list_a] :
( ( ( OP2 @ X @ T2 )
= X )
& ( ( OP2 @ T2 @ X )
= X ) )
& ! [Y: list_a] :
( ! [T3: list_a] :
( ( ( OP2 @ Y @ T3 )
= Y )
& ( ( OP2 @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_64_partial__magma_Ointro,axiom,
! [OP2: a > a > a] :
( ? [X4: a] :
( ! [T3: a] :
( ( ( OP2 @ X4 @ T3 )
= X4 )
& ( ( OP2 @ T3 @ X4 )
= X4 ) )
& ! [Y4: a] :
( ! [T2: a] :
( ( ( OP2 @ Y4 @ T2 )
= Y4 )
& ( ( OP2 @ T2 @ Y4 )
= Y4 ) )
=> ( Y4 = X4 ) ) )
=> ( partial_magma_a @ OP2 ) ) ).
% partial_magma.intro
thf(fact_65_partial__magma_Ointro,axiom,
! [OP2: list_a > list_a > list_a] :
( ? [X4: list_a] :
( ! [T3: list_a] :
( ( ( OP2 @ X4 @ T3 )
= X4 )
& ( ( OP2 @ T3 @ X4 )
= X4 ) )
& ! [Y4: list_a] :
( ! [T2: list_a] :
( ( ( OP2 @ Y4 @ T2 )
= Y4 )
& ( ( OP2 @ T2 @ Y4 )
= Y4 ) )
=> ( Y4 = X4 ) ) )
=> ( partial_magma_list_a @ OP2 ) ) ).
% partial_magma.intro
thf(fact_66_paths__in__rts_OResid1x_Osimps_I3_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Resid1x_a @ Resid @ T @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) )
= ( paths_in_Resid1x_a @ Resid @ ( Resid @ T @ U ) @ ( cons_a @ V @ Va2 ) ) ) ) ).
% paths_in_rts.Resid1x.simps(3)
thf(fact_67_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_68_paths__in__rts_OResid1x_Osimps_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Resid1x_a @ Resid @ T @ ( cons_a @ U @ nil_a ) )
= ( Resid @ T @ U ) ) ) ).
% paths_in_rts.Resid1x.simps(2)
thf(fact_69_paths__in__rts_OResid1x__as__Resid,axiom,
! [Resid: a > a > a,T: a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ U2 )
= ( cons_a @ ( paths_in_Resid1x_a @ Resid @ T @ U2 ) @ nil_a ) ) ) ) ).
% paths_in_rts.Resid1x_as_Resid
thf(fact_70_paths__in__rts_OResid__cons_H,axiom,
! [Resid: a > a > a,T4: list_a,T: a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ U2 )
= ( cons_a @ ( paths_in_Resid1x_a @ Resid @ T @ U2 ) @ ( paths_in_Resid_a @ Resid @ T4 @ ( paths_in_Resid_a @ Resid @ U2 @ ( cons_a @ T @ nil_a ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_cons'
thf(fact_71_paths__in__rts_OTrgs_Ocases,axiom,
! [Resid: a > a > a,X2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( X2 != nil_a )
=> ( ! [T3: a] :
( X2
!= ( cons_a @ T3 @ nil_a ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( X2
!= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ).
% paths_in_rts.Trgs.cases
thf(fact_72_paths__in__rts_OCon__sym,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a )
= ( ( paths_in_Resid_a @ Resid @ U2 @ T4 )
!= nil_a ) ) ) ).
% paths_in_rts.Con_sym
thf(fact_73_paths__in__rts_OResid_Osimps_I1_J,axiom,
! [Resid: a > a > a,Uu: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Resid_a @ Resid @ nil_a @ Uu )
= nil_a ) ) ).
% paths_in_rts.Resid.simps(1)
thf(fact_74_paths__in__rts_OCube_I1_J,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,V3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U2 ) @ ( paths_in_Resid_a @ Resid @ V3 @ U2 ) )
!= nil_a )
= ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ V3 ) @ ( paths_in_Resid_a @ Resid @ U2 @ V3 ) )
!= nil_a ) ) ) ).
% paths_in_rts.Cube(1)
thf(fact_75_paths__in__rts_OCube_I2_J,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,V3: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U2 ) @ ( paths_in_Resid_a @ Resid @ V3 @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U2 ) @ ( paths_in_Resid_a @ Resid @ V3 @ U2 ) )
= ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ V3 ) @ ( paths_in_Resid_a @ Resid @ U2 @ V3 ) ) ) ) ) ).
% paths_in_rts.Cube(2)
thf(fact_76_paths__in__rts_OCon__initial__right,axiom,
! [Resid: a > a > a,T4: list_a,U: a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a ) ) ) ).
% paths_in_rts.Con_initial_right
thf(fact_77_paths__in__rts_OCon__initial__left,axiom,
! [Resid: a > a > a,T: a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ U2 )
!= nil_a ) ) ) ).
% paths_in_rts.Con_initial_left
thf(fact_78_paths__in__rts_OResid_Osimps_I2_J,axiom,
! [Resid: a > a > a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ V @ Va2 ) @ nil_a )
= nil_a ) ) ).
% paths_in_rts.Resid.simps(2)
thf(fact_79_paths__in__rts_OResid__cons_I2_J,axiom,
! [Resid: a > a > a,U2: list_a,T4: list_a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ U2 ) )
= ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) ) @ U2 ) ) ) ) ) ).
% paths_in_rts.Resid_cons(2)
thf(fact_80_paths__in__rts_OResid__rec_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_a ) )
= ( cons_a @ ( Resid @ T @ U ) @ nil_a ) ) ) ) ).
% paths_in_rts.Resid_rec(1)
thf(fact_81_paths__in__rts_OResid__rec_I2_J,axiom,
! [Resid: a > a > a,T4: list_a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ nil_a ) )
= ( cons_a @ ( Resid @ T @ U ) @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ ( Resid @ U @ T ) @ nil_a ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_rec(2)
thf(fact_82_paths__in__rts_OResid__rec_I3_J,axiom,
! [Resid: a > a > a,U2: list_a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ U2 ) )
= ( paths_in_Resid_a @ Resid @ ( cons_a @ ( Resid @ T @ U ) @ nil_a ) @ U2 ) ) ) ) ) ).
% paths_in_rts.Resid_rec(3)
thf(fact_83_paths__in__rts_OCon__consI_I1_J,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,T: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ U2 )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( paths_in_Resid_a @ Resid @ U2 @ ( cons_a @ T @ nil_a ) ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ U2 )
!= nil_a ) ) ) ) ) ) ).
% paths_in_rts.Con_consI(1)
thf(fact_84_paths__in__rts_OCon__consI_I2_J,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ U2 ) )
!= nil_a ) ) ) ) ) ) ).
% paths_in_rts.Con_consI(2)
thf(fact_85_paths__in__rts_OCon__cons_I1_J,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,T: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ U2 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ U2 )
!= nil_a )
& ( ( paths_in_Resid_a @ Resid @ T4 @ ( paths_in_Resid_a @ Resid @ U2 @ ( cons_a @ T @ nil_a ) ) )
!= nil_a ) ) ) ) ) ) ).
% paths_in_rts.Con_cons(1)
thf(fact_86_paths__in__rts_OCon__cons_I2_J,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ U2 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) ) @ U2 )
!= nil_a ) ) ) ) ) ) ).
% paths_in_rts.Con_cons(2)
thf(fact_87_not__Cons__self2,axiom,
! [X2: a,Xs: list_a] :
( ( cons_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_88_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_89_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_90_paths__in__rts_OResid__cons_I1_J,axiom,
! [Resid: a > a > a,U2: list_a,T: a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ U2 )
= ( append_a @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ U2 ) @ ( paths_in_Resid_a @ Resid @ T4 @ ( paths_in_Resid_a @ Resid @ U2 @ ( cons_a @ T @ nil_a ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_cons(1)
thf(fact_91_paths__in__rts_OResid__rec_I4_J,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ U2 ) )
= ( append_a @ ( paths_in_Resid_a @ Resid @ ( cons_a @ ( Resid @ T @ U ) @ nil_a ) @ U2 ) @ ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ ( Resid @ U @ T ) @ nil_a ) ) @ ( paths_in_Resid_a @ Resid @ U2 @ ( cons_a @ ( Resid @ T @ U ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_rec(4)
thf(fact_92_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_93_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a] : ( P @ ( cons_a @ X @ Xs2 ) @ nil_a )
=> ( ! [Y4: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y4 @ Ys2 ) )
=> ( ! [X: a,Xs2: list_a,Y4: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_94_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y3: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_95_remdups__adj_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X: a] :
( X2
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,Y4: a,Xs2: list_a] :
( X2
!= ( cons_a @ X @ ( cons_a @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_96_transpose_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X2
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X: a,Xs2: list_a,Xss: list_list_a] :
( X2
!= ( cons_list_a @ ( cons_a @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_97_list_Oexhaust,axiom,
! [Y2: list_a] :
( ( Y2 != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y2
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_98_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_99_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_100_Cons__eq__appendI,axiom,
! [X2: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_101_append__Cons,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X2 @ Xs ) @ Ys )
= ( cons_a @ X2 @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_102_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_103_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_104_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_105_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_106_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X2: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X2 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X2 @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X2 @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_107_Cons__eq__append__conv,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_108_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y4: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y4 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_109_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_110_Residx1__as__Resid,axiom,
! [T4: list_a,U: a] :
( ( paths_in_Residx1_a @ resid @ T4 @ U )
= ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) ) ) ).
% Residx1_as_Resid
thf(fact_111_residuation__axioms,axiom,
residuation_list_a @ ( paths_in_Resid_a @ resid ) ).
% residuation_axioms
thf(fact_112_Resid_Osimps_I3_J,axiom,
! [T: a,U: a] :
( ( ( con_a @ resid @ T @ U )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_a ) )
= ( cons_a @ ( resid @ T @ U ) @ nil_a ) ) )
& ( ~ ( con_a @ resid @ T @ U )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_a ) )
= nil_a ) ) ) ).
% Resid.simps(3)
thf(fact_113_Con__rec_I1_J,axiom,
! [T: a,U: a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_a ) )
!= nil_a )
= ( con_a @ resid @ T @ U ) ) ).
% Con_rec(1)
thf(fact_114_Con__rec_I2_J,axiom,
! [T4: list_a,T: a,U: a] :
( ( T4 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ nil_a ) )
!= nil_a )
= ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ ( resid @ U @ T ) @ nil_a ) )
!= nil_a ) ) ) ) ).
% Con_rec(2)
thf(fact_115_Con__rec_I3_J,axiom,
! [U2: list_a,T: a,U: a] :
( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ U2 ) )
!= nil_a )
= ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T @ U ) @ nil_a ) @ U2 )
!= nil_a ) ) ) ) ).
% Con_rec(3)
thf(fact_116_Con__rec_I4_J,axiom,
! [T4: list_a,U2: list_a,T: a,U: a] :
( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ U2 ) )
!= nil_a )
= ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ ( resid @ U @ T ) @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( cons_a @ ( resid @ T @ U ) @ nil_a ) @ U2 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ ( resid @ U @ T ) @ nil_a ) ) @ ( paths_in_Resid_a @ resid @ U2 @ ( cons_a @ ( resid @ T @ U ) @ nil_a ) ) )
!= nil_a ) ) ) ) ) ).
% Con_rec(4)
thf(fact_117_Resid1x_Oelims,axiom,
! [X2: a,Xa: list_a,Y2: a] :
( ( ( paths_in_Resid1x_a @ resid @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_a )
=> ( Y2
!= ( partial_null_a @ resid ) ) )
=> ( ! [U3: a] :
( ( Xa
= ( cons_a @ U3 @ nil_a ) )
=> ( Y2
!= ( resid @ X2 @ U3 ) ) )
=> ~ ! [U3: a,V2: a,Va: list_a] :
( ( Xa
= ( cons_a @ U3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_in_Resid1x_a @ resid @ ( resid @ X2 @ U3 ) @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ).
% Resid1x.elims
thf(fact_118_Trgs__Resid__sym__Arr__single,axiom,
! [T4: list_a,U: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) ) )
= ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ U @ nil_a ) @ T4 ) ) ) ) ).
% Trgs_Resid_sym_Arr_single
thf(fact_119_length__Resid,axiom,
! [T4: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
=> ( ( size_size_list_a @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) )
= ( size_size_list_a @ T4 ) ) ) ).
% length_Resid
thf(fact_120_Arr__Resid__single,axiom,
! [T4: list_a,U: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) ) ) ) ).
% Arr_Resid_single
thf(fact_121_R_Ocon__sym,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ U @ T ) ) ).
% R.con_sym
thf(fact_122_R_Oresid__reflects__con,axiom,
! [T: a,V: a,U: a] :
( ( con_a @ resid @ T @ V )
=> ( ( con_a @ resid @ U @ V )
=> ( ( con_a @ resid @ ( resid @ T @ V ) @ ( resid @ U @ V ) )
=> ( con_a @ resid @ T @ U ) ) ) ) ).
% R.resid_reflects_con
thf(fact_123_R_Ocon__imp__arr__resid,axiom,
! [T: a,U: a] :
( ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ ( resid @ T @ U ) @ ( resid @ T @ U ) )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_imp_arr_resid
thf(fact_124_R_Ocon__sym__ax,axiom,
! [T: a,U: a] :
( ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ U @ T )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_sym_ax
thf(fact_125_R_Ocube__ax,axiom,
! [V: a,T: a,U: a] :
( ( ( resid @ ( resid @ V @ T ) @ ( resid @ U @ T ) )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ ( resid @ V @ T ) @ ( resid @ U @ T ) )
= ( resid @ ( resid @ V @ U ) @ ( resid @ T @ U ) ) ) ) ).
% R.cube_ax
thf(fact_126_R_Onull__eqI,axiom,
! [N: a] :
( ! [T3: a] :
( ( ( resid @ N @ T3 )
= N )
& ( ( resid @ T3 @ N )
= N ) )
=> ( N
= ( partial_null_a @ resid ) ) ) ).
% R.null_eqI
thf(fact_127_Arr_Osimps_I1_J,axiom,
~ ( paths_in_Arr_a @ resid @ nil_a ) ).
% Arr.simps(1)
thf(fact_128_Trgs_Osimps_I3_J,axiom,
! [T: a,V: a,Va2: list_a] :
( ( paths_in_Trgs_a @ resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) )
= ( paths_in_Trgs_a @ resid @ ( cons_a @ V @ Va2 ) ) ) ).
% Trgs.simps(3)
thf(fact_129_Residx1_Osimps_I1_J,axiom,
! [U: a] :
( ( paths_in_Residx1_a @ resid @ nil_a @ U )
= nil_a ) ).
% Residx1.simps(1)
thf(fact_130_Trgs__are__con,axiom,
! [B: a,T4: list_a,B2: a] :
( ( member_a @ B @ ( paths_in_Trgs_a @ resid @ T4 ) )
=> ( ( member_a @ B2 @ ( paths_in_Trgs_a @ resid @ T4 ) )
=> ( con_a @ resid @ B @ B2 ) ) ) ).
% Trgs_are_con
thf(fact_131_R_OconE,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) ) ) ).
% R.conE
thf(fact_132_R_Ocon__def,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
= ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_def
thf(fact_133_Resid1x__null,axiom,
! [T4: list_a] :
( ( paths_in_Resid1x_a @ resid @ ( partial_null_a @ resid ) @ T4 )
= ( partial_null_a @ resid ) ) ).
% Resid1x_null
thf(fact_134_Con__implies__Arr_I2_J,axiom,
! [T4: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ U2 ) ) ).
% Con_implies_Arr(2)
thf(fact_135_Con__implies__Arr_I1_J,axiom,
! [T4: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ T4 ) ) ).
% Con_implies_Arr(1)
thf(fact_136_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_137_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_138_Con__imp__Arr__Resid,axiom,
! [T4: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
=> ( paths_in_Arr_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) ) ) ).
% Con_imp_Arr_Resid
thf(fact_139_Resid1x_Osimps_I1_J,axiom,
! [T: a] :
( ( paths_in_Resid1x_a @ resid @ T @ nil_a )
= ( partial_null_a @ resid ) ) ).
% Resid1x.simps(1)
thf(fact_140_Residx1_Osimps_I3_J,axiom,
! [T: a,U: a,V: a,Va2: list_a] :
( ( ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ U @ T ) )
!= nil_a ) )
=> ( ( paths_in_Residx1_a @ resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ U )
= ( cons_a @ ( resid @ T @ U ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ U @ T ) )
!= nil_a ) )
=> ( ( paths_in_Residx1_a @ resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ U )
= nil_a ) ) ) ).
% Residx1.simps(3)
thf(fact_141_Residx1_Osimps_I2_J,axiom,
! [T: a,U: a] :
( ( ( con_a @ resid @ T @ U )
=> ( ( paths_in_Residx1_a @ resid @ ( cons_a @ T @ nil_a ) @ U )
= ( cons_a @ ( resid @ T @ U ) @ nil_a ) ) )
& ( ~ ( con_a @ resid @ T @ U )
=> ( ( paths_in_Residx1_a @ resid @ ( cons_a @ T @ nil_a ) @ U )
= nil_a ) ) ) ).
% Residx1.simps(2)
thf(fact_142_Con__sym1,axiom,
! [T4: list_a,U: a] :
( ( ( paths_in_Residx1_a @ resid @ T4 @ U )
!= nil_a )
= ( ( paths_in_Resid1x_a @ resid @ U @ T4 )
!= ( partial_null_a @ resid ) ) ) ).
% Con_sym1
thf(fact_143_Resid_Osimps_I5_J,axiom,
! [T: a,U: a,V: a,Va2: list_a] :
( ( ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ U @ T ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ nil_a ) )
= ( cons_a @ ( resid @ T @ U ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ U @ T ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ nil_a ) )
= nil_a ) ) ) ).
% Resid.simps(5)
thf(fact_144_Resid_Osimps_I4_J,axiom,
! [T: a,U: a,V: a,Va2: list_a] :
( ( ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T @ U ) @ ( cons_a @ V @ Va2 ) )
!= ( partial_null_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ ( resid @ T @ U ) @ ( cons_a @ V @ Va2 ) ) @ nil_a ) ) )
& ( ~ ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T @ U ) @ ( cons_a @ V @ Va2 ) )
!= ( partial_null_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) )
= nil_a ) ) ) ).
% Resid.simps(4)
thf(fact_145_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_146_Resid_Osimps_I7_J,axiom,
! [T: a,U: a,V: a,Va2: list_a,Vb: a,Vc: list_a] :
( ( ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T @ U ) @ ( cons_a @ V @ Va2 ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ U @ T ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ T @ U ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ ( resid @ T @ U ) @ ( cons_a @ V @ Va2 ) ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ U @ T ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T @ U ) @ ( cons_a @ V @ Va2 ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ U @ T ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ T @ U ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) )
= nil_a ) ) ) ).
% Resid.simps(7)
thf(fact_147_Resid_Osimps_I6_J,axiom,
! [T: a,U: a,Vb: a,Vc: list_a,V: a,Va2: list_a] :
( ( ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T @ U ) @ ( cons_a @ Vb @ Vc ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ U @ T ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ T @ U ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ ( cons_a @ Vb @ Vc ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ resid @ ( resid @ T @ U ) @ ( cons_a @ Vb @ Vc ) ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ U @ T ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_a @ resid @ T @ U )
& ( ( paths_in_Resid1x_a @ resid @ ( resid @ T @ U ) @ ( cons_a @ Vb @ Vc ) )
!= ( partial_null_a @ resid ) )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Residx1_a @ resid @ ( cons_a @ V @ Va2 ) @ ( resid @ U @ T ) ) @ ( paths_in_Residx1_a @ resid @ ( cons_a @ Vb @ Vc ) @ ( resid @ T @ U ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ ( cons_a @ Vb @ Vc ) ) )
= nil_a ) ) ) ).
% Resid.simps(6)
thf(fact_148_R_Onull__is__zero_I2_J,axiom,
! [T: a] :
( ( resid @ T @ ( partial_null_a @ resid ) )
= ( partial_null_a @ resid ) ) ).
% R.null_is_zero(2)
thf(fact_149_R_Onull__is__zero_I1_J,axiom,
! [T: a] :
( ( resid @ ( partial_null_a @ resid ) @ T )
= ( partial_null_a @ resid ) ) ).
% R.null_is_zero(1)
thf(fact_150_R_OconI,axiom,
! [T: a,U: a] :
( ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) )
=> ( con_a @ resid @ T @ U ) ) ).
% R.conI
thf(fact_151_Trgs__Resid__sym,axiom,
! [T4: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
=> ( ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) )
= ( paths_in_Trgs_a @ resid @ ( paths_in_Resid_a @ resid @ U2 @ T4 ) ) ) ) ).
% Trgs_Resid_sym
thf(fact_152_Trgs__append,axiom,
! [U2: list_a,T4: list_a] :
( ( U2 != nil_a )
=> ( ( paths_in_Trgs_a @ resid @ ( append_a @ T4 @ U2 ) )
= ( paths_in_Trgs_a @ resid @ U2 ) ) ) ).
% Trgs_append
thf(fact_153_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_154_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_155_residuation_Ocon__imp__arr__resid,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U ) )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.con_imp_arr_resid
thf(fact_156_residuation_Ocon__imp__arr__resid,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_list_a @ Resid ) )
=> ( ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U ) )
!= ( partial_null_list_a @ Resid ) ) ) ) ).
% residuation.con_imp_arr_resid
thf(fact_157_paths__in__rts_OResidx1_Ocong,axiom,
paths_in_Residx1_a = paths_in_Residx1_a ).
% paths_in_rts.Residx1.cong
thf(fact_158_partial__magma_Onull_Ocong,axiom,
partial_null_a = partial_null_a ).
% partial_magma.null.cong
thf(fact_159_partial__magma_Onull_Ocong,axiom,
partial_null_list_a = partial_null_list_a ).
% partial_magma.null.cong
thf(fact_160_residuation_Ocon__sym__ax,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ U @ T )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.con_sym_ax
thf(fact_161_residuation_Ocon__sym__ax,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_list_a @ Resid ) )
=> ( ( Resid @ U @ T )
!= ( partial_null_list_a @ Resid ) ) ) ) ).
% residuation.con_sym_ax
thf(fact_162_paths__in__rts_OTrgs_Ocong,axiom,
paths_in_Trgs_a = paths_in_Trgs_a ).
% paths_in_rts.Trgs.cong
thf(fact_163_paths__in__rts_OArr_Ocong,axiom,
paths_in_Arr_a = paths_in_Arr_a ).
% paths_in_rts.Arr.cong
thf(fact_164_residuation_Ocon_Ocong,axiom,
con_a = con_a ).
% residuation.con.cong
thf(fact_165_residuation_Ocon_Ocong,axiom,
con_list_a = con_list_a ).
% residuation.con.cong
thf(fact_166_residuation_Ocube__ax,axiom,
! [Resid: a > a > a,V: a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ).
% residuation.cube_ax
thf(fact_167_residuation_Ocube__ax,axiom,
! [Resid: list_a > list_a > list_a,V: list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
!= ( partial_null_list_a @ Resid ) )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ).
% residuation.cube_ax
thf(fact_168_residuation_Ocon__sym,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( con_a @ Resid @ U @ T ) ) ) ).
% residuation.con_sym
thf(fact_169_residuation_Ocon__sym,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T @ U )
=> ( con_list_a @ Resid @ U @ T ) ) ) ).
% residuation.con_sym
thf(fact_170_residuation_Ocon__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
= ( ( Resid @ T @ U )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.con_def
thf(fact_171_residuation_Ocon__def,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T @ U )
= ( ( Resid @ T @ U )
!= ( partial_null_list_a @ Resid ) ) ) ) ).
% residuation.con_def
thf(fact_172_residuation_Ocube,axiom,
! [Resid: list_a > list_a > list_a,V: list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ).
% residuation.cube
thf(fact_173_residuation_Ocube,axiom,
! [Resid: a > a > a,V: a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ).
% residuation.cube
thf(fact_174_residuation_OconI,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_a @ Resid ) )
=> ( con_a @ Resid @ T @ U ) ) ) ).
% residuation.conI
thf(fact_175_residuation_OconI,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_list_a @ Resid ) )
=> ( con_list_a @ Resid @ T @ U ) ) ) ).
% residuation.conI
thf(fact_176_residuation_OconE,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( Resid @ T @ U )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.conE
thf(fact_177_residuation_OconE,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T @ U )
=> ( ( Resid @ T @ U )
!= ( partial_null_list_a @ Resid ) ) ) ) ).
% residuation.conE
thf(fact_178_paths__in__rts_OTrgs__are__con,axiom,
! [Resid: list_a > list_a > list_a,B: list_a,T4: list_list_a,B2: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( member_list_a @ B @ ( paths_in_Trgs_list_a @ Resid @ T4 ) )
=> ( ( member_list_a @ B2 @ ( paths_in_Trgs_list_a @ Resid @ T4 ) )
=> ( con_list_a @ Resid @ B @ B2 ) ) ) ) ).
% paths_in_rts.Trgs_are_con
thf(fact_179_paths__in__rts_OTrgs__are__con,axiom,
! [Resid: a > a > a,B: a,T4: list_a,B2: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( member_a @ B @ ( paths_in_Trgs_a @ Resid @ T4 ) )
=> ( ( member_a @ B2 @ ( paths_in_Trgs_a @ Resid @ T4 ) )
=> ( con_a @ Resid @ B @ B2 ) ) ) ) ).
% paths_in_rts.Trgs_are_con
thf(fact_180_partial__magma_Onull__eqI,axiom,
! [OP2: a > a > a,N: a] :
( ( partial_magma_a @ OP2 )
=> ( ! [T3: a] :
( ( ( OP2 @ N @ T3 )
= N )
& ( ( OP2 @ T3 @ N )
= N ) )
=> ( N
= ( partial_null_a @ OP2 ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_181_partial__magma_Onull__eqI,axiom,
! [OP2: list_a > list_a > list_a,N: list_a] :
( ( partial_magma_list_a @ OP2 )
=> ( ! [T3: list_a] :
( ( ( OP2 @ N @ T3 )
= N )
& ( ( OP2 @ T3 @ N )
= N ) )
=> ( N
= ( partial_null_list_a @ OP2 ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_182_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP2: a > a > a,T: a] :
( ( partial_magma_a @ OP2 )
=> ( ( OP2 @ ( partial_null_a @ OP2 ) @ T )
= ( partial_null_a @ OP2 ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_183_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP2: list_a > list_a > list_a,T: list_a] :
( ( partial_magma_list_a @ OP2 )
=> ( ( OP2 @ ( partial_null_list_a @ OP2 ) @ T )
= ( partial_null_list_a @ OP2 ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_184_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP2: a > a > a,T: a] :
( ( partial_magma_a @ OP2 )
=> ( ( OP2 @ T @ ( partial_null_a @ OP2 ) )
= ( partial_null_a @ OP2 ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_185_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP2: list_a > list_a > list_a,T: list_a] :
( ( partial_magma_list_a @ OP2 )
=> ( ( OP2 @ T @ ( partial_null_list_a @ OP2 ) )
= ( partial_null_list_a @ OP2 ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_186_paths__in__rts_OResidx1_Osimps_I3_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a,V: list_a,Va2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_list_a ) )
=> ( ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ V @ Va2 ) ) @ U )
= ( cons_list_a @ ( Resid @ T @ U ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_list_a ) )
=> ( ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ V @ Va2 ) ) @ U )
= nil_list_a ) ) ) ) ).
% paths_in_rts.Residx1.simps(3)
thf(fact_187_paths__in__rts_OResidx1_Osimps_I3_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_a ) )
=> ( ( paths_in_Residx1_a @ Resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ U )
= ( cons_a @ ( Resid @ T @ U ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_a ) )
=> ( ( paths_in_Residx1_a @ Resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ U )
= nil_a ) ) ) ) ).
% paths_in_rts.Residx1.simps(3)
thf(fact_188_paths__in__rts_OResidx1_Osimps_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( con_list_a @ Resid @ T @ U )
=> ( ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ U )
= ( cons_list_a @ ( Resid @ T @ U ) @ nil_list_a ) ) )
& ( ~ ( con_list_a @ Resid @ T @ U )
=> ( ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ U )
= nil_list_a ) ) ) ) ).
% paths_in_rts.Residx1.simps(2)
thf(fact_189_paths__in__rts_OResidx1_Osimps_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( con_a @ Resid @ T @ U )
=> ( ( paths_in_Residx1_a @ Resid @ ( cons_a @ T @ nil_a ) @ U )
= ( cons_a @ ( Resid @ T @ U ) @ nil_a ) ) )
& ( ~ ( con_a @ Resid @ T @ U )
=> ( ( paths_in_Residx1_a @ Resid @ ( cons_a @ T @ nil_a ) @ U )
= nil_a ) ) ) ) ).
% paths_in_rts.Residx1.simps(2)
thf(fact_190_paths__in__rts_OResid_Osimps_I7_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a,V: list_a,Va2: list_list_a,Vb: list_a,Vc: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T @ U ) @ ( cons_list_a @ V @ Va2 ) )
!= ( partial_null_list_a @ Resid ) )
& ( ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ U @ T ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ T @ U ) ) )
!= nil_list_a ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ Vb @ Vc ) ) @ ( cons_list_a @ U @ ( cons_list_a @ V @ Va2 ) ) )
= ( cons_list_a @ ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T @ U ) @ ( cons_list_a @ V @ Va2 ) ) @ ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ U @ T ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T @ U ) @ ( cons_list_a @ V @ Va2 ) )
!= ( partial_null_list_a @ Resid ) )
& ( ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ U @ T ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ T @ U ) ) )
!= nil_list_a ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ Vb @ Vc ) ) @ ( cons_list_a @ U @ ( cons_list_a @ V @ Va2 ) ) )
= nil_list_a ) ) ) ) ).
% paths_in_rts.Resid.simps(7)
thf(fact_191_paths__in__rts_OResid_Osimps_I7_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,Va2: list_a,Vb: a,Vc: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Resid1x_a @ Resid @ ( Resid @ T @ U ) @ ( cons_a @ V @ Va2 ) )
!= ( partial_null_a @ Resid ) )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ U @ T ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ T @ U ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ Resid @ ( Resid @ T @ U ) @ ( cons_a @ V @ Va2 ) ) @ ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ U @ T ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Resid1x_a @ Resid @ ( Resid @ T @ U ) @ ( cons_a @ V @ Va2 ) )
!= ( partial_null_a @ Resid ) )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ U @ T ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ T @ U ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) )
= nil_a ) ) ) ) ).
% paths_in_rts.Resid.simps(7)
thf(fact_192_paths__in__rts_OResid_Osimps_I6_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a,Vb: list_a,Vc: list_list_a,V: list_a,Va2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T @ U ) @ ( cons_list_a @ Vb @ Vc ) )
!= ( partial_null_list_a @ Resid ) )
& ( ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ U @ T ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ T @ U ) ) )
!= nil_list_a ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ V @ Va2 ) ) @ ( cons_list_a @ U @ ( cons_list_a @ Vb @ Vc ) ) )
= ( cons_list_a @ ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T @ U ) @ ( cons_list_a @ Vb @ Vc ) ) @ ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ U @ T ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T @ U ) @ ( cons_list_a @ Vb @ Vc ) )
!= ( partial_null_list_a @ Resid ) )
& ( ( paths_8620460302779588466list_a @ Resid @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ U @ T ) ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ Vb @ Vc ) @ ( Resid @ T @ U ) ) )
!= nil_list_a ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ V @ Va2 ) ) @ ( cons_list_a @ U @ ( cons_list_a @ Vb @ Vc ) ) )
= nil_list_a ) ) ) ) ).
% paths_in_rts.Resid.simps(6)
thf(fact_193_paths__in__rts_OResid_Osimps_I6_J,axiom,
! [Resid: a > a > a,T: a,U: a,Vb: a,Vc: list_a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Resid1x_a @ Resid @ ( Resid @ T @ U ) @ ( cons_a @ Vb @ Vc ) )
!= ( partial_null_a @ Resid ) )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ U @ T ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ T @ U ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ ( cons_a @ Vb @ Vc ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ Resid @ ( Resid @ T @ U ) @ ( cons_a @ Vb @ Vc ) ) @ ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ U @ T ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Resid1x_a @ Resid @ ( Resid @ T @ U ) @ ( cons_a @ Vb @ Vc ) )
!= ( partial_null_a @ Resid ) )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ U @ T ) ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ Vb @ Vc ) @ ( Resid @ T @ U ) ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ ( cons_a @ Vb @ Vc ) ) )
= nil_a ) ) ) ) ).
% paths_in_rts.Resid.simps(6)
thf(fact_194_paths__in__rts_OCon__sym1,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_3541054012941122297list_a @ Resid @ T4 @ U )
!= nil_list_a )
= ( ( paths_1777230443808135851list_a @ Resid @ U @ T4 )
!= ( partial_null_list_a @ Resid ) ) ) ) ).
% paths_in_rts.Con_sym1
thf(fact_195_paths__in__rts_OCon__sym1,axiom,
! [Resid: a > a > a,T4: list_a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Residx1_a @ Resid @ T4 @ U )
!= nil_a )
= ( ( paths_in_Resid1x_a @ Resid @ U @ T4 )
!= ( partial_null_a @ Resid ) ) ) ) ).
% paths_in_rts.Con_sym1
thf(fact_196_residuation_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( partial_magma_a @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_197_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_198_paths__in__rts_OResid_Osimps_I5_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a,V: list_a,Va2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_list_a ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ V @ Va2 ) ) @ ( cons_list_a @ U @ nil_list_a ) )
= ( cons_list_a @ ( Resid @ T @ U ) @ ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_3541054012941122297list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_list_a ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ V @ Va2 ) ) @ ( cons_list_a @ U @ nil_list_a ) )
= nil_list_a ) ) ) ) ).
% paths_in_rts.Resid.simps(5)
thf(fact_199_paths__in__rts_OResid_Osimps_I5_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ nil_a ) )
= ( cons_a @ ( Resid @ T @ U ) @ ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Residx1_a @ Resid @ ( cons_a @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_a ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ nil_a ) )
= nil_a ) ) ) ) ).
% paths_in_rts.Resid.simps(5)
thf(fact_200_paths__in__rts_OArr_Osimps_I1_J,axiom,
! [Resid: a > a > a] :
( ( paths_in_rts_a @ Resid )
=> ~ ( paths_in_Arr_a @ Resid @ nil_a ) ) ).
% paths_in_rts.Arr.simps(1)
thf(fact_201_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 )
=> ( ! [X: a,Xs2: list_a,Y4: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_202_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 )
=> ( ! [X: a,Xs2: list_a,Y4: 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 @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_203_paths__in__rts_OResid1x__null,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_1777230443808135851list_a @ Resid @ ( partial_null_list_a @ Resid ) @ T4 )
= ( partial_null_list_a @ Resid ) ) ) ).
% paths_in_rts.Resid1x_null
thf(fact_204_paths__in__rts_OResid1x__null,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Resid1x_a @ Resid @ ( partial_null_a @ Resid ) @ T4 )
= ( partial_null_a @ Resid ) ) ) ).
% paths_in_rts.Resid1x_null
thf(fact_205_paths__in__rts_OTrgs_Osimps_I3_J,axiom,
! [Resid: a > a > a,T: a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Trgs_a @ Resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) )
= ( paths_in_Trgs_a @ Resid @ ( cons_a @ V @ Va2 ) ) ) ) ).
% paths_in_rts.Trgs.simps(3)
thf(fact_206_paths__in__rts_OResid_Osimps_I4_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a,V: list_a,Va2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T @ U ) @ ( cons_list_a @ V @ Va2 ) )
!= ( partial_null_list_a @ Resid ) ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ ( cons_list_a @ U @ ( cons_list_a @ V @ Va2 ) ) )
= ( cons_list_a @ ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T @ U ) @ ( cons_list_a @ V @ Va2 ) ) @ nil_list_a ) ) )
& ( ~ ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T @ U ) @ ( cons_list_a @ V @ Va2 ) )
!= ( partial_null_list_a @ Resid ) ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ ( cons_list_a @ U @ ( cons_list_a @ V @ Va2 ) ) )
= nil_list_a ) ) ) ) ).
% paths_in_rts.Resid.simps(4)
thf(fact_207_paths__in__rts_OResid_Osimps_I4_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Resid1x_a @ Resid @ ( Resid @ T @ U ) @ ( cons_a @ V @ Va2 ) )
!= ( partial_null_a @ Resid ) ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) )
= ( cons_a @ ( paths_in_Resid1x_a @ Resid @ ( Resid @ T @ U ) @ ( cons_a @ V @ Va2 ) ) @ nil_a ) ) )
& ( ~ ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Resid1x_a @ Resid @ ( Resid @ T @ U ) @ ( cons_a @ V @ Va2 ) )
!= ( partial_null_a @ Resid ) ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) )
= nil_a ) ) ) ) ).
% paths_in_rts.Resid.simps(4)
thf(fact_208_paths__in__rts_OResidx1_Osimps_I1_J,axiom,
! [Resid: a > a > a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Residx1_a @ Resid @ nil_a @ U )
= nil_a ) ) ).
% paths_in_rts.Residx1.simps(1)
thf(fact_209_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_210_paths__in__rts_OCon__implies__Arr_I2_J,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a )
=> ( paths_in_Arr_a @ Resid @ U2 ) ) ) ).
% paths_in_rts.Con_implies_Arr(2)
thf(fact_211_paths__in__rts_OCon__implies__Arr_I1_J,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a )
=> ( paths_in_Arr_a @ Resid @ T4 ) ) ) ).
% paths_in_rts.Con_implies_Arr(1)
thf(fact_212_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_213_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_214_paths__in__rts_OCon__imp__Arr__Resid,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a )
=> ( paths_in_Arr_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U2 ) ) ) ) ).
% paths_in_rts.Con_imp_Arr_Resid
thf(fact_215_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,X: a,Xs3: list_a,Y4: a,Ys5: list_a] :
( ( X != Y4 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_216_paths__in__rts_Olength__Resid,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a )
=> ( ( size_size_list_a @ ( paths_in_Resid_a @ Resid @ T4 @ U2 ) )
= ( size_size_list_a @ T4 ) ) ) ) ).
% paths_in_rts.length_Resid
thf(fact_217_paths__in__rts_OResid1x_Osimps_I1_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_1777230443808135851list_a @ Resid @ T @ nil_list_a )
= ( partial_null_list_a @ Resid ) ) ) ).
% paths_in_rts.Resid1x.simps(1)
thf(fact_218_paths__in__rts_OResid1x_Osimps_I1_J,axiom,
! [Resid: a > a > a,T: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Resid1x_a @ Resid @ T @ nil_a )
= ( partial_null_a @ Resid ) ) ) ).
% paths_in_rts.Resid1x.simps(1)
thf(fact_219_paths__in__rts_OTrgs__Resid__sym,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a )
=> ( ( paths_in_Trgs_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U2 ) )
= ( paths_in_Trgs_a @ Resid @ ( paths_in_Resid_a @ Resid @ U2 @ T4 ) ) ) ) ) ).
% paths_in_rts.Trgs_Resid_sym
thf(fact_220_paths__in__rts_OTrgs__append,axiom,
! [Resid: a > a > a,U2: list_a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( U2 != nil_a )
=> ( ( paths_in_Trgs_a @ Resid @ ( append_a @ T4 @ U2 ) )
= ( paths_in_Trgs_a @ Resid @ U2 ) ) ) ) ).
% paths_in_rts.Trgs_append
thf(fact_221_paths__in__rts_OCon__rec_I4_J,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a,T: list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( T4 != nil_list_a )
=> ( ( U2 != nil_list_a )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ T4 ) @ ( cons_list_a @ U @ U2 ) )
!= nil_list_a )
= ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ ( Resid @ U @ T ) @ nil_list_a ) )
!= nil_list_a )
& ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ ( Resid @ T @ U ) @ nil_list_a ) @ U2 )
!= nil_list_a )
& ( ( paths_8620460302779588466list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ ( Resid @ U @ T ) @ nil_list_a ) ) @ ( paths_8620460302779588466list_a @ Resid @ U2 @ ( cons_list_a @ ( Resid @ T @ U ) @ nil_list_a ) ) )
!= nil_list_a ) ) ) ) ) ) ).
% paths_in_rts.Con_rec(4)
thf(fact_222_paths__in__rts_OCon__rec_I4_J,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ U2 ) )
!= nil_a )
= ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ ( Resid @ U @ T ) @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ Resid @ ( cons_a @ ( Resid @ T @ U ) @ nil_a ) @ U2 )
!= nil_a )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ ( Resid @ U @ T ) @ nil_a ) ) @ ( paths_in_Resid_a @ Resid @ U2 @ ( cons_a @ ( Resid @ T @ U ) @ nil_a ) ) )
!= nil_a ) ) ) ) ) ) ).
% paths_in_rts.Con_rec(4)
thf(fact_223_paths__in__rts_OCon__rec_I3_J,axiom,
! [Resid: list_a > list_a > list_a,U2: list_list_a,T: list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( U2 != nil_list_a )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ ( cons_list_a @ U @ U2 ) )
!= nil_list_a )
= ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ ( Resid @ T @ U ) @ nil_list_a ) @ U2 )
!= nil_list_a ) ) ) ) ) ).
% paths_in_rts.Con_rec(3)
thf(fact_224_paths__in__rts_OCon__rec_I3_J,axiom,
! [Resid: a > a > a,U2: list_a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( U2 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ U2 ) )
!= nil_a )
= ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Resid_a @ Resid @ ( cons_a @ ( Resid @ T @ U ) @ nil_a ) @ U2 )
!= nil_a ) ) ) ) ) ).
% paths_in_rts.Con_rec(3)
thf(fact_225_paths__in__rts_OCon__rec_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,T: list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( T4 != nil_list_a )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ T4 ) @ ( cons_list_a @ U @ nil_list_a ) )
!= nil_list_a )
= ( ( con_list_a @ Resid @ T @ U )
& ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ ( Resid @ U @ T ) @ nil_list_a ) )
!= nil_list_a ) ) ) ) ) ).
% paths_in_rts.Con_rec(2)
thf(fact_226_paths__in__rts_OCon__rec_I2_J,axiom,
! [Resid: a > a > a,T4: list_a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ T4 ) @ ( cons_a @ U @ nil_a ) )
!= nil_a )
= ( ( con_a @ Resid @ T @ U )
& ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ ( Resid @ U @ T ) @ nil_a ) )
!= nil_a ) ) ) ) ) ).
% paths_in_rts.Con_rec(2)
thf(fact_227_paths__in__rts_OCon__rec_I1_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ ( cons_list_a @ U @ nil_list_a ) )
!= nil_list_a )
= ( con_list_a @ Resid @ T @ U ) ) ) ).
% paths_in_rts.Con_rec(1)
thf(fact_228_paths__in__rts_OCon__rec_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_a ) )
!= nil_a )
= ( con_a @ Resid @ T @ U ) ) ) ).
% paths_in_rts.Con_rec(1)
thf(fact_229_paths__in__rts_OResid_Osimps_I3_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( con_list_a @ Resid @ T @ U )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ ( cons_list_a @ U @ nil_list_a ) )
= ( cons_list_a @ ( Resid @ T @ U ) @ nil_list_a ) ) )
& ( ~ ( con_list_a @ Resid @ T @ U )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ ( cons_list_a @ U @ nil_list_a ) )
= nil_list_a ) ) ) ) ).
% paths_in_rts.Resid.simps(3)
thf(fact_230_paths__in__rts_OResid_Osimps_I3_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( con_a @ Resid @ T @ U )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_a ) )
= ( cons_a @ ( Resid @ T @ U ) @ nil_a ) ) )
& ( ~ ( con_a @ Resid @ T @ U )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_a ) )
= nil_a ) ) ) ) ).
% paths_in_rts.Resid.simps(3)
thf(fact_231_paths__in__rts_OArr__Resid__single,axiom,
! [Resid: a > a > a,T4: list_a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( paths_in_Arr_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) ) ) ) ) ).
% paths_in_rts.Arr_Resid_single
thf(fact_232_paths__in__rts_OResid1x_Oelims,axiom,
! [Resid: list_a > list_a > list_a,X2: list_a,Xa: list_list_a,Y2: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_1777230443808135851list_a @ Resid @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_list_a )
=> ( Y2
!= ( partial_null_list_a @ Resid ) ) )
=> ( ! [U3: list_a] :
( ( Xa
= ( cons_list_a @ U3 @ nil_list_a ) )
=> ( Y2
!= ( Resid @ X2 @ U3 ) ) )
=> ~ ! [U3: list_a,V2: list_a,Va: list_list_a] :
( ( Xa
= ( cons_list_a @ U3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_1777230443808135851list_a @ Resid @ ( Resid @ X2 @ U3 ) @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid1x.elims
thf(fact_233_paths__in__rts_OResid1x_Oelims,axiom,
! [Resid: a > a > a,X2: a,Xa: list_a,Y2: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid1x_a @ Resid @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_a )
=> ( Y2
!= ( partial_null_a @ Resid ) ) )
=> ( ! [U3: a] :
( ( Xa
= ( cons_a @ U3 @ nil_a ) )
=> ( Y2
!= ( Resid @ X2 @ U3 ) ) )
=> ~ ! [U3: a,V2: a,Va: list_a] :
( ( Xa
= ( cons_a @ U3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_in_Resid1x_a @ Resid @ ( Resid @ X2 @ U3 ) @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid1x.elims
thf(fact_234_paths__in__rts_OTrgs__Resid__sym__Arr__single,axiom,
! [Resid: a > a > a,T4: list_a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Trgs_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) ) )
= ( paths_in_Trgs_a @ Resid @ ( paths_in_Resid_a @ Resid @ ( cons_a @ U @ nil_a ) @ T4 ) ) ) ) ) ).
% paths_in_rts.Trgs_Resid_sym_Arr_single
thf(fact_235_paths__in__rts_OResidx1__as__Resid,axiom,
! [Resid: a > a > a,T4: list_a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Residx1_a @ Resid @ T4 @ U )
= ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) ) ) ) ).
% paths_in_rts.Residx1_as_Resid
thf(fact_236_Resid1x__as__Resid_H,axiom,
! [T: a,U2: list_a] :
( ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid1x_a @ resid @ T @ U2 )
= ( hd_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ U2 ) ) ) )
& ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T @ nil_a ) @ U2 )
= nil_a )
=> ( ( paths_in_Resid1x_a @ resid @ T @ U2 )
= ( partial_null_a @ resid ) ) ) ) ).
% Resid1x_as_Resid'
thf(fact_237_R_Ojoinable__implies__con,axiom,
! [T: a,U: a] :
( ( joinable_a @ resid @ T @ U )
=> ( con_a @ resid @ T @ U ) ) ).
% R.joinable_implies_con
thf(fact_238_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_239_length__Residx1,axiom,
! [T4: list_a,U: a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( paths_in_Residx1_a @ resid @ T4 @ U ) ) @ ( size_size_list_a @ T4 ) ) ).
% length_Residx1
thf(fact_240_Trgs_Osimps_I1_J,axiom,
( ( paths_in_Trgs_a @ resid @ nil_a )
= bot_bot_set_a ) ).
% Trgs.simps(1)
thf(fact_241_Arr__appendE_092_060_094sub_062P,axiom,
! [T4: list_a,U2: list_a] :
( ( paths_in_Arr_a @ resid @ ( append_a @ T4 @ U2 ) )
=> ( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ~ ( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Arr_a @ resid @ U2 )
=> ( ( paths_in_Trgs_a @ resid @ T4 )
!= ( paths_in_Srcs_a @ resid @ U2 ) ) ) ) ) ) ) ).
% Arr_appendE\<^sub>P
thf(fact_242_sources__cons,axiom,
! [T: a,T4: list_a] :
( ( paths_in_Arr_a @ resid @ ( cons_a @ T @ T4 ) )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T @ T4 ) )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ ( cons_a @ T @ nil_a ) ) ) ) ).
% sources_cons
thf(fact_243_Srcs__Resid__single__Arr,axiom,
! [U: a,T4: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ U @ nil_a ) @ T4 )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( paths_in_Resid_a @ resid @ ( cons_a @ U @ nil_a ) @ T4 ) )
= ( paths_in_Trgs_a @ resid @ T4 ) ) ) ).
% Srcs_Resid_single_Arr
thf(fact_244_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_245_R_Oresiduation__axioms,axiom,
residuation_a @ resid ).
% R.residuation_axioms
thf(fact_246_con__def,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
= ( ( paths_in_Resid_a @ resid @ T @ U )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% con_def
thf(fact_247_conE,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( paths_in_Resid_a @ resid @ T @ U )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% conE
thf(fact_248_null__eqI,axiom,
! [N: list_a] :
( ! [T3: list_a] :
( ( ( paths_in_Resid_a @ resid @ N @ T3 )
= N )
& ( ( paths_in_Resid_a @ resid @ T3 @ N )
= N ) )
=> ( N
= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% null_eqI
thf(fact_249_cube__ax,axiom,
! [V: list_a,T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ T ) @ ( paths_in_Resid_a @ resid @ U @ T ) )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ T ) @ ( paths_in_Resid_a @ resid @ U @ T ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ U ) @ ( paths_in_Resid_a @ resid @ T @ U ) ) ) ) ).
% cube_ax
thf(fact_250_con__sym__ax,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ U @ T )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% con_sym_ax
thf(fact_251_con__imp__arr__resid,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ) ).
% con_imp_arr_resid
thf(fact_252_resid__reflects__con,axiom,
! [T: list_a,V: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ V )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U @ V )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ V ) @ ( paths_in_Resid_a @ resid @ U @ V ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ) ) ).
% resid_reflects_con
thf(fact_253_con__sym,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U @ T ) ) ).
% con_sym
thf(fact_254_null__char,axiom,
( ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) )
= nil_a ) ).
% null_char
thf(fact_255_con__char,axiom,
! [T4: list_a,U2: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U2 )
= ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a ) ) ).
% con_char
thf(fact_256_sources__are__con,axiom,
! [A: list_a,T: list_a,A3: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ( member_list_a @ A3 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ A3 ) ) ) ).
% sources_are_con
thf(fact_257_Srcs__are__con,axiom,
! [A: a,T4: list_a,A3: a] :
( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( ( member_a @ A3 @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( con_a @ resid @ A @ A3 ) ) ) ).
% Srcs_are_con
thf(fact_258_Con__imp__eq__Srcs,axiom,
! [T4: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ U2 ) ) ) ).
% Con_imp_eq_Srcs
thf(fact_259_Srcs_Osimps_I1_J,axiom,
( ( paths_in_Srcs_a @ resid @ nil_a )
= bot_bot_set_a ) ).
% Srcs.simps(1)
thf(fact_260_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_261_conI,axiom,
! [T: list_a,U: list_a] :
( ( ( paths_in_Resid_a @ resid @ T @ U )
!= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ).
% conI
thf(fact_262_null__is__zero_I1_J,axiom,
! [T: list_a] :
( ( paths_in_Resid_a @ resid @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) @ T )
= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ).
% null_is_zero(1)
thf(fact_263_null__is__zero_I2_J,axiom,
! [T: list_a] :
( ( paths_in_Resid_a @ resid @ T @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) )
= ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) ).
% null_is_zero(2)
thf(fact_264_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_265_conI_092_060_094sub_062P,axiom,
! [T4: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U2 ) ) ).
% conI\<^sub>P
thf(fact_266_Srcs__append,axiom,
! [T4: list_a,U2: list_a] :
( ( T4 != nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( append_a @ T4 @ U2 ) )
= ( paths_in_Srcs_a @ resid @ T4 ) ) ) ).
% Srcs_append
thf(fact_267_Srcs__Resid,axiom,
! [T4: list_a,U2: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) )
= ( paths_in_Trgs_a @ resid @ U2 ) ) ) ).
% Srcs_Resid
thf(fact_268_paths__in__rts_OSrcs__simp_092_060_094sub_062P,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 )
= ( sources_list_a @ Resid @ ( hd_list_a @ T4 ) ) ) ) ) ).
% paths_in_rts.Srcs_simp\<^sub>P
thf(fact_269_paths__in__rts_OSrcs__simp_092_060_094sub_062P,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 )
= ( sources_a @ Resid @ ( hd_a @ T4 ) ) ) ) ) ).
% paths_in_rts.Srcs_simp\<^sub>P
thf(fact_270_rts_Osources_Ocong,axiom,
sources_list_a = sources_list_a ).
% rts.sources.cong
thf(fact_271_rts_Osources_Ocong,axiom,
sources_a = sources_a ).
% rts.sources.cong
thf(fact_272_rts_Ojoinable_Ocong,axiom,
joinable_a = joinable_a ).
% rts.joinable.cong
thf(fact_273_rts_Ojoinable_Ocong,axiom,
joinable_list_a = joinable_list_a ).
% rts.joinable.cong
thf(fact_274_paths__in__rts_OSrcs_Ocong,axiom,
paths_in_Srcs_a = paths_in_Srcs_a ).
% paths_in_rts.Srcs.cong
thf(fact_275_paths__in__rts_OSrcs_Oelims,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a,Y2: set_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_in_Srcs_list_a @ Resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_list_a )
=> ( Y2 != bot_bot_set_list_a ) )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( Y2
!= ( sources_list_a @ Resid @ T3 ) ) )
=> ~ ! [T3: list_a] :
( ? [V2: list_a,Va: list_list_a] :
( X2
= ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ( Y2
!= ( sources_list_a @ Resid @ T3 ) ) ) ) ) ) ) ).
% paths_in_rts.Srcs.elims
thf(fact_276_paths__in__rts_OSrcs_Oelims,axiom,
! [Resid: a > a > a,X2: list_a,Y2: set_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Srcs_a @ Resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> ( Y2 != bot_bot_set_a ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y2
!= ( sources_a @ Resid @ T3 ) ) )
=> ~ ! [T3: a] :
( ? [V2: a,Va: list_a] :
( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y2
!= ( sources_a @ Resid @ T3 ) ) ) ) ) ) ) ).
% paths_in_rts.Srcs.elims
thf(fact_277_paths__in__rts_OSrcs_Osimps_I3_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,V: list_a,Va2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ V @ Va2 ) ) )
= ( sources_list_a @ Resid @ T ) ) ) ).
% paths_in_rts.Srcs.simps(3)
thf(fact_278_paths__in__rts_OSrcs_Osimps_I3_J,axiom,
! [Resid: a > a > a,T: a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Srcs_a @ Resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) )
= ( sources_a @ Resid @ T ) ) ) ).
% paths_in_rts.Srcs.simps(3)
thf(fact_279_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_280_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_281_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_282_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_283_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_284_paths__in__rts_OSrcs_Osimps_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) )
= ( sources_list_a @ Resid @ T ) ) ) ).
% paths_in_rts.Srcs.simps(2)
thf(fact_285_paths__in__rts_OSrcs_Osimps_I2_J,axiom,
! [Resid: a > a > a,T: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Srcs_a @ Resid @ ( cons_a @ T @ nil_a ) )
= ( sources_a @ Resid @ T ) ) ) ).
% paths_in_rts.Srcs.simps(2)
thf(fact_286_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_287_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_288_paths__in__rts_OSrcs__are__con,axiom,
! [Resid: list_a > list_a > list_a,A: list_a,T4: list_list_a,A3: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( member_list_a @ A @ ( paths_in_Srcs_list_a @ Resid @ T4 ) )
=> ( ( member_list_a @ A3 @ ( paths_in_Srcs_list_a @ Resid @ T4 ) )
=> ( con_list_a @ Resid @ A @ A3 ) ) ) ) ).
% paths_in_rts.Srcs_are_con
thf(fact_289_paths__in__rts_OSrcs__are__con,axiom,
! [Resid: a > a > a,A: a,T4: list_a,A3: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ Resid @ T4 ) )
=> ( ( member_a @ A3 @ ( paths_in_Srcs_a @ Resid @ T4 ) )
=> ( con_a @ Resid @ A @ A3 ) ) ) ) ).
% paths_in_rts.Srcs_are_con
thf(fact_290_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_291_paths__in__rts_OconI_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a )
=> ( con_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ U2 ) ) ) ).
% paths_in_rts.conI\<^sub>P
thf(fact_292_paths__in__rts_Ocon__char,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( con_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ U2 )
= ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a ) ) ) ).
% paths_in_rts.con_char
thf(fact_293_paths__in__rts_OCon__imp__eq__Srcs,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a )
=> ( ( paths_in_Srcs_a @ Resid @ T4 )
= ( paths_in_Srcs_a @ Resid @ U2 ) ) ) ) ).
% paths_in_rts.Con_imp_eq_Srcs
thf(fact_294_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_295_paths__in__rts_OSrcs__append,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( paths_in_Srcs_a @ Resid @ ( append_a @ T4 @ U2 ) )
= ( paths_in_Srcs_a @ Resid @ T4 ) ) ) ) ).
% paths_in_rts.Srcs_append
thf(fact_296_paths__in__rts_Olength__Residx1,axiom,
! [Resid: a > a > a,T4: list_a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ord_less_eq_nat @ ( size_size_list_a @ ( paths_in_Residx1_a @ Resid @ T4 @ U ) ) @ ( size_size_list_a @ T4 ) ) ) ).
% paths_in_rts.length_Residx1
thf(fact_297_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_298_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_299_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_300_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_301_paths__in__rts_OSrcs__Resid,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a )
=> ( ( paths_in_Srcs_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ U2 ) )
= ( paths_in_Trgs_a @ Resid @ U2 ) ) ) ) ).
% paths_in_rts.Srcs_Resid
thf(fact_302_paths__in__rts_Osources__cons,axiom,
! [Resid: a > a > a,T: a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ ( cons_a @ T @ T4 ) )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ Resid ) @ ( cons_a @ T @ T4 ) )
= ( sources_list_a @ ( paths_in_Resid_a @ Resid ) @ ( cons_a @ T @ nil_a ) ) ) ) ) ).
% paths_in_rts.sources_cons
thf(fact_303_paths__in__rts_OResid__Arr__Src,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,A: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Arr_list_a @ Resid @ T4 )
=> ( ( member_list_a @ A @ ( paths_in_Srcs_list_a @ Resid @ T4 ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ A @ nil_list_a ) )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_Arr_Src
thf(fact_304_paths__in__rts_OResid__Arr__Src,axiom,
! [Resid: a > a > a,T4: list_a,A: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( 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 ) ) ) ) ).
% paths_in_rts.Resid_Arr_Src
thf(fact_305_paths__in__rts_OSrcs__Resid__single__Arr,axiom,
! [Resid: a > a > a,U: a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ U @ nil_a ) @ T4 )
!= nil_a )
=> ( ( paths_in_Srcs_a @ Resid @ ( paths_in_Resid_a @ Resid @ ( cons_a @ U @ nil_a ) @ T4 ) )
= ( paths_in_Trgs_a @ Resid @ T4 ) ) ) ) ).
% paths_in_rts.Srcs_Resid_single_Arr
thf(fact_306_paths__in__rts_OArr__appendE_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ ( append_a @ T4 @ U2 ) )
=> ( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ~ ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( ( paths_in_Arr_a @ Resid @ U2 )
=> ( ( paths_in_Trgs_a @ Resid @ T4 )
!= ( paths_in_Srcs_a @ Resid @ U2 ) ) ) ) ) ) ) ) ).
% paths_in_rts.Arr_appendE\<^sub>P
thf(fact_307_paths__in__rts_OResid1x__as__Resid_H,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ U2 )
!= nil_list_a )
=> ( ( paths_1777230443808135851list_a @ Resid @ T @ U2 )
= ( hd_list_a @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ U2 ) ) ) )
& ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ U2 )
= nil_list_a )
=> ( ( paths_1777230443808135851list_a @ Resid @ T @ U2 )
= ( partial_null_list_a @ Resid ) ) ) ) ) ).
% paths_in_rts.Resid1x_as_Resid'
thf(fact_308_paths__in__rts_OResid1x__as__Resid_H,axiom,
! [Resid: a > a > a,T: a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid1x_a @ Resid @ T @ U2 )
= ( hd_a @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ U2 ) ) ) )
& ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T @ nil_a ) @ U2 )
= nil_a )
=> ( ( paths_in_Resid1x_a @ Resid @ T @ U2 )
= ( partial_null_a @ Resid ) ) ) ) ) ).
% paths_in_rts.Resid1x_as_Resid'
thf(fact_309_Resid__cons__ind,axiom,
! [T4: list_a,U2: list_a,N: nat] :
( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_size_list_a @ T4 ) @ ( size_size_list_a @ U2 ) ) @ N )
=> ( ! [T2: a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T4 ) @ U2 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U2 )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U2 @ ( cons_a @ T2 @ nil_a ) ) )
!= nil_a ) ) )
& ! [U4: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U4 @ U2 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U4 @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U4 @ nil_a ) ) @ U2 )
!= nil_a ) ) )
& ! [T2: a] :
( ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T4 ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ T4 ) @ U2 )
= ( append_a @ ( paths_in_Resid_a @ resid @ ( cons_a @ T2 @ nil_a ) @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ ( paths_in_Resid_a @ resid @ U2 @ ( cons_a @ T2 @ nil_a ) ) ) ) ) )
& ! [U4: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U4 @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U4 @ U2 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U4 @ nil_a ) ) @ U2 ) ) ) ) ) ) ) ).
% Resid_cons_ind
thf(fact_310_Arr__append__iff_092_060_094sub_062P,axiom,
! [T4: list_a,U2: list_a] :
( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( paths_in_Arr_a @ resid @ ( append_a @ T4 @ U2 ) )
= ( ( paths_in_Arr_a @ resid @ T4 )
& ( paths_in_Arr_a @ resid @ U2 )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U2 ) ) ) ) ) ) ).
% Arr_append_iff\<^sub>P
thf(fact_311_seq__char,axiom,
! [T4: list_a,U2: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U2 )
= ( ( paths_in_Arr_a @ resid @ T4 )
& ( paths_in_Arr_a @ resid @ U2 )
& ( ( paths_in_Trgs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ U2 ) ) ) ) ).
% seq_char
thf(fact_312_length__Resid__ind,axiom,
! [T4: list_a,U2: list_a,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_size_list_a @ T4 ) @ ( size_size_list_a @ U2 ) ) @ N )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
=> ( ( size_size_list_a @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) )
= ( size_size_list_a @ T4 ) ) ) ) ).
% length_Resid_ind
thf(fact_313_Cube__ind,axiom,
! [T4: list_a,U2: list_a,V3: list_a,N: nat] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= 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 @ U2 ) ) @ ( size_size_list_a @ V3 ) ) @ N )
=> ( ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V3 @ T4 ) @ ( paths_in_Resid_a @ resid @ U2 @ T4 ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) )
!= nil_a ) )
& ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V3 @ T4 ) @ ( paths_in_Resid_a @ resid @ U2 @ T4 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V3 @ T4 ) @ ( paths_in_Resid_a @ resid @ U2 @ T4 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V3 @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) ) ) ) ) ) ) ) ).
% Cube_ind
thf(fact_314_Con__sym__ind,axiom,
! [T4: list_a,U2: list_a,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_size_list_a @ T4 ) @ ( size_size_list_a @ U2 ) ) @ N )
=> ( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ U2 @ T4 )
!= nil_a ) ) ) ).
% Con_sym_ind
thf(fact_315_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_316_R_Osource__is__ide,axiom,
! [A: a,T: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ide_a @ resid @ A ) ) ).
% R.source_is_ide
thf(fact_317_R_Osources__are__cong,axiom,
! [A: a,T: a,A3: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( ( ide_a @ resid @ ( resid @ A @ A3 ) )
& ( ide_a @ resid @ ( resid @ A3 @ A ) ) ) ) ) ).
% R.sources_are_cong
thf(fact_318_R_Osources__cong__closed,axiom,
! [A: a,T: a,A3: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( ( ide_a @ resid @ ( resid @ A @ A3 ) )
& ( ide_a @ resid @ ( resid @ A3 @ A ) ) )
=> ( member_a @ A3 @ ( sources_a @ resid @ T ) ) ) ) ).
% R.sources_cong_closed
thf(fact_319_R_Ocong__symmetric,axiom,
! [T: a,U: a] :
( ( ( ide_a @ resid @ ( resid @ T @ U ) )
& ( ide_a @ resid @ ( resid @ U @ T ) ) )
=> ( ( ide_a @ resid @ ( resid @ U @ T ) )
& ( ide_a @ resid @ ( resid @ T @ U ) ) ) ) ).
% R.cong_symmetric
thf(fact_320_R_Ocong__transitive,axiom,
! [T: a,U: a,V: a] :
( ( ( ide_a @ resid @ ( resid @ T @ U ) )
& ( ide_a @ resid @ ( resid @ U @ T ) ) )
=> ( ( ( ide_a @ resid @ ( resid @ U @ V ) )
& ( ide_a @ resid @ ( resid @ V @ U ) ) )
=> ( ( ide_a @ resid @ ( resid @ T @ V ) )
& ( ide_a @ resid @ ( resid @ V @ T ) ) ) ) ) ).
% R.cong_transitive
thf(fact_321_R_Oide__backward__stable,axiom,
! [A: a,T: a] :
( ( ide_a @ resid @ A )
=> ( ( ide_a @ resid @ ( resid @ T @ A ) )
=> ( ide_a @ resid @ T ) ) ) ).
% R.ide_backward_stable
thf(fact_322_R_Oprfx__transitive,axiom,
! [T: a,U: a,V: a] :
( ( ide_a @ resid @ ( resid @ T @ U ) )
=> ( ( ide_a @ resid @ ( resid @ U @ V ) )
=> ( ide_a @ resid @ ( resid @ T @ V ) ) ) ) ).
% R.prfx_transitive
thf(fact_323_R_Osources__are__con,axiom,
! [A: a,T: a,A3: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( con_a @ resid @ A @ A3 ) ) ) ).
% R.sources_are_con
thf(fact_324_joinable__implies__con,axiom,
! [T: list_a,U: list_a] :
( ( joinable_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ).
% joinable_implies_con
thf(fact_325_Srcs_Osimps_I3_J,axiom,
! [T: a,V: a,Va2: list_a] :
( ( paths_in_Srcs_a @ resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) )
= ( sources_a @ resid @ T ) ) ).
% Srcs.simps(3)
thf(fact_326_R_Oin__sourcesE,axiom,
! [A: a,T: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ~ ( ( ide_a @ resid @ A )
=> ~ ( con_a @ resid @ T @ A ) ) ) ).
% R.in_sourcesE
thf(fact_327_R_Osources__con__closed,axiom,
! [A: a,T: a,A3: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( ide_a @ resid @ A3 )
=> ( ( con_a @ resid @ A @ A3 )
=> ( member_a @ A3 @ ( sources_a @ resid @ T ) ) ) ) ) ).
% R.sources_con_closed
thf(fact_328_R_Ocong__subst__left_I2_J,axiom,
! [T: a,T6: a,U: a] :
( ( ( ide_a @ resid @ ( resid @ T @ T6 ) )
& ( ide_a @ resid @ ( resid @ T6 @ T ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ T @ U ) @ ( resid @ T6 @ U ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ T6 @ U ) @ ( resid @ T @ U ) ) ) ) ) ) ).
% R.cong_subst_left(2)
thf(fact_329_R_Ocong__subst__left_I1_J,axiom,
! [T: a,T6: a,U: a] :
( ( ( ide_a @ resid @ ( resid @ T @ T6 ) )
& ( ide_a @ resid @ ( resid @ T6 @ T ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ T6 @ U ) ) ) ).
% R.cong_subst_left(1)
thf(fact_330_R_Ocong__subst__right_I2_J,axiom,
! [U: a,U5: a,T: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U5 ) )
& ( ide_a @ resid @ ( resid @ U5 @ U ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ T @ U ) @ ( resid @ T @ U5 ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ T @ U5 ) @ ( resid @ T @ U ) ) ) ) ) ) ).
% R.cong_subst_right(2)
thf(fact_331_R_Ocong__subst__right_I1_J,axiom,
! [U: a,U5: a,T: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U5 ) )
& ( ide_a @ resid @ ( resid @ U5 @ U ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ T @ U5 ) ) ) ).
% R.cong_subst_right(1)
thf(fact_332_R_Ocon__imp__coinitial__ax,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ? [A4: a] :
( ( ide_a @ resid @ A4 )
& ( con_a @ resid @ A4 @ T )
& ( con_a @ resid @ A4 @ U ) ) ) ).
% R.con_imp_coinitial_ax
thf(fact_333_R_Ocon__target,axiom,
! [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 ) ) ) ) ).
% R.con_target
thf(fact_334_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_335_R_OideE,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ~ ( ( con_a @ resid @ A @ A )
=> ( ( resid @ A @ A )
!= A ) ) ) ).
% R.ideE
thf(fact_336_R_Oide__def,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
= ( ( con_a @ resid @ A @ A )
& ( ( resid @ A @ A )
= A ) ) ) ).
% R.ide_def
thf(fact_337_R_Oide__imp__con__iff__cong,axiom,
! [T: a,U: a] :
( ( ide_a @ resid @ T )
=> ( ( ide_a @ resid @ U )
=> ( ( con_a @ resid @ T @ U )
= ( ( ide_a @ resid @ ( resid @ T @ U ) )
& ( ide_a @ resid @ ( resid @ U @ T ) ) ) ) ) ) ).
% R.ide_imp_con_iff_cong
thf(fact_338_R_Oprfx__implies__con,axiom,
! [T: a,U: a] :
( ( ide_a @ resid @ ( resid @ T @ U ) )
=> ( con_a @ resid @ T @ U ) ) ).
% R.prfx_implies_con
thf(fact_339_R_Oresid__arr__ide,axiom,
! [A: a,T: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ T @ A )
=> ( ( resid @ T @ A )
= T ) ) ) ).
% R.resid_arr_ide
thf(fact_340_R_Oresid__ide__arr,axiom,
! [A: a,T: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ A @ T )
=> ( ide_a @ resid @ ( resid @ A @ T ) ) ) ) ).
% R.resid_ide_arr
thf(fact_341_Srcs_Osimps_I2_J,axiom,
! [T: a] :
( ( paths_in_Srcs_a @ resid @ ( cons_a @ T @ nil_a ) )
= ( sources_a @ resid @ T ) ) ).
% Srcs.simps(2)
thf(fact_342_Srcs__con__closed,axiom,
! [A: a,T4: list_a,A3: a] :
( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( ( ide_a @ resid @ A3 )
=> ( ( con_a @ resid @ A @ A3 )
=> ( member_a @ A3 @ ( paths_in_Srcs_a @ resid @ T4 ) ) ) ) ) ).
% Srcs_con_closed
thf(fact_343_Trgs__con__closed,axiom,
! [B: a,T4: list_a,B2: a] :
( ( member_a @ B @ ( paths_in_Trgs_a @ resid @ T4 ) )
=> ( ( ide_a @ resid @ B2 )
=> ( ( con_a @ resid @ B @ B2 )
=> ( member_a @ B2 @ ( paths_in_Trgs_a @ resid @ T4 ) ) ) ) ) ).
% Trgs_con_closed
thf(fact_344_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_345_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_346_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_347_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_348_Srcs_Oelims,axiom,
! [X2: list_a,Y2: set_a] :
( ( ( paths_in_Srcs_a @ resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> ( Y2 != bot_bot_set_a ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y2
!= ( sources_a @ resid @ T3 ) ) )
=> ~ ! [T3: a] :
( ? [V2: a,Va: list_a] :
( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y2
!= ( sources_a @ resid @ T3 ) ) ) ) ) ) ).
% Srcs.elims
thf(fact_349_seq__implies__Trgs__eq__Srcs,axiom,
! [T4: list_a,U2: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Arr_a @ resid @ U2 )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U2 ) )
=> ( ( paths_in_Trgs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ U2 ) ) ) ) ) ).
% seq_implies_Trgs_eq_Srcs
thf(fact_350_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_351_R_Oin__sourcesI,axiom,
! [A: a,T: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ T @ A )
=> ( member_a @ A @ ( sources_a @ resid @ T ) ) ) ) ).
% R.in_sourcesI
thf(fact_352_R_OideI,axiom,
! [A: a] :
( ( con_a @ resid @ A @ A )
=> ( ( ( resid @ A @ A )
= A )
=> ( ide_a @ resid @ A ) ) ) ).
% R.ideI
thf(fact_353_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_354_Arr__appendI_092_060_094sub_062P,axiom,
! [T4: list_a,U2: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Arr_a @ resid @ U2 )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U2 ) )
=> ( paths_in_Arr_a @ resid @ ( append_a @ T4 @ U2 ) ) ) ) ) ).
% Arr_appendI\<^sub>P
thf(fact_355_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_356_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_357_residuation_Oide_Ocong,axiom,
ide_a = ide_a ).
% residuation.ide.cong
thf(fact_358_residuation_Oide_Ocong,axiom,
ide_list_a = ide_list_a ).
% residuation.ide.cong
thf(fact_359_rts_Oseq_Ocong,axiom,
seq_list_a = seq_list_a ).
% rts.seq.cong
thf(fact_360_rts_Oseq_Ocong,axiom,
seq_a = seq_a ).
% rts.seq.cong
thf(fact_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_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_369_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_370_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_371_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: list_a > list_a > list_a,A: list_a,T4: list_list_a,A3: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( member_list_a @ A @ ( paths_in_Srcs_list_a @ Resid @ T4 ) )
=> ( ( ide_list_a @ Resid @ A3 )
=> ( ( con_list_a @ Resid @ A @ A3 )
=> ( member_list_a @ A3 @ ( paths_in_Srcs_list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_372_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: a > a > a,A: a,T4: list_a,A3: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ Resid @ T4 ) )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( con_a @ Resid @ A @ A3 )
=> ( member_a @ A3 @ ( paths_in_Srcs_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_373_paths__in__rts_OTrgs__con__closed,axiom,
! [Resid: list_a > list_a > list_a,B: list_a,T4: list_list_a,B2: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( member_list_a @ B @ ( paths_in_Trgs_list_a @ Resid @ T4 ) )
=> ( ( ide_list_a @ Resid @ B2 )
=> ( ( con_list_a @ Resid @ B @ B2 )
=> ( member_list_a @ B2 @ ( paths_in_Trgs_list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Trgs_con_closed
thf(fact_374_paths__in__rts_OTrgs__con__closed,axiom,
! [Resid: a > a > a,B: a,T4: list_a,B2: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( member_a @ B @ ( paths_in_Trgs_a @ Resid @ T4 ) )
=> ( ( ide_a @ Resid @ B2 )
=> ( ( con_a @ Resid @ B @ B2 )
=> ( member_a @ B2 @ ( paths_in_Trgs_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Trgs_con_closed
thf(fact_375_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_376_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_377_paths__in__rts_OResid__single__ide_I2_J,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_8620460302779588466list_a @ Resid @ ( cons_list_a @ A @ nil_list_a ) @ T4 )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ A @ nil_list_a ) )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_single_ide(2)
thf(fact_378_paths__in__rts_OResid__single__ide_I2_J,axiom,
! [Resid: a > a > a,A: a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( 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 ) ) ) ) ).
% paths_in_rts.Resid_single_ide(2)
thf(fact_379_paths__in__rts_Oseq__implies__Trgs__eq__Srcs,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( ( paths_in_Arr_a @ Resid @ U2 )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ Resid @ T4 ) @ ( paths_in_Srcs_a @ Resid @ U2 ) )
=> ( ( paths_in_Trgs_a @ Resid @ T4 )
= ( paths_in_Srcs_a @ Resid @ U2 ) ) ) ) ) ) ).
% paths_in_rts.seq_implies_Trgs_eq_Srcs
thf(fact_380_paths__in__rts_OCube__ind,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,V3: list_a,N: nat] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= 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 @ U2 ) ) @ ( size_size_list_a @ V3 ) ) @ N )
=> ( ( ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ V3 @ T4 ) @ ( paths_in_Resid_a @ Resid @ U2 @ T4 ) )
!= nil_a )
= ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ V3 @ U2 ) @ ( paths_in_Resid_a @ Resid @ T4 @ U2 ) )
!= nil_a ) )
& ( ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ V3 @ T4 ) @ ( paths_in_Resid_a @ Resid @ U2 @ T4 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ V3 @ T4 ) @ ( paths_in_Resid_a @ Resid @ U2 @ T4 ) )
= ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ V3 @ U2 ) @ ( paths_in_Resid_a @ Resid @ T4 @ U2 ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Cube_ind
thf(fact_381_paths__in__rts_OCon__sym__ind,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,N: nat] :
( ( paths_in_rts_a @ Resid )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_size_list_a @ T4 ) @ ( size_size_list_a @ U2 ) ) @ N )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a )
= ( ( paths_in_Resid_a @ Resid @ U2 @ T4 )
!= nil_a ) ) ) ) ).
% paths_in_rts.Con_sym_ind
thf(fact_382_paths__in__rts_Olength__Resid__ind,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,N: nat] :
( ( paths_in_rts_a @ Resid )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_size_list_a @ T4 ) @ ( size_size_list_a @ U2 ) ) @ N )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ U2 )
!= nil_a )
=> ( ( size_size_list_a @ ( paths_in_Resid_a @ Resid @ T4 @ U2 ) )
= ( size_size_list_a @ T4 ) ) ) ) ) ).
% paths_in_rts.length_Resid_ind
thf(fact_383_paths__in__rts_OCon__single__ideI_I2_J,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_in_Arr_list_a @ Resid @ T4 )
=> ( ( member_list_a @ A @ ( paths_in_Srcs_list_a @ Resid @ T4 ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ A @ nil_list_a ) )
!= nil_list_a ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(2)
thf(fact_384_paths__in__rts_OCon__single__ideI_I2_J,axiom,
! [Resid: a > a > a,A: a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( 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 ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(2)
thf(fact_385_paths__in__rts_OCon__single__ideI_I1_J,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_in_Arr_list_a @ Resid @ T4 )
=> ( ( member_list_a @ A @ ( paths_in_Srcs_list_a @ Resid @ T4 ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ A @ nil_list_a ) @ T4 )
!= nil_list_a ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(1)
thf(fact_386_paths__in__rts_OCon__single__ideI_I1_J,axiom,
! [Resid: a > a > a,A: a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( 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 ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(1)
thf(fact_387_paths__in__rts_OCon__single__ide__ind,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_8620460302779588466list_a @ Resid @ ( cons_list_a @ A @ nil_list_a ) @ T4 )
!= nil_list_a )
= ( ( paths_in_Arr_list_a @ Resid @ T4 )
& ( member_list_a @ A @ ( paths_in_Srcs_list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Con_single_ide_ind
thf(fact_388_paths__in__rts_OCon__single__ide__ind,axiom,
! [Resid: a > a > a,A: a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( 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 ) ) ) ) ) ) ).
% paths_in_rts.Con_single_ide_ind
thf(fact_389_paths__in__rts_OArr__appendI_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( ( paths_in_Arr_a @ Resid @ U2 )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ Resid @ T4 ) @ ( paths_in_Srcs_a @ Resid @ U2 ) )
=> ( paths_in_Arr_a @ Resid @ ( append_a @ T4 @ U2 ) ) ) ) ) ) ).
% paths_in_rts.Arr_appendI\<^sub>P
thf(fact_390_paths__in__rts_Oseq__char,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ U2 )
= ( ( paths_in_Arr_a @ Resid @ T4 )
& ( paths_in_Arr_a @ Resid @ U2 )
& ( ( paths_in_Trgs_a @ Resid @ T4 )
= ( paths_in_Srcs_a @ Resid @ U2 ) ) ) ) ) ).
% paths_in_rts.seq_char
thf(fact_391_paths__in__rts_OArr__append__iff_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( paths_in_Arr_a @ Resid @ ( append_a @ T4 @ U2 ) )
= ( ( paths_in_Arr_a @ Resid @ T4 )
& ( paths_in_Arr_a @ Resid @ U2 )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ Resid @ T4 ) @ ( paths_in_Srcs_a @ Resid @ U2 ) ) ) ) ) ) ) ).
% paths_in_rts.Arr_append_iff\<^sub>P
thf(fact_392_paths__in__rts_OResid__cons__ind,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a,N: nat] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_size_list_a @ T4 ) @ ( size_size_list_a @ U2 ) ) @ N )
=> ( ! [T2: a] :
( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T2 @ T4 ) @ U2 )
!= nil_a )
= ( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T2 @ nil_a ) @ U2 )
!= nil_a )
& ( ( paths_in_Resid_a @ Resid @ T4 @ ( paths_in_Resid_a @ Resid @ U2 @ ( cons_a @ T2 @ nil_a ) ) )
!= nil_a ) ) )
& ! [U4: a] :
( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U4 @ U2 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U4 @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U4 @ nil_a ) ) @ U2 )
!= nil_a ) ) )
& ! [T2: a] :
( ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T2 @ T4 ) @ U2 )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ T2 @ T4 ) @ U2 )
= ( append_a @ ( paths_in_Resid_a @ Resid @ ( cons_a @ T2 @ nil_a ) @ U2 ) @ ( paths_in_Resid_a @ Resid @ T4 @ ( paths_in_Resid_a @ Resid @ U2 @ ( cons_a @ T2 @ nil_a ) ) ) ) ) )
& ! [U4: a] :
( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U4 @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U4 @ U2 ) )
= ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U4 @ nil_a ) ) @ U2 ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_cons_ind
thf(fact_393_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_394_R_Ocong__implies__coterminal,axiom,
! [U: a,U5: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U5 ) )
& ( ide_a @ resid @ ( resid @ U5 @ U ) ) )
=> ( coterminal_a @ resid @ U @ U5 ) ) ).
% R.cong_implies_coterminal
thf(fact_395_seq__char_H,axiom,
! [T4: list_a,U2: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U2 )
= ( ( paths_in_Arr_a @ resid @ T4 )
& ( paths_in_Arr_a @ resid @ U2 )
& ( ( inf_inf_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U2 ) )
!= bot_bot_set_a ) ) ) ).
% seq_char'
thf(fact_396_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_397_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_398_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_399_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_400_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_401_prfx__transitive,axiom,
! [T: list_a,U: list_a,V: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ V ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ V ) ) ) ) ).
% prfx_transitive
thf(fact_402_ide__backward__stable,axiom,
! [A: list_a,T: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ A ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ).
% ide_backward_stable
thf(fact_403_cong__transitive,axiom,
! [T: list_a,U: list_a,V: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ T ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ V ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ U ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ V ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ T ) ) ) ) ) ).
% cong_transitive
thf(fact_404_cong__symmetric,axiom,
! [T: list_a,U: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ T ) ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ T ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) ) ) ) ).
% cong_symmetric
thf(fact_405_R_Ocong__respects__seq,axiom,
! [T: a,U: a,T6: a,U5: a] :
( ( seq_a @ resid @ T @ U )
=> ( ( ( ide_a @ resid @ ( resid @ T @ T6 ) )
& ( ide_a @ resid @ ( resid @ T6 @ T ) ) )
=> ( ( ( ide_a @ resid @ ( resid @ U @ U5 ) )
& ( ide_a @ resid @ ( resid @ U5 @ U ) ) )
=> ( seq_a @ resid @ T6 @ U5 ) ) ) ) ).
% R.cong_respects_seq
thf(fact_406_resid__ide__arr,axiom,
! [A: list_a,T: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ T )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A @ T ) ) ) ) ).
% resid_ide_arr
thf(fact_407_resid__arr__ide,axiom,
! [A: list_a,T: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ A )
=> ( ( paths_in_Resid_a @ resid @ T @ A )
= T ) ) ) ).
% resid_arr_ide
thf(fact_408_prfx__implies__con,axiom,
! [T: list_a,U: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ).
% prfx_implies_con
thf(fact_409_ide__imp__con__iff__cong,axiom,
! [T: list_a,U: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ U )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
= ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ T ) ) ) ) ) ) ).
% ide_imp_con_iff_cong
thf(fact_410_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_411_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_412_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_413_con__target,axiom,
! [T: list_a,U: list_a,V: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U @ V )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) @ ( paths_in_Resid_a @ resid @ V @ U ) ) ) ) ).
% con_target
thf(fact_414_con__imp__coinitial__ax,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ? [A4: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A4 )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A4 @ T )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A4 @ U ) ) ) ).
% con_imp_coinitial_ax
thf(fact_415_cong__subst__right_I1_J,axiom,
! [U: list_a,U5: list_a,T: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ U5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U5 @ U ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U5 ) ) ) ).
% cong_subst_right(1)
thf(fact_416_cong__subst__right_I2_J,axiom,
! [U: list_a,U5: list_a,T: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ U5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U5 @ U ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) @ ( paths_in_Resid_a @ resid @ T @ U5 ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U5 ) @ ( paths_in_Resid_a @ resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_417_cong__subst__left_I1_J,axiom,
! [T: list_a,T6: list_a,U: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ T6 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T6 @ T ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T6 @ U ) ) ) ).
% cong_subst_left(1)
thf(fact_418_cong__subst__left_I2_J,axiom,
! [T: list_a,T6: list_a,U: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ T6 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T6 @ T ) ) )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T @ U ) @ ( paths_in_Resid_a @ resid @ T6 @ U ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T6 @ U ) @ ( paths_in_Resid_a @ resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_419_sources__cong__closed,axiom,
! [A: list_a,T: list_a,A3: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A @ A3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A3 @ A ) ) )
=> ( member_list_a @ A3 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ) ).
% sources_cong_closed
thf(fact_420_sources__are__cong,axiom,
! [A: list_a,T: list_a,A3: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ( member_list_a @ A3 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A @ A3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A3 @ A ) ) ) ) ) ).
% sources_are_cong
thf(fact_421_source__is__ide,axiom,
! [A: list_a,T: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A ) ) ).
% source_is_ide
thf(fact_422_cong__respects__seq,axiom,
! [T: list_a,U: list_a,T6: list_a,U5: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ T6 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T6 @ T ) ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ U5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U5 @ U ) ) )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T6 @ U5 ) ) ) ) ).
% cong_respects_seq
thf(fact_423_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_424_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_425_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_426_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_427_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_428_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_429_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_430_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_431_subset__antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_432_subsetI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ( member_list_a @ X @ B3 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_433_subsetI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_a @ X @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_434_sources__con__closed,axiom,
! [A: list_a,T: list_a,A3: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A3 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ A @ A3 )
=> ( member_list_a @ A3 @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ) ) ).
% sources_con_closed
thf(fact_435_in__sourcesE,axiom,
! [A: list_a,T: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ~ ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ~ ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ A ) ) ) ).
% in_sourcesE
thf(fact_436_IntI,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B3 )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_437_IntI,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( ( member_list_a @ C @ B3 )
=> ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_438_Int__iff,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_439_Int__iff,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
= ( ( member_list_a @ C @ A2 )
& ( member_list_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_440_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_441_R_Osources__eqI,axiom,
! [T: a,T6: a] :
( ( ( inf_inf_set_a @ ( sources_a @ resid @ T ) @ ( sources_a @ resid @ T6 ) )
!= bot_bot_set_a )
=> ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ T6 ) ) ) ).
% R.sources_eqI
thf(fact_442_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_443_R_Ocon__imp__common__source,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( ( inf_inf_set_a @ ( sources_a @ resid @ T ) @ ( sources_a @ resid @ U ) )
!= bot_bot_set_a ) ) ).
% R.con_imp_common_source
thf(fact_444_Int__subset__iff,axiom,
! [C2: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B3 ) )
= ( ( ord_le8861187494160871172list_a @ C2 @ A2 )
& ( ord_le8861187494160871172list_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_445_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_446_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_447_in__sourcesI,axiom,
! [A: list_a,T: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ A )
=> ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ) ).
% in_sourcesI
thf(fact_448_seqI_092_060_094sub_062P,axiom,
! [T4: list_a,U2: list_a] :
( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( paths_in_Arr_a @ resid @ U2 )
=> ( ( ( inf_inf_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U2 ) )
!= bot_bot_set_a )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T4 @ U2 ) ) ) ) ).
% seqI\<^sub>P
thf(fact_449_Int__mono,axiom,
! [A2: set_list_a,C2: set_list_a,B3: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ D )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ ( inf_inf_set_list_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_450_Int__mono,axiom,
! [A2: set_a,C2: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_451_Int__emptyI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ~ ( member_a @ X @ B3 ) )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_452_Int__emptyI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ~ ( member_list_a @ X @ B3 ) )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_453_Int__lower1,axiom,
! [A2: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_454_Int__lower1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_455_Int__lower2,axiom,
! [A2: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_456_Int__lower2,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_457_Int__absorb1,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_458_Int__absorb1,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_459_Int__absorb2,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_460_Int__absorb2,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_461_Int__greatest,axiom,
! [C2: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ C2 @ B3 )
=> ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_462_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_463_disjoint__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_464_disjoint__iff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ~ ( member_list_a @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_465_Int__empty__left,axiom,
! [B3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B3 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_466_Int__empty__left,axiom,
! [B3: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B3 )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_467_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_468_Int__empty__right,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_469_Int__Collect__mono,axiom,
! [A2: set_list_a,B3: set_list_a,P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B3 @ ( collect_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_470_Int__Collect__mono,axiom,
! [A2: set_a,B3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_471_disjoint__iff__not__equal,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B3 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_472_disjoint__iff__not__equal,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ! [Y3: list_a] :
( ( member_list_a @ Y3 @ B3 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_473_IntE,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_474_IntE,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ~ ( ( member_list_a @ C @ A2 )
=> ~ ( member_list_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_475_IntD1,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_476_IntD1,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ( member_list_a @ C @ A2 ) ) ).
% IntD1
thf(fact_477_IntD2,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ( member_a @ C @ B3 ) ) ).
% IntD2
thf(fact_478_IntD2,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ( member_list_a @ C @ B3 ) ) ).
% IntD2
thf(fact_479_Int__assoc,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_480_Int__assoc,axiom,
! [A2: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ C2 )
= ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_481_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_482_Int__absorb,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_483_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A5 ) ) ) ).
% Int_commute
thf(fact_484_Int__commute,axiom,
( inf_inf_set_list_a
= ( ^ [A5: set_list_a,B4: set_list_a] : ( inf_inf_set_list_a @ B4 @ A5 ) ) ) ).
% Int_commute
thf(fact_485_Int__left__absorb,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_486_Int__left__absorb,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ A2 @ B3 ) )
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_487_Int__left__commute,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C2 ) )
= ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_488_Int__left__commute,axiom,
! [A2: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ B3 @ C2 ) )
= ( inf_inf_set_list_a @ B3 @ ( inf_inf_set_list_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_489_rts_Ocoterminal_Ocong,axiom,
coterminal_a = coterminal_a ).
% rts.coterminal.cong
thf(fact_490_rts_Ocoterminal_Ocong,axiom,
coterminal_list_a = coterminal_list_a ).
% rts.coterminal.cong
thf(fact_491_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_492_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_493_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_494_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_495_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_496_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_497_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_498_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_499_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_500_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y4: list_a] :
~ ( member_list_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_501_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_502_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_503_Collect__mono__iff,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
= ( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_504_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_505_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_506_subset__trans,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_507_Collect__mono,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X: list_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_508_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_509_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_510_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B4: set_list_a] :
! [T5: list_a] :
( ( member_list_a @ T5 @ A5 )
=> ( member_list_a @ T5 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_511_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
! [T5: a] :
( ( member_a @ T5 @ A5 )
=> ( member_a @ T5 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_512_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_513_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_514_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B4: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A5 )
=> ( member_list_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_515_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_516_equalityE,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_517_subsetD,axiom,
! [A2: set_list_a,B3: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_518_subsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_519_in__mono,axiom,
! [A2: set_list_a,B3: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_520_in__mono,axiom,
! [A2: set_a,B3: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_521_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_522_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_523_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_524_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_525_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_526_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_527_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_528_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_529_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_530_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_531_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_532_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_533_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_534_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_535_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_536_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_537_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_538_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_539_paths__in__rts_OseqI_092_060_094sub_062P,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Arr_list_a @ Resid @ T4 )
=> ( ( paths_in_Arr_list_a @ Resid @ U2 )
=> ( ( ( inf_inf_set_list_a @ ( paths_in_Trgs_list_a @ Resid @ T4 ) @ ( paths_in_Srcs_list_a @ Resid @ U2 ) )
!= bot_bot_set_list_a )
=> ( seq_list_list_a @ ( paths_8620460302779588466list_a @ Resid ) @ T4 @ U2 ) ) ) ) ) ).
% paths_in_rts.seqI\<^sub>P
thf(fact_540_paths__in__rts_OseqI_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( ( paths_in_Arr_a @ Resid @ U2 )
=> ( ( ( inf_inf_set_a @ ( paths_in_Trgs_a @ Resid @ T4 ) @ ( paths_in_Srcs_a @ Resid @ U2 ) )
!= bot_bot_set_a )
=> ( seq_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ U2 ) ) ) ) ) ).
% paths_in_rts.seqI\<^sub>P
thf(fact_541_paths__in__rts_Oseq__char_H,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( seq_list_list_a @ ( paths_8620460302779588466list_a @ Resid ) @ T4 @ U2 )
= ( ( paths_in_Arr_list_a @ Resid @ T4 )
& ( paths_in_Arr_list_a @ Resid @ U2 )
& ( ( inf_inf_set_list_a @ ( paths_in_Trgs_list_a @ Resid @ T4 ) @ ( paths_in_Srcs_list_a @ Resid @ U2 ) )
!= bot_bot_set_list_a ) ) ) ) ).
% paths_in_rts.seq_char'
thf(fact_542_paths__in__rts_Oseq__char_H,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( seq_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 @ U2 )
= ( ( paths_in_Arr_a @ Resid @ T4 )
& ( paths_in_Arr_a @ Resid @ U2 )
& ( ( inf_inf_set_a @ ( paths_in_Trgs_a @ Resid @ T4 ) @ ( paths_in_Srcs_a @ Resid @ U2 ) )
!= bot_bot_set_a ) ) ) ) ).
% paths_in_rts.seq_char'
thf(fact_543_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_544_cong__implies__coterminal,axiom,
! [U: list_a,U5: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ U5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U5 @ U ) ) )
=> ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ U @ U5 ) ) ).
% cong_implies_coterminal
thf(fact_545_R_Ocoinitial__def,axiom,
! [T: a,U: a] :
( ( coinitial_a @ resid @ T @ U )
= ( ( inf_inf_set_a @ ( sources_a @ resid @ T ) @ ( sources_a @ resid @ U ) )
!= bot_bot_set_a ) ) ).
% R.coinitial_def
thf(fact_546_inf__bot__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_547_inf__bot__left,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X2 )
= bot_bot_set_list_a ) ).
% inf_bot_left
thf(fact_548_inf__bot__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_549_inf__bot__right,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% inf_bot_right
thf(fact_550_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_551_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X2 )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_552_sources__eqI,axiom,
! [T: list_a,T6: list_a] :
( ( ( inf_inf_set_list_a @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) )
!= bot_bot_set_list_a )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) ) ) ).
% sources_eqI
thf(fact_553_R_Ocoinitial__ide__are__cong,axiom,
! [A: a,A3: a] :
( ( ide_a @ resid @ A )
=> ( ( ide_a @ resid @ A3 )
=> ( ( coinitial_a @ resid @ A @ A3 )
=> ( ( ide_a @ resid @ ( resid @ A @ A3 ) )
& ( ide_a @ resid @ ( resid @ A3 @ A ) ) ) ) ) ) ).
% R.coinitial_ide_are_cong
thf(fact_554_R_Ocong__implies__coinitial,axiom,
! [U: a,U5: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U5 ) )
& ( ide_a @ resid @ ( resid @ U5 @ U ) ) )
=> ( coinitial_a @ resid @ U @ U5 ) ) ).
% R.cong_implies_coinitial
thf(fact_555_R_Ocon__imp__coinitial,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( coinitial_a @ resid @ T @ U ) ) ).
% R.con_imp_coinitial
thf(fact_556_inf__right__idem,axiom,
! [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_set_a @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_557_inf__right__idem,axiom,
! [X2: set_list_a,Y2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_set_list_a @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_558_inf_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ B )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_559_inf_Oright__idem,axiom,
! [A: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B ) @ B )
= ( inf_inf_set_list_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_560_inf__left__idem,axiom,
! [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y2 ) )
= ( inf_inf_set_a @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_561_inf__left__idem,axiom,
! [X2: set_list_a,Y2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ X2 @ Y2 ) )
= ( inf_inf_set_list_a @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_562_inf_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_563_inf_Oleft__idem,axiom,
! [A: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ A @ B ) )
= ( inf_inf_set_list_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_564_inf__idem,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_565_inf__idem,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_566_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_567_inf_Oidem,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_568_con__imp__common__source,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( inf_inf_set_list_a @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U ) )
!= bot_bot_set_list_a ) ) ).
% con_imp_common_source
thf(fact_569_R_Ojoinable__implies__coinitial,axiom,
! [T: a,U: a] :
( ( joinable_a @ resid @ T @ U )
=> ( coinitial_a @ resid @ T @ U ) ) ).
% R.joinable_implies_coinitial
thf(fact_570_inf_Obounded__iff,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B @ C ) )
= ( ( ord_le8861187494160871172list_a @ A @ B )
& ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_571_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_572_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_573_le__inf__iff,axiom,
! [X2: set_list_a,Y2: set_list_a,Z3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z3 ) )
= ( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
& ( ord_le8861187494160871172list_a @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_574_le__inf__iff,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z3 ) )
= ( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_575_le__inf__iff,axiom,
! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z3 ) )
= ( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( ord_less_eq_set_a @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_576_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_577_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_578_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_579_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_580_rts_Ocoinitial_Ocong,axiom,
coinitial_a = coinitial_a ).
% rts.coinitial.cong
thf(fact_581_rts_Ocoinitial_Ocong,axiom,
coinitial_list_a = coinitial_list_a ).
% rts.coinitial.cong
thf(fact_582_inf__left__commute,axiom,
! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z3 ) )
= ( inf_inf_set_a @ Y2 @ ( inf_inf_set_a @ X2 @ Z3 ) ) ) ).
% inf_left_commute
thf(fact_583_inf__left__commute,axiom,
! [X2: set_list_a,Y2: set_list_a,Z3: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z3 ) )
= ( inf_inf_set_list_a @ Y2 @ ( inf_inf_set_list_a @ X2 @ Z3 ) ) ) ).
% inf_left_commute
thf(fact_584_inf_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_585_inf_Oleft__commute,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ B @ ( inf_inf_set_list_a @ A @ C ) )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_586_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_a,K: set_a,B: set_a,A: set_a] :
( ( B3
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_587_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_list_a,K: set_list_a,B: set_list_a,A: set_list_a] :
( ( B3
= ( inf_inf_set_list_a @ K @ B ) )
=> ( ( inf_inf_set_list_a @ A @ B3 )
= ( inf_inf_set_list_a @ K @ ( inf_inf_set_list_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_588_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_589_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_list_a,K: set_list_a,A: set_list_a,B: set_list_a] :
( ( A2
= ( inf_inf_set_list_a @ K @ A ) )
=> ( ( inf_inf_set_list_a @ A2 @ B )
= ( inf_inf_set_list_a @ K @ ( inf_inf_set_list_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_590_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_591_inf__commute,axiom,
( inf_inf_set_list_a
= ( ^ [X3: set_list_a,Y3: set_list_a] : ( inf_inf_set_list_a @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_592_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A6 ) ) ) ).
% inf.commute
thf(fact_593_inf_Ocommute,axiom,
( inf_inf_set_list_a
= ( ^ [A6: set_list_a,B5: set_list_a] : ( inf_inf_set_list_a @ B5 @ A6 ) ) ) ).
% inf.commute
thf(fact_594_inf__assoc,axiom,
! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Z3 )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z3 ) ) ) ).
% inf_assoc
thf(fact_595_inf__assoc,axiom,
! [X2: set_list_a,Y2: set_list_a,Z3: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ Z3 )
= ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z3 ) ) ) ).
% inf_assoc
thf(fact_596_inf_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_597_inf_Oassoc,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B ) @ C )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_598_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_599_inf__sup__aci_I1_J,axiom,
( inf_inf_set_list_a
= ( ^ [X3: set_list_a,Y3: set_list_a] : ( inf_inf_set_list_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_600_inf__sup__aci_I2_J,axiom,
! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Z3 )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z3 ) ) ) ).
% inf_sup_aci(2)
thf(fact_601_inf__sup__aci_I2_J,axiom,
! [X2: set_list_a,Y2: set_list_a,Z3: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ Z3 )
= ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z3 ) ) ) ).
% inf_sup_aci(2)
thf(fact_602_inf__sup__aci_I3_J,axiom,
! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z3 ) )
= ( inf_inf_set_a @ Y2 @ ( inf_inf_set_a @ X2 @ Z3 ) ) ) ).
% inf_sup_aci(3)
thf(fact_603_inf__sup__aci_I3_J,axiom,
! [X2: set_list_a,Y2: set_list_a,Z3: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z3 ) )
= ( inf_inf_set_list_a @ Y2 @ ( inf_inf_set_list_a @ X2 @ Z3 ) ) ) ).
% inf_sup_aci(3)
thf(fact_604_inf__sup__aci_I4_J,axiom,
! [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y2 ) )
= ( inf_inf_set_a @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_605_inf__sup__aci_I4_J,axiom,
! [X2: set_list_a,Y2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ X2 @ Y2 ) )
= ( inf_inf_set_list_a @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_606_inf_OcoboundedI2,axiom,
! [B: set_list_a,C: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_607_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_608_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_609_inf_OcoboundedI1,axiom,
! [A: set_list_a,C: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_610_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_611_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_612_inf_Oabsorb__iff2,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B5: set_list_a,A6: set_list_a] :
( ( inf_inf_set_list_a @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_613_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A6: nat] :
( ( inf_inf_nat @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_614_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A6: set_a] :
( ( inf_inf_set_a @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_615_inf_Oabsorb__iff1,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B5: set_list_a] :
( ( inf_inf_set_list_a @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_616_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
( ( inf_inf_nat @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_617_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ( inf_inf_set_a @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_618_inf_Ocobounded2,axiom,
! [A: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_619_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_620_inf_Ocobounded2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_621_inf_Ocobounded1,axiom,
! [A: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_622_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_623_inf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_624_inf_Oorder__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B5: set_list_a] :
( A6
= ( inf_inf_set_list_a @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_625_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
( A6
= ( inf_inf_nat @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_626_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( A6
= ( inf_inf_set_a @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_627_inf__greatest,axiom,
! [X2: set_list_a,Y2: set_list_a,Z3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Z3 )
=> ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_628_inf__greatest,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_629_inf__greatest,axiom,
! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_630_inf_OboundedI,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ C )
=> ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_631_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_632_inf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_633_inf_OboundedE,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B @ C ) )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B )
=> ~ ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_634_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_635_inf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_636_inf__absorb2,axiom,
! [Y2: set_list_a,X2: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y2 @ X2 )
=> ( ( inf_inf_set_list_a @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_637_inf__absorb2,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_638_inf__absorb2,axiom,
! [Y2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( inf_inf_set_a @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_639_inf__absorb1,axiom,
! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ( inf_inf_set_list_a @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_640_inf__absorb1,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_641_inf__absorb1,axiom,
! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( inf_inf_set_a @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_642_inf_Oabsorb2,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( inf_inf_set_list_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_643_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_644_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_645_inf_Oabsorb1,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( inf_inf_set_list_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_646_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_647_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_648_le__iff__inf,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X3: set_list_a,Y3: set_list_a] :
( ( inf_inf_set_list_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_649_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( inf_inf_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_650_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_651_inf__unique,axiom,
! [F: set_list_a > set_list_a > set_list_a,X2: set_list_a,Y2: set_list_a] :
( ! [X: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: set_list_a,Y4: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y4 )
=> ( ( ord_le8861187494160871172list_a @ X @ Z )
=> ( ord_le8861187494160871172list_a @ X @ ( F @ Y4 @ Z ) ) ) )
=> ( ( inf_inf_set_list_a @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_652_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y2: nat] :
( ! [X: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( F @ Y4 @ Z ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_653_inf__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y2: set_a] :
( ! [X: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( F @ Y4 @ Z ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_654_inf_OorderI,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A
= ( inf_inf_set_list_a @ A @ B ) )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% inf.orderI
thf(fact_655_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_656_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_657_inf_OorderE,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( A
= ( inf_inf_set_list_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_658_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_659_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_660_le__infI2,axiom,
! [B: set_list_a,X2: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ X2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_661_le__infI2,axiom,
! [B: nat,X2: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_662_le__infI2,axiom,
! [B: set_a,X2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_663_le__infI1,axiom,
! [A: set_list_a,X2: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ X2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_664_le__infI1,axiom,
! [A: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_665_le__infI1,axiom,
! [A: set_a,X2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_666_inf__mono,axiom,
! [A: set_list_a,C: set_list_a,B: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C )
=> ( ( ord_le8861187494160871172list_a @ B @ D2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B ) @ ( inf_inf_set_list_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_667_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_668_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_669_le__infI,axiom,
! [X2: set_list_a,A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ A )
=> ( ( ord_le8861187494160871172list_a @ X2 @ B )
=> ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_670_le__infI,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A )
=> ( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_671_le__infI,axiom,
! [X2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ B )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_672_le__infE,axiom,
! [X2: set_list_a,A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ A @ B ) )
=> ~ ( ( ord_le8861187494160871172list_a @ X2 @ A )
=> ~ ( ord_le8861187494160871172list_a @ X2 @ B ) ) ) ).
% le_infE
thf(fact_673_le__infE,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A )
=> ~ ( ord_less_eq_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_674_le__infE,axiom,
! [X2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X2 @ A )
=> ~ ( ord_less_eq_set_a @ X2 @ B ) ) ) ).
% le_infE
thf(fact_675_inf__le2,axiom,
! [X2: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_676_inf__le2,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_677_inf__le2,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_678_inf__le1,axiom,
! [X2: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_679_inf__le1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_680_inf__le1,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_681_inf__sup__ord_I1_J,axiom,
! [X2: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_682_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_683_inf__sup__ord_I1_J,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_684_inf__sup__ord_I2_J,axiom,
! [X2: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_685_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_686_inf__sup__ord_I2_J,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_687_joinable__implies__coinitial,axiom,
! [T: list_a,U: list_a] :
( ( joinable_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ).
% joinable_implies_coinitial
thf(fact_688_composable__imp__seq,axiom,
! [T: list_a,U: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ).
% composable_imp_seq
thf(fact_689_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_690_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_691_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_692_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_693_cong__implies__coinitial,axiom,
! [U: list_a,U5: list_a] :
( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ U5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U5 @ U ) ) )
=> ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ U @ U5 ) ) ).
% cong_implies_coinitial
thf(fact_694_coinitial__ide__are__cong,axiom,
! [A: list_a,A3: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A3 )
=> ( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ A @ A3 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A @ A3 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ A3 @ A ) ) ) ) ) ) ).
% coinitial_ide_are_cong
thf(fact_695_con__imp__coinitial,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ).
% con_imp_coinitial
thf(fact_696_coinitial__def,axiom,
! [T: list_a,U: list_a] :
( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
= ( ( inf_inf_set_list_a @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U ) )
!= bot_bot_set_list_a ) ) ).
% coinitial_def
thf(fact_697_rts_Ocomposable_Ocong,axiom,
composable_list_a = composable_list_a ).
% rts.composable.cong
thf(fact_698_rts_Ocomposable_Ocong,axiom,
composable_a = composable_a ).
% rts.composable.cong
thf(fact_699_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_700_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_701_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_702_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_703_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_704_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A6: nat,B5: nat] : ( plus_plus_nat @ B5 @ A6 ) ) ) ).
% add.commute
thf(fact_705_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_706_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_707_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_708_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_709_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_710_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
? [C3: nat] :
( B5
= ( plus_plus_nat @ A6 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_711_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_712_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_713_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_714_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_715_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_716_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_717_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_718_Ide__append__iff_092_060_094sub_062P,axiom,
! [T4: list_a,U2: list_a] :
( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( paths_in_Ide_a @ resid @ ( append_a @ T4 @ U2 ) )
= ( ( paths_in_Ide_a @ resid @ T4 )
& ( paths_in_Ide_a @ resid @ U2 )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U2 ) ) ) ) ) ) ).
% Ide_append_iff\<^sub>P
thf(fact_719_reflects__con,axiom,
! [T: a,U: a] :
( ( ( paths_in_Resid_a @ resid @ ( if_list_a @ ( arr_a @ resid @ T ) @ ( cons_a @ T @ nil_a ) @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) @ ( if_list_a @ ( arr_a @ resid @ U ) @ ( cons_a @ U @ nil_a ) @ ( partial_null_list_a @ ( paths_in_Resid_a @ resid ) ) ) )
!= nil_a )
=> ( con_a @ resid @ T @ U ) ) ).
% reflects_con
thf(fact_720_R_Ocomposable__imp__seq,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( seq_a @ resid @ T @ U ) ) ).
% R.composable_imp_seq
thf(fact_721_Ide_Osimps_I1_J,axiom,
~ ( paths_in_Ide_a @ resid @ nil_a ) ).
% Ide.simps(1)
thf(fact_722_R_Ocong__reflexive,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ( ide_a @ resid @ ( resid @ T @ T ) )
& ( ide_a @ resid @ ( resid @ T @ T ) ) ) ) ).
% R.cong_reflexive
thf(fact_723_R_Oide__implies__arr,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( arr_a @ resid @ A ) ) ).
% R.ide_implies_arr
thf(fact_724_R_Oprfx__reflexive,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ide_a @ resid @ ( resid @ T @ T ) ) ) ).
% R.prfx_reflexive
thf(fact_725_R_Ocon__implies__arr_I2_J,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( arr_a @ resid @ U ) ) ).
% R.con_implies_arr(2)
thf(fact_726_R_Ocon__implies__arr_I1_J,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( arr_a @ resid @ T ) ) ).
% R.con_implies_arr(1)
thf(fact_727_R_OarrE,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( con_a @ resid @ T @ T ) ) ).
% R.arrE
thf(fact_728_R_Oarr__def,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
= ( con_a @ resid @ T @ T ) ) ).
% R.arr_def
thf(fact_729_R_Oarr__resid,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( arr_a @ resid @ ( resid @ T @ U ) ) ) ).
% R.arr_resid
thf(fact_730_R_Oarr__resid__iff__con,axiom,
! [T: a,U: a] :
( ( arr_a @ resid @ ( resid @ T @ U ) )
= ( con_a @ resid @ T @ U ) ) ).
% R.arr_resid_iff_con
thf(fact_731_Ide__implies__Arr,axiom,
! [T4: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
=> ( paths_in_Arr_a @ resid @ T4 ) ) ).
% Ide_implies_Arr
thf(fact_732_R_Onot__arr__null,axiom,
~ ( arr_a @ resid @ ( partial_null_a @ resid ) ) ).
% R.not_arr_null
thf(fact_733_R_OcomposableD_I2_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( arr_a @ resid @ U ) ) ).
% R.composableD(2)
thf(fact_734_R_OcomposableD_I1_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( arr_a @ resid @ T ) ) ).
% R.composableD(1)
thf(fact_735_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_736_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_737_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_738_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_739_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_740_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_741_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_742_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_743_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_744_R_Oarr__iff__has__source,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
= ( ( sources_a @ resid @ T )
!= bot_bot_set_a ) ) ).
% R.arr_iff_has_source
thf(fact_745_R_OcoinitialE,axiom,
! [T: a,U: a] :
( ( coinitial_a @ resid @ T @ U )
=> ~ ( ( arr_a @ resid @ T )
=> ( ( arr_a @ resid @ U )
=> ( ( sources_a @ resid @ T )
!= ( sources_a @ resid @ U ) ) ) ) ) ).
% R.coinitialE
thf(fact_746_R_Ocoinitial__iff,axiom,
! [T: a,T6: a] :
( ( coinitial_a @ resid @ T @ T6 )
= ( ( arr_a @ resid @ T )
& ( arr_a @ resid @ T6 )
& ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ T6 ) ) ) ) ).
% R.coinitial_iff
thf(fact_747_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_748_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_749_Ide_Osimps_I2_J,axiom,
! [T: a] :
( ( paths_in_Ide_a @ resid @ ( cons_a @ T @ nil_a ) )
= ( ide_a @ resid @ T ) ) ).
% Ide.simps(2)
thf(fact_750_Arr_Osimps_I2_J,axiom,
! [T: a] :
( ( paths_in_Arr_a @ resid @ ( cons_a @ T @ nil_a ) )
= ( arr_a @ resid @ T ) ) ).
% Arr.simps(2)
thf(fact_751_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_752_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_753_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_754_R_OarrI,axiom,
! [T: a] :
( ( con_a @ resid @ T @ T )
=> ( arr_a @ resid @ T ) ) ).
% R.arrI
thf(fact_755_R_OcoinitialI,axiom,
! [T: a,U: a] :
( ( arr_a @ resid @ T )
=> ( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( coinitial_a @ resid @ T @ U ) ) ) ).
% R.coinitialI
thf(fact_756_Ide__appendI_092_060_094sub_062P,axiom,
! [T4: list_a,U2: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
=> ( ( paths_in_Ide_a @ resid @ U2 )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ resid @ T4 ) @ ( paths_in_Srcs_a @ resid @ U2 ) )
=> ( paths_in_Ide_a @ resid @ ( append_a @ T4 @ U2 ) ) ) ) ) ).
% Ide_appendI\<^sub>P
thf(fact_757_paths__in__rts_OIde_Ocong,axiom,
paths_in_Ide_a = paths_in_Ide_a ).
% paths_in_rts.Ide.cong
thf(fact_758_residuation_Oarr_Ocong,axiom,
arr_a = arr_a ).
% residuation.arr.cong
thf(fact_759_residuation_Oarr_Ocong,axiom,
arr_list_a = arr_list_a ).
% residuation.arr.cong
thf(fact_760_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_761_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_762_residuation_Oarr__resid__iff__con,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ ( Resid @ T @ U ) )
= ( con_a @ Resid @ T @ U ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_763_residuation_Oarr__resid__iff__con,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( arr_list_a @ Resid @ ( Resid @ T @ U ) )
= ( con_list_a @ Resid @ T @ U ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_764_residuation_Oarr__resid,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( arr_a @ Resid @ ( Resid @ T @ U ) ) ) ) ).
% residuation.arr_resid
thf(fact_765_residuation_Oarr__resid,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T @ U )
=> ( arr_list_a @ Resid @ ( Resid @ T @ U ) ) ) ) ).
% residuation.arr_resid
thf(fact_766_residuation_Oarr__def,axiom,
! [Resid: a > a > a,T: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ T )
= ( con_a @ Resid @ T @ T ) ) ) ).
% residuation.arr_def
thf(fact_767_residuation_Oarr__def,axiom,
! [Resid: list_a > list_a > list_a,T: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( arr_list_a @ Resid @ T )
= ( con_list_a @ Resid @ T @ T ) ) ) ).
% residuation.arr_def
thf(fact_768_residuation_OarrI,axiom,
! [Resid: a > a > a,T: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ T )
=> ( arr_a @ Resid @ T ) ) ) ).
% residuation.arrI
thf(fact_769_residuation_OarrI,axiom,
! [Resid: list_a > list_a > list_a,T: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T @ T )
=> ( arr_list_a @ Resid @ T ) ) ) ).
% residuation.arrI
thf(fact_770_residuation_OarrE,axiom,
! [Resid: a > a > a,T: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( con_a @ Resid @ T @ T ) ) ) ).
% residuation.arrE
thf(fact_771_residuation_OarrE,axiom,
! [Resid: list_a > list_a > list_a,T: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( arr_list_a @ Resid @ T )
=> ( con_list_a @ Resid @ T @ T ) ) ) ).
% residuation.arrE
thf(fact_772_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( arr_a @ Resid @ T ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_773_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T @ U )
=> ( arr_list_a @ Resid @ T ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_774_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( arr_a @ Resid @ U ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_775_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( con_list_a @ Resid @ T @ U )
=> ( arr_list_a @ Resid @ U ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_776_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_777_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_778_paths__in__rts_OIde_Osimps_I1_J,axiom,
! [Resid: a > a > a] :
( ( paths_in_rts_a @ Resid )
=> ~ ( paths_in_Ide_a @ Resid @ nil_a ) ) ).
% paths_in_rts.Ide.simps(1)
thf(fact_779_paths__in__rts_OIde__implies__Arr,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ T4 )
=> ( paths_in_Arr_a @ Resid @ T4 ) ) ) ).
% paths_in_rts.Ide_implies_Arr
thf(fact_780_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_781_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_782_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_783_order__antisym__conv,axiom,
! [Y2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_784_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_785_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_786_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_787_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_788_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_789_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_790_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_791_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_792_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_793_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_794_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_795_order__eq__refl,axiom,
! [X2: set_a,Y2: set_a] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_796_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_797_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_798_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_799_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_800_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_801_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_802_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_803_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_804_order__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ A6 @ B5 )
& ( ord_less_eq_nat @ B5 @ A6 ) ) ) ) ).
% order_eq_iff
thf(fact_805_order__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A6 ) ) ) ) ).
% order_eq_iff
thf(fact_806_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_807_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_808_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_809_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_810_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_811_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_812_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A6 )
& ( ord_less_eq_nat @ A6 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_813_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A6 )
& ( ord_less_eq_set_a @ A6 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_814_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B6: nat] :
( ( ord_less_eq_nat @ A4 @ B6 )
=> ( P @ A4 @ B6 ) )
=> ( ! [A4: nat,B6: nat] :
( ( P @ B6 @ A4 )
=> ( P @ A4 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_815_order__trans,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_816_order__trans,axiom,
! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_817_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_818_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_819_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_820_order__antisym,axiom,
! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_821_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_822_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_823_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_824_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_825_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_826_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_827_le__cases3,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_828_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_829_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_830_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_831_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_832_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_833_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_834_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_835_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_836_paths__in__rts_OArr_Osimps_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Arr_list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) )
= ( arr_list_a @ Resid @ T ) ) ) ).
% paths_in_rts.Arr.simps(2)
thf(fact_837_paths__in__rts_OArr_Osimps_I2_J,axiom,
! [Resid: a > a > a,T: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ ( cons_a @ T @ nil_a ) )
= ( arr_a @ Resid @ T ) ) ) ).
% paths_in_rts.Arr.simps(2)
thf(fact_838_paths__in__rts_OIde_Osimps_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Ide_list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) )
= ( ide_list_a @ Resid @ T ) ) ) ).
% paths_in_rts.Ide.simps(2)
thf(fact_839_paths__in__rts_OIde_Osimps_I2_J,axiom,
! [Resid: a > a > a,T: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ ( cons_a @ T @ nil_a ) )
= ( ide_a @ Resid @ T ) ) ) ).
% paths_in_rts.Ide.simps(2)
thf(fact_840_paths__in__rts_OCon__Ide__iff,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_Arr_a @ Resid @ T4 )
& ( ( paths_in_Srcs_a @ Resid @ T4 )
= ( paths_in_Srcs_a @ Resid @ A2 ) ) ) ) ) ) ).
% paths_in_rts.Con_Ide_iff
thf(fact_841_paths__in__rts_OCon__IdeI_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_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 ) ) ) ) ) ).
% paths_in_rts.Con_IdeI(1)
thf(fact_842_paths__in__rts_OCon__IdeI_I2_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_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 ) ) ) ) ) ).
% paths_in_rts.Con_IdeI(2)
thf(fact_843_paths__in__rts_Oreflects__con,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ ( if_list_list_a @ ( arr_list_a @ Resid @ T ) @ ( cons_list_a @ T @ nil_list_a ) @ ( partia5616633150074602416list_a @ ( paths_8620460302779588466list_a @ Resid ) ) ) @ ( if_list_list_a @ ( arr_list_a @ Resid @ U ) @ ( cons_list_a @ U @ nil_list_a ) @ ( partia5616633150074602416list_a @ ( paths_8620460302779588466list_a @ Resid ) ) ) )
!= nil_list_a )
=> ( con_list_a @ Resid @ T @ U ) ) ) ).
% paths_in_rts.reflects_con
thf(fact_844_paths__in__rts_Oreflects__con,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ ( if_list_a @ ( arr_a @ Resid @ T ) @ ( cons_a @ T @ nil_a ) @ ( partial_null_list_a @ ( paths_in_Resid_a @ Resid ) ) ) @ ( if_list_a @ ( arr_a @ Resid @ U ) @ ( cons_a @ U @ nil_a ) @ ( partial_null_list_a @ ( paths_in_Resid_a @ Resid ) ) ) )
!= nil_a )
=> ( con_a @ Resid @ T @ U ) ) ) ).
% paths_in_rts.reflects_con
thf(fact_845_paths__in__rts_OIde__appendI_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ T4 )
=> ( ( paths_in_Ide_a @ Resid @ U2 )
=> ( ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ Resid @ T4 ) @ ( paths_in_Srcs_a @ Resid @ U2 ) )
=> ( paths_in_Ide_a @ Resid @ ( append_a @ T4 @ U2 ) ) ) ) ) ) ).
% paths_in_rts.Ide_appendI\<^sub>P
thf(fact_846_paths__in__rts_OIde__append__iff_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T4: list_a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( U2 != nil_a )
=> ( ( paths_in_Ide_a @ Resid @ ( append_a @ T4 @ U2 ) )
= ( ( paths_in_Ide_a @ Resid @ T4 )
& ( paths_in_Ide_a @ Resid @ U2 )
& ( ord_less_eq_set_a @ ( paths_in_Trgs_a @ Resid @ T4 ) @ ( paths_in_Srcs_a @ Resid @ U2 ) ) ) ) ) ) ) ).
% paths_in_rts.Ide_append_iff\<^sub>P
thf(fact_847_bot_Oextremum__uniqueI,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
=> ( A = bot_bot_set_list_a ) ) ).
% bot.extremum_uniqueI
thf(fact_848_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_849_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_850_bot_Oextremum__unique,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% bot.extremum_unique
thf(fact_851_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_852_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_853_bot_Oextremum,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% bot.extremum
thf(fact_854_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_855_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_856_Arr_Oelims_I3_J,axiom,
! [X2: list_a] :
( ~ ( paths_in_Arr_a @ resid @ X2 )
=> ( ( X2 != nil_a )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( arr_a @ resid @ T3 ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( arr_a @ resid @ T3 )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% Arr.elims(3)
thf(fact_857_Arr_Oelims_I2_J,axiom,
! [X2: list_a] :
( ( paths_in_Arr_a @ resid @ X2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ~ ( arr_a @ resid @ T3 ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ~ ( ( arr_a @ resid @ T3 )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ).
% Arr.elims(2)
thf(fact_858_Arr_Oelims_I1_J,axiom,
! [X2: list_a,Y2: $o] :
( ( ( paths_in_Arr_a @ resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> Y2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y2
= ( ~ ( arr_a @ resid @ T3 ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y2
= ( ~ ( ( arr_a @ resid @ T3 )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% Arr.elims(1)
thf(fact_859_Ide_Oelims_I3_J,axiom,
! [X2: list_a] :
( ~ ( paths_in_Ide_a @ resid @ X2 )
=> ( ( X2 != nil_a )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ide_a @ resid @ T3 ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( ide_a @ resid @ T3 )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% Ide.elims(3)
thf(fact_860_prfx__reflexive,axiom,
! [T: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ T ) ) ) ).
% prfx_reflexive
thf(fact_861_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_862_cong__reflexive,axiom,
! [T: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ T ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ T ) ) ) ) ).
% cong_reflexive
thf(fact_863_arr__resid__iff__con,axiom,
! [T: list_a,U: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ).
% arr_resid_iff_con
thf(fact_864_arr__resid,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) ) ) ).
% arr_resid
thf(fact_865_arr__def,axiom,
! [T: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ T ) ) ).
% arr_def
thf(fact_866_arrE,axiom,
! [T: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ T ) ) ).
% arrE
thf(fact_867_con__implies__arr_I1_J,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ).
% con_implies_arr(1)
thf(fact_868_con__implies__arr_I2_J,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ).
% con_implies_arr(2)
thf(fact_869_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_870_R_Otargets__cong__closed,axiom,
! [B: a,T: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( ( ide_a @ resid @ ( resid @ B @ B2 ) )
& ( ide_a @ resid @ ( resid @ B2 @ B ) ) )
=> ( member_a @ B2 @ ( targets_a @ resid @ T ) ) ) ) ).
% R.targets_cong_closed
thf(fact_871_R_Otargets__are__cong,axiom,
! [B: a,T: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ resid @ T ) )
=> ( ( ide_a @ resid @ ( resid @ B @ B2 ) )
& ( ide_a @ resid @ ( resid @ B2 @ B ) ) ) ) ) ).
% R.targets_are_cong
thf(fact_872_R_Otarget__is__ide,axiom,
! [A: a,T: a] :
( ( member_a @ A @ ( targets_a @ resid @ T ) )
=> ( ide_a @ resid @ A ) ) ).
% R.target_is_ide
thf(fact_873_R_Otargets__resid__sym,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( ( targets_a @ resid @ ( resid @ T @ U ) )
= ( targets_a @ resid @ ( resid @ U @ T ) ) ) ) ).
% R.targets_resid_sym
thf(fact_874_R_Otargets__are__con,axiom,
! [B: a,T: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ resid @ T ) )
=> ( con_a @ resid @ B @ B2 ) ) ) ).
% R.targets_are_con
thf(fact_875_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_876_R_Oresid__source__in__targets,axiom,
! [A: a,T: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( member_a @ ( resid @ A @ T ) @ ( targets_a @ resid @ T ) ) ) ).
% R.resid_source_in_targets
thf(fact_877_composableD_I1_J,axiom,
! [T: list_a,U: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ).
% composableD(1)
thf(fact_878_composableD_I2_J,axiom,
! [T: list_a,U: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ).
% composableD(2)
thf(fact_879_R_Otargets__con__closed,axiom,
! [B: a,T: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( ide_a @ resid @ B2 )
=> ( ( con_a @ resid @ B @ B2 )
=> ( member_a @ B2 @ ( targets_a @ resid @ T ) ) ) ) ) ).
% R.targets_con_closed
thf(fact_880_R_Oarr__iff__has__target,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
= ( ( targets_a @ resid @ T )
!= bot_bot_set_a ) ) ).
% R.arr_iff_has_target
thf(fact_881_coinitial__iff,axiom,
! [T: list_a,T6: list_a] :
( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T @ T6 )
= ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
& ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T6 )
& ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) ) ) ) ).
% coinitial_iff
thf(fact_882_coinitialE,axiom,
! [T: list_a,U: list_a] :
( ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ~ ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T )
!= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ) ) ) ).
% coinitialE
thf(fact_883_arr__iff__has__source,axiom,
! [T: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T )
!= bot_bot_set_list_a ) ) ).
% arr_iff_has_source
thf(fact_884_R_OcoterminalE,axiom,
! [T: a,U: a] :
( ( coterminal_a @ resid @ T @ U )
=> ~ ( ( arr_a @ resid @ T )
=> ( ( arr_a @ resid @ U )
=> ( ( targets_a @ resid @ T )
!= ( targets_a @ resid @ U ) ) ) ) ) ).
% R.coterminalE
thf(fact_885_R_Ocoterminal__iff,axiom,
! [T: a,T6: a] :
( ( coterminal_a @ resid @ T @ T6 )
= ( ( arr_a @ resid @ T )
& ( arr_a @ resid @ T6 )
& ( ( targets_a @ resid @ T )
= ( targets_a @ resid @ T6 ) ) ) ) ).
% R.coterminal_iff
thf(fact_886_R_Otargets__eqI,axiom,
! [T: a,T6: a] :
( ( ( inf_inf_set_a @ ( targets_a @ resid @ T ) @ ( targets_a @ resid @ T6 ) )
!= bot_bot_set_a )
=> ( ( targets_a @ resid @ T )
= ( targets_a @ resid @ T6 ) ) ) ).
% R.targets_eqI
thf(fact_887_R_OcomposableD_I3_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( ( targets_a @ resid @ T )
= ( sources_a @ resid @ U ) ) ) ).
% R.composableD(3)
thf(fact_888_Trgs_Osimps_I2_J,axiom,
! [T: a] :
( ( paths_in_Trgs_a @ resid @ ( cons_a @ T @ nil_a ) )
= ( targets_a @ resid @ T ) ) ).
% Trgs.simps(2)
thf(fact_889_R_OseqE,axiom,
! [T: a,U: a] :
( ( seq_a @ resid @ T @ U )
=> ~ ( ( arr_a @ resid @ T )
=> ( ( arr_a @ resid @ U )
=> ( ( targets_a @ resid @ T )
!= ( sources_a @ resid @ U ) ) ) ) ) ).
% R.seqE
thf(fact_890_R_Oseq__def,axiom,
! [T: a,U: a] :
( ( seq_a @ resid @ T @ U )
= ( ( arr_a @ resid @ T )
& ( arr_a @ resid @ U )
& ( ( targets_a @ resid @ T )
= ( sources_a @ resid @ U ) ) ) ) ).
% R.seq_def
thf(fact_891_Srcs__Resid__Arr__single,axiom,
! [T4: list_a,U: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Srcs_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U @ nil_a ) ) )
= ( targets_a @ resid @ U ) ) ) ).
% Srcs_Resid_Arr_single
thf(fact_892_R_Ocoterminal__def,axiom,
! [T: a,U: a] :
( ( coterminal_a @ resid @ T @ U )
= ( ( inf_inf_set_a @ ( targets_a @ resid @ T ) @ ( targets_a @ resid @ U ) )
!= bot_bot_set_a ) ) ).
% R.coterminal_def
thf(fact_893_Trgs_Oelims,axiom,
! [X2: list_a,Y2: set_a] :
( ( ( paths_in_Trgs_a @ resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> ( Y2 != bot_bot_set_a ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y2
!= ( targets_a @ resid @ T3 ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_in_Trgs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ).
% Trgs.elims
thf(fact_894_Ide_Osimps_I3_J,axiom,
! [T: a,V: a,Va2: list_a] :
( ( paths_in_Ide_a @ resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) )
= ( ( ide_a @ resid @ T )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V @ Va2 ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V @ Va2 ) ) ) ) ) ).
% Ide.simps(3)
thf(fact_895_Arr_Osimps_I3_J,axiom,
! [T: a,V: a,Va2: list_a] :
( ( paths_in_Arr_a @ resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) )
= ( ( arr_a @ resid @ T )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V @ Va2 ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V @ Va2 ) ) ) ) ) ).
% Arr.simps(3)
thf(fact_896_Ide_Oelims_I1_J,axiom,
! [X2: list_a,Y2: $o] :
( ( ( paths_in_Ide_a @ resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> Y2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y2
= ( ~ ( ide_a @ resid @ T3 ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y2
= ( ~ ( ( ide_a @ resid @ T3 )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% Ide.elims(1)
thf(fact_897_Ide_Oelims_I2_J,axiom,
! [X2: list_a] :
( ( paths_in_Ide_a @ resid @ X2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ~ ( ide_a @ resid @ T3 ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ~ ( ( ide_a @ resid @ T3 )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ).
% Ide.elims(2)
thf(fact_898_arrI,axiom,
! [T: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ T )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ).
% arrI
thf(fact_899_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_900_R_Osources__resid,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( ( sources_a @ resid @ ( resid @ T @ U ) )
= ( targets_a @ resid @ U ) ) ) ).
% R.sources_resid
thf(fact_901_coinitialI,axiom,
! [T: list_a,U: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U ) )
=> ( coinitial_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ) ).
% coinitialI
thf(fact_902_R_OcoterminalI,axiom,
! [T: a,U: a] :
( ( arr_a @ resid @ T )
=> ( ( ( targets_a @ resid @ T )
= ( targets_a @ resid @ U ) )
=> ( coterminal_a @ resid @ T @ U ) ) ) ).
% R.coterminalI
thf(fact_903_R_OseqI,axiom,
! [T: a,U: a] :
( ( arr_a @ resid @ T )
=> ( ( arr_a @ resid @ U )
=> ( ( ( targets_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( seq_a @ resid @ T @ U ) ) ) ) ).
% R.seqI
thf(fact_904_Arr__consI_092_060_094sub_062P,axiom,
! [T: a,U2: list_a] :
( ( arr_a @ resid @ T )
=> ( ( paths_in_Arr_a @ resid @ U2 )
=> ( ( ord_less_eq_set_a @ ( targets_a @ resid @ T ) @ ( paths_in_Srcs_a @ resid @ U2 ) )
=> ( paths_in_Arr_a @ resid @ ( cons_a @ T @ U2 ) ) ) ) ) ).
% Arr_consI\<^sub>P
thf(fact_905_rts_Otargets_Ocong,axiom,
targets_a = targets_a ).
% rts.targets.cong
thf(fact_906_rts_Otargets_Ocong,axiom,
targets_list_a = targets_list_a ).
% rts.targets.cong
thf(fact_907_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: a > a > a,NN: set_a,V: a,V4: a,W: a,W2: a,T: a,T6: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( member_a @ V @ NN )
=> ( ( member_a @ V4 @ NN )
=> ( ( member_a @ W @ NN )
=> ( ( member_a @ W2 @ NN )
=> ( ( ( sources_a @ Resid @ V )
= ( sources_a @ Resid @ W ) )
=> ( ( ( sources_a @ Resid @ V4 )
= ( sources_a @ Resid @ W2 ) )
=> ( ( ( targets_a @ Resid @ W )
= ( targets_a @ Resid @ W2 ) )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V4 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T6 @ W2 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_908_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,V: list_a,V4: list_a,W: list_a,W2: list_a,T: list_a,T6: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( member_list_a @ V @ NN )
=> ( ( member_list_a @ V4 @ NN )
=> ( ( member_list_a @ W @ NN )
=> ( ( member_list_a @ W2 @ NN )
=> ( ( ( sources_list_a @ Resid @ V )
= ( sources_list_a @ Resid @ W ) )
=> ( ( ( sources_list_a @ Resid @ V4 )
= ( sources_list_a @ Resid @ W2 ) )
=> ( ( ( targets_list_a @ Resid @ W )
= ( targets_list_a @ Resid @ W2 ) )
=> ( ( ( member_list_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V4 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_list_a @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T6 @ W2 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_909_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_910_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_911_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,U5: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ U5 @ NN )
=> ( ( ( sources_a @ Resid @ U )
= ( sources_a @ Resid @ U5 ) )
=> ( ( ( targets_a @ Resid @ U )
= ( targets_a @ Resid @ U5 ) )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U5 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T @ U5 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_912_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T: list_a,U: list_a,U5: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( arr_list_a @ Resid @ T )
=> ( ( member_list_a @ U @ NN )
=> ( ( member_list_a @ U5 @ NN )
=> ( ( ( sources_list_a @ Resid @ U )
= ( sources_list_a @ Resid @ U5 ) )
=> ( ( ( targets_list_a @ Resid @ U )
= ( targets_list_a @ Resid @ U5 ) )
=> ( ( ( sources_list_a @ Resid @ T )
= ( sources_list_a @ Resid @ U ) )
=> ( ( member_list_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U5 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T @ U5 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_913_paths__in__rts_OTrgs_Osimps_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Trgs_list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) )
= ( targets_list_a @ Resid @ T ) ) ) ).
% paths_in_rts.Trgs.simps(2)
thf(fact_914_paths__in__rts_OTrgs_Osimps_I2_J,axiom,
! [Resid: a > a > a,T: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Trgs_a @ Resid @ ( cons_a @ T @ nil_a ) )
= ( targets_a @ Resid @ T ) ) ) ).
% paths_in_rts.Trgs.simps(2)
thf(fact_915_paths__in__rts_OSrcs__Resid__Arr__single,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ nil_list_a ) )
!= nil_list_a )
=> ( ( paths_in_Srcs_list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ nil_list_a ) ) )
= ( targets_list_a @ Resid @ U ) ) ) ) ).
% paths_in_rts.Srcs_Resid_Arr_single
thf(fact_916_paths__in__rts_OSrcs__Resid__Arr__single,axiom,
! [Resid: a > a > a,T4: list_a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) )
!= nil_a )
=> ( ( paths_in_Srcs_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U @ nil_a ) ) )
= ( targets_a @ Resid @ U ) ) ) ) ).
% paths_in_rts.Srcs_Resid_Arr_single
thf(fact_917_paths__in__rts_OTrgs_Oelims,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a,Y2: set_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_in_Trgs_list_a @ Resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_list_a )
=> ( Y2 != bot_bot_set_list_a ) )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( Y2
!= ( targets_list_a @ Resid @ T3 ) ) )
=> ~ ! [T3: list_a,V2: list_a,Va: list_list_a] :
( ( X2
= ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_in_Trgs_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Trgs.elims
thf(fact_918_paths__in__rts_OTrgs_Oelims,axiom,
! [Resid: a > a > a,X2: list_a,Y2: set_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Trgs_a @ Resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> ( Y2 != bot_bot_set_a ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y2
!= ( targets_a @ Resid @ T3 ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_in_Trgs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Trgs.elims
thf(fact_919_paths__in__rts_OArr__consI_092_060_094sub_062P,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( arr_list_a @ Resid @ T )
=> ( ( paths_in_Arr_list_a @ Resid @ U2 )
=> ( ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T ) @ ( paths_in_Srcs_list_a @ Resid @ U2 ) )
=> ( paths_in_Arr_list_a @ Resid @ ( cons_list_a @ T @ U2 ) ) ) ) ) ) ).
% paths_in_rts.Arr_consI\<^sub>P
thf(fact_920_paths__in__rts_OArr__consI_092_060_094sub_062P,axiom,
! [Resid: a > a > a,T: a,U2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( paths_in_Arr_a @ Resid @ U2 )
=> ( ( ord_less_eq_set_a @ ( targets_a @ Resid @ T ) @ ( paths_in_Srcs_a @ Resid @ U2 ) )
=> ( paths_in_Arr_a @ Resid @ ( cons_a @ T @ U2 ) ) ) ) ) ) ).
% paths_in_rts.Arr_consI\<^sub>P
thf(fact_921_paths__in__rts_OArr_Osimps_I3_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,V: list_a,Va2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Arr_list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ V @ Va2 ) ) )
= ( ( arr_list_a @ Resid @ T )
& ( paths_in_Arr_list_a @ Resid @ ( cons_list_a @ V @ Va2 ) )
& ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T ) @ ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ V @ Va2 ) ) ) ) ) ) ).
% paths_in_rts.Arr.simps(3)
thf(fact_922_paths__in__rts_OArr_Osimps_I3_J,axiom,
! [Resid: a > a > a,T: a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) )
= ( ( arr_a @ Resid @ T )
& ( paths_in_Arr_a @ Resid @ ( cons_a @ V @ Va2 ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V @ Va2 ) ) ) ) ) ) ).
% paths_in_rts.Arr.simps(3)
thf(fact_923_paths__in__rts_OIde_Osimps_I3_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,V: list_a,Va2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Ide_list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ V @ Va2 ) ) )
= ( ( ide_list_a @ Resid @ T )
& ( paths_in_Ide_list_a @ Resid @ ( cons_list_a @ V @ Va2 ) )
& ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T ) @ ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ V @ Va2 ) ) ) ) ) ) ).
% paths_in_rts.Ide.simps(3)
thf(fact_924_paths__in__rts_OIde_Osimps_I3_J,axiom,
! [Resid: a > a > a,T: a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) )
= ( ( ide_a @ Resid @ T )
& ( paths_in_Ide_a @ Resid @ ( cons_a @ V @ Va2 ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V @ Va2 ) ) ) ) ) ) ).
% paths_in_rts.Ide.simps(3)
thf(fact_925_paths__in__rts_OArr_Oelims_I1_J,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a,Y2: $o] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_in_Arr_list_a @ Resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_list_a )
=> Y2 )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( Y2
= ( ~ ( arr_list_a @ Resid @ T3 ) ) ) )
=> ~ ! [T3: list_a,V2: list_a,Va: list_list_a] :
( ( X2
= ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ( Y2
= ( ~ ( ( arr_list_a @ Resid @ T3 )
& ( paths_in_Arr_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) )
& ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T3 ) @ ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Arr.elims(1)
thf(fact_926_paths__in__rts_OArr_Oelims_I1_J,axiom,
! [Resid: a > a > a,X2: list_a,Y2: $o] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Arr_a @ Resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> Y2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y2
= ( ~ ( arr_a @ Resid @ T3 ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y2
= ( ~ ( ( arr_a @ Resid @ T3 )
& ( paths_in_Arr_a @ Resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T3 ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Arr.elims(1)
thf(fact_927_paths__in__rts_OArr_Oelims_I2_J,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Arr_list_a @ Resid @ X2 )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ~ ( arr_list_a @ Resid @ T3 ) )
=> ~ ! [T3: list_a,V2: list_a,Va: list_list_a] :
( ( X2
= ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ~ ( ( arr_list_a @ Resid @ T3 )
& ( paths_in_Arr_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) )
& ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T3 ) @ ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Arr.elims(2)
thf(fact_928_paths__in__rts_OArr_Oelims_I2_J,axiom,
! [Resid: a > a > a,X2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ X2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ~ ( arr_a @ Resid @ T3 ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ~ ( ( arr_a @ Resid @ T3 )
& ( paths_in_Arr_a @ Resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T3 ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Arr.elims(2)
thf(fact_929_paths__in__rts_OArr_Oelims_I3_J,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ~ ( paths_in_Arr_list_a @ Resid @ X2 )
=> ( ( X2 != nil_list_a )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( arr_list_a @ Resid @ T3 ) )
=> ~ ! [T3: list_a,V2: list_a,Va: list_list_a] :
( ( X2
= ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ( ( arr_list_a @ Resid @ T3 )
& ( paths_in_Arr_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) )
& ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T3 ) @ ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Arr.elims(3)
thf(fact_930_paths__in__rts_OArr_Oelims_I3_J,axiom,
! [Resid: a > a > a,X2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ~ ( paths_in_Arr_a @ Resid @ X2 )
=> ( ( X2 != nil_a )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( arr_a @ Resid @ T3 ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( arr_a @ Resid @ T3 )
& ( paths_in_Arr_a @ Resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T3 ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Arr.elims(3)
thf(fact_931_paths__in__rts_OIde_Oelims_I1_J,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a,Y2: $o] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_in_Ide_list_a @ Resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_list_a )
=> Y2 )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( Y2
= ( ~ ( ide_list_a @ Resid @ T3 ) ) ) )
=> ~ ! [T3: list_a,V2: list_a,Va: list_list_a] :
( ( X2
= ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ( Y2
= ( ~ ( ( ide_list_a @ Resid @ T3 )
& ( paths_in_Ide_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) )
& ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T3 ) @ ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(1)
thf(fact_932_paths__in__rts_OIde_Oelims_I1_J,axiom,
! [Resid: a > a > a,X2: list_a,Y2: $o] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Ide_a @ Resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> Y2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y2
= ( ~ ( ide_a @ Resid @ T3 ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y2
= ( ~ ( ( ide_a @ Resid @ T3 )
& ( paths_in_Ide_a @ Resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T3 ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(1)
thf(fact_933_paths__in__rts_OIde_Oelims_I2_J,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Ide_list_a @ Resid @ X2 )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ~ ( ide_list_a @ Resid @ T3 ) )
=> ~ ! [T3: list_a,V2: list_a,Va: list_list_a] :
( ( X2
= ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ~ ( ( ide_list_a @ Resid @ T3 )
& ( paths_in_Ide_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) )
& ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T3 ) @ ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(2)
thf(fact_934_paths__in__rts_OIde_Oelims_I2_J,axiom,
! [Resid: a > a > a,X2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ X2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ~ ( ide_a @ Resid @ T3 ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ~ ( ( ide_a @ Resid @ T3 )
& ( paths_in_Ide_a @ Resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T3 ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(2)
thf(fact_935_paths__in__rts_OIde_Oelims_I3_J,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ~ ( paths_in_Ide_list_a @ Resid @ X2 )
=> ( ( X2 != nil_list_a )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( ide_list_a @ Resid @ T3 ) )
=> ~ ! [T3: list_a,V2: list_a,Va: list_list_a] :
( ( X2
= ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ( ( ide_list_a @ Resid @ T3 )
& ( paths_in_Ide_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) )
& ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T3 ) @ ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(3)
thf(fact_936_paths__in__rts_OIde_Oelims_I3_J,axiom,
! [Resid: a > a > a,X2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ~ ( paths_in_Ide_a @ Resid @ X2 )
=> ( ( X2 != nil_a )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ide_a @ Resid @ T3 ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( ide_a @ Resid @ T3 )
& ( paths_in_Ide_a @ Resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T3 ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(3)
thf(fact_937_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_938_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_939_coterminal__def,axiom,
! [T: list_a,U: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
= ( ( inf_inf_set_list_a @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U ) )
!= bot_bot_set_list_a ) ) ).
% coterminal_def
thf(fact_940_target__is__ide,axiom,
! [A: list_a,T: list_a] :
( ( member_list_a @ A @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A ) ) ).
% target_is_ide
thf(fact_941_targets__are__cong,axiom,
! [B: list_a,T: list_a,B2: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ( member_list_a @ B2 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B @ B2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B2 @ B ) ) ) ) ) ).
% targets_are_cong
thf(fact_942_targets__cong__closed,axiom,
! [B: list_a,T: list_a,B2: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B @ B2 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ B2 @ B ) ) )
=> ( member_list_a @ B2 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ) ).
% targets_cong_closed
thf(fact_943_targets__are__con,axiom,
! [B: list_a,T: list_a,B2: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ( member_list_a @ B2 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ B @ B2 ) ) ) ).
% targets_are_con
thf(fact_944_targets__resid__sym,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ T ) ) ) ) ).
% targets_resid_sym
thf(fact_945_resid__source__in__targets,axiom,
! [A: list_a,T: list_a] :
( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( member_list_a @ ( paths_in_Resid_a @ resid @ A @ T ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ).
% resid_source_in_targets
thf(fact_946_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_947_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_948_targets__con__closed,axiom,
! [B: list_a,T: list_a,B2: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ B2 )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ B @ B2 )
=> ( member_list_a @ B2 @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ) ) ).
% targets_con_closed
thf(fact_949_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_950_arr__iff__has__target,axiom,
! [T: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T )
!= bot_bot_set_list_a ) ) ).
% arr_iff_has_target
thf(fact_951_coterminal__iff,axiom,
! [T: list_a,T6: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T @ T6 )
= ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
& ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T6 )
& ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) ) ) ) ).
% coterminal_iff
thf(fact_952_coterminalE,axiom,
! [T: list_a,U: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ~ ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T )
!= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ) ) ) ).
% coterminalE
thf(fact_953_composableD_I3_J,axiom,
! [T: list_a,U: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ) ).
% composableD(3)
thf(fact_954_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_955_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_956_targets__eqI,axiom,
! [T: list_a,T6: list_a] :
( ( ( inf_inf_set_list_a @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) )
!= bot_bot_set_list_a )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T6 ) ) ) ).
% targets_eqI
thf(fact_957_seq__def,axiom,
! [T: list_a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
= ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
& ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U )
& ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ) ) ).
% seq_def
thf(fact_958_seqE,axiom,
! [T: list_a,U: list_a] :
( ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ~ ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T )
!= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ) ) ) ).
% seqE
thf(fact_959_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_960_set__empty,axiom,
! [Xs: list_list_a] :
( ( ( set_list_a2 @ Xs )
= bot_bot_set_list_a )
= ( Xs = nil_list_a ) ) ).
% set_empty
thf(fact_961_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_962_set__empty2,axiom,
! [Xs: list_list_a] :
( ( bot_bot_set_list_a
= ( set_list_a2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% set_empty2
thf(fact_963_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_964_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_965_last__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% last_snoc
thf(fact_966_sources__resid,axiom,
! [T: list_a,U: list_a] :
( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ) ).
% sources_resid
thf(fact_967_coterminalI,axiom,
! [T: list_a,U: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ U ) )
=> ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ) ).
% coterminalI
thf(fact_968_seqI,axiom,
! [T: list_a,U: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ U )
=> ( ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U ) )
=> ( seq_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U ) ) ) ) ).
% seqI
thf(fact_969_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_970_subset__code_I1_J,axiom,
! [Xs: list_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B3 )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_971_subset__code_I1_J,axiom,
! [Xs: list_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B3 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_972_last__in__set,axiom,
! [As: list_list_a] :
( ( As != nil_list_a )
=> ( member_list_a @ ( last_list_a @ As ) @ ( set_list_a2 @ As ) ) ) ).
% last_in_set
thf(fact_973_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_974_set__ConsD,axiom,
! [Y2: list_a,X2: list_a,Xs: list_list_a] :
( ( member_list_a @ Y2 @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_list_a @ Y2 @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_975_set__ConsD,axiom,
! [Y2: a,X2: a,Xs: list_a] :
( ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_a @ Y2 @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_976_list_Oset__cases,axiom,
! [E: list_a,A: list_list_a] :
( ( member_list_a @ E @ ( set_list_a2 @ A ) )
=> ( ! [Z22: list_list_a] :
( A
!= ( cons_list_a @ E @ Z22 ) )
=> ~ ! [Z1: list_a,Z22: list_list_a] :
( ( A
= ( cons_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_a @ E @ ( set_list_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_977_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_978_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_979_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_980_list_Oset__intros_I2_J,axiom,
! [Y2: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a @ Y2 @ ( set_list_a2 @ X22 ) )
=> ( member_list_a @ Y2 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_981_list_Oset__intros_I2_J,axiom,
! [Y2: a,X22: list_a,X21: a] :
( ( member_a @ Y2 @ ( set_a2 @ X22 ) )
=> ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_982_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_983_empty__set,axiom,
( bot_bot_set_list_a
= ( set_list_a2 @ nil_list_a ) ) ).
% empty_set
thf(fact_984_set__subset__Cons,axiom,
! [Xs: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_985_split__list,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_986_split__list,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_987_split__list__last,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_988_split__list__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_989_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_990_split__list__first,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_991_split__list__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_992_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_993_append__Cons__eq__iff,axiom,
! [X2: list_a,Xs: list_list_a,Ys: list_list_a,Xs4: list_list_a,Ys6: list_list_a] :
( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys ) )
=> ( ( ( append_list_a @ Xs @ ( cons_list_a @ X2 @ Ys ) )
= ( append_list_a @ Xs4 @ ( cons_list_a @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_994_append__Cons__eq__iff,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys6: list_a] :
( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_995_in__set__conv__decomp,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_996_in__set__conv__decomp,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_997_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_a,X: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_998_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_999_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_a,X: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1000_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1001_in__set__conv__decomp__last,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1002_in__set__conv__decomp__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1003_in__set__conv__decomp__first,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1004_in__set__conv__decomp__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1005_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1006_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1007_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_1008_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_1009_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_1010_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1011_last_Osimps,axiom,
! [Xs: list_a,X2: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_1012_last__ConsL,axiom,
! [Xs: list_a,X2: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_1013_last__ConsR,axiom,
! [Xs: list_a,X2: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_1014_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_1015_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_1016_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_1017_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_1018_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_1019_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_1020_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_1021_paths__in__rts_OIde__imp__sources__eq__targets,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ T4 )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 )
= ( targets_list_a @ ( paths_in_Resid_a @ Resid ) @ T4 ) ) ) ) ).
% paths_in_rts.Ide_imp_sources_eq_targets
thf(fact_1022_paths__in__rts_Oset__Arr__subset__arr,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 )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ T4 ) @ ( collect_list_a @ ( arr_list_a @ Resid ) ) ) ) ) ).
% paths_in_rts.set_Arr_subset_arr
thf(fact_1023_paths__in__rts_Oset__Arr__subset__arr,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ T4 )
=> ( ord_less_eq_set_a @ ( set_a2 @ T4 ) @ ( collect_a @ ( arr_a @ Resid ) ) ) ) ) ).
% paths_in_rts.set_Arr_subset_arr
thf(fact_1024_paths__in__rts_Oset__Ide__subset__ide,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 )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ T4 ) @ ( collect_list_a @ ( ide_list_a @ Resid ) ) ) ) ) ).
% paths_in_rts.set_Ide_subset_ide
thf(fact_1025_paths__in__rts_Oset__Ide__subset__ide,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ T4 )
=> ( ord_less_eq_set_a @ ( set_a2 @ T4 ) @ ( collect_a @ ( ide_a @ Resid ) ) ) ) ) ).
% paths_in_rts.set_Ide_subset_ide
thf(fact_1026_paths__in__rts_OTrgs__simp_092_060_094sub_062P,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 )
= ( targets_list_a @ Resid @ ( last_list_a @ T4 ) ) ) ) ) ).
% paths_in_rts.Trgs_simp\<^sub>P
thf(fact_1027_paths__in__rts_OTrgs__simp_092_060_094sub_062P,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 )
= ( targets_a @ Resid @ ( last_a @ T4 ) ) ) ) ) ).
% paths_in_rts.Trgs_simp\<^sub>P
thf(fact_1028_paths__in__rts_OIdeI,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 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ T4 ) @ ( collect_list_a @ ( ide_list_a @ Resid ) ) )
=> ( paths_in_Ide_list_a @ Resid @ T4 ) ) ) ) ).
% paths_in_rts.IdeI
thf(fact_1029_paths__in__rts_OIdeI,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( 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 ) ) ) ) ).
% paths_in_rts.IdeI
thf(fact_1030_paths__in__rts_OIde__char,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 )
= ( ( paths_in_Arr_list_a @ Resid @ T4 )
& ( ord_le8861187494160871172list_a @ ( set_list_a2 @ T4 ) @ ( collect_list_a @ ( ide_list_a @ Resid ) ) ) ) ) ) ).
% paths_in_rts.Ide_char
thf(fact_1031_paths__in__rts_OIde__char,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( 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 ) ) ) ) ) ) ).
% paths_in_rts.Ide_char
thf(fact_1032_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_a @ ( hd_a @ T4 ) @ bot_bot_set_a ) )
=> ( paths_in_Ide_a @ resid @ T4 ) ) ) ) ).
% const_ide_is_Ide
thf(fact_1033_join__of__arr__src_I2_J,axiom,
! [T: list_a,A: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ A @ T ) ) ) ).
% join_of_arr_src(2)
thf(fact_1034_join__of__arr__src_I1_J,axiom,
! [T: list_a,A: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ A @ T @ T ) ) ) ).
% join_of_arr_src(1)
thf(fact_1035_join__of__symmetric,axiom,
! [T: list_a,U: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ T @ V ) ) ).
% join_of_symmetric
thf(fact_1036_insert__absorb2,axiom,
! [X2: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A2 ) )
= ( insert_a @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_1037_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1038_insert__iff,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
= ( ( A = B )
| ( member_list_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1039_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_1040_insertCI,axiom,
! [A: list_a,B3: set_list_a,B: list_a] :
( ( ~ ( member_list_a @ A @ B3 )
=> ( A = B ) )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_1041_join__of__un__upto__cong,axiom,
! [T: list_a,U: list_a,V: list_a,V4: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V )
=> ( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V4 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ V4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V4 @ V ) ) ) ) ) ).
% join_of_un_upto_cong
thf(fact_1042_join__of__resid,axiom,
! [T: list_a,U: list_a,W: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ W )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ V @ W )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ V ) @ ( paths_in_Resid_a @ resid @ U @ V ) @ ( paths_in_Resid_a @ resid @ W @ V ) ) ) ) ).
% join_of_resid
thf(fact_1043_con__with__join__of__iff_I1_J,axiom,
! [T: list_a,U: list_a,W: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ W )
=> ( ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ U @ V )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ U ) @ ( paths_in_Resid_a @ resid @ T @ U ) ) )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W @ V ) ) ) ).
% con_with_join_of_iff(1)
thf(fact_1044_con__with__join__of__iff_I2_J,axiom,
! [T: list_a,U: list_a,W: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ W )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W @ V )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ V )
& ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ T ) @ ( paths_in_Resid_a @ resid @ U @ T ) ) ) ) ) ).
% con_with_join_of_iff(2)
thf(fact_1045_sources__join__of_I1_J,axiom,
! [T: list_a,U: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V ) ) ) ).
% sources_join_of(1)
thf(fact_1046_sources__join__of_I2_J,axiom,
! [T: list_a,U: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V ) ) ) ).
% sources_join_of(2)
thf(fact_1047_targets__join__of_I2_J,axiom,
! [T: list_a,U: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ T ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ V ) ) ) ).
% targets_join_of(2)
thf(fact_1048_targets__join__of_I1_J,axiom,
! [T: list_a,U: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ V ) ) ) ).
% targets_join_of(1)
thf(fact_1049_join__of__arr__self,axiom,
! [T: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ T @ T ) ) ).
% join_of_arr_self
thf(fact_1050_joinable__def,axiom,
! [T: list_a,U: list_a] :
( ( joinable_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
= ( ? [X5: list_a] : ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ X5 ) ) ) ).
% joinable_def
thf(fact_1051_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_1052_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_1053_insert__subset,axiom,
! [X2: list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X2 @ A2 ) @ B3 )
= ( ( member_list_a @ X2 @ B3 )
& ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_1054_insert__subset,axiom,
! [X2: a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A2 ) @ B3 )
= ( ( member_a @ X2 @ B3 )
& ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_1055_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1056_Int__insert__right__if1,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1057_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1058_Int__insert__right__if0,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1059_insert__inter__insert,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_1060_insert__inter__insert,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ ( insert_list_a @ A @ B3 ) )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_1061_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B3: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1062_Int__insert__left__if1,axiom,
! [A: list_a,C2: set_list_a,B3: set_list_a] :
( ( member_list_a @ A @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B3 ) @ C2 )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1063_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B3: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1064_Int__insert__left__if0,axiom,
! [A: list_a,C2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ A @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_list_a @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1065_singleton__insert__inj__eq_H,axiom,
! [A: list_a,A2: set_list_a,B: list_a] :
( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1066_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1067_singleton__insert__inj__eq,axiom,
! [B: list_a,A: list_a,A2: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1068_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1069_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1070_disjoint__insert_I2_J,axiom,
! [A2: set_a,B: a,B3: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B @ B3 ) ) )
= ( ~ ( member_a @ B @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1071_disjoint__insert_I2_J,axiom,
! [A2: set_list_a,B: list_a,B3: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ B @ B3 ) ) )
= ( ~ ( member_list_a @ B @ A2 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1072_disjoint__insert_I1_J,axiom,
! [B3: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B3 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B3 )
& ( ( inf_inf_set_a @ B3 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1073_disjoint__insert_I1_J,axiom,
! [B3: set_list_a,A: list_a,A2: set_list_a] :
( ( ( inf_inf_set_list_a @ B3 @ ( insert_list_a @ A @ A2 ) )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A @ B3 )
& ( ( inf_inf_set_list_a @ B3 @ A2 )
= bot_bot_set_list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1074_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B3 ) )
= ( ~ ( member_a @ A @ B3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1075_insert__disjoint_I2_J,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ B3 ) )
= ( ~ ( member_list_a @ A @ B3 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1076_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B3 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B3 )
& ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1077_insert__disjoint_I1_J,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ B3 )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A @ B3 )
& ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1078_subset__singletonD,axiom,
! [A2: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
=> ( ( A2 = bot_bot_set_list_a )
| ( A2
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_1079_subset__singletonD,axiom,
! [A2: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1080_subset__singleton__iff,axiom,
! [X6: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ X6 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( ( X6 = bot_bot_set_list_a )
| ( X6
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1081_subset__singleton__iff,axiom,
! [X6: set_a,A: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1082_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1083_singleton__inject,axiom,
! [A: list_a,B: list_a] :
( ( ( insert_list_a @ A @ bot_bot_set_list_a )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1084_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_1085_insert__not__empty,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ A2 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_1086_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1087_doubleton__eq__iff,axiom,
! [A: list_a,B: list_a,C: list_a,D2: list_a] :
( ( ( insert_list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D2 @ bot_bot_set_list_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1088_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1089_singleton__iff,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1090_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1091_singletonD,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1092_rts_Ojoin__of_Ocong,axiom,
join_of_list_a = join_of_list_a ).
% rts.join_of.cong
thf(fact_1093_rts_Ojoin__of_Ocong,axiom,
join_of_a = join_of_a ).
% rts.join_of.cong
thf(fact_1094_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B7: set_a] :
( ( A2
= ( insert_a @ A @ B7 ) )
& ~ ( member_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1095_mk__disjoint__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ? [B7: set_list_a] :
( ( A2
= ( insert_list_a @ A @ B7 ) )
& ~ ( member_list_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1096_insert__commute,axiom,
! [X2: a,Y2: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ Y2 @ A2 ) )
= ( insert_a @ Y2 @ ( insert_a @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_1097_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C5: set_a] :
( ( A2
= ( insert_a @ B @ C5 ) )
& ~ ( member_a @ B @ C5 )
& ( B3
= ( insert_a @ A @ C5 ) )
& ~ ( member_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1098_insert__eq__iff,axiom,
! [A: list_a,A2: set_list_a,B: list_a,B3: set_list_a] :
( ~ ( member_list_a @ A @ A2 )
=> ( ~ ( member_list_a @ B @ B3 )
=> ( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C5: set_list_a] :
( ( A2
= ( insert_list_a @ B @ C5 ) )
& ~ ( member_list_a @ B @ C5 )
& ( B3
= ( insert_list_a @ A @ C5 ) )
& ~ ( member_list_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1099_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1100_insert__absorb,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1101_insert__ident,axiom,
! [X2: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ~ ( member_a @ X2 @ B3 )
=> ( ( ( insert_a @ X2 @ A2 )
= ( insert_a @ X2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_1102_insert__ident,axiom,
! [X2: list_a,A2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ X2 @ A2 )
=> ( ~ ( member_list_a @ X2 @ B3 )
=> ( ( ( insert_list_a @ X2 @ A2 )
= ( insert_list_a @ X2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_1103_Set_Oset__insert,axiom,
! [X2: a,A2: set_a] :
( ( member_a @ X2 @ A2 )
=> ~ ! [B7: set_a] :
( ( A2
= ( insert_a @ X2 @ B7 ) )
=> ( member_a @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1104_Set_Oset__insert,axiom,
! [X2: list_a,A2: set_list_a] :
( ( member_list_a @ X2 @ A2 )
=> ~ ! [B7: set_list_a] :
( ( A2
= ( insert_list_a @ X2 @ B7 ) )
=> ( member_list_a @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1105_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_1106_insertI2,axiom,
! [A: list_a,B3: set_list_a,B: list_a] :
( ( member_list_a @ A @ B3 )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_1107_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_1108_insertI1,axiom,
! [A: list_a,B3: set_list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ B3 ) ) ).
% insertI1
thf(fact_1109_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_1110_insertE,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_list_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_1111_Int__insert__left,axiom,
! [A: a,C2: set_a,B3: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1112_Int__insert__left,axiom,
! [A: list_a,C2: set_list_a,B3: set_list_a] :
( ( ( member_list_a @ A @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B3 ) @ C2 )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ B3 @ C2 ) ) ) )
& ( ~ ( member_list_a @ A @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_list_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1113_Int__insert__right,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B3 ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_1114_Int__insert__right,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ( ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) )
& ( ~ ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_1115_subset__insertI2,axiom,
! [A2: set_a,B3: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_1116_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_1117_subset__insert,axiom,
! [X2: list_a,A2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ X2 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ B3 ) )
= ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_1118_subset__insert,axiom,
! [X2: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B3 ) )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_1119_insert__mono,axiom,
! [C2: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_1120_paths__in__rts_Oconst__ide__is__Ide,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( T4 != nil_list_a )
=> ( ( ide_list_a @ Resid @ ( hd_list_a @ T4 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ T4 ) @ ( insert_list_a @ ( hd_list_a @ T4 ) @ bot_bot_set_list_a ) )
=> ( paths_in_Ide_list_a @ Resid @ T4 ) ) ) ) ) ).
% paths_in_rts.const_ide_is_Ide
thf(fact_1121_paths__in__rts_Oconst__ide__is__Ide,axiom,
! [Resid: a > a > a,T4: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( T4 != nil_a )
=> ( ( ide_a @ Resid @ ( hd_a @ T4 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ T4 ) @ ( insert_a @ ( hd_a @ T4 ) @ bot_bot_set_a ) )
=> ( paths_in_Ide_a @ Resid @ T4 ) ) ) ) ) ).
% paths_in_rts.const_ide_is_Ide
thf(fact_1122_in__targetsE,axiom,
! [B: list_a,T: list_a] :
( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ~ ( ( 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 ) @ T ) @ B ) ) ) ).
% in_targetsE
thf(fact_1123_coterminal__iff__con__trg,axiom,
! [T: list_a,U: list_a] :
( ( coterminal_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ) ).
% coterminal_iff_con_trg
thf(fact_1124_R_Ojoin__of__symmetric,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( join_of_a @ resid @ U @ T @ V ) ) ).
% R.join_of_symmetric
thf(fact_1125_R_Ojoin__of__un__upto__cong,axiom,
! [T: a,U: a,V: a,V4: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( join_of_a @ resid @ T @ U @ V4 )
=> ( ( ide_a @ resid @ ( resid @ V @ V4 ) )
& ( ide_a @ resid @ ( resid @ V4 @ V ) ) ) ) ) ).
% R.join_of_un_upto_cong
thf(fact_1126_R_Ocon__with__join__of__iff_I2_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( ( con_a @ resid @ T @ V )
& ( con_a @ resid @ ( resid @ V @ T ) @ ( resid @ U @ T ) ) ) ) ) ).
% R.con_with_join_of_iff(2)
thf(fact_1127_R_Ocon__with__join__of__iff_I1_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W )
=> ( ( ( con_a @ resid @ U @ V )
& ( con_a @ resid @ ( resid @ V @ U ) @ ( resid @ T @ U ) ) )
=> ( con_a @ resid @ W @ V ) ) ) ).
% R.con_with_join_of_iff(1)
thf(fact_1128_R_Ojoin__of__resid,axiom,
! [T: a,U: a,W: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ V @ W )
=> ( join_of_a @ resid @ ( resid @ T @ V ) @ ( resid @ U @ V ) @ ( resid @ W @ V ) ) ) ) ).
% R.join_of_resid
thf(fact_1129_R_Ojoin__of__arr__self,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( join_of_a @ resid @ T @ T @ T ) ) ).
% R.join_of_arr_self
thf(fact_1130_R_Otargets__join__of_I1_J,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( targets_a @ resid @ ( resid @ T @ U ) )
= ( targets_a @ resid @ V ) ) ) ).
% R.targets_join_of(1)
thf(fact_1131_R_Otargets__join__of_I2_J,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( targets_a @ resid @ ( resid @ U @ T ) )
= ( targets_a @ resid @ V ) ) ) ).
% R.targets_join_of(2)
thf(fact_1132_R_Osources__join__of_I2_J,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( sources_a @ resid @ U )
= ( sources_a @ resid @ V ) ) ) ).
% R.sources_join_of(2)
thf(fact_1133_R_Osources__join__of_I1_J,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ V ) ) ) ).
% R.sources_join_of(1)
thf(fact_1134_R_Ojoinable__def,axiom,
! [T: a,U: a] :
( ( joinable_a @ resid @ T @ U )
= ( ? [X5: a] : ( join_of_a @ resid @ T @ U @ X5 ) ) ) ).
% R.joinable_def
thf(fact_1135_R_Ojoin__of__arr__src_I2_J,axiom,
! [T: a,A: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( join_of_a @ resid @ T @ A @ T ) ) ) ).
% R.join_of_arr_src(2)
thf(fact_1136_R_Ojoin__of__arr__src_I1_J,axiom,
! [T: a,A: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( join_of_a @ resid @ A @ T @ T ) ) ) ).
% R.join_of_arr_src(1)
thf(fact_1137_trg__def,axiom,
! [T: list_a] :
( ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T )
= ( paths_in_Resid_a @ resid @ T @ T ) ) ).
% trg_def
thf(fact_1138_ide__trg,axiom,
! [T: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ).
% ide_trg
thf(fact_1139_trg__in__targets,axiom,
! [T: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( member_list_a @ ( trg_list_a @ ( paths_in_Resid_a @ resid ) @ T ) @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ).
% trg_in_targets
thf(fact_1140_in__targetsI,axiom,
! [B: list_a,T: 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 ) @ T ) @ B )
=> ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ) ).
% in_targetsI
thf(fact_1141_residuation_Otrg_Ocong,axiom,
trg_list_a = trg_list_a ).
% residuation.trg.cong
thf(fact_1142_residuation_Otrg_Ocong,axiom,
trg_a = trg_a ).
% residuation.trg.cong
thf(fact_1143_residuation_Otrg__def,axiom,
! [Resid: list_a > list_a > list_a,T: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( trg_list_a @ Resid @ T )
= ( Resid @ T @ T ) ) ) ).
% residuation.trg_def
thf(fact_1144_residuation_Otrg__def,axiom,
! [Resid: a > a > a,T: a] :
( ( residuation_a @ Resid )
=> ( ( trg_a @ Resid @ T )
= ( Resid @ T @ T ) ) ) ).
% residuation.trg_def
thf(fact_1145_residuation_Oresid__arr__self,axiom,
! [Resid: list_a > list_a > list_a,T: list_a] :
( ( residuation_list_a @ Resid )
=> ( ( Resid @ T @ T )
= ( trg_list_a @ Resid @ T ) ) ) ).
% residuation.resid_arr_self
thf(fact_1146_residuation_Oresid__arr__self,axiom,
! [Resid: a > a > a,T: a] :
( ( residuation_a @ Resid )
=> ( ( Resid @ T @ T )
= ( trg_a @ Resid @ T ) ) ) ).
% residuation.resid_arr_self
thf(fact_1147_R_Oin__targetsE,axiom,
! [B: a,T: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ~ ( ( ide_a @ resid @ B )
=> ~ ( con_a @ resid @ ( trg_a @ resid @ T ) @ B ) ) ) ).
% R.in_targetsE
thf(fact_1148_Cong__iff__cong,axiom,
! [T: list_a,U: list_a] :
( ( normal4889798360446511898list_a @ ( paths_in_Resid_a @ resid ) @ ( collect_list_a @ ( ide_list_a @ ( paths_in_Resid_a @ resid ) ) ) @ T @ U )
= ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ U ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ T ) ) ) ) ).
% Cong_iff_cong
thf(fact_1149_R_Otrg__def,axiom,
! [T: a] :
( ( trg_a @ resid @ T )
= ( resid @ T @ T ) ) ).
% R.trg_def
thf(fact_1150_R_Oide__trg,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ide_a @ resid @ ( trg_a @ resid @ T ) ) ) ).
% R.ide_trg
thf(fact_1151_R_Otrg__in__targets,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( member_a @ ( trg_a @ resid @ T ) @ ( targets_a @ resid @ T ) ) ) ).
% R.trg_in_targets
thf(fact_1152_R_Ocoterminal__iff__con__trg,axiom,
! [T: a,U: a] :
( ( coterminal_a @ resid @ T @ U )
= ( con_a @ resid @ ( trg_a @ resid @ T ) @ ( trg_a @ resid @ U ) ) ) ).
% R.coterminal_iff_con_trg
thf(fact_1153_R_Oin__targetsI,axiom,
! [B: a,T: a] :
( ( ide_a @ resid @ B )
=> ( ( con_a @ resid @ ( trg_a @ resid @ T ) @ B )
=> ( member_a @ B @ ( targets_a @ resid @ T ) ) ) ) ).
% R.in_targetsI
thf(fact_1154_normal__sub__rts_OCong_Ocong,axiom,
normal4889798360446511898list_a = normal4889798360446511898list_a ).
% normal_sub_rts.Cong.cong
thf(fact_1155_normal__sub__rts_OCong_Ocong,axiom,
normal_sub_Cong_a = normal_sub_Cong_a ).
% normal_sub_rts.Cong.cong
thf(fact_1156_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,T6: a,U5: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( ( ( sources_a @ Resid @ T6 )
= ( sources_a @ Resid @ U5 ) )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U5 )
=> ( ( con_a @ Resid @ T @ U )
= ( con_a @ Resid @ T6 @ U5 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_1157_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T: list_a,U: list_a,T6: list_a,U5: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( ( sources_list_a @ Resid @ T )
= ( sources_list_a @ Resid @ U ) )
=> ( ( ( sources_list_a @ Resid @ T6 )
= ( sources_list_a @ Resid @ U5 ) )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T @ T6 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ U @ U5 )
=> ( ( con_list_a @ Resid @ T @ U )
= ( con_list_a @ Resid @ T6 @ U5 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_1158_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,U: a,U5: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U5 )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( ( sources_a @ Resid @ T6 )
= ( sources_a @ Resid @ U5 ) )
=> ( con_a @ Resid @ T6 @ U5 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_1159_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T: list_a,T6: list_a,U: list_a,U5: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T @ T6 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ U @ U5 )
=> ( ( con_list_a @ Resid @ T @ U )
=> ( ( ( sources_list_a @ Resid @ T6 )
= ( sources_list_a @ Resid @ U5 ) )
=> ( con_list_a @ Resid @ T6 @ U5 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_1160_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,U: a,U5: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U5 )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( ( sources_a @ Resid @ T6 )
= ( sources_a @ Resid @ U5 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ ( Resid @ T @ U ) @ ( Resid @ T6 @ U5 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_1161_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,T: list_a,T6: list_a,U: list_a,U5: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ T @ T6 )
=> ( ( normal4889798360446511898list_a @ Resid @ NN @ U @ U5 )
=> ( ( con_list_a @ Resid @ T @ U )
=> ( ( ( sources_list_a @ Resid @ T6 )
= ( sources_list_a @ Resid @ U5 ) )
=> ( normal4889798360446511898list_a @ Resid @ NN @ ( Resid @ T @ U ) @ ( Resid @ T6 @ U5 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_1162_rts__axioms_Ointro,axiom,
! [Resid: list_a > list_a > list_a] :
( ! [T3: list_a] :
( ( arr_list_a @ Resid @ T3 )
=> ( ide_list_a @ Resid @ ( trg_list_a @ Resid @ T3 ) ) )
=> ( ! [A4: list_a,T3: list_a] :
( ( ide_list_a @ Resid @ A4 )
=> ( ( con_list_a @ Resid @ T3 @ A4 )
=> ( ( Resid @ T3 @ A4 )
= T3 ) ) )
=> ( ! [A4: list_a,T3: list_a] :
( ( ide_list_a @ Resid @ A4 )
=> ( ( con_list_a @ Resid @ A4 @ T3 )
=> ( ide_list_a @ Resid @ ( Resid @ A4 @ T3 ) ) ) )
=> ( ! [T3: list_a,U3: list_a] :
( ( con_list_a @ Resid @ T3 @ U3 )
=> ? [A7: list_a] :
( ( ide_list_a @ Resid @ A7 )
& ( con_list_a @ Resid @ A7 @ T3 )
& ( con_list_a @ Resid @ A7 @ U3 ) ) )
=> ( ! [T3: list_a,U3: list_a,V2: list_a] :
( ( ide_list_a @ Resid @ ( Resid @ T3 @ U3 ) )
=> ( ( con_list_a @ Resid @ U3 @ V2 )
=> ( con_list_a @ Resid @ ( Resid @ T3 @ U3 ) @ ( Resid @ V2 @ U3 ) ) ) )
=> ( rts_axioms_list_a @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_1163_rts__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a] :
( ( arr_a @ Resid @ T3 )
=> ( ide_a @ Resid @ ( trg_a @ Resid @ T3 ) ) )
=> ( ! [A4: a,T3: a] :
( ( ide_a @ Resid @ A4 )
=> ( ( con_a @ Resid @ T3 @ A4 )
=> ( ( Resid @ T3 @ A4 )
= T3 ) ) )
=> ( ! [A4: a,T3: a] :
( ( ide_a @ Resid @ A4 )
=> ( ( con_a @ Resid @ A4 @ T3 )
=> ( ide_a @ Resid @ ( Resid @ A4 @ T3 ) ) ) )
=> ( ! [T3: a,U3: a] :
( ( con_a @ Resid @ T3 @ U3 )
=> ? [A7: a] :
( ( ide_a @ Resid @ A7 )
& ( con_a @ Resid @ A7 @ T3 )
& ( con_a @ Resid @ A7 @ U3 ) ) )
=> ( ! [T3: a,U3: a,V2: a] :
( ( ide_a @ Resid @ ( Resid @ T3 @ U3 ) )
=> ( ( con_a @ Resid @ U3 @ V2 )
=> ( con_a @ Resid @ ( Resid @ T3 @ U3 ) @ ( Resid @ V2 @ U3 ) ) ) )
=> ( rts_axioms_a @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_1164_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 ) ) )
& ! [A6: list_a,T5: list_a] :
( ( ide_list_a @ Resid2 @ A6 )
=> ( ( con_list_a @ Resid2 @ T5 @ A6 )
=> ( ( Resid2 @ T5 @ A6 )
= T5 ) ) )
& ! [A6: list_a,T5: list_a] :
( ( ide_list_a @ Resid2 @ A6 )
=> ( ( con_list_a @ Resid2 @ A6 @ T5 )
=> ( ide_list_a @ Resid2 @ ( Resid2 @ A6 @ T5 ) ) ) )
& ! [T5: list_a,U6: list_a] :
( ( con_list_a @ Resid2 @ T5 @ U6 )
=> ? [A6: list_a] :
( ( ide_list_a @ Resid2 @ A6 )
& ( con_list_a @ Resid2 @ A6 @ T5 )
& ( con_list_a @ Resid2 @ A6 @ U6 ) ) )
& ! [T5: list_a,U6: list_a,V5: list_a] :
( ( ide_list_a @ Resid2 @ ( Resid2 @ T5 @ U6 ) )
=> ( ( con_list_a @ Resid2 @ U6 @ V5 )
=> ( con_list_a @ Resid2 @ ( Resid2 @ T5 @ U6 ) @ ( Resid2 @ V5 @ U6 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_1165_rts__axioms__def,axiom,
( rts_axioms_a
= ( ^ [Resid2: a > a > a] :
( ! [T5: a] :
( ( arr_a @ Resid2 @ T5 )
=> ( ide_a @ Resid2 @ ( trg_a @ Resid2 @ T5 ) ) )
& ! [A6: a,T5: a] :
( ( ide_a @ Resid2 @ A6 )
=> ( ( con_a @ Resid2 @ T5 @ A6 )
=> ( ( Resid2 @ T5 @ A6 )
= T5 ) ) )
& ! [A6: a,T5: a] :
( ( ide_a @ Resid2 @ A6 )
=> ( ( con_a @ Resid2 @ A6 @ T5 )
=> ( ide_a @ Resid2 @ ( Resid2 @ A6 @ T5 ) ) ) )
& ! [T5: a,U6: a] :
( ( con_a @ Resid2 @ T5 @ U6 )
=> ? [A6: a] :
( ( ide_a @ Resid2 @ A6 )
& ( con_a @ Resid2 @ A6 @ T5 )
& ( con_a @ Resid2 @ A6 @ U6 ) ) )
& ! [T5: a,U6: a,V5: a] :
( ( ide_a @ Resid2 @ ( Resid2 @ T5 @ U6 ) )
=> ( ( con_a @ Resid2 @ U6 @ V5 )
=> ( con_a @ Resid2 @ ( Resid2 @ T5 @ U6 ) @ ( Resid2 @ V5 @ U6 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_1166_R_OCong__iff__cong,axiom,
! [T: a,U: a] :
( ( normal_sub_Cong_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) @ T @ U )
= ( ( ide_a @ resid @ ( resid @ T @ U ) )
& ( ide_a @ resid @ ( resid @ U @ T ) ) ) ) ).
% R.Cong_iff_cong
thf(fact_1167_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_1168_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_1169_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_1170_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_1171_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_1172_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_1173_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_1174_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_1175_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_1176_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_1177_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_1178_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_1179_list_Oset__sel_I2_J,axiom,
! [A: list_list_a,X2: list_a] :
( ( A != nil_list_a )
=> ( ( member_list_a @ X2 @ ( set_list_a2 @ ( tl_list_a @ A ) ) )
=> ( member_list_a @ X2 @ ( set_list_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1180_list_Oset__sel_I2_J,axiom,
! [A: list_a,X2: a] :
( ( A != nil_a )
=> ( ( member_a @ X2 @ ( set_a2 @ ( tl_a @ A ) ) )
=> ( member_a @ X2 @ ( set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1181_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_1182_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_1183_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_1184_composite__of__arr__target,axiom,
! [T: list_a,B: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( member_list_a @ B @ ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ B @ T ) ) ) ).
% composite_of_arr_target
thf(fact_1185_composite__of__source__arr,axiom,
! [T: list_a,A: list_a] :
( ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ T )
=> ( ( member_list_a @ A @ ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ T ) )
=> ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ A @ T @ T ) ) ) ).
% composite_of_source_arr
thf(fact_1186_composite__ofE,axiom,
! [U: list_a,T: list_a,V: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ T @ V )
=> ~ ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ V ) )
=> ~ ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ U ) @ T ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ ( paths_in_Resid_a @ resid @ V @ U ) ) ) ) ) ) ).
% composite_ofE
thf(fact_1187_composite__of__cancel__left,axiom,
! [T: list_a,U: list_a,V: list_a,U5: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U5 @ V )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ U5 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U5 @ U ) ) ) ) ) ).
% composite_of_cancel_left
thf(fact_1188_composite__of__def,axiom,
! [U: list_a,T: list_a,V: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ T @ V )
= ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ V ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ U ) @ T ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ ( paths_in_Resid_a @ resid @ V @ U ) ) ) ) ) ).
% composite_of_def
thf(fact_1189_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_1190_composite__of__unq__upto__cong,axiom,
! [U: list_a,T: list_a,V: list_a,V4: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ T @ V )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ T @ V4 )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ V4 ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V4 @ V ) ) ) ) ) ).
% composite_of_unq_upto_cong
thf(fact_1191_con__prfx__composite__of_I2_J,axiom,
! [T: list_a,U: list_a,W: list_a,V: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ W )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W @ V )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ V ) ) ) ).
% con_prfx_composite_of(2)
thf(fact_1192_con__prfx__composite__of_I1_J,axiom,
! [T: list_a,U: list_a,W: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ W )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ W ) ) ).
% con_prfx_composite_of(1)
thf(fact_1193_resid__composite__of_I4_J,axiom,
! [T: list_a,U: list_a,W: list_a,V: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ W )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W @ V )
=> ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ V ) @ ( paths_in_Resid_a @ resid @ U @ ( paths_in_Resid_a @ resid @ V @ T ) ) @ ( paths_in_Resid_a @ resid @ W @ V ) ) ) ) ).
% resid_composite_of(4)
thf(fact_1194_resid__composite__of_I2_J,axiom,
! [T: list_a,U: list_a,W: list_a,V: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ W )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W @ V )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ T ) @ U ) ) ) ).
% resid_composite_of(2)
thf(fact_1195_resid__composite__of_I1_J,axiom,
! [T: list_a,U: list_a,W: list_a,V: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ W )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W @ V )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ V @ T ) @ ( paths_in_Resid_a @ resid @ W @ T ) ) ) ) ).
% resid_composite_of(1)
thf(fact_1196_bounded__imp__con,axiom,
! [T: list_a,U: list_a,V: list_a,T6: list_a,U5: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T6 @ U5 @ V )
=> ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ T6 ) ) ) ).
% bounded_imp_con
thf(fact_1197_con__composite__of__iff,axiom,
! [T: list_a,U: list_a,V: list_a,W: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W @ V )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ W @ T ) @ U ) ) ) ).
% con_composite_of_iff
thf(fact_1198_sources__composite__of,axiom,
! [U: list_a,T: list_a,V: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ T @ V )
=> ( ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ V )
= ( sources_list_a @ ( paths_in_Resid_a @ resid ) @ U ) ) ) ).
% sources_composite_of
thf(fact_1199_targets__composite__of,axiom,
! [U: list_a,T: list_a,V: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ T @ V )
=> ( ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ V )
= ( targets_list_a @ ( paths_in_Resid_a @ resid ) @ T ) ) ) ).
% targets_composite_of
thf(fact_1200_arr__composite__of,axiom,
! [U: list_a,T: list_a,V: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ T @ V )
=> ( arr_list_a @ ( paths_in_Resid_a @ resid ) @ V ) ) ).
% arr_composite_of
thf(fact_1201_join__of__def,axiom,
! [T: list_a,U: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V )
= ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ ( paths_in_Resid_a @ resid @ U @ T ) @ V )
& ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ ( paths_in_Resid_a @ resid @ T @ U ) @ V ) ) ) ).
% join_of_def
thf(fact_1202_join__ofE,axiom,
! [T: list_a,U: list_a,V: list_a] :
( ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V )
=> ~ ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ ( paths_in_Resid_a @ resid @ U @ T ) @ V )
=> ~ ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ ( paths_in_Resid_a @ resid @ T @ U ) @ V ) ) ) ).
% join_ofE
thf(fact_1203_composable__def,axiom,
! [T: list_a,U: list_a] :
( ( composable_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U )
= ( ? [X5: list_a] : ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ X5 ) ) ) ).
% composable_def
thf(fact_1204_resid__composite__of_I3_J,axiom,
! [T: list_a,U: list_a,W: list_a,V: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ W )
=> ( ( con_list_a @ ( paths_in_Resid_a @ resid ) @ W @ V )
=> ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ W ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ T ) @ U ) ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ T ) @ U ) @ ( paths_in_Resid_a @ resid @ V @ W ) ) ) ) ) ) ).
% resid_composite_of(3)
thf(fact_1205_composite__of__arr__ide,axiom,
! [B: list_a,T: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ B )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ B @ T )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ T ) @ B ) ) ) ).
% composite_of_arr_ide
thf(fact_1206_composite__of__ide__arr,axiom,
! [A: list_a,T: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ A )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ A @ T @ T )
= ( con_list_a @ ( paths_in_Resid_a @ resid ) @ T @ A ) ) ) ).
% composite_of_ide_arr
thf(fact_1207_composite__ofI,axiom,
! [U: list_a,V: list_a,T: list_a] :
( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ U @ V ) )
=> ( ( ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V @ U ) @ T ) )
& ( ide_list_a @ ( paths_in_Resid_a @ resid ) @ ( paths_in_Resid_a @ resid @ T @ ( paths_in_Resid_a @ resid @ V @ U ) ) ) )
=> ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ T @ V ) ) ) ).
% composite_ofI
thf(fact_1208_join__ofI,axiom,
! [T: list_a,U: list_a,V: list_a] :
( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ ( paths_in_Resid_a @ resid @ U @ T ) @ V )
=> ( ( composite_of_list_a @ ( paths_in_Resid_a @ resid ) @ U @ ( paths_in_Resid_a @ resid @ T @ U ) @ V )
=> ( join_of_list_a @ ( paths_in_Resid_a @ resid ) @ T @ U @ V ) ) ) ).
% join_ofI
thf(fact_1209_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a,T6: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ U @ T @ T6 )
=> ( ( member_a @ U @ NN )
=> ( normal_sub_Cong_a @ Resid @ NN @ T6 @ T ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_1210_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,U: list_a,T: list_a,T6: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( composite_of_list_a @ Resid @ U @ T @ T6 )
=> ( ( member_list_a @ U @ NN )
=> ( normal4889798360446511898list_a @ Resid @ NN @ T6 @ T ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_1211_rts_Ocomposite__of_Ocong,axiom,
composite_of_list_a = composite_of_list_a ).
% rts.composite_of.cong
thf(fact_1212_rts_Ocomposite__of_Ocong,axiom,
composite_of_a = composite_of_a ).
% rts.composite_of.cong
thf(fact_1213_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,T6: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ T6 )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ ( Resid @ T6 @ T ) @ NN )
& ( member_a @ ( Resid @ T @ T6 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_1214_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,T: list_a,U: list_a,T6: list_a] :
( ( cohere6429906645900029933list_a @ Resid @ NN )
=> ( ( composite_of_list_a @ Resid @ T @ U @ T6 )
=> ( ( member_list_a @ U @ NN )
=> ( ( member_list_a @ ( Resid @ T6 @ T ) @ NN )
& ( member_list_a @ ( Resid @ T @ T6 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_1215_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 ) )
& ! [A6: list_a] :
( ( ide_list_a @ Resid2 @ A6 )
=> ( member_list_a @ A6 @ NN2 ) )
& ! [U6: list_a,T5: list_a] :
( ( member_list_a @ U6 @ NN2 )
=> ( ( coinitial_list_a @ Resid2 @ T5 @ U6 )
=> ( member_list_a @ ( Resid2 @ U6 @ T5 ) @ NN2 ) ) )
& ! [U6: list_a,T5: list_a] :
( ( member_list_a @ U6 @ NN2 )
=> ( ( member_list_a @ ( Resid2 @ T5 @ U6 ) @ NN2 )
=> ( member_list_a @ T5 @ NN2 ) ) )
& ! [U6: list_a,T5: list_a] :
( ( member_list_a @ U6 @ NN2 )
=> ( ( seq_list_a @ Resid2 @ U6 @ T5 )
=> ? [X5: list_a] : ( composite_of_list_a @ Resid2 @ U6 @ T5 @ X5 ) ) )
& ! [U6: list_a,T5: list_a] :
( ( member_list_a @ U6 @ NN2 )
=> ( ( seq_list_a @ Resid2 @ T5 @ U6 )
=> ? [X5: list_a] : ( composite_of_list_a @ Resid2 @ T5 @ U6 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_1216_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 ) )
& ! [A6: a] :
( ( ide_a @ Resid2 @ A6 )
=> ( member_a @ A6 @ NN2 ) )
& ! [U6: a,T5: a] :
( ( member_a @ U6 @ NN2 )
=> ( ( coinitial_a @ Resid2 @ T5 @ U6 )
=> ( member_a @ ( Resid2 @ U6 @ T5 ) @ NN2 ) ) )
& ! [U6: a,T5: a] :
( ( member_a @ U6 @ NN2 )
=> ( ( member_a @ ( Resid2 @ T5 @ U6 ) @ NN2 )
=> ( member_a @ T5 @ NN2 ) ) )
& ! [U6: a,T5: a] :
( ( member_a @ U6 @ NN2 )
=> ( ( seq_a @ Resid2 @ U6 @ T5 )
=> ? [X5: a] : ( composite_of_a @ Resid2 @ U6 @ T5 @ X5 ) ) )
& ! [U6: a,T5: a] :
( ( member_a @ U6 @ NN2 )
=> ( ( seq_a @ Resid2 @ T5 @ U6 )
=> ? [X5: a] : ( composite_of_a @ Resid2 @ T5 @ U6 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_1217_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_list_a,Resid: list_a > list_a > list_a] :
( ! [T3: list_a] :
( ( member_list_a @ T3 @ NN )
=> ( arr_list_a @ Resid @ T3 ) )
=> ( ! [A4: list_a] :
( ( ide_list_a @ Resid @ A4 )
=> ( member_list_a @ A4 @ NN ) )
=> ( ! [U3: list_a,T3: list_a] :
( ( member_list_a @ U3 @ NN )
=> ( ( coinitial_list_a @ Resid @ T3 @ U3 )
=> ( member_list_a @ ( Resid @ U3 @ T3 ) @ NN ) ) )
=> ( ! [U3: list_a,T3: list_a] :
( ( member_list_a @ U3 @ NN )
=> ( ( member_list_a @ ( Resid @ T3 @ U3 ) @ NN )
=> ( member_list_a @ T3 @ NN ) ) )
=> ( ! [U3: list_a,T3: list_a] :
( ( member_list_a @ U3 @ NN )
=> ( ( seq_list_a @ Resid @ U3 @ T3 )
=> ? [X_1: list_a] : ( composite_of_list_a @ Resid @ U3 @ T3 @ X_1 ) ) )
=> ( ! [U3: list_a,T3: list_a] :
( ( member_list_a @ U3 @ NN )
=> ( ( seq_list_a @ Resid @ T3 @ U3 )
=> ? [X_1: list_a] : ( composite_of_list_a @ Resid @ T3 @ U3 @ X_1 ) ) )
=> ( normal2939518615156061708list_a @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_1218_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_a,Resid: a > a > a] :
( ! [T3: a] :
( ( member_a @ T3 @ NN )
=> ( arr_a @ Resid @ T3 ) )
=> ( ! [A4: a] :
( ( ide_a @ Resid @ A4 )
=> ( member_a @ A4 @ NN ) )
=> ( ! [U3: a,T3: a] :
( ( member_a @ U3 @ NN )
=> ( ( coinitial_a @ Resid @ T3 @ U3 )
=> ( member_a @ ( Resid @ U3 @ T3 ) @ NN ) ) )
=> ( ! [U3: a,T3: a] :
( ( member_a @ U3 @ NN )
=> ( ( member_a @ ( Resid @ T3 @ U3 ) @ NN )
=> ( member_a @ T3 @ NN ) ) )
=> ( ! [U3: a,T3: a] :
( ( member_a @ U3 @ NN )
=> ( ( seq_a @ Resid @ U3 @ T3 )
=> ? [X_1: a] : ( composite_of_a @ Resid @ U3 @ T3 @ X_1 ) ) )
=> ( ! [U3: a,T3: a] :
( ( member_a @ U3 @ NN )
=> ( ( seq_a @ Resid @ T3 @ U3 )
=> ? [X_1: a] : ( composite_of_a @ Resid @ T3 @ U3 @ X_1 ) ) )
=> ( normal7698203753654205830ioms_a @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_1219_R_Ocomposite__of__unq__upto__cong,axiom,
! [U: a,T: a,V: a,V4: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ( ( composite_of_a @ resid @ U @ T @ V4 )
=> ( ( ide_a @ resid @ ( resid @ V @ V4 ) )
& ( ide_a @ resid @ ( resid @ V4 @ V ) ) ) ) ) ).
% R.composite_of_unq_upto_cong
thf(fact_1220_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_1221_R_Ocomposite__of__def,axiom,
! [U: a,T: a,V: a] :
( ( composite_of_a @ resid @ U @ T @ V )
= ( ( ide_a @ resid @ ( resid @ U @ V ) )
& ( ide_a @ resid @ ( resid @ ( resid @ V @ U ) @ T ) )
& ( ide_a @ resid @ ( resid @ T @ ( resid @ V @ U ) ) ) ) ) ).
% R.composite_of_def
thf(fact_1222_R_Ocomposite__of__cancel__left,axiom,
! [T: a,U: a,V: a,U5: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( composite_of_a @ resid @ T @ U5 @ V )
=> ( ( ide_a @ resid @ ( resid @ U @ U5 ) )
& ( ide_a @ resid @ ( resid @ U5 @ U ) ) ) ) ) ).
% R.composite_of_cancel_left
thf(fact_1223_R_Ocomposite__ofE,axiom,
! [U: a,T: a,V: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ~ ( ( ide_a @ resid @ ( resid @ U @ V ) )
=> ~ ( ( ide_a @ resid @ ( resid @ ( resid @ V @ U ) @ T ) )
& ( ide_a @ resid @ ( resid @ T @ ( resid @ V @ U ) ) ) ) ) ) ).
% R.composite_ofE
thf(fact_1224_R_Ocon__composite__of__iff,axiom,
! [T: a,U: a,V: a,W: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( con_a @ resid @ W @ V )
= ( con_a @ resid @ ( resid @ W @ T ) @ U ) ) ) ).
% R.con_composite_of_iff
thf(fact_1225_R_Obounded__imp__con,axiom,
! [T: a,U: a,V: a,T6: a,U5: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( composite_of_a @ resid @ T6 @ U5 @ V )
=> ( con_a @ resid @ T @ T6 ) ) ) ).
% R.bounded_imp_con
thf(fact_1226_R_Oresid__composite__of_I1_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( con_a @ resid @ ( resid @ V @ T ) @ ( resid @ W @ T ) ) ) ) ).
% R.resid_composite_of(1)
thf(fact_1227_R_Oresid__composite__of_I2_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( con_a @ resid @ ( resid @ V @ T ) @ U ) ) ) ).
% R.resid_composite_of(2)
thf(fact_1228_R_Oresid__composite__of_I4_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( composite_of_a @ resid @ ( resid @ T @ V ) @ ( resid @ U @ ( resid @ V @ T ) ) @ ( resid @ W @ V ) ) ) ) ).
% R.resid_composite_of(4)
thf(fact_1229_R_Ocon__prfx__composite__of_I1_J,axiom,
! [T: a,U: a,W: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( con_a @ resid @ T @ W ) ) ).
% R.con_prfx_composite_of(1)
thf(fact_1230_R_Ocon__prfx__composite__of_I2_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( con_a @ resid @ T @ V ) ) ) ).
% R.con_prfx_composite_of(2)
thf(fact_1231_R_Oarr__composite__of,axiom,
! [U: a,T: a,V: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ( arr_a @ resid @ V ) ) ).
% R.arr_composite_of
thf(fact_1232_R_Otargets__composite__of,axiom,
! [U: a,T: a,V: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ( ( targets_a @ resid @ V )
= ( targets_a @ resid @ T ) ) ) ).
% R.targets_composite_of
thf(fact_1233_R_Osources__composite__of,axiom,
! [U: a,T: a,V: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ( ( sources_a @ resid @ V )
= ( sources_a @ resid @ U ) ) ) ).
% R.sources_composite_of
thf(fact_1234_R_Ojoin__of__def,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
= ( ( composite_of_a @ resid @ T @ ( resid @ U @ T ) @ V )
& ( composite_of_a @ resid @ U @ ( resid @ T @ U ) @ V ) ) ) ).
% R.join_of_def
thf(fact_1235_R_Ojoin__ofE,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ~ ( ( composite_of_a @ resid @ T @ ( resid @ U @ T ) @ V )
=> ~ ( composite_of_a @ resid @ U @ ( resid @ T @ U ) @ V ) ) ) ).
% R.join_ofE
thf(fact_1236_R_Ocomposable__def,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
= ( ? [X5: a] : ( composite_of_a @ resid @ T @ U @ X5 ) ) ) ).
% R.composable_def
thf(fact_1237_R_Ocomposite__of__ide__arr,axiom,
! [A: a,T: a] :
( ( ide_a @ resid @ A )
=> ( ( composite_of_a @ resid @ A @ T @ T )
= ( con_a @ resid @ T @ A ) ) ) ).
% R.composite_of_ide_arr
thf(fact_1238_R_Ocomposite__of__arr__ide,axiom,
! [B: a,T: a] :
( ( ide_a @ resid @ B )
=> ( ( composite_of_a @ resid @ T @ B @ T )
= ( con_a @ resid @ ( resid @ T @ T ) @ B ) ) ) ).
% R.composite_of_arr_ide
thf(fact_1239_R_Oresid__composite__of_I3_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ V @ W ) @ ( resid @ ( resid @ V @ T ) @ U ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ ( resid @ V @ T ) @ U ) @ ( resid @ V @ W ) ) ) ) ) ) ).
% R.resid_composite_of(3)
thf(fact_1240_R_Ocomposite__of__arr__target,axiom,
! [T: a,B: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( composite_of_a @ resid @ T @ B @ T ) ) ) ).
% R.composite_of_arr_target
thf(fact_1241_R_Ocomposite__of__source__arr,axiom,
! [T: a,A: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( composite_of_a @ resid @ A @ T @ T ) ) ) ).
% R.composite_of_source_arr
thf(fact_1242_R_Ocomposite__ofI,axiom,
! [U: a,V: a,T: a] :
( ( ide_a @ resid @ ( resid @ U @ V ) )
=> ( ( ( ide_a @ resid @ ( resid @ ( resid @ V @ U ) @ T ) )
& ( ide_a @ resid @ ( resid @ T @ ( resid @ V @ U ) ) ) )
=> ( composite_of_a @ resid @ U @ T @ V ) ) ) ).
% R.composite_ofI
thf(fact_1243_R_Ojoin__ofI,axiom,
! [T: a,U: a,V: a] :
( ( composite_of_a @ resid @ T @ ( resid @ U @ T ) @ V )
=> ( ( composite_of_a @ resid @ U @ ( resid @ T @ U ) @ V )
=> ( join_of_a @ resid @ T @ U @ V ) ) ) ).
% R.join_ofI
thf(fact_1244_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_1245_the__elem__eq,axiom,
! [X2: list_a] :
( ( the_elem_list_a @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_1246_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_1247_bot__empty__eq,axiom,
( bot_bot_list_a_o
= ( ^ [X3: list_a] : ( member_list_a @ X3 @ bot_bot_set_list_a ) ) ) ).
% bot_empty_eq
thf(fact_1248_the__elem__set,axiom,
! [X2: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_1249_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1250_Collect__empty__eq__bot,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( P = bot_bot_list_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1251_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A5: set_a] :
( A5
= ( insert_a @ ( the_elem_a @ A5 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1252_is__singleton__the__elem,axiom,
( is_singleton_list_a
= ( ^ [A5: set_list_a] :
( A5
= ( insert_list_a @ ( the_elem_list_a @ A5 ) @ bot_bot_set_list_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1253_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_1254_is__singletonI,axiom,
! [X2: list_a] : ( is_singleton_list_a @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ).
% is_singletonI
thf(fact_1255_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X: a,Y4: a] :
( ( member_a @ X @ A2 )
=> ( ( member_a @ Y4 @ A2 )
=> ( X = Y4 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1256_is__singletonI_H,axiom,
! [A2: set_list_a] :
( ( A2 != bot_bot_set_list_a )
=> ( ! [X: list_a,Y4: list_a] :
( ( member_list_a @ X @ A2 )
=> ( ( member_list_a @ Y4 @ A2 )
=> ( X = Y4 ) ) )
=> ( is_singleton_list_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1257_is__singletonE,axiom,
! [A2: set_a] :
( ( is_singleton_a @ A2 )
=> ~ ! [X: a] :
( A2
!= ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_1258_is__singletonE,axiom,
! [A2: set_list_a] :
( ( is_singleton_list_a @ A2 )
=> ~ ! [X: list_a] :
( A2
!= ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ).
% is_singletonE
thf(fact_1259_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A5: set_a] :
? [X3: a] :
( A5
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_1260_is__singleton__def,axiom,
( is_singleton_list_a
= ( ^ [A5: set_list_a] :
? [X3: list_a] :
( A5
= ( insert_list_a @ X3 @ bot_bot_set_list_a ) ) ) ) ).
% is_singleton_def
thf(fact_1261_Arr_Opelims_I3_J,axiom,
! [X2: list_a] :
( ~ ( paths_in_Arr_a @ resid @ X2 )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ~ ( accp_list_a @ paths_in_Arr_rel_a @ nil_a ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ ( cons_a @ T3 @ nil_a ) )
=> ( arr_a @ resid @ T3 ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( arr_a @ resid @ T3 )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% Arr.pelims(3)
thf(fact_1262_Arr_Opelims_I2_J,axiom,
! [X2: list_a] :
( ( paths_in_Arr_a @ resid @ X2 )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ X2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ ( cons_a @ T3 @ nil_a ) )
=> ~ ( arr_a @ resid @ T3 ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ~ ( ( arr_a @ resid @ T3 )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% Arr.pelims(2)
thf(fact_1263_Arr_Opelims_I1_J,axiom,
! [X2: list_a,Y2: $o] :
( ( ( paths_in_Arr_a @ resid @ X2 )
= Y2 )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ~ Y2
=> ~ ( accp_list_a @ paths_in_Arr_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y2
= ( arr_a @ resid @ T3 ) )
=> ~ ( accp_list_a @ paths_in_Arr_rel_a @ ( cons_a @ T3 @ nil_a ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( Y2
= ( ( arr_a @ resid @ T3 )
& ( paths_in_Arr_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) )
=> ~ ( accp_list_a @ paths_in_Arr_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% Arr.pelims(1)
thf(fact_1264_Srcs_Opelims,axiom,
! [X2: list_a,Y2: set_a] :
( ( ( paths_in_Srcs_a @ resid @ X2 )
= Y2 )
=> ( ( accp_list_a @ paths_in_Srcs_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y2 = bot_bot_set_a )
=> ~ ( accp_list_a @ paths_in_Srcs_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y2
= ( sources_a @ resid @ T3 ) )
=> ~ ( accp_list_a @ paths_in_Srcs_rel_a @ ( cons_a @ T3 @ nil_a ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( Y2
= ( sources_a @ resid @ T3 ) )
=> ~ ( accp_list_a @ paths_in_Srcs_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% Srcs.pelims
thf(fact_1265_Trgs_Opelims,axiom,
! [X2: list_a,Y2: set_a] :
( ( ( paths_in_Trgs_a @ resid @ X2 )
= Y2 )
=> ( ( accp_list_a @ paths_in_Trgs_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y2 = bot_bot_set_a )
=> ~ ( accp_list_a @ paths_in_Trgs_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y2
= ( targets_a @ resid @ T3 ) )
=> ~ ( accp_list_a @ paths_in_Trgs_rel_a @ ( cons_a @ T3 @ nil_a ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( Y2
= ( paths_in_Trgs_a @ resid @ ( cons_a @ V2 @ Va ) ) )
=> ~ ( accp_list_a @ paths_in_Trgs_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% Trgs.pelims
thf(fact_1266_paths__in__rts_OArr_Opelims_I1_J,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a,Y2: $o] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_in_Arr_list_a @ Resid @ X2 )
= Y2 )
=> ( ( accp_list_list_a @ paths_7232963753683178177list_a @ X2 )
=> ( ( ( X2 = nil_list_a )
=> ( ~ Y2
=> ~ ( accp_list_list_a @ paths_7232963753683178177list_a @ nil_list_a ) ) )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( ( Y2
= ( arr_list_a @ Resid @ T3 ) )
=> ~ ( accp_list_list_a @ paths_7232963753683178177list_a @ ( cons_list_a @ T3 @ nil_list_a ) ) ) )
=> ~ ! [T3: list_a,V2: list_a,Va: list_list_a] :
( ( X2
= ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ( ( Y2
= ( ( arr_list_a @ Resid @ T3 )
& ( paths_in_Arr_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) )
& ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T3 ) @ ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) ) ) ) )
=> ~ ( accp_list_list_a @ paths_7232963753683178177list_a @ ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Arr.pelims(1)
thf(fact_1267_paths__in__rts_OArr_Opelims_I1_J,axiom,
! [Resid: a > a > a,X2: list_a,Y2: $o] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Arr_a @ Resid @ X2 )
= Y2 )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ~ Y2
=> ~ ( accp_list_a @ paths_in_Arr_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y2
= ( arr_a @ Resid @ T3 ) )
=> ~ ( accp_list_a @ paths_in_Arr_rel_a @ ( cons_a @ T3 @ nil_a ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( Y2
= ( ( arr_a @ Resid @ T3 )
& ( paths_in_Arr_a @ Resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T3 ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) )
=> ~ ( accp_list_a @ paths_in_Arr_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Arr.pelims(1)
thf(fact_1268_paths__in__rts_OArr_Opelims_I2_J,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Arr_list_a @ Resid @ X2 )
=> ( ( accp_list_list_a @ paths_7232963753683178177list_a @ X2 )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( ( accp_list_list_a @ paths_7232963753683178177list_a @ ( cons_list_a @ T3 @ nil_list_a ) )
=> ~ ( arr_list_a @ Resid @ T3 ) ) )
=> ~ ! [T3: list_a,V2: list_a,Va: list_list_a] :
( ( X2
= ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ( ( accp_list_list_a @ paths_7232963753683178177list_a @ ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ~ ( ( arr_list_a @ Resid @ T3 )
& ( paths_in_Arr_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) )
& ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T3 ) @ ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Arr.pelims(2)
thf(fact_1269_paths__in__rts_OArr_Opelims_I2_J,axiom,
! [Resid: a > a > a,X2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Arr_a @ Resid @ X2 )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ X2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ ( cons_a @ T3 @ nil_a ) )
=> ~ ( arr_a @ Resid @ T3 ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ~ ( ( arr_a @ Resid @ T3 )
& ( paths_in_Arr_a @ Resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T3 ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Arr.pelims(2)
thf(fact_1270_Ide_Opelims_I1_J,axiom,
! [X2: list_a,Y2: $o] :
( ( ( paths_in_Ide_a @ resid @ X2 )
= Y2 )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ~ Y2
=> ~ ( accp_list_a @ paths_in_Ide_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y2
= ( ide_a @ resid @ T3 ) )
=> ~ ( accp_list_a @ paths_in_Ide_rel_a @ ( cons_a @ T3 @ nil_a ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( Y2
= ( ( ide_a @ resid @ T3 )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) )
=> ~ ( accp_list_a @ paths_in_Ide_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% Ide.pelims(1)
thf(fact_1271_Ide_Opelims_I2_J,axiom,
! [X2: list_a] :
( ( paths_in_Ide_a @ resid @ X2 )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ X2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ ( cons_a @ T3 @ nil_a ) )
=> ~ ( ide_a @ resid @ T3 ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ~ ( ( ide_a @ resid @ T3 )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% Ide.pelims(2)
thf(fact_1272_Ide_Opelims_I3_J,axiom,
! [X2: list_a] :
( ~ ( paths_in_Ide_a @ resid @ X2 )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ~ ( accp_list_a @ paths_in_Ide_rel_a @ nil_a ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ ( cons_a @ T3 @ nil_a ) )
=> ( ide_a @ resid @ T3 ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( ide_a @ resid @ T3 )
& ( paths_in_Ide_a @ resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ resid @ T3 ) @ ( paths_in_Srcs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% Ide.pelims(3)
thf(fact_1273_paths__in__rts_OIde_Opelims_I1_J,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a,Y2: $o] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_in_Ide_list_a @ Resid @ X2 )
= Y2 )
=> ( ( accp_list_list_a @ paths_4942548060537684810list_a @ X2 )
=> ( ( ( X2 = nil_list_a )
=> ( ~ Y2
=> ~ ( accp_list_list_a @ paths_4942548060537684810list_a @ nil_list_a ) ) )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( ( Y2
= ( ide_list_a @ Resid @ T3 ) )
=> ~ ( accp_list_list_a @ paths_4942548060537684810list_a @ ( cons_list_a @ T3 @ nil_list_a ) ) ) )
=> ~ ! [T3: list_a,V2: list_a,Va: list_list_a] :
( ( X2
= ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ( ( Y2
= ( ( ide_list_a @ Resid @ T3 )
& ( paths_in_Ide_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) )
& ( ord_le8861187494160871172list_a @ ( targets_list_a @ Resid @ T3 ) @ ( paths_in_Srcs_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) ) ) ) )
=> ~ ( accp_list_list_a @ paths_4942548060537684810list_a @ ( cons_list_a @ T3 @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.pelims(1)
thf(fact_1274_paths__in__rts_OIde_Opelims_I1_J,axiom,
! [Resid: a > a > a,X2: list_a,Y2: $o] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Ide_a @ Resid @ X2 )
= Y2 )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ~ Y2
=> ~ ( accp_list_a @ paths_in_Ide_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y2
= ( ide_a @ Resid @ T3 ) )
=> ~ ( accp_list_a @ paths_in_Ide_rel_a @ ( cons_a @ T3 @ nil_a ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( Y2
= ( ( ide_a @ Resid @ T3 )
& ( paths_in_Ide_a @ Resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T3 ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) )
=> ~ ( accp_list_a @ paths_in_Ide_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.pelims(1)
thf(fact_1275_paths__in__rts_OIde_Opelims_I2_J,axiom,
! [Resid: a > a > a,X2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( paths_in_Ide_a @ Resid @ X2 )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ X2 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ ( cons_a @ T3 @ nil_a ) )
=> ~ ( ide_a @ Resid @ T3 ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ~ ( ( ide_a @ Resid @ T3 )
& ( paths_in_Ide_a @ Resid @ ( cons_a @ V2 @ Va ) )
& ( ord_less_eq_set_a @ ( targets_a @ Resid @ T3 ) @ ( paths_in_Srcs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.pelims(2)
thf(fact_1276_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_1277_R_Oidentities__form__normal__sub__rts,axiom,
normal_sub_rts_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ).
% R.identities_form_normal_sub_rts
% Helper facts (5)
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y2: list_a] :
( ( if_list_a @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y2: list_a] :
( ( if_list_a @ $true @ X2 @ Y2 )
= X2 ) ).
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,
! [X2: list_list_a,Y2: list_list_a] :
( ( if_list_list_a @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: list_list_a,Y2: list_list_a] :
( ( if_list_list_a @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
( ( paths_in_Resid_a @ resid @ ta @ ( append_a @ ( cons_a @ v @ va ) @ ua ) )
!= nil_a ) ).
thf(conj_1,conjecture,
( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ta @ ( cons_a @ v @ va ) ) @ ua )
!= nil_a ) ).
%------------------------------------------------------------------------------