TPTP Problem File: SLH0691^1.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_05103_200899__14213426_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1523 ( 396 unt; 240 typ; 0 def)
% Number of atoms : 4638 (2055 equ; 0 cnn)
% Maximal formula atoms : 18 ( 3 avg)
% Number of connectives : 18600 ( 913 ~; 45 |; 495 &;14872 @)
% ( 0 <=>;2275 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 1482 (1482 >; 0 *; 0 +; 0 <<)
% Number of symbols : 218 ( 215 usr; 25 con; 0-4 aty)
% Number of variables : 3950 ( 84 ^;3749 !; 117 ?;3950 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:48:07.611
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_Pr5382606609415531783list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
produc1473018763691903991list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
produc7709606177366032167list_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
list_P321204300973800749list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr4048851178543822343list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_M_062_Itf__a_M_Eo_J_J_Mt__List__Olist_Itf__a_J_J,type,
produc5032551385658279741list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
list_P4541805568828049459list_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
list_P2210424090985720871st_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr2070066670564046349list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
set_Pr8962057229576493569st_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
produc8685980395799941037list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
produc2579390645249093025st_a_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
list_P1396940483166286381od_a_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__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 (215)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
gcd_Gcd_nat: set_nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
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_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( a > list_a ) > list_a ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Odistinct__adj_001tf__a,type,
distinct_adj_a: list_a > $o ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
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_Olex_001t__List__Olist_Itf__a_J,type,
lex_list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olex_001tf__a,type,
lex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
cons_list_list_a: list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
cons_P5184657343811988189list_a: produc9164743771328383783list_a > list_P321204300973800749list_a > list_P321204300973800749list_a ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
cons_P2018802349718741079st_a_a: produc2579390645249093025st_a_a > list_P2210424090985720871st_a_a > list_P2210424090985720871st_a_a ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
cons_P8125392100269589091list_a: produc8685980395799941037list_a > list_P4541805568828049459list_a > list_P4541805568828049459list_a ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
cons_P7316939126706565853od_a_a: product_prod_a_a > list_P1396940483166286381od_a_a > list_P1396940483166286381od_a_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
nil_Pr3188421586756112173list_a: list_P321204300973800749list_a ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
nil_Pr523822031547952295st_a_a: list_P2210424090985720871st_a_a ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
nil_Pr6630411782098800307list_a: list_P4541805568828049459list_a ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
nil_Product_prod_a_a: list_P1396940483166286381od_a_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_Osize__list_001tf__a,type,
size_list_a: ( a > nat ) > list_a > nat ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olistrel1_001t__List__Olist_Itf__a_J,type,
listrel1_list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olistrel1_001tf__a,type,
listrel1_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Oremdups__adj_001tf__a,type,
remdups_adj_a: list_a > list_a ).
thf(sy_c_List_Oremdups__adj__rel_001tf__a,type,
remdups_adj_rel_a: list_a > list_a > $o ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Oshuffles_001tf__a,type,
shuffles_a: list_a > list_a > set_list_a ).
thf(sy_c_List_Oshuffles__rel_001tf__a,type,
shuffles_rel_a: produc9164743771328383783list_a > produc9164743771328383783list_a > $o ).
thf(sy_c_List_Osplice_001tf__a,type,
splice_a: list_a > list_a > list_a ).
thf(sy_c_List_Osplice__rel_001tf__a,type,
splice_rel_a: produc9164743771328383783list_a > produc9164743771328383783list_a > $o ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
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_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_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
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_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
ord_le7857023143581076903list_a: set_Pr4048851178543822343list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
ord_le2808437291371905441st_a_a: set_Pr8962057229576493569st_a_a > set_Pr8962057229576493569st_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
ord_le5139818769214234029list_a: set_Pr2070066670564046349list_a > set_Pr2070066670564046349list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
produc8111569692950616493list_a: ( a > a > $o ) > list_a > produc5032551385658279741list_a ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc8643929849434629545list_a: ( a > a ) > produc9164743771328383783list_a > produc1473018763691903991list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
produc8696003437204565271list_a: list_list_a > list_list_a > produc7709606177366032167list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001tf__a,type,
produc4781227316648555537st_a_a: list_a > a > produc2579390645249093025st_a_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__List__Olist_Itf__a_J,type,
produc6670463072477821725list_a: a > list_a > produc8685980395799941037list_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001t__List__Olist_Itf__a_J,type,
cohere6429906645900029933list_a: ( list_a > list_a > list_a ) > set_list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
cohere5499257456005325328list_a: ( produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a ) > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
cohere8414546589650652426st_a_a: ( produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a ) > set_Pr8962057229576493569st_a_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
cohere5297764303346724630list_a: ( produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a ) > set_Pr2070066670564046349list_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
cohere3906135489564710160od_a_a: ( product_prod_a_a > product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_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_Ocoherent__normal__sub__rts__axioms_001tf__a,type,
cohere4894532172567702276ioms_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oconfluent__rts_001tf__a,type,
confluent_rts_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oconfluent__rts__axioms_001tf__a,type,
conflu3014480972103220363ioms_a: ( a > a > 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_001tf__a,type,
normal_sub_Cong_a: ( a > a > a ) > set_a > a > 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_001tf__a,type,
partial_magma_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_Onull_001t__List__Olist_Itf__a_J,type,
partial_null_list_a: ( list_a > list_a > list_a ) > list_a ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_Onull_001tf__a,type,
partial_null_a: ( a > a > a ) > a ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__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_001t__Nat__Onat,type,
paths_in_rts_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
paths_893215811548098956list_a: ( produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a ) > $o ).
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target2287938126545038671list_a: ( produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a ) > produc8685980395799941037list_a > set_Pr2070066670564046349list_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
target5293506191220573129od_a_a: ( product_prod_a_a > product_prod_a_a > product_prod_a_a ) > product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001tf__a,type,
targets_a: ( a > a > a ) > a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001tf__a,type,
rts_axioms_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites__axioms_001tf__a,type,
rts_wi2614412583573296275ioms_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__joins__axioms_001tf__a,type,
rts_wi560353115624263628ioms_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__axioms_001tf__a_001tf__a,type,
simula3868467710248865958ms_a_a: ( a > a > a ) > ( a > 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_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
collec943055143889122450list_a: ( produc9164743771328383783list_a > $o ) > set_Pr4048851178543822343list_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
collec3957028472668211340st_a_a: ( produc2579390645249093025st_a_a > $o ) > set_Pr8962057229576493569st_a_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
collec840246186364283544list_a: ( produc8685980395799941037list_a > $o ) > set_Pr2070066670564046349list_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec3336397797384452498od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_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_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
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_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
accp_P7377042638478740784list_a: ( produc9164743771328383783list_a > produc9164743771328383783list_a > $o ) > produc9164743771328383783list_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
accp_P3213725926765619766list_a: ( produc8685980395799941037list_a > produc8685980395799941037list_a > $o ) > produc8685980395799941037list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member1318342207407915856list_a: produc7709606177366032167list_a > set_Pr5382606609415531783list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member8006451231845903178st_a_a: produc2579390645249093025st_a_a > set_Pr8962057229576493569st_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member4889668945541975382list_a: produc8685980395799941037list_a > set_Pr2070066670564046349list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_T_H____,type,
t: list_a ).
thf(sy_v_Ta____,type,
ta: list_a ).
thf(sy_v_U_H____,type,
u: list_a ).
thf(sy_v_Ua____,type,
ua: list_a ).
thf(sy_v_V_H____,type,
v: list_a ).
thf(sy_v_Va____,type,
va: list_a ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_resid,type,
resid: a > a > a ).
thf(sy_v_t____,type,
t2: a ).
thf(sy_v_u____,type,
u2: a ).
thf(sy_v_v____,type,
v2: a ).
% Relevant facts (1277)
thf(fact_0_Resid1x__rel_Ocong,axiom,
paths_6492648068886854876_rel_a = paths_6492648068886854876_rel_a ).
% Resid1x_rel.cong
thf(fact_1_T_H,axiom,
t = nil_a ).
% T'
thf(fact_2_V_H,axiom,
v = nil_a ).
% V'
thf(fact_3_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_4_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_5_U_H,axiom,
u != nil_a ).
% U'
thf(fact_6_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_7_V,axiom,
( va
= ( cons_a @ v2 @ v ) ) ).
% V
thf(fact_8_Resid_Osimps_I1_J,axiom,
! [Uu: list_a] :
( ( paths_in_Resid_a @ resid @ nil_a @ Uu )
= nil_a ) ).
% Resid.simps(1)
thf(fact_9_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_10_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_11_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_12_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_13_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_14_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_15_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_16_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_17_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_18_Resid__rel_Ocong,axiom,
paths_in_Resid_rel_a = paths_in_Resid_rel_a ).
% Resid_rel.cong
thf(fact_19__092_060open_062_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091t_093_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_Iu_A_092_At_A_D_AU_H_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091t_A_092_Au_093_J_A_092_060noteq_062_A_091_093_J_A_061_A_I_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091t_093_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091u_A_092_At_093_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IU_H_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091t_A_092_Au_093_J_A_092_060noteq_062_A_091_093_J_092_060close_062,axiom,
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ( cons_a @ t2 @ nil_a ) ) @ ( cons_a @ ( resid @ u2 @ t2 ) @ ( paths_in_Resid_a @ resid @ u @ ( cons_a @ ( resid @ t2 @ u2 ) @ nil_a ) ) ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ( cons_a @ t2 @ nil_a ) ) @ ( cons_a @ ( resid @ u2 @ t2 ) @ nil_a ) ) @ ( paths_in_Resid_a @ resid @ u @ ( cons_a @ ( resid @ t2 @ u2 ) @ nil_a ) ) )
!= nil_a ) ) ).
% \<open>((V \<^sup>*\\<^sup>* [t]) \<^sup>*\\<^sup>* (u \ t # U' \<^sup>*\\<^sup>* [t \ u]) \<noteq> []) = (((V \<^sup>*\\<^sup>* [t]) \<^sup>*\\<^sup>* [u \ t]) \<^sup>*\\<^sup>* (U' \<^sup>*\\<^sup>* [t \ u]) \<noteq> [])\<close>
thf(fact_20_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_21_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_22_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_23_paths__in__rts__axioms,axiom,
paths_in_rts_a @ resid ).
% paths_in_rts_axioms
thf(fact_24_U,axiom,
( ua
= ( cons_a @ u2 @ u ) ) ).
% U
thf(fact_25_paths__in__rts_OResid_Ocong,axiom,
paths_in_Resid_a = paths_in_Resid_a ).
% paths_in_rts.Resid.cong
thf(fact_26_Con__VT,axiom,
( ( paths_in_Resid_a @ resid @ va @ ta )
!= nil_a ) ).
% Con_VT
thf(fact_27_R_Opartial__magma__axioms,axiom,
partial_magma_a @ resid ).
% R.partial_magma_axioms
thf(fact_28_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_29_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_30_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_31_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_32_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_33_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_34_mem__Collect__eq,axiom,
! [A: produc9164743771328383783list_a,P: produc9164743771328383783list_a > $o] :
( ( member8191768239178080336list_a @ A @ ( collec943055143889122450list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_35_mem__Collect__eq,axiom,
! [A: produc2579390645249093025st_a_a,P: produc2579390645249093025st_a_a > $o] :
( ( member8006451231845903178st_a_a @ A @ ( collec3957028472668211340st_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_36_mem__Collect__eq,axiom,
! [A: produc8685980395799941037list_a,P: produc8685980395799941037list_a > $o] :
( ( member4889668945541975382list_a @ A @ ( collec840246186364283544list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_37_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_38_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_39_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_40_Collect__mem__eq,axiom,
! [A2: set_Pr4048851178543822343list_a] :
( ( collec943055143889122450list_a
@ ^ [X3: produc9164743771328383783list_a] : ( member8191768239178080336list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_41_Collect__mem__eq,axiom,
! [A2: set_Pr8962057229576493569st_a_a] :
( ( collec3957028472668211340st_a_a
@ ^ [X3: produc2579390645249093025st_a_a] : ( member8006451231845903178st_a_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_42_Collect__mem__eq,axiom,
! [A2: set_Pr2070066670564046349list_a] :
( ( collec840246186364283544list_a
@ ^ [X3: produc8685980395799941037list_a] : ( member4889668945541975382list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_44_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_45_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_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_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_48_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_49_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_50_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_51_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_52_R_Ocon__sym,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ U @ T ) ) ).
% R.con_sym
thf(fact_53_Con__TU,axiom,
( ( paths_in_Resid_a @ resid @ ta @ ua )
!= nil_a ) ).
% Con_TU
thf(fact_54_Residx1_Osimps_I1_J,axiom,
! [U: a] :
( ( paths_in_Residx1_a @ resid @ nil_a @ U )
= nil_a ) ).
% Residx1.simps(1)
thf(fact_55_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_56_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_57_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_58_T,axiom,
( ta
= ( cons_a @ t2 @ t ) ) ).
% T
thf(fact_59_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_60_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_61_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_62_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_63_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_64__092_060open_062V_A_061_A_091_093_A_092_060Longrightarrow_062_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_092_060noteq_062_A_091_093_J_A_061_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060noteq_062_A_091_093_J_A_092_060and_062_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_092_060noteq_062_A_091_093_A_092_060longrightarrow_062_A_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_061_A_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_J_092_060close_062,axiom,
( ( va = nil_a )
=> ( ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ta ) @ ( paths_in_Resid_a @ resid @ ua @ ta ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ua ) @ ( paths_in_Resid_a @ resid @ ta @ ua ) )
!= nil_a ) )
& ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ta ) @ ( paths_in_Resid_a @ resid @ ua @ ta ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ta ) @ ( paths_in_Resid_a @ resid @ ua @ ta ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ua ) @ ( paths_in_Resid_a @ resid @ ta @ ua ) ) ) ) ) ) ).
% \<open>V = [] \<Longrightarrow> ((V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) \<noteq> []) = ((V \<^sup>*\\<^sup>* U) \<^sup>*\\<^sup>* (T \<^sup>*\\<^sup>* U) \<noteq> []) \<and> ((V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) \<noteq> [] \<longrightarrow> (V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) = (V \<^sup>*\\<^sup>* U) \<^sup>*\\<^sup>* (T \<^sup>*\\<^sup>* U))\<close>
thf(fact_65__092_060open_062U_A_061_A_091_093_A_092_060Longrightarrow_062_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_092_060noteq_062_A_091_093_J_A_061_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060noteq_062_A_091_093_J_A_092_060and_062_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_092_060noteq_062_A_091_093_A_092_060longrightarrow_062_A_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_061_A_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_J_092_060close_062,axiom,
( ( ua = nil_a )
=> ( ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ta ) @ ( paths_in_Resid_a @ resid @ ua @ ta ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ua ) @ ( paths_in_Resid_a @ resid @ ta @ ua ) )
!= nil_a ) )
& ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ta ) @ ( paths_in_Resid_a @ resid @ ua @ ta ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ta ) @ ( paths_in_Resid_a @ resid @ ua @ ta ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ua ) @ ( paths_in_Resid_a @ resid @ ta @ ua ) ) ) ) ) ) ).
% \<open>U = [] \<Longrightarrow> ((V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) \<noteq> []) = ((V \<^sup>*\\<^sup>* U) \<^sup>*\\<^sup>* (T \<^sup>*\\<^sup>* U) \<noteq> []) \<and> ((V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) \<noteq> [] \<longrightarrow> (V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) = (V \<^sup>*\\<^sup>* U) \<^sup>*\\<^sup>* (T \<^sup>*\\<^sup>* U))\<close>
thf(fact_66__092_060open_062T_A_061_A_091_093_A_092_060Longrightarrow_062_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_092_060noteq_062_A_091_093_J_A_061_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060noteq_062_A_091_093_J_A_092_060and_062_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_092_060noteq_062_A_091_093_A_092_060longrightarrow_062_A_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_061_A_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_J_092_060close_062,axiom,
( ( ta = nil_a )
=> ( ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ta ) @ ( paths_in_Resid_a @ resid @ ua @ ta ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ua ) @ ( paths_in_Resid_a @ resid @ ta @ ua ) )
!= nil_a ) )
& ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ta ) @ ( paths_in_Resid_a @ resid @ ua @ ta ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ta ) @ ( paths_in_Resid_a @ resid @ ua @ ta ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ua ) @ ( paths_in_Resid_a @ resid @ ta @ ua ) ) ) ) ) ) ).
% \<open>T = [] \<Longrightarrow> ((V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) \<noteq> []) = ((V \<^sup>*\\<^sup>* U) \<^sup>*\\<^sup>* (T \<^sup>*\\<^sup>* U) \<noteq> []) \<and> ((V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) \<noteq> [] \<longrightarrow> (V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) = (V \<^sup>*\\<^sup>* U) \<^sup>*\\<^sup>* (T \<^sup>*\\<^sup>* U))\<close>
thf(fact_67_calculation,axiom,
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ta ) @ ( paths_in_Resid_a @ resid @ ua @ ta ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ( cons_a @ t2 @ nil_a ) ) @ ( cons_a @ ( resid @ u2 @ t2 ) @ nil_a ) ) @ ( paths_in_Resid_a @ resid @ u @ ( cons_a @ ( resid @ t2 @ u2 ) @ nil_a ) ) )
!= nil_a ) ) ).
% calculation
thf(fact_68__092_060open_062_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_092_060noteq_062_A_091_093_J_A_061_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091t_093_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_Iu_A_092_At_A_D_AU_H_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_091t_A_092_Au_093_J_A_092_060noteq_062_A_091_093_J_092_060close_062,axiom,
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ta ) @ ( paths_in_Resid_a @ resid @ ua @ ta ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ( cons_a @ t2 @ nil_a ) ) @ ( cons_a @ ( resid @ u2 @ t2 ) @ ( paths_in_Resid_a @ resid @ u @ ( cons_a @ ( resid @ t2 @ u2 ) @ nil_a ) ) ) )
!= nil_a ) ) ).
% \<open>((V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) \<noteq> []) = ((V \<^sup>*\\<^sup>* [t]) \<^sup>*\\<^sup>* (u \ t # U' \<^sup>*\\<^sup>* [t \ u]) \<noteq> [])\<close>
thf(fact_69_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_70_residuation_Ocon_Ocong,axiom,
con_a = con_a ).
% residuation.con.cong
thf(fact_71_paths__in__rts_OTrgs_Ocong,axiom,
paths_in_Trgs_a = paths_in_Trgs_a ).
% paths_in_rts.Trgs.cong
thf(fact_72_paths__in__rts_OResid1x_Ocong,axiom,
paths_in_Resid1x_a = paths_in_Resid1x_a ).
% paths_in_rts.Resid1x.cong
thf(fact_73_paths__in__rts_OResidx1_Ocong,axiom,
paths_in_Residx1_a = paths_in_Residx1_a ).
% paths_in_rts.Residx1.cong
thf(fact_74_paths__in__rts_OTrgs__are__con,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,B: produc9164743771328383783list_a,T4: list_P321204300973800749list_a,B2: produc9164743771328383783list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ( member8191768239178080336list_a @ B @ ( paths_8434672324333973362list_a @ Resid @ T4 ) )
=> ( ( member8191768239178080336list_a @ B2 @ ( paths_8434672324333973362list_a @ Resid @ T4 ) )
=> ( con_Pr6815052424635904531list_a @ Resid @ B @ B2 ) ) ) ) ).
% paths_in_rts.Trgs_are_con
thf(fact_75_paths__in__rts_OTrgs__are__con,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,B: produc2579390645249093025st_a_a,T4: list_P2210424090985720871st_a_a,B2: produc2579390645249093025st_a_a] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ( member8006451231845903178st_a_a @ B @ ( paths_2794423126656725868st_a_a @ Resid @ T4 ) )
=> ( ( member8006451231845903178st_a_a @ B2 @ ( paths_2794423126656725868st_a_a @ Resid @ T4 ) )
=> ( con_Pr7383801431968512653st_a_a @ Resid @ B @ B2 ) ) ) ) ).
% paths_in_rts.Trgs_are_con
thf(fact_76_paths__in__rts_OTrgs__are__con,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,B: produc8685980395799941037list_a,T4: list_P4541805568828049459list_a,B2: produc8685980395799941037list_a] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ( member4889668945541975382list_a @ B @ ( paths_8901012877207573880list_a @ Resid @ T4 ) )
=> ( ( member4889668945541975382list_a @ B2 @ ( paths_8901012877207573880list_a @ Resid @ T4 ) )
=> ( con_Pr4267019145664584857list_a @ Resid @ B @ B2 ) ) ) ) ).
% paths_in_rts.Trgs_are_con
thf(fact_77_paths__in__rts_OTrgs__are__con,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,B: product_prod_a_a,T4: list_P1396940483166286381od_a_a,B2: product_prod_a_a] :
( ( paths_2703364527051407500od_a_a @ Resid )
=> ( ( member1426531477525435216od_a_a @ B @ ( paths_4392315070405530738od_a_a @ Resid @ T4 ) )
=> ( ( member1426531477525435216od_a_a @ B2 @ ( paths_4392315070405530738od_a_a @ Resid @ T4 ) )
=> ( con_Product_prod_a_a @ Resid @ B @ B2 ) ) ) ) ).
% paths_in_rts.Trgs_are_con
thf(fact_78_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_79_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_80_partial__magma__def,axiom,
( partial_magma_a
= ( ^ [OP: a > a > a] :
? [X3: a] :
( ! [T5: a] :
( ( ( OP @ X3 @ T5 )
= X3 )
& ( ( OP @ T5 @ X3 )
= X3 ) )
& ! [Y2: a] :
( ! [T5: a] :
( ( ( OP @ Y2 @ T5 )
= Y2 )
& ( ( OP @ T5 @ Y2 )
= Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_81_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_82_partial__magma_Ointro,axiom,
! [OP2: a > a > a] :
( ? [X4: a] :
( ! [T3: a] :
( ( ( OP2 @ X4 @ T3 )
= X4 )
& ( ( OP2 @ T3 @ X4 )
= X4 ) )
& ! [Y3: a] :
( ! [T2: a] :
( ( ( OP2 @ Y3 @ T2 )
= Y3 )
& ( ( OP2 @ T2 @ Y3 )
= Y3 ) )
=> ( Y3 = X4 ) ) )
=> ( partial_magma_a @ OP2 ) ) ).
% partial_magma.intro
thf(fact_83_paths__in__rts_OResidx1_Osimps_I1_J,axiom,
! [Resid: list_a > list_a > list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_3541054012941122297list_a @ Resid @ nil_list_a @ U )
= nil_list_a ) ) ).
% paths_in_rts.Residx1.simps(1)
thf(fact_84_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_85_paths__in__rts_OTrgs_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_Trgs_list_a @ Resid @ ( cons_list_a @ T @ ( cons_list_a @ V @ Va2 ) ) )
= ( paths_in_Trgs_list_a @ Resid @ ( cons_list_a @ V @ Va2 ) ) ) ) ).
% paths_in_rts.Trgs.simps(3)
thf(fact_86_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_87_paths__in__rts_OResid1x_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 )
=> ( ( paths_1777230443808135851list_a @ Resid @ T @ ( cons_list_a @ U @ ( cons_list_a @ V @ Va2 ) ) )
= ( paths_1777230443808135851list_a @ Resid @ ( Resid @ T @ U ) @ ( cons_list_a @ V @ Va2 ) ) ) ) ).
% paths_in_rts.Resid1x.simps(3)
thf(fact_88_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_89_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_90_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_91_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_92_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_93_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_94_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_95_paths__in__rts_OResid1x_Osimps_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_1777230443808135851list_a @ Resid @ T @ ( cons_list_a @ U @ nil_list_a ) )
= ( Resid @ T @ U ) ) ) ).
% paths_in_rts.Resid1x.simps(2)
thf(fact_96_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_97_paths__in__rts_OTrgs__Resid__sym,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 )
!= nil_list_a )
=> ( ( paths_in_Trgs_list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 ) )
= ( paths_in_Trgs_list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ U2 @ T4 ) ) ) ) ) ).
% paths_in_rts.Trgs_Resid_sym
thf(fact_98_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_99_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_100_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_101_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_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_paths__in__rts_OResid__cons_I1_J,axiom,
! [Resid: list_a > list_a > list_a,U2: list_list_a,T: list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( U2 != nil_list_a )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ T4 ) @ U2 )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ T4 ) @ U2 )
= ( append_list_a @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ U2 ) @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( paths_8620460302779588466list_a @ Resid @ U2 @ ( cons_list_a @ T @ nil_list_a ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_cons(1)
thf(fact_110_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_111_paths__in__rts_OResid__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 )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ T4 ) @ ( cons_list_a @ U @ U2 ) )
= ( append_list_a @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ ( Resid @ T @ U ) @ nil_list_a ) @ U2 ) @ ( 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 ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_rec(4)
thf(fact_112_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_113_paths__in__rts_OResidx1__as__Resid,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 )
= ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ nil_list_a ) ) ) ) ).
% paths_in_rts.Residx1_as_Resid
thf(fact_114_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_115_paths__in__rts_OTrgs__Resid__sym__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_Trgs_list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ nil_list_a ) ) )
= ( paths_in_Trgs_list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ U @ nil_list_a ) @ T4 ) ) ) ) ) ).
% paths_in_rts.Trgs_Resid_sym_Arr_single
thf(fact_116_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_117_paths__in__rts_OResid1x__as__Resid,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_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ U2 )
= ( cons_list_a @ ( paths_1777230443808135851list_a @ Resid @ T @ U2 ) @ nil_list_a ) ) ) ) ).
% paths_in_rts.Resid1x_as_Resid
thf(fact_118_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_119_paths__in__rts_OResid__cons_H,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,T: list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( T4 != nil_list_a )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ T4 ) @ U2 )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ T4 ) @ U2 )
= ( cons_list_a @ ( paths_1777230443808135851list_a @ Resid @ T @ U2 ) @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( paths_8620460302779588466list_a @ Resid @ U2 @ ( cons_list_a @ T @ nil_list_a ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_cons'
thf(fact_120_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_121_paths__in__rts_OTrgs_Ocases,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( X2 != nil_list_a )
=> ( ! [T3: list_a] :
( X2
!= ( 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 ) ) ) ) ) ) ).
% paths_in_rts.Trgs.cases
thf(fact_122_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_123_paths__in__rts_OCon__sym,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 )
!= nil_list_a )
= ( ( paths_8620460302779588466list_a @ Resid @ U2 @ T4 )
!= nil_list_a ) ) ) ).
% paths_in_rts.Con_sym
thf(fact_124_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_125_paths__in__rts_OResid_Osimps_I1_J,axiom,
! [Resid: list_a > list_a > list_a,Uu: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_8620460302779588466list_a @ Resid @ nil_list_a @ Uu )
= nil_list_a ) ) ).
% paths_in_rts.Resid.simps(1)
thf(fact_126_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_127_paths__in__rts_OCon__initial__right,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U: list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ U2 ) )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ nil_list_a ) )
!= nil_list_a ) ) ) ).
% paths_in_rts.Con_initial_right
thf(fact_128_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_129_paths__in__rts_OCon__initial__left,axiom,
! [Resid: list_a > list_a > list_a,T: list_a,T4: list_list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ T4 ) @ U2 )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ U2 )
!= nil_list_a ) ) ) ).
% paths_in_rts.Con_initial_left
thf(fact_130_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_131_paths__in__rts_OResid_Osimps_I2_J,axiom,
! [Resid: list_a > list_a > list_a,V: list_a,Va2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ V @ Va2 ) @ nil_list_a )
= nil_list_a ) ) ).
% paths_in_rts.Resid.simps(2)
thf(fact_132_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_133_paths__in__rts_OResid__cons_I2_J,axiom,
! [Resid: list_a > list_a > list_a,U2: list_list_a,T4: list_list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( U2 != nil_list_a )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ U2 ) )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ U2 ) )
= ( paths_8620460302779588466list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ nil_list_a ) ) @ U2 ) ) ) ) ) ).
% paths_in_rts.Resid_cons(2)
thf(fact_134_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_135_paths__in__rts_OResid__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 )
=> ( ( 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 ) ) ) ) ).
% paths_in_rts.Resid_rec(1)
thf(fact_136_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_137_paths__in__rts_OResid__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 )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ T4 ) @ ( cons_list_a @ U @ nil_list_a ) )
= ( cons_list_a @ ( Resid @ T @ U ) @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ ( Resid @ U @ T ) @ nil_list_a ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_rec(2)
thf(fact_138_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_139_paths__in__rts_OResid__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 )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ ( cons_list_a @ U @ U2 ) )
= ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ ( Resid @ T @ U ) @ nil_list_a ) @ U2 ) ) ) ) ) ).
% paths_in_rts.Resid_rec(3)
thf(fact_140_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_141_paths__in__rts_OCon__consI_I1_J,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a,T: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( T4 != nil_list_a )
=> ( ( U2 != nil_list_a )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ U2 )
!= nil_list_a )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( paths_8620460302779588466list_a @ Resid @ U2 @ ( cons_list_a @ T @ nil_list_a ) ) )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ T4 ) @ U2 )
!= nil_list_a ) ) ) ) ) ) ).
% paths_in_rts.Con_consI(1)
thf(fact_142_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_143_paths__in__rts_OCon__consI_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( T4 != nil_list_a )
=> ( ( U2 != nil_list_a )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ nil_list_a ) )
!= nil_list_a )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ nil_list_a ) ) @ U2 )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ U2 ) )
!= nil_list_a ) ) ) ) ) ) ).
% paths_in_rts.Con_consI(2)
thf(fact_144_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_145_paths__in__rts_OCon__cons_I1_J,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a,T: 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 ) @ U2 )
!= nil_list_a )
= ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T @ nil_list_a ) @ U2 )
!= nil_list_a )
& ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( paths_8620460302779588466list_a @ Resid @ U2 @ ( cons_list_a @ T @ nil_list_a ) ) )
!= nil_list_a ) ) ) ) ) ) ).
% paths_in_rts.Con_cons(1)
thf(fact_146_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_147_paths__in__rts_OCon__cons_I2_J,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( T4 != nil_list_a )
=> ( ( U2 != nil_list_a )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ U2 ) )
!= nil_list_a )
= ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ nil_list_a ) )
!= nil_list_a )
& ( ( paths_8620460302779588466list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ nil_list_a ) ) @ U2 )
!= nil_list_a ) ) ) ) ) ) ).
% paths_in_rts.Con_cons(2)
thf(fact_148_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_149_append1__eq__conv,axiom,
! [Xs: list_list_a,X2: list_a,Ys: list_list_a,Y4: list_a] :
( ( ( append_list_a @ Xs @ ( cons_list_a @ X2 @ nil_list_a ) )
= ( append_list_a @ Ys @ ( cons_list_a @ Y4 @ nil_list_a ) ) )
= ( ( Xs = Ys )
& ( X2 = Y4 ) ) ) ).
% append1_eq_conv
thf(fact_150_append1__eq__conv,axiom,
! [Xs: list_a,X2: a,Ys: list_a,Y4: a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y4 @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X2 = Y4 ) ) ) ).
% append1_eq_conv
thf(fact_151_append__is__Nil__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= nil_list_a )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% append_is_Nil_conv
thf(fact_152_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_153_Nil__is__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( nil_list_a
= ( append_list_a @ Xs @ Ys ) )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% Nil_is_append_conv
thf(fact_154_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_155_self__append__conv2,axiom,
! [Y4: list_list_a,Xs: list_list_a] :
( ( Y4
= ( append_list_a @ Xs @ Y4 ) )
= ( Xs = nil_list_a ) ) ).
% self_append_conv2
thf(fact_156_self__append__conv2,axiom,
! [Y4: list_a,Xs: list_a] :
( ( Y4
= ( append_a @ Xs @ Y4 ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_157_append__self__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_a ) ) ).
% append_self_conv2
thf(fact_158_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_159_self__append__conv,axiom,
! [Y4: list_list_a,Ys: list_list_a] :
( ( Y4
= ( append_list_a @ Y4 @ Ys ) )
= ( Ys = nil_list_a ) ) ).
% self_append_conv
thf(fact_160_self__append__conv,axiom,
! [Y4: list_a,Ys: list_a] :
( ( Y4
= ( append_a @ Y4 @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_161_append__self__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_a ) ) ).
% append_self_conv
thf(fact_162_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_163_append__Nil2,axiom,
! [Xs: list_list_a] :
( ( append_list_a @ Xs @ nil_list_a )
= Xs ) ).
% append_Nil2
thf(fact_164_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_165_append_Oright__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ A @ nil_list_a )
= A ) ).
% append.right_neutral
thf(fact_166_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_167_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_168_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_169_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_170_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_171_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_172_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_173_R_OconE,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) ) ) ).
% R.conE
thf(fact_174_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_175_Resid1x__null,axiom,
! [T4: list_a] :
( ( paths_in_Resid1x_a @ resid @ ( partial_null_a @ resid ) @ T4 )
= ( partial_null_a @ resid ) ) ).
% Resid1x_null
thf(fact_176_list_Oinject,axiom,
! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( ( cons_list_a @ X21 @ X22 )
= ( cons_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_177_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_178_same__append__eq,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_179_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_180_append__same__eq,axiom,
! [Ys: list_list_a,Xs: list_list_a,Zs: list_list_a] :
( ( ( append_list_a @ Ys @ Xs )
= ( append_list_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_181_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_182_append__assoc,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( append_list_a @ ( append_list_a @ Xs @ Ys ) @ Zs )
= ( append_list_a @ Xs @ ( append_list_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_183_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_184_append_Oassoc,axiom,
! [A: list_list_a,B: list_list_a,C: list_list_a] :
( ( append_list_a @ ( append_list_a @ A @ B ) @ C )
= ( append_list_a @ A @ ( append_list_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_185_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_186_Resid1x_Osimps_I1_J,axiom,
! [T: a] :
( ( paths_in_Resid1x_a @ resid @ T @ nil_a )
= ( partial_null_a @ resid ) ) ).
% Resid1x.simps(1)
thf(fact_187_Resid1x_Oelims,axiom,
! [X2: a,Xa: list_a,Y4: a] :
( ( ( paths_in_Resid1x_a @ resid @ X2 @ Xa )
= Y4 )
=> ( ( ( Xa = nil_a )
=> ( Y4
!= ( partial_null_a @ resid ) ) )
=> ( ! [U3: a] :
( ( Xa
= ( cons_a @ U3 @ nil_a ) )
=> ( Y4
!= ( resid @ X2 @ U3 ) ) )
=> ~ ! [U3: a,V2: a,Va: list_a] :
( ( Xa
= ( cons_a @ U3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y4
!= ( paths_in_Resid1x_a @ resid @ ( resid @ X2 @ U3 ) @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ).
% Resid1x.elims
thf(fact_188_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_189_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_190_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_191_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_192_R_OconI,axiom,
! [T: a,U: a] :
( ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) )
=> ( con_a @ resid @ T @ U ) ) ).
% R.conI
thf(fact_193_partial__magma_Onull_Ocong,axiom,
partial_null_a = partial_null_a ).
% partial_magma.null.cong
thf(fact_194_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_195_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_196_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_197_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_198_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_199_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_200_not__Cons__self2,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( cons_list_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_201_not__Cons__self2,axiom,
! [X2: a,Xs: list_a] :
( ( cons_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_202_append__eq__append__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,Ts: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Zs @ Ts ) )
= ( ? [Us: list_list_a] :
( ( ( Xs
= ( append_list_a @ Zs @ Us ) )
& ( ( append_list_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_list_a @ Xs @ Us )
= Zs )
& ( Ys
= ( append_list_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_203_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_204_append__eq__appendI,axiom,
! [Xs: list_list_a,Xs1: list_list_a,Zs: list_list_a,Ys: list_list_a,Us2: list_list_a] :
( ( ( append_list_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_a @ Xs1 @ Us2 ) )
=> ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_205_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_206_paths__in__rts_OResid1x_Oelims,axiom,
! [Resid: list_a > list_a > list_a,X2: list_a,Xa: list_list_a,Y4: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_1777230443808135851list_a @ Resid @ X2 @ Xa )
= Y4 )
=> ( ( ( Xa = nil_list_a )
=> ( Y4
!= ( partial_null_list_a @ Resid ) ) )
=> ( ! [U3: list_a] :
( ( Xa
= ( cons_list_a @ U3 @ nil_list_a ) )
=> ( Y4
!= ( Resid @ X2 @ U3 ) ) )
=> ~ ! [U3: list_a,V2: list_a,Va: list_list_a] :
( ( Xa
= ( cons_list_a @ U3 @ ( cons_list_a @ V2 @ Va ) ) )
=> ( Y4
!= ( paths_1777230443808135851list_a @ Resid @ ( Resid @ X2 @ U3 ) @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid1x.elims
thf(fact_207_paths__in__rts_OResid1x_Oelims,axiom,
! [Resid: a > a > a,X2: a,Xa: list_a,Y4: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Resid1x_a @ Resid @ X2 @ Xa )
= Y4 )
=> ( ( ( Xa = nil_a )
=> ( Y4
!= ( partial_null_a @ Resid ) ) )
=> ( ! [U3: a] :
( ( Xa
= ( cons_a @ U3 @ nil_a ) )
=> ( Y4
!= ( Resid @ X2 @ U3 ) ) )
=> ~ ! [U3: a,V2: a,Va: list_a] :
( ( Xa
= ( cons_a @ U3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y4
!= ( paths_in_Resid1x_a @ Resid @ ( Resid @ X2 @ U3 ) @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid1x.elims
thf(fact_208_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_209_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_210_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_211_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_212_list_Odistinct_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( nil_list_a
!= ( cons_list_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_213_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_214_list_OdiscI,axiom,
! [List: list_list_a,X21: list_a,X22: list_list_a] :
( ( List
= ( cons_list_a @ X21 @ X22 ) )
=> ( List != nil_list_a ) ) ).
% list.discI
thf(fact_215_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_216_list_Oexhaust,axiom,
! [Y4: list_list_a] :
( ( Y4 != nil_list_a )
=> ~ ! [X212: list_a,X222: list_list_a] :
( Y4
!= ( cons_list_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_217_list_Oexhaust,axiom,
! [Y4: list_a] :
( ( Y4 != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y4
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_218_transpose_Ocases,axiom,
! [X2: list_list_list_a] :
( ( X2 != nil_list_list_a )
=> ( ! [Xss: list_list_list_a] :
( X2
!= ( cons_list_list_a @ nil_list_a @ Xss ) )
=> ~ ! [X: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
( X2
!= ( cons_list_list_a @ ( cons_list_a @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_219_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_220_remdups__adj_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [X: list_a] :
( X2
!= ( cons_list_a @ X @ nil_list_a ) )
=> ~ ! [X: list_a,Y3: list_a,Xs2: list_list_a] :
( X2
!= ( cons_list_a @ X @ ( cons_list_a @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_221_remdups__adj_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X: a] :
( X2
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,Y3: a,Xs2: list_a] :
( X2
!= ( cons_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_222_neq__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
= ( ? [Y2: list_a,Ys2: list_list_a] :
( Xs
= ( cons_list_a @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_223_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y2: a,Ys2: list_a] :
( Xs
= ( cons_a @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_224_list__induct2_H,axiom,
! [P: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( P @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a] : ( P @ ( cons_a @ X @ Xs2 ) @ nil_list_a )
=> ( ! [Y3: list_a,Ys3: list_list_a] : ( P @ nil_a @ ( cons_list_a @ Y3 @ Ys3 ) )
=> ( ! [X: a,Xs2: list_a,Y3: list_a,Ys3: list_list_a] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_225_list__induct2_H,axiom,
! [P: list_list_a > list_a > $o,Xs: list_list_a,Ys: list_a] :
( ( P @ nil_list_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a] : ( P @ ( cons_list_a @ X @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P @ nil_list_a @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X: list_a,Xs2: list_list_a,Y3: a,Ys3: list_a] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_226_list__induct2_H,axiom,
! [P: list_list_a > list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a] : ( P @ ( cons_list_a @ X @ Xs2 ) @ nil_list_a )
=> ( ! [Y3: list_a,Ys3: list_list_a] : ( P @ nil_list_a @ ( cons_list_a @ Y3 @ Ys3 ) )
=> ( ! [X: list_a,Xs2: list_list_a,Y3: list_a,Ys3: list_list_a] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_227_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a] : ( P @ ( cons_a @ X @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys3: list_a] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_228_list__nonempty__induct,axiom,
! [Xs: list_list_a,P: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X: list_a] : ( P @ ( cons_list_a @ X @ nil_list_a ) )
=> ( ! [X: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_a @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_229_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_230_append__Cons,axiom,
! [X2: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( append_list_a @ ( cons_list_a @ X2 @ Xs ) @ Ys )
= ( cons_list_a @ X2 @ ( append_list_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_231_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_232_Cons__eq__appendI,axiom,
! [X2: list_a,Xs1: list_list_a,Ys: list_list_a,Xs: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_a @ Xs1 @ Zs ) )
=> ( ( cons_list_a @ X2 @ Xs )
= ( append_list_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_233_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_234_append__Nil,axiom,
! [Ys: list_list_a] :
( ( append_list_a @ nil_list_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_235_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_236_append_Oleft__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ nil_list_a @ A )
= A ) ).
% append.left_neutral
thf(fact_237_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_238_eq__Nil__appendI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_a @ nil_list_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_239_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_240_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_241_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_242_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_243_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_244_rev__induct,axiom,
! [P: list_list_a > $o,Xs: list_list_a] :
( ( P @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_list_a @ Xs2 @ ( cons_list_a @ X @ nil_list_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_245_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_246_rev__exhaust,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ~ ! [Ys3: list_list_a,Y3: list_a] :
( Xs
!= ( append_list_a @ Ys3 @ ( cons_list_a @ Y3 @ nil_list_a ) ) ) ) ).
% rev_exhaust
thf(fact_247_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys3: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys3 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_248_Cons__eq__append__conv,axiom,
! [X2: list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X2 @ Xs )
= ( append_list_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_a )
& ( ( cons_list_a @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_list_a] :
( ( ( cons_list_a @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_list_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_249_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_250_append__eq__Cons__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,X2: list_a,Xs: list_list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( cons_list_a @ X2 @ Xs ) )
= ( ( ( Ys = nil_list_a )
& ( Zs
= ( cons_list_a @ X2 @ Xs ) ) )
| ? [Ys4: list_list_a] :
( ( Ys
= ( cons_list_a @ X2 @ Ys4 ) )
& ( ( append_list_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_251_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_252_rev__nonempty__induct,axiom,
! [Xs: list_list_a,P: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X: list_a] : ( P @ ( cons_list_a @ X @ nil_list_a ) )
=> ( ! [X: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_list_a @ Xs2 @ ( cons_list_a @ X @ nil_list_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_253_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_254_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_255_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_256_Trgs_Osimps_I1_J,axiom,
( ( paths_in_Trgs_a @ resid @ nil_a )
= bot_bot_set_a ) ).
% Trgs.simps(1)
thf(fact_257_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_258_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_259_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_260_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_261_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_262_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_263_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_264_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_265_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_266_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_267_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_268_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_269_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_270_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_271_R_Oide__def,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
= ( ( con_a @ resid @ A @ A )
& ( ( resid @ A @ A )
= A ) ) ) ).
% R.ide_def
thf(fact_272_R_OideE,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ~ ( ( con_a @ resid @ A @ A )
=> ( ( resid @ A @ A )
!= A ) ) ) ).
% R.ideE
thf(fact_273_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_274_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_275_R_Ocon__imp__coinitial__ax,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ? [A3: a] :
( ( ide_a @ resid @ A3 )
& ( con_a @ resid @ A3 @ T )
& ( con_a @ resid @ A3 @ U ) ) ) ).
% R.con_imp_coinitial_ax
thf(fact_276_R_Ocong__subst__right_I1_J,axiom,
! [U: a,U4: a,T: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ T @ U4 ) ) ) ).
% R.cong_subst_right(1)
thf(fact_277_R_Ocong__subst__right_I2_J,axiom,
! [U: a,U4: a,T: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ T @ U ) @ ( resid @ T @ U4 ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ T @ U4 ) @ ( resid @ T @ U ) ) ) ) ) ) ).
% R.cong_subst_right(2)
thf(fact_278_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_279_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_280_Arr_Osimps_I1_J,axiom,
~ ( paths_in_Arr_a @ resid @ nil_a ) ).
% Arr.simps(1)
thf(fact_281_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_282_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_283_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_284_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_285_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_286_Srcs__are__con,axiom,
! [A: a,T4: list_a,A4: a] :
( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( ( member_a @ A4 @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( con_a @ resid @ A @ A4 ) ) ) ).
% Srcs_are_con
thf(fact_287_R_Ocong__implies__coinitial,axiom,
! [U: a,U4: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U ) ) )
=> ( coinitial_a @ resid @ U @ U4 ) ) ).
% R.cong_implies_coinitial
thf(fact_288_R_Ocoinitial__ide__are__cong,axiom,
! [A: a,A4: a] :
( ( ide_a @ resid @ A )
=> ( ( ide_a @ resid @ A4 )
=> ( ( coinitial_a @ resid @ A @ A4 )
=> ( ( ide_a @ resid @ ( resid @ A @ A4 ) )
& ( ide_a @ resid @ ( resid @ A4 @ A ) ) ) ) ) ) ).
% R.coinitial_ide_are_cong
thf(fact_289_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_290_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_291_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_292_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_293_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_294_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_295_Srcs__con__closed,axiom,
! [A: a,T4: list_a,A4: a] :
( ( member_a @ A @ ( paths_in_Srcs_a @ resid @ T4 ) )
=> ( ( ide_a @ resid @ A4 )
=> ( ( con_a @ resid @ A @ A4 )
=> ( member_a @ A4 @ ( paths_in_Srcs_a @ resid @ T4 ) ) ) ) ) ).
% Srcs_con_closed
thf(fact_296_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_297_Srcs_Osimps_I1_J,axiom,
( ( paths_in_Srcs_a @ resid @ nil_a )
= bot_bot_set_a ) ).
% Srcs.simps(1)
thf(fact_298_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_299_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_300_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_301_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_302_Trgs_Oelims,axiom,
! [X2: list_a,Y4: set_a] :
( ( ( paths_in_Trgs_a @ resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_a )
=> ( Y4 != bot_bot_set_a ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y4
!= ( targets_a @ resid @ T3 ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y4
!= ( paths_in_Trgs_a @ resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ).
% Trgs.elims
thf(fact_303_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_304_append__eq__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
| ( ( size_s349497388124573686list_a @ Us2 )
= ( size_s349497388124573686list_a @ Vs ) ) )
=> ( ( ( append_list_a @ Xs @ Us2 )
= ( append_list_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_305_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_306_hd__append2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( hd_list_a @ Xs ) ) ) ).
% hd_append2
thf(fact_307_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_308_R_OideI,axiom,
! [A: a] :
( ( con_a @ resid @ A @ A )
=> ( ( ( resid @ A @ A )
= A )
=> ( ide_a @ resid @ A ) ) ) ).
% R.ideI
thf(fact_309_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_310_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_311_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_312_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_313_paths__in__rts_OArr__has__Src,axiom,
! [Resid: nat > nat > nat,T4: list_nat] :
( ( paths_in_rts_nat @ Resid )
=> ( ( paths_in_Arr_nat @ Resid @ T4 )
=> ( ( paths_in_Srcs_nat @ Resid @ T4 )
!= bot_bot_set_nat ) ) ) ).
% paths_in_rts.Arr_has_Src
thf(fact_314_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_315_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_316_neq__if__length__neq,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
!= ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_317_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_318_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_a] :
( ( size_s349497388124573686list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_319_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_320_rts_Otargets_Ocong,axiom,
targets_a = targets_a ).
% rts.targets.cong
thf(fact_321_rts_Ojoinable_Ocong,axiom,
joinable_a = joinable_a ).
% rts.joinable.cong
thf(fact_322_rts_Ocoinitial_Ocong,axiom,
coinitial_a = coinitial_a ).
% rts.coinitial.cong
thf(fact_323_residuation_Oide_Ocong,axiom,
ide_a = ide_a ).
% residuation.ide.cong
thf(fact_324_paths__in__rts_OArr_Ocong,axiom,
paths_in_Arr_a = paths_in_Arr_a ).
% paths_in_rts.Arr.cong
thf(fact_325_paths__in__rts_OSrcs_Ocong,axiom,
paths_in_Srcs_a = paths_in_Srcs_a ).
% paths_in_rts.Srcs.cong
thf(fact_326_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,A: produc9164743771328383783list_a,T4: list_P321204300973800749list_a,A4: produc9164743771328383783list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ( member8191768239178080336list_a @ A @ ( paths_6857623006468044397list_a @ Resid @ T4 ) )
=> ( ( ide_Pr1708403647435852235list_a @ Resid @ A4 )
=> ( ( con_Pr6815052424635904531list_a @ Resid @ A @ A4 )
=> ( member8191768239178080336list_a @ A4 @ ( paths_6857623006468044397list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_327_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,A: produc2579390645249093025st_a_a,T4: list_P2210424090985720871st_a_a,A4: produc2579390645249093025st_a_a] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ( member8006451231845903178st_a_a @ A @ ( paths_332773396847195111st_a_a @ Resid @ T4 ) )
=> ( ( ide_Pr4006845958993808965st_a_a @ Resid @ A4 )
=> ( ( con_Pr7383801431968512653st_a_a @ Resid @ A @ A4 )
=> ( member8006451231845903178st_a_a @ A4 @ ( paths_332773396847195111st_a_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_328_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,A: produc8685980395799941037list_a,T4: list_P4541805568828049459list_a,A4: produc8685980395799941037list_a] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ( member4889668945541975382list_a @ A @ ( paths_6439363147398043123list_a @ Resid @ T4 ) )
=> ( ( ide_Pr890063672689881169list_a @ Resid @ A4 )
=> ( ( con_Pr4267019145664584857list_a @ Resid @ A @ A4 )
=> ( member4889668945541975382list_a @ A4 @ ( paths_6439363147398043123list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_329_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,A: product_prod_a_a,T4: list_P1396940483166286381od_a_a,A4: product_prod_a_a] :
( ( paths_2703364527051407500od_a_a @ Resid )
=> ( ( member1426531477525435216od_a_a @ A @ ( paths_2488790558265101933od_a_a @ Resid @ T4 ) )
=> ( ( ide_Product_prod_a_a @ Resid @ A4 )
=> ( ( con_Product_prod_a_a @ Resid @ A @ A4 )
=> ( member1426531477525435216od_a_a @ A4 @ ( paths_2488790558265101933od_a_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_330_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: list_a > list_a > list_a,A: list_a,T4: list_list_a,A4: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( member_list_a @ A @ ( paths_in_Srcs_list_a @ Resid @ T4 ) )
=> ( ( ide_list_a @ Resid @ A4 )
=> ( ( con_list_a @ Resid @ A @ A4 )
=> ( member_list_a @ A4 @ ( paths_in_Srcs_list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_331_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: a > a > a,A: a,T4: list_a,A4: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ Resid @ T4 ) )
=> ( ( ide_a @ Resid @ A4 )
=> ( ( con_a @ Resid @ A @ A4 )
=> ( member_a @ A4 @ ( paths_in_Srcs_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_332_paths__in__rts_OSrcs_Osimps_I1_J,axiom,
! [Resid: nat > nat > nat] :
( ( paths_in_rts_nat @ Resid )
=> ( ( paths_in_Srcs_nat @ Resid @ nil_nat )
= bot_bot_set_nat ) ) ).
% paths_in_rts.Srcs.simps(1)
thf(fact_333_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_334_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_335_list_Osel_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( ( hd_list_a @ ( cons_list_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_336_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_337_paths__in__rts_OArr__has__Trg,axiom,
! [Resid: nat > nat > nat,T4: list_nat] :
( ( paths_in_rts_nat @ Resid )
=> ( ( paths_in_Arr_nat @ Resid @ T4 )
=> ( ( paths_in_Trgs_nat @ Resid @ T4 )
!= bot_bot_set_nat ) ) ) ).
% paths_in_rts.Arr_has_Trg
thf(fact_338_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_339_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_340_paths__in__rts_OCon__single__ide__ind,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,A: produc9164743771328383783list_a,T4: list_P321204300973800749list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ( ide_Pr1708403647435852235list_a @ Resid @ A )
=> ( ( ( paths_2295606913481735701list_a @ Resid @ ( cons_P5184657343811988189list_a @ A @ nil_Pr3188421586756112173list_a ) @ T4 )
!= nil_Pr3188421586756112173list_a )
= ( ( paths_5744671163884122927list_a @ Resid @ T4 )
& ( member8191768239178080336list_a @ A @ ( paths_6857623006468044397list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Con_single_ide_ind
thf(fact_341_paths__in__rts_OCon__single__ide__ind,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,A: produc2579390645249093025st_a_a,T4: list_P2210424090985720871st_a_a] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ( ide_Pr4006845958993808965st_a_a @ Resid @ A )
=> ( ( ( paths_8255835059299053455st_a_a @ Resid @ ( cons_P2018802349718741079st_a_a @ A @ nil_Pr523822031547952295st_a_a ) @ T4 )
!= nil_Pr523822031547952295st_a_a )
= ( ( paths_424260355915577769st_a_a @ Resid @ T4 )
& ( member8006451231845903178st_a_a @ A @ ( paths_332773396847195111st_a_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Con_single_ide_ind
thf(fact_342_paths__in__rts_OCon__single__ide__ind,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,A: produc8685980395799941037list_a,T4: list_P4541805568828049459list_a] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ( ide_Pr890063672689881169list_a @ Resid @ A )
=> ( ( ( paths_5139052772995125659list_a @ Resid @ ( cons_P8125392100269589091list_a @ A @ nil_Pr6630411782098800307list_a ) @ T4 )
!= nil_Pr6630411782098800307list_a )
= ( ( paths_6530850106466425781list_a @ Resid @ T4 )
& ( member4889668945541975382list_a @ A @ ( paths_6439363147398043123list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Con_single_ide_ind
thf(fact_343_paths__in__rts_OCon__single__ide__ind,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,A: product_prod_a_a,T4: list_P1396940483166286381od_a_a] :
( ( paths_2703364527051407500od_a_a @ Resid )
=> ( ( ide_Product_prod_a_a @ Resid @ A )
=> ( ( ( paths_4544677030372982293od_a_a @ Resid @ ( cons_P7316939126706565853od_a_a @ A @ nil_Product_prod_a_a ) @ T4 )
!= nil_Product_prod_a_a )
= ( ( paths_2884821253938355503od_a_a @ Resid @ T4 )
& ( member1426531477525435216od_a_a @ A @ ( paths_2488790558265101933od_a_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Con_single_ide_ind
thf(fact_344_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_345_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_346_paths__in__rts_OCon__single__ideI_I1_J,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,A: produc9164743771328383783list_a,T4: list_P321204300973800749list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ( ide_Pr1708403647435852235list_a @ Resid @ A )
=> ( ( paths_5744671163884122927list_a @ Resid @ T4 )
=> ( ( member8191768239178080336list_a @ A @ ( paths_6857623006468044397list_a @ Resid @ T4 ) )
=> ( ( paths_2295606913481735701list_a @ Resid @ ( cons_P5184657343811988189list_a @ A @ nil_Pr3188421586756112173list_a ) @ T4 )
!= nil_Pr3188421586756112173list_a ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(1)
thf(fact_347_paths__in__rts_OCon__single__ideI_I1_J,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,A: produc2579390645249093025st_a_a,T4: list_P2210424090985720871st_a_a] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ( ide_Pr4006845958993808965st_a_a @ Resid @ A )
=> ( ( paths_424260355915577769st_a_a @ Resid @ T4 )
=> ( ( member8006451231845903178st_a_a @ A @ ( paths_332773396847195111st_a_a @ Resid @ T4 ) )
=> ( ( paths_8255835059299053455st_a_a @ Resid @ ( cons_P2018802349718741079st_a_a @ A @ nil_Pr523822031547952295st_a_a ) @ T4 )
!= nil_Pr523822031547952295st_a_a ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(1)
thf(fact_348_paths__in__rts_OCon__single__ideI_I1_J,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,A: produc8685980395799941037list_a,T4: list_P4541805568828049459list_a] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ( ide_Pr890063672689881169list_a @ Resid @ A )
=> ( ( paths_6530850106466425781list_a @ Resid @ T4 )
=> ( ( member4889668945541975382list_a @ A @ ( paths_6439363147398043123list_a @ Resid @ T4 ) )
=> ( ( paths_5139052772995125659list_a @ Resid @ ( cons_P8125392100269589091list_a @ A @ nil_Pr6630411782098800307list_a ) @ T4 )
!= nil_Pr6630411782098800307list_a ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(1)
thf(fact_349_paths__in__rts_OCon__single__ideI_I1_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,A: product_prod_a_a,T4: list_P1396940483166286381od_a_a] :
( ( paths_2703364527051407500od_a_a @ Resid )
=> ( ( ide_Product_prod_a_a @ Resid @ A )
=> ( ( paths_2884821253938355503od_a_a @ Resid @ T4 )
=> ( ( member1426531477525435216od_a_a @ A @ ( paths_2488790558265101933od_a_a @ Resid @ T4 ) )
=> ( ( paths_4544677030372982293od_a_a @ Resid @ ( cons_P7316939126706565853od_a_a @ A @ nil_Product_prod_a_a ) @ T4 )
!= nil_Product_prod_a_a ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(1)
thf(fact_350_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_351_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_352_paths__in__rts_OCon__single__ideI_I2_J,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,A: produc9164743771328383783list_a,T4: list_P321204300973800749list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ( ide_Pr1708403647435852235list_a @ Resid @ A )
=> ( ( paths_5744671163884122927list_a @ Resid @ T4 )
=> ( ( member8191768239178080336list_a @ A @ ( paths_6857623006468044397list_a @ Resid @ T4 ) )
=> ( ( paths_2295606913481735701list_a @ Resid @ T4 @ ( cons_P5184657343811988189list_a @ A @ nil_Pr3188421586756112173list_a ) )
!= nil_Pr3188421586756112173list_a ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(2)
thf(fact_353_paths__in__rts_OCon__single__ideI_I2_J,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,A: produc2579390645249093025st_a_a,T4: list_P2210424090985720871st_a_a] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ( ide_Pr4006845958993808965st_a_a @ Resid @ A )
=> ( ( paths_424260355915577769st_a_a @ Resid @ T4 )
=> ( ( member8006451231845903178st_a_a @ A @ ( paths_332773396847195111st_a_a @ Resid @ T4 ) )
=> ( ( paths_8255835059299053455st_a_a @ Resid @ T4 @ ( cons_P2018802349718741079st_a_a @ A @ nil_Pr523822031547952295st_a_a ) )
!= nil_Pr523822031547952295st_a_a ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(2)
thf(fact_354_paths__in__rts_OCon__single__ideI_I2_J,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,A: produc8685980395799941037list_a,T4: list_P4541805568828049459list_a] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ( ide_Pr890063672689881169list_a @ Resid @ A )
=> ( ( paths_6530850106466425781list_a @ Resid @ T4 )
=> ( ( member4889668945541975382list_a @ A @ ( paths_6439363147398043123list_a @ Resid @ T4 ) )
=> ( ( paths_5139052772995125659list_a @ Resid @ T4 @ ( cons_P8125392100269589091list_a @ A @ nil_Pr6630411782098800307list_a ) )
!= nil_Pr6630411782098800307list_a ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(2)
thf(fact_355_paths__in__rts_OCon__single__ideI_I2_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,A: product_prod_a_a,T4: list_P1396940483166286381od_a_a] :
( ( paths_2703364527051407500od_a_a @ Resid )
=> ( ( ide_Product_prod_a_a @ Resid @ A )
=> ( ( paths_2884821253938355503od_a_a @ Resid @ T4 )
=> ( ( member1426531477525435216od_a_a @ A @ ( paths_2488790558265101933od_a_a @ Resid @ T4 ) )
=> ( ( paths_4544677030372982293od_a_a @ Resid @ T4 @ ( cons_P7316939126706565853od_a_a @ A @ nil_Product_prod_a_a ) )
!= nil_Product_prod_a_a ) ) ) ) ) ).
% paths_in_rts.Con_single_ideI(2)
thf(fact_356_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_357_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_358_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_359_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_360_paths__in__rts_OTrgs_Oelims,axiom,
! [Resid: nat > nat > nat,X2: list_nat,Y4: set_nat] :
( ( paths_in_rts_nat @ Resid )
=> ( ( ( paths_in_Trgs_nat @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_nat )
=> ( Y4 != bot_bot_set_nat ) )
=> ( ! [T3: nat] :
( ( X2
= ( cons_nat @ T3 @ nil_nat ) )
=> ( Y4
!= ( targets_nat @ Resid @ T3 ) ) )
=> ~ ! [T3: nat,V2: nat,Va: list_nat] :
( ( X2
= ( cons_nat @ T3 @ ( cons_nat @ V2 @ Va ) ) )
=> ( Y4
!= ( paths_in_Trgs_nat @ Resid @ ( cons_nat @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Trgs.elims
thf(fact_361_paths__in__rts_OTrgs_Oelims,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a,Y4: set_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_in_Trgs_list_a @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_list_a )
=> ( Y4 != bot_bot_set_list_a ) )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( Y4
!= ( 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 ) ) )
=> ( Y4
!= ( paths_in_Trgs_list_a @ Resid @ ( cons_list_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Trgs.elims
thf(fact_362_paths__in__rts_OTrgs_Oelims,axiom,
! [Resid: a > a > a,X2: list_a,Y4: set_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Trgs_a @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_a )
=> ( Y4 != bot_bot_set_a ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y4
!= ( targets_a @ Resid @ T3 ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y4
!= ( paths_in_Trgs_a @ Resid @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Trgs.elims
thf(fact_363_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,P: list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys3: list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_364_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,P: list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y3: list_a,Ys3: list_list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_365_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,P: list_a > list_list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y3: list_a,Ys3: list_list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys3 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_366_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,P: list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y3: a,Ys3: list_a,Z: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_367_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,P: list_list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y3: a,Ys3: list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_368_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,P: list_list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y3: list_a,Ys3: list_list_a,Z: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_369_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,P: list_list_a > list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_list_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y3: list_a,Ys3: list_list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys3 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_370_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,Y3: a,Ys3: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_371_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_a,P: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y3: list_a,Ys3: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_372_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_a,P: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y3: a,Ys3: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_373_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_list_a,P: list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y3: list_a,Ys3: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_374_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,Y3: a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_375_paths__in__rts_OResid__Arr__Src,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,T4: list_P321204300973800749list_a,A: produc9164743771328383783list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ( paths_5744671163884122927list_a @ Resid @ T4 )
=> ( ( member8191768239178080336list_a @ A @ ( paths_6857623006468044397list_a @ Resid @ T4 ) )
=> ( ( paths_2295606913481735701list_a @ Resid @ T4 @ ( cons_P5184657343811988189list_a @ A @ nil_Pr3188421586756112173list_a ) )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_Arr_Src
thf(fact_376_paths__in__rts_OResid__Arr__Src,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,T4: list_P2210424090985720871st_a_a,A: produc2579390645249093025st_a_a] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ( paths_424260355915577769st_a_a @ Resid @ T4 )
=> ( ( member8006451231845903178st_a_a @ A @ ( paths_332773396847195111st_a_a @ Resid @ T4 ) )
=> ( ( paths_8255835059299053455st_a_a @ Resid @ T4 @ ( cons_P2018802349718741079st_a_a @ A @ nil_Pr523822031547952295st_a_a ) )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_Arr_Src
thf(fact_377_paths__in__rts_OResid__Arr__Src,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,T4: list_P4541805568828049459list_a,A: produc8685980395799941037list_a] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ( paths_6530850106466425781list_a @ Resid @ T4 )
=> ( ( member4889668945541975382list_a @ A @ ( paths_6439363147398043123list_a @ Resid @ T4 ) )
=> ( ( paths_5139052772995125659list_a @ Resid @ T4 @ ( cons_P8125392100269589091list_a @ A @ nil_Pr6630411782098800307list_a ) )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_Arr_Src
thf(fact_378_paths__in__rts_OResid__Arr__Src,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,T4: list_P1396940483166286381od_a_a,A: product_prod_a_a] :
( ( paths_2703364527051407500od_a_a @ Resid )
=> ( ( paths_2884821253938355503od_a_a @ Resid @ T4 )
=> ( ( member1426531477525435216od_a_a @ A @ ( paths_2488790558265101933od_a_a @ Resid @ T4 ) )
=> ( ( paths_4544677030372982293od_a_a @ Resid @ T4 @ ( cons_P7316939126706565853od_a_a @ A @ nil_Product_prod_a_a ) )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_Arr_Src
thf(fact_379_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_380_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_381_hd__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( Xs = nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( hd_list_a @ Ys ) ) )
& ( ( Xs != nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( hd_list_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_382_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_383_longest__common__prefix,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
? [Ps: list_list_a,Xs3: list_list_a,Ys5: list_list_a] :
( ( Xs
= ( append_list_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_list_a @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_list_a )
| ( Ys5 = nil_list_a )
| ( ( hd_list_a @ Xs3 )
!= ( hd_list_a @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_384_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_385_paths__in__rts_OSrcs__are__con,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,A: produc9164743771328383783list_a,T4: list_P321204300973800749list_a,A4: produc9164743771328383783list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ( member8191768239178080336list_a @ A @ ( paths_6857623006468044397list_a @ Resid @ T4 ) )
=> ( ( member8191768239178080336list_a @ A4 @ ( paths_6857623006468044397list_a @ Resid @ T4 ) )
=> ( con_Pr6815052424635904531list_a @ Resid @ A @ A4 ) ) ) ) ).
% paths_in_rts.Srcs_are_con
thf(fact_386_paths__in__rts_OSrcs__are__con,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,A: produc2579390645249093025st_a_a,T4: list_P2210424090985720871st_a_a,A4: produc2579390645249093025st_a_a] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ( member8006451231845903178st_a_a @ A @ ( paths_332773396847195111st_a_a @ Resid @ T4 ) )
=> ( ( member8006451231845903178st_a_a @ A4 @ ( paths_332773396847195111st_a_a @ Resid @ T4 ) )
=> ( con_Pr7383801431968512653st_a_a @ Resid @ A @ A4 ) ) ) ) ).
% paths_in_rts.Srcs_are_con
thf(fact_387_paths__in__rts_OSrcs__are__con,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,A: produc8685980395799941037list_a,T4: list_P4541805568828049459list_a,A4: produc8685980395799941037list_a] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ( member4889668945541975382list_a @ A @ ( paths_6439363147398043123list_a @ Resid @ T4 ) )
=> ( ( member4889668945541975382list_a @ A4 @ ( paths_6439363147398043123list_a @ Resid @ T4 ) )
=> ( con_Pr4267019145664584857list_a @ Resid @ A @ A4 ) ) ) ) ).
% paths_in_rts.Srcs_are_con
thf(fact_388_paths__in__rts_OSrcs__are__con,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,A: product_prod_a_a,T4: list_P1396940483166286381od_a_a,A4: product_prod_a_a] :
( ( paths_2703364527051407500od_a_a @ Resid )
=> ( ( member1426531477525435216od_a_a @ A @ ( paths_2488790558265101933od_a_a @ Resid @ T4 ) )
=> ( ( member1426531477525435216od_a_a @ A4 @ ( paths_2488790558265101933od_a_a @ Resid @ T4 ) )
=> ( con_Product_prod_a_a @ Resid @ A @ A4 ) ) ) ) ).
% paths_in_rts.Srcs_are_con
thf(fact_389_paths__in__rts_OSrcs__are__con,axiom,
! [Resid: list_a > list_a > list_a,A: list_a,T4: list_list_a,A4: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( member_list_a @ A @ ( paths_in_Srcs_list_a @ Resid @ T4 ) )
=> ( ( member_list_a @ A4 @ ( paths_in_Srcs_list_a @ Resid @ T4 ) )
=> ( con_list_a @ Resid @ A @ A4 ) ) ) ) ).
% paths_in_rts.Srcs_are_con
thf(fact_390_paths__in__rts_OSrcs__are__con,axiom,
! [Resid: a > a > a,A: a,T4: list_a,A4: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( member_a @ A @ ( paths_in_Srcs_a @ Resid @ T4 ) )
=> ( ( member_a @ A4 @ ( paths_in_Srcs_a @ Resid @ T4 ) )
=> ( con_a @ Resid @ A @ A4 ) ) ) ) ).
% paths_in_rts.Srcs_are_con
thf(fact_391_paths__in__rts_OArr_Osimps_I1_J,axiom,
! [Resid: list_a > list_a > list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ~ ( paths_in_Arr_list_a @ Resid @ nil_list_a ) ) ).
% paths_in_rts.Arr.simps(1)
thf(fact_392_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_393_paths__in__rts_OTrgs__con__closed,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,B: produc9164743771328383783list_a,T4: list_P321204300973800749list_a,B2: produc9164743771328383783list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ( member8191768239178080336list_a @ B @ ( paths_8434672324333973362list_a @ Resid @ T4 ) )
=> ( ( ide_Pr1708403647435852235list_a @ Resid @ B2 )
=> ( ( con_Pr6815052424635904531list_a @ Resid @ B @ B2 )
=> ( member8191768239178080336list_a @ B2 @ ( paths_8434672324333973362list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Trgs_con_closed
thf(fact_394_paths__in__rts_OTrgs__con__closed,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,B: produc2579390645249093025st_a_a,T4: list_P2210424090985720871st_a_a,B2: produc2579390645249093025st_a_a] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ( member8006451231845903178st_a_a @ B @ ( paths_2794423126656725868st_a_a @ Resid @ T4 ) )
=> ( ( ide_Pr4006845958993808965st_a_a @ Resid @ B2 )
=> ( ( con_Pr7383801431968512653st_a_a @ Resid @ B @ B2 )
=> ( member8006451231845903178st_a_a @ B2 @ ( paths_2794423126656725868st_a_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Trgs_con_closed
thf(fact_395_paths__in__rts_OTrgs__con__closed,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,B: produc8685980395799941037list_a,T4: list_P4541805568828049459list_a,B2: produc8685980395799941037list_a] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ( member4889668945541975382list_a @ B @ ( paths_8901012877207573880list_a @ Resid @ T4 ) )
=> ( ( ide_Pr890063672689881169list_a @ Resid @ B2 )
=> ( ( con_Pr4267019145664584857list_a @ Resid @ B @ B2 )
=> ( member4889668945541975382list_a @ B2 @ ( paths_8901012877207573880list_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Trgs_con_closed
thf(fact_396_paths__in__rts_OTrgs__con__closed,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,B: product_prod_a_a,T4: list_P1396940483166286381od_a_a,B2: product_prod_a_a] :
( ( paths_2703364527051407500od_a_a @ Resid )
=> ( ( member1426531477525435216od_a_a @ B @ ( paths_4392315070405530738od_a_a @ Resid @ T4 ) )
=> ( ( ide_Product_prod_a_a @ Resid @ B2 )
=> ( ( con_Product_prod_a_a @ Resid @ B @ B2 )
=> ( member1426531477525435216od_a_a @ B2 @ ( paths_4392315070405530738od_a_a @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Trgs_con_closed
thf(fact_397_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_398_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_399_same__length__different,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != Ys )
=> ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ? [Pre: list_list_a,X: list_a,Xs3: list_list_a,Y3: list_a,Ys5: list_list_a] :
( ( X != Y3 )
& ( Xs
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ X @ nil_list_a ) @ Xs3 ) ) )
& ( Ys
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ Y3 @ nil_list_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_400_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,Y3: a,Ys5: list_a] :
( ( X != Y3 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_401_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_402_paths__in__rts_Olength__Resid,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 )
!= nil_list_a )
=> ( ( size_s349497388124573686list_a @ ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 ) )
= ( size_s349497388124573686list_a @ T4 ) ) ) ) ).
% paths_in_rts.length_Resid
thf(fact_403_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_404_paths__in__rts_OCon__imp__eq__Srcs,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 )
!= nil_list_a )
=> ( ( paths_in_Srcs_list_a @ Resid @ T4 )
= ( paths_in_Srcs_list_a @ Resid @ U2 ) ) ) ) ).
% paths_in_rts.Con_imp_eq_Srcs
thf(fact_405_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_406_paths__in__rts_OCon__imp__Arr__Resid,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 )
!= nil_list_a )
=> ( paths_in_Arr_list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 ) ) ) ) ).
% paths_in_rts.Con_imp_Arr_Resid
thf(fact_407_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_408_paths__in__rts_OArr__iff__Con__self,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_8620460302779588466list_a @ Resid @ T4 @ T4 )
!= nil_list_a ) ) ) ).
% paths_in_rts.Arr_iff_Con_self
thf(fact_409_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_410_paths__in__rts_OCon__Arr__self,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_8620460302779588466list_a @ Resid @ T4 @ T4 )
!= nil_list_a ) ) ) ).
% paths_in_rts.Con_Arr_self
thf(fact_411_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_412_paths__in__rts_OTrgs_Osimps_I1_J,axiom,
! [Resid: nat > nat > nat] :
( ( paths_in_rts_nat @ Resid )
=> ( ( paths_in_Trgs_nat @ Resid @ nil_nat )
= bot_bot_set_nat ) ) ).
% paths_in_rts.Trgs.simps(1)
thf(fact_413_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_414_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_415_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_416_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_417_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_418_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_419_paths__in__rts_OArr__Resid__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_Arr_list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U @ nil_list_a ) ) ) ) ) ).
% paths_in_rts.Arr_Resid_single
thf(fact_420_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_421_paths__in__rts_OSrcs__Resid,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 )
!= nil_list_a )
=> ( ( paths_in_Srcs_list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 ) )
= ( paths_in_Trgs_list_a @ Resid @ U2 ) ) ) ) ).
% paths_in_rts.Srcs_Resid
thf(fact_422_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_423_paths__in__rts_OSrcs__Resid__single__Arr,axiom,
! [Resid: list_a > list_a > list_a,U: list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ U @ nil_list_a ) @ T4 )
!= nil_list_a )
=> ( ( paths_in_Srcs_list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ U @ nil_list_a ) @ T4 ) )
= ( paths_in_Trgs_list_a @ Resid @ T4 ) ) ) ) ).
% paths_in_rts.Srcs_Resid_single_Arr
thf(fact_424_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_425_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_426_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_427_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_428_R_Ocong__implies__coterminal,axiom,
! [U: a,U4: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U ) ) )
=> ( coterminal_a @ resid @ U @ U4 ) ) ).
% R.cong_implies_coterminal
thf(fact_429_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_430_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_431_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_432_Con__IdeI_I2_J,axiom,
! [A2: list_a,T4: list_a] :
( ( paths_in_Ide_a @ resid @ A2 )
=> ( ( paths_in_Arr_a @ resid @ T4 )
=> ( ( ( paths_in_Srcs_a @ resid @ T4 )
= ( paths_in_Srcs_a @ resid @ A2 ) )
=> ( ( paths_in_Resid_a @ resid @ T4 @ A2 )
!= nil_a ) ) ) ) ).
% Con_IdeI(2)
thf(fact_433_Srcs_Oelims,axiom,
! [X2: list_a,Y4: set_a] :
( ( ( paths_in_Srcs_a @ resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_a )
=> ( Y4 != bot_bot_set_a ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y4
!= ( sources_a @ resid @ T3 ) ) )
=> ~ ! [T3: a] :
( ? [V2: a,Va: list_a] :
( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y4
!= ( sources_a @ resid @ T3 ) ) ) ) ) ) ).
% Srcs.elims
thf(fact_434_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_435_Ide_Osimps_I1_J,axiom,
~ ( paths_in_Ide_a @ resid @ nil_a ) ).
% Ide.simps(1)
thf(fact_436_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_437_R_Osources__are__cong,axiom,
! [A: a,T: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T ) )
=> ( ( ide_a @ resid @ ( resid @ A @ A4 ) )
& ( ide_a @ resid @ ( resid @ A4 @ A ) ) ) ) ) ).
% R.sources_are_cong
thf(fact_438_R_Osources__cong__closed,axiom,
! [A: a,T: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( ( ide_a @ resid @ ( resid @ A @ A4 ) )
& ( ide_a @ resid @ ( resid @ A4 @ A ) ) )
=> ( member_a @ A4 @ ( sources_a @ resid @ T ) ) ) ) ).
% R.sources_cong_closed
thf(fact_439_R_Osources__are__con,axiom,
! [A: a,T: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T ) )
=> ( con_a @ resid @ A @ A4 ) ) ) ).
% R.sources_are_con
thf(fact_440_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_441_Ide__implies__Arr,axiom,
! [T4: list_a] :
( ( paths_in_Ide_a @ resid @ T4 )
=> ( paths_in_Arr_a @ resid @ T4 ) ) ).
% Ide_implies_Arr
thf(fact_442_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_443_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_444_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_445_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_446_R_Osources__con__closed,axiom,
! [A: a,T: a,A4: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( ide_a @ resid @ A4 )
=> ( ( con_a @ resid @ A @ A4 )
=> ( member_a @ A4 @ ( sources_a @ resid @ T ) ) ) ) ) ).
% R.sources_con_closed
thf(fact_447_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_448_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_449_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_450_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_451_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_452_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_453_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_454_last__appendR,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( Ys != nil_list_a )
=> ( ( last_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( last_list_a @ Ys ) ) ) ).
% last_appendR
thf(fact_455_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_456_last__appendL,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( Ys = nil_list_a )
=> ( ( last_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( last_list_a @ Xs ) ) ) ).
% last_appendL
thf(fact_457_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_458_last__snoc,axiom,
! [Xs: list_list_a,X2: list_a] :
( ( last_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X2 @ nil_list_a ) ) )
= X2 ) ).
% last_snoc
thf(fact_459_last__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% last_snoc
thf(fact_460_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_461_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_462_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,NN: set_Pr4048851178543822343list_a,V: produc9164743771328383783list_a,V3: produc9164743771328383783list_a,W: produc9164743771328383783list_a,W2: produc9164743771328383783list_a,T: produc9164743771328383783list_a,T6: produc9164743771328383783list_a] :
( ( cohere5499257456005325328list_a @ Resid @ NN )
=> ( ( member8191768239178080336list_a @ V @ NN )
=> ( ( member8191768239178080336list_a @ V3 @ NN )
=> ( ( member8191768239178080336list_a @ W @ NN )
=> ( ( member8191768239178080336list_a @ W2 @ NN )
=> ( ( ( source4107297218376970195list_a @ Resid @ V )
= ( source4107297218376970195list_a @ Resid @ W ) )
=> ( ( ( source4107297218376970195list_a @ Resid @ V3 )
= ( source4107297218376970195list_a @ Resid @ W2 ) )
=> ( ( ( target2700282342086602697list_a @ Resid @ W )
= ( target2700282342086602697list_a @ Resid @ W2 ) )
=> ( ( ( member8191768239178080336list_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V3 ) ) @ NN )
& ( member8191768239178080336list_a @ ( Resid @ ( Resid @ T6 @ V3 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member8191768239178080336list_a @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T6 @ W2 ) ) @ NN )
& ( member8191768239178080336list_a @ ( Resid @ ( Resid @ T6 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_463_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,NN: set_Pr8962057229576493569st_a_a,V: produc2579390645249093025st_a_a,V3: produc2579390645249093025st_a_a,W: produc2579390645249093025st_a_a,W2: produc2579390645249093025st_a_a,T: produc2579390645249093025st_a_a,T6: produc2579390645249093025st_a_a] :
( ( cohere8414546589650652426st_a_a @ Resid @ NN )
=> ( ( member8006451231845903178st_a_a @ V @ NN )
=> ( ( member8006451231845903178st_a_a @ V3 @ NN )
=> ( ( member8006451231845903178st_a_a @ W @ NN )
=> ( ( member8006451231845903178st_a_a @ W2 @ NN )
=> ( ( ( source8862560589801822285st_a_a @ Resid @ V )
= ( source8862560589801822285st_a_a @ Resid @ W ) )
=> ( ( ( source8862560589801822285st_a_a @ Resid @ V3 )
= ( source8862560589801822285st_a_a @ Resid @ W2 ) )
=> ( ( ( target5404720412848966467st_a_a @ Resid @ W )
= ( target5404720412848966467st_a_a @ Resid @ W2 ) )
=> ( ( ( member8006451231845903178st_a_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V3 ) ) @ NN )
& ( member8006451231845903178st_a_a @ ( Resid @ ( Resid @ T6 @ V3 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member8006451231845903178st_a_a @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T6 @ W2 ) ) @ NN )
& ( member8006451231845903178st_a_a @ ( Resid @ ( Resid @ T6 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_464_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,NN: set_Pr2070066670564046349list_a,V: produc8685980395799941037list_a,V3: produc8685980395799941037list_a,W: produc8685980395799941037list_a,W2: produc8685980395799941037list_a,T: produc8685980395799941037list_a,T6: produc8685980395799941037list_a] :
( ( cohere5297764303346724630list_a @ Resid @ NN )
=> ( ( member4889668945541975382list_a @ V @ NN )
=> ( ( member4889668945541975382list_a @ V3 @ NN )
=> ( ( member4889668945541975382list_a @ W @ NN )
=> ( ( member4889668945541975382list_a @ W2 @ NN )
=> ( ( ( source5745778303497894489list_a @ Resid @ V )
= ( source5745778303497894489list_a @ Resid @ W ) )
=> ( ( ( source5745778303497894489list_a @ Resid @ V3 )
= ( source5745778303497894489list_a @ Resid @ W2 ) )
=> ( ( ( target2287938126545038671list_a @ Resid @ W )
= ( target2287938126545038671list_a @ Resid @ W2 ) )
=> ( ( ( member4889668945541975382list_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V3 ) ) @ NN )
& ( member4889668945541975382list_a @ ( Resid @ ( Resid @ T6 @ V3 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member4889668945541975382list_a @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T6 @ W2 ) ) @ NN )
& ( member4889668945541975382list_a @ ( Resid @ ( Resid @ T6 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_465_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,V: product_prod_a_a,V3: product_prod_a_a,W: product_prod_a_a,W2: product_prod_a_a,T: product_prod_a_a,T6: product_prod_a_a] :
( ( cohere3906135489564710160od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ V @ NN )
=> ( ( member1426531477525435216od_a_a @ V3 @ NN )
=> ( ( member1426531477525435216od_a_a @ W @ NN )
=> ( ( member1426531477525435216od_a_a @ W2 @ NN )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ V )
= ( source6950040787684646355od_a_a @ Resid @ W ) )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ V3 )
= ( source6950040787684646355od_a_a @ Resid @ W2 ) )
=> ( ( ( target5293506191220573129od_a_a @ Resid @ W )
= ( target5293506191220573129od_a_a @ Resid @ W2 ) )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V3 ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T6 @ V3 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T6 @ W2 ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T6 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_466_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: list_a > list_a > list_a,NN: set_list_a,V: list_a,V3: 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 @ V3 @ 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 @ V3 )
= ( sources_list_a @ Resid @ W2 ) )
=> ( ( ( targets_list_a @ Resid @ W )
= ( targets_list_a @ Resid @ W2 ) )
=> ( ( ( member_list_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V3 ) ) @ NN )
& ( member_list_a @ ( Resid @ ( Resid @ T6 @ V3 ) @ ( 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_467_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: a > a > a,NN: set_a,V: a,V3: a,W: a,W2: a,T: a,T6: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( member_a @ V @ NN )
=> ( ( member_a @ V3 @ NN )
=> ( ( member_a @ W @ NN )
=> ( ( member_a @ W2 @ NN )
=> ( ( ( sources_a @ Resid @ V )
= ( sources_a @ Resid @ W ) )
=> ( ( ( sources_a @ Resid @ V3 )
= ( sources_a @ Resid @ W2 ) )
=> ( ( ( targets_a @ Resid @ W )
= ( targets_a @ Resid @ W2 ) )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V3 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ V3 ) @ ( 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_468_rts_Ocoterminal_Ocong,axiom,
coterminal_a = coterminal_a ).
% rts.coterminal.cong
thf(fact_469_paths__in__rts_OIde_Ocong,axiom,
paths_in_Ide_a = paths_in_Ide_a ).
% paths_in_rts.Ide.cong
thf(fact_470_rts_Osources_Ocong,axiom,
sources_a = sources_a ).
% rts.sources.cong
thf(fact_471_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_472_impossible__Cons,axiom,
! [Xs: list_list_a,Ys: list_list_a,X2: list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs
!= ( cons_list_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_473_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_474_last_Osimps,axiom,
! [Xs: list_list_a,X2: list_a] :
( ( ( Xs = nil_list_a )
=> ( ( last_list_a @ ( cons_list_a @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_list_a )
=> ( ( last_list_a @ ( cons_list_a @ X2 @ Xs ) )
= ( last_list_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_475_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_476_last__ConsL,axiom,
! [Xs: list_list_a,X2: list_a] :
( ( Xs = nil_list_a )
=> ( ( last_list_a @ ( cons_list_a @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_477_last__ConsL,axiom,
! [Xs: list_a,X2: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_478_last__ConsR,axiom,
! [Xs: list_list_a,X2: list_a] :
( ( Xs != nil_list_a )
=> ( ( last_list_a @ ( cons_list_a @ X2 @ Xs ) )
= ( last_list_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_479_last__ConsR,axiom,
! [Xs: list_a,X2: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_480_longest__common__suffix,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
? [Ss: list_list_a,Xs3: list_list_a,Ys5: list_list_a] :
( ( Xs
= ( append_list_a @ Xs3 @ Ss ) )
& ( Ys
= ( append_list_a @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_list_a )
| ( Ys5 = nil_list_a )
| ( ( last_list_a @ Xs3 )
!= ( last_list_a @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_481_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_482_last__append,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( ( Ys = nil_list_a )
=> ( ( last_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( last_list_a @ Xs ) ) )
& ( ( Ys != nil_list_a )
=> ( ( last_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( last_list_a @ Ys ) ) ) ) ).
% last_append
thf(fact_483_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_484_hd__Nil__eq__last,axiom,
( ( hd_list_a @ nil_list_a )
= ( last_list_a @ nil_list_a ) ) ).
% hd_Nil_eq_last
thf(fact_485_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_486_paths__in__rts_OIde_Osimps_I1_J,axiom,
! [Resid: list_a > list_a > list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ~ ( paths_in_Ide_list_a @ Resid @ nil_list_a ) ) ).
% paths_in_rts.Ide.simps(1)
thf(fact_487_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_488_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_489_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_490_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_491_paths__in__rts_Olength__Residx1,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U: list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ ( paths_3541054012941122297list_a @ Resid @ T4 @ U ) ) @ ( size_s349497388124573686list_a @ T4 ) ) ) ).
% paths_in_rts.length_Residx1
thf(fact_492_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_493_paths__in__rts_OResid__Ide_I1_J,axiom,
! [Resid: list_a > list_a > list_a,A2: list_list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Ide_list_a @ Resid @ A2 )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ A2 @ T4 )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ T4 @ A2 )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_Ide(1)
thf(fact_494_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_495_paths__in__rts_OResid__Arr__Ide__ind,axiom,
! [Resid: list_a > list_a > list_a,A2: list_list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Ide_list_a @ Resid @ A2 )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ A2 )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ T4 @ A2 )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_Arr_Ide_ind
thf(fact_496_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_497_paths__in__rts_OResid__Ide__Arr__ind,axiom,
! [Resid: list_a > list_a > list_a,A2: list_list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Ide_list_a @ Resid @ A2 )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ A2 @ T4 )
!= nil_list_a )
=> ( paths_in_Ide_list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ A2 @ T4 ) ) ) ) ) ).
% paths_in_rts.Resid_Ide_Arr_ind
thf(fact_498_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_499_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_500_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_501_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_502_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_503_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_504_paths__in__rts_OSrcs_Oelims,axiom,
! [Resid: nat > nat > nat,X2: list_nat,Y4: set_nat] :
( ( paths_in_rts_nat @ Resid )
=> ( ( ( paths_in_Srcs_nat @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_nat )
=> ( Y4 != bot_bot_set_nat ) )
=> ( ! [T3: nat] :
( ( X2
= ( cons_nat @ T3 @ nil_nat ) )
=> ( Y4
!= ( sources_nat @ Resid @ T3 ) ) )
=> ~ ! [T3: nat] :
( ? [V2: nat,Va: list_nat] :
( X2
= ( cons_nat @ T3 @ ( cons_nat @ V2 @ Va ) ) )
=> ( Y4
!= ( sources_nat @ Resid @ T3 ) ) ) ) ) ) ) ).
% paths_in_rts.Srcs.elims
thf(fact_505_paths__in__rts_OSrcs_Oelims,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a,Y4: set_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_in_Srcs_list_a @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_list_a )
=> ( Y4 != bot_bot_set_list_a ) )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( Y4
!= ( 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 ) ) )
=> ( Y4
!= ( sources_list_a @ Resid @ T3 ) ) ) ) ) ) ) ).
% paths_in_rts.Srcs.elims
thf(fact_506_paths__in__rts_OSrcs_Oelims,axiom,
! [Resid: a > a > a,X2: list_a,Y4: set_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Srcs_a @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_a )
=> ( Y4 != bot_bot_set_a ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y4
!= ( sources_a @ Resid @ T3 ) ) )
=> ~ ! [T3: a] :
( ? [V2: a,Va: list_a] :
( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y4
!= ( sources_a @ Resid @ T3 ) ) ) ) ) ) ) ).
% paths_in_rts.Srcs.elims
thf(fact_507_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_508_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_509_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_510_paths__in__rts_OCon__IdeI_I2_J,axiom,
! [Resid: list_a > list_a > list_a,A2: list_list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Ide_list_a @ Resid @ A2 )
=> ( ( paths_in_Arr_list_a @ Resid @ T4 )
=> ( ( ( paths_in_Srcs_list_a @ Resid @ T4 )
= ( paths_in_Srcs_list_a @ Resid @ A2 ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ T4 @ A2 )
!= nil_list_a ) ) ) ) ) ).
% paths_in_rts.Con_IdeI(2)
thf(fact_511_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_512_paths__in__rts_OCon__IdeI_I1_J,axiom,
! [Resid: list_a > list_a > list_a,A2: list_list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Ide_list_a @ Resid @ A2 )
=> ( ( paths_in_Arr_list_a @ Resid @ T4 )
=> ( ( ( paths_in_Srcs_list_a @ Resid @ T4 )
= ( paths_in_Srcs_list_a @ Resid @ A2 ) )
=> ( ( paths_8620460302779588466list_a @ Resid @ A2 @ T4 )
!= nil_list_a ) ) ) ) ) ).
% paths_in_rts.Con_IdeI(1)
thf(fact_513_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_514_paths__in__rts_OCon__Ide__iff,axiom,
! [Resid: list_a > list_a > list_a,A2: list_list_a,T4: list_list_a] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( paths_in_Ide_list_a @ Resid @ A2 )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ A2 @ T4 )
!= nil_list_a )
= ( ( paths_in_Arr_list_a @ Resid @ T4 )
& ( ( paths_in_Srcs_list_a @ Resid @ T4 )
= ( paths_in_Srcs_list_a @ Resid @ A2 ) ) ) ) ) ) ).
% paths_in_rts.Con_Ide_iff
thf(fact_515_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_516_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 ) ) )
& ! [U5: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U5 @ U2 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U5 @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U5 @ 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 ) ) ) ) ) )
& ! [U5: a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U5 @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U5 @ U2 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ T4 @ ( cons_a @ U5 @ nil_a ) ) @ U2 ) ) ) ) ) ) ) ).
% Resid_cons_ind
thf(fact_517_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_518_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_519_Ide_Oelims_I1_J,axiom,
! [X2: list_a,Y4: $o] :
( ( ( paths_in_Ide_a @ resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_a )
=> Y4 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y4
= ( ~ ( ide_a @ resid @ T3 ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y4
= ( ~ ( ( 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_520_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_521_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_522_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_523_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_524_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_525_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_526_length__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( size_s349497388124573686list_a @ ( append_list_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( size_s349497388124573686list_a @ Ys ) ) ) ).
% length_append
thf(fact_527_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_528_ind,axiom,
! [T4: list_a,U2: list_a,V4: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ V4 @ 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 @ V4 ) ) @ na )
=> ( ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V4 @ T4 ) @ ( paths_in_Resid_a @ resid @ U2 @ T4 ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V4 @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) )
!= nil_a ) )
& ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V4 @ T4 ) @ ( paths_in_Resid_a @ resid @ U2 @ T4 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V4 @ T4 ) @ ( paths_in_Resid_a @ resid @ U2 @ T4 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V4 @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) ) ) ) ) ) ) ) ).
% ind
thf(fact_529_rts_Ocomposable_Ocong,axiom,
composable_a = composable_a ).
% rts.composable.cong
thf(fact_530_paths__in__rts_OSrcs__are__ide,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,T4: list_P4541805568828049459list_a] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ord_le5139818769214234029list_a @ ( paths_6439363147398043123list_a @ Resid @ T4 ) @ ( collec840246186364283544list_a @ ( ide_Pr890063672689881169list_a @ Resid ) ) ) ) ).
% paths_in_rts.Srcs_are_ide
thf(fact_531_paths__in__rts_OSrcs__are__ide,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,T4: list_P2210424090985720871st_a_a] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ord_le2808437291371905441st_a_a @ ( paths_332773396847195111st_a_a @ Resid @ T4 ) @ ( collec3957028472668211340st_a_a @ ( ide_Pr4006845958993808965st_a_a @ Resid ) ) ) ) ).
% paths_in_rts.Srcs_are_ide
thf(fact_532_paths__in__rts_OSrcs__are__ide,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,T4: list_P321204300973800749list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ord_le7857023143581076903list_a @ ( paths_6857623006468044397list_a @ Resid @ T4 ) @ ( collec943055143889122450list_a @ ( ide_Pr1708403647435852235list_a @ Resid ) ) ) ) ).
% paths_in_rts.Srcs_are_ide
thf(fact_533_paths__in__rts_OSrcs__are__ide,axiom,
! [Resid: nat > nat > nat,T4: list_nat] :
( ( paths_in_rts_nat @ Resid )
=> ( ord_less_eq_set_nat @ ( paths_in_Srcs_nat @ Resid @ T4 ) @ ( collect_nat @ ( ide_nat @ Resid ) ) ) ) ).
% paths_in_rts.Srcs_are_ide
thf(fact_534_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_535_paths__in__rts_OTrgs__are__ide,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,T4: list_P4541805568828049459list_a] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ord_le5139818769214234029list_a @ ( paths_8901012877207573880list_a @ Resid @ T4 ) @ ( collec840246186364283544list_a @ ( ide_Pr890063672689881169list_a @ Resid ) ) ) ) ).
% paths_in_rts.Trgs_are_ide
thf(fact_536_paths__in__rts_OTrgs__are__ide,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,T4: list_P2210424090985720871st_a_a] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ord_le2808437291371905441st_a_a @ ( paths_2794423126656725868st_a_a @ Resid @ T4 ) @ ( collec3957028472668211340st_a_a @ ( ide_Pr4006845958993808965st_a_a @ Resid ) ) ) ) ).
% paths_in_rts.Trgs_are_ide
thf(fact_537_paths__in__rts_OTrgs__are__ide,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,T4: list_P321204300973800749list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ord_le7857023143581076903list_a @ ( paths_8434672324333973362list_a @ Resid @ T4 ) @ ( collec943055143889122450list_a @ ( ide_Pr1708403647435852235list_a @ Resid ) ) ) ) ).
% paths_in_rts.Trgs_are_ide
thf(fact_538_paths__in__rts_OTrgs__are__ide,axiom,
! [Resid: nat > nat > nat,T4: list_nat] :
( ( paths_in_rts_nat @ Resid )
=> ( ord_less_eq_set_nat @ ( paths_in_Trgs_nat @ Resid @ T4 ) @ ( collect_nat @ ( ide_nat @ Resid ) ) ) ) ).
% paths_in_rts.Trgs_are_ide
thf(fact_539_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_540_paths__in__rts_Olength__Resid__ind,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a,N: nat] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s349497388124573686list_a @ T4 ) @ ( size_s349497388124573686list_a @ U2 ) ) @ N )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 )
!= nil_list_a )
=> ( ( size_s349497388124573686list_a @ ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 ) )
= ( size_s349497388124573686list_a @ T4 ) ) ) ) ) ).
% paths_in_rts.length_Resid_ind
thf(fact_541_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_542_paths__in__rts_OCon__sym__ind,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a,N: nat] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s349497388124573686list_a @ T4 ) @ ( size_s349497388124573686list_a @ U2 ) ) @ N )
=> ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ U2 )
!= nil_list_a )
= ( ( paths_8620460302779588466list_a @ Resid @ U2 @ T4 )
!= nil_list_a ) ) ) ) ).
% paths_in_rts.Con_sym_ind
thf(fact_543_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_544_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_545_paths__in__rts_OIde_Osimps_I3_J,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,T: produc8685980395799941037list_a,V: produc8685980395799941037list_a,Va2: list_P4541805568828049459list_a] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ( paths_6928703355489960172list_a @ Resid @ ( cons_P8125392100269589091list_a @ T @ ( cons_P8125392100269589091list_a @ V @ Va2 ) ) )
= ( ( ide_Pr890063672689881169list_a @ Resid @ T )
& ( paths_6928703355489960172list_a @ Resid @ ( cons_P8125392100269589091list_a @ V @ Va2 ) )
& ( ord_le5139818769214234029list_a @ ( target2287938126545038671list_a @ Resid @ T ) @ ( paths_6439363147398043123list_a @ Resid @ ( cons_P8125392100269589091list_a @ V @ Va2 ) ) ) ) ) ) ).
% paths_in_rts.Ide.simps(3)
thf(fact_546_paths__in__rts_OIde_Osimps_I3_J,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,T: produc2579390645249093025st_a_a,V: produc2579390645249093025st_a_a,Va2: list_P2210424090985720871st_a_a] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ( paths_822113604939112160st_a_a @ Resid @ ( cons_P2018802349718741079st_a_a @ T @ ( cons_P2018802349718741079st_a_a @ V @ Va2 ) ) )
= ( ( ide_Pr4006845958993808965st_a_a @ Resid @ T )
& ( paths_822113604939112160st_a_a @ Resid @ ( cons_P2018802349718741079st_a_a @ V @ Va2 ) )
& ( ord_le2808437291371905441st_a_a @ ( target5404720412848966467st_a_a @ Resid @ T ) @ ( paths_332773396847195111st_a_a @ Resid @ ( cons_P2018802349718741079st_a_a @ V @ Va2 ) ) ) ) ) ) ).
% paths_in_rts.Ide.simps(3)
thf(fact_547_paths__in__rts_OIde_Osimps_I3_J,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,T: produc9164743771328383783list_a,V: produc9164743771328383783list_a,Va2: list_P321204300973800749list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ( paths_5975756276716474598list_a @ Resid @ ( cons_P5184657343811988189list_a @ T @ ( cons_P5184657343811988189list_a @ V @ Va2 ) ) )
= ( ( ide_Pr1708403647435852235list_a @ Resid @ T )
& ( paths_5975756276716474598list_a @ Resid @ ( cons_P5184657343811988189list_a @ V @ Va2 ) )
& ( ord_le7857023143581076903list_a @ ( target2700282342086602697list_a @ Resid @ T ) @ ( paths_6857623006468044397list_a @ Resid @ ( cons_P5184657343811988189list_a @ V @ Va2 ) ) ) ) ) ) ).
% paths_in_rts.Ide.simps(3)
thf(fact_548_paths__in__rts_OIde_Osimps_I3_J,axiom,
! [Resid: nat > nat > nat,T: nat,V: nat,Va2: list_nat] :
( ( paths_in_rts_nat @ Resid )
=> ( ( paths_in_Ide_nat @ Resid @ ( cons_nat @ T @ ( cons_nat @ V @ Va2 ) ) )
= ( ( ide_nat @ Resid @ T )
& ( paths_in_Ide_nat @ Resid @ ( cons_nat @ V @ Va2 ) )
& ( ord_less_eq_set_nat @ ( targets_nat @ Resid @ T ) @ ( paths_in_Srcs_nat @ Resid @ ( cons_nat @ V @ Va2 ) ) ) ) ) ) ).
% paths_in_rts.Ide.simps(3)
thf(fact_549_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_550_paths__in__rts_OResid__cons__ind,axiom,
! [Resid: list_a > list_a > list_a,T4: list_list_a,U2: list_list_a,N: nat] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( T4 != nil_list_a )
=> ( ( U2 != nil_list_a )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s349497388124573686list_a @ T4 ) @ ( size_s349497388124573686list_a @ U2 ) ) @ N )
=> ( ! [T2: list_a] :
( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T2 @ T4 ) @ U2 )
!= nil_list_a )
= ( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T2 @ nil_list_a ) @ U2 )
!= nil_list_a )
& ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( paths_8620460302779588466list_a @ Resid @ U2 @ ( cons_list_a @ T2 @ nil_list_a ) ) )
!= nil_list_a ) ) )
& ! [U5: list_a] :
( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U5 @ U2 ) )
!= nil_list_a )
= ( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U5 @ nil_list_a ) )
!= nil_list_a )
& ( ( paths_8620460302779588466list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U5 @ nil_list_a ) ) @ U2 )
!= nil_list_a ) ) )
& ! [T2: list_a] :
( ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T2 @ T4 ) @ U2 )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T2 @ T4 ) @ U2 )
= ( append_list_a @ ( paths_8620460302779588466list_a @ Resid @ ( cons_list_a @ T2 @ nil_list_a ) @ U2 ) @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( paths_8620460302779588466list_a @ Resid @ U2 @ ( cons_list_a @ T2 @ nil_list_a ) ) ) ) ) )
& ! [U5: list_a] :
( ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U5 @ U2 ) )
!= nil_list_a )
=> ( ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U5 @ U2 ) )
= ( paths_8620460302779588466list_a @ Resid @ ( paths_8620460302779588466list_a @ Resid @ T4 @ ( cons_list_a @ U5 @ nil_list_a ) ) @ U2 ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_cons_ind
thf(fact_551_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 ) ) )
& ! [U5: a] :
( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U5 @ U2 ) )
!= nil_a )
= ( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U5 @ nil_a ) )
!= nil_a )
& ( ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U5 @ 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 ) ) ) ) ) )
& ! [U5: a] :
( ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U5 @ U2 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U5 @ U2 ) )
= ( paths_in_Resid_a @ Resid @ ( paths_in_Resid_a @ Resid @ T4 @ ( cons_a @ U5 @ nil_a ) ) @ U2 ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_cons_ind
thf(fact_552_paths__in__rts_OIde_Oelims_I1_J,axiom,
! [Resid: list_a > list_a > list_a,X2: list_list_a,Y4: $o] :
( ( paths_in_rts_list_a @ Resid )
=> ( ( ( paths_in_Ide_list_a @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_list_a )
=> Y4 )
=> ( ! [T3: list_a] :
( ( X2
= ( cons_list_a @ T3 @ nil_list_a ) )
=> ( Y4
= ( ~ ( 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 ) ) )
=> ( Y4
= ( ~ ( ( 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_553_paths__in__rts_OIde_Oelims_I1_J,axiom,
! [Resid: produc8685980395799941037list_a > produc8685980395799941037list_a > produc8685980395799941037list_a,X2: list_P4541805568828049459list_a,Y4: $o] :
( ( paths_1115770337333439634list_a @ Resid )
=> ( ( ( paths_6928703355489960172list_a @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_Pr6630411782098800307list_a )
=> Y4 )
=> ( ! [T3: produc8685980395799941037list_a] :
( ( X2
= ( cons_P8125392100269589091list_a @ T3 @ nil_Pr6630411782098800307list_a ) )
=> ( Y4
= ( ~ ( ide_Pr890063672689881169list_a @ Resid @ T3 ) ) ) )
=> ~ ! [T3: produc8685980395799941037list_a,V2: produc8685980395799941037list_a,Va: list_P4541805568828049459list_a] :
( ( X2
= ( cons_P8125392100269589091list_a @ T3 @ ( cons_P8125392100269589091list_a @ V2 @ Va ) ) )
=> ( Y4
= ( ~ ( ( ide_Pr890063672689881169list_a @ Resid @ T3 )
& ( paths_6928703355489960172list_a @ Resid @ ( cons_P8125392100269589091list_a @ V2 @ Va ) )
& ( ord_le5139818769214234029list_a @ ( target2287938126545038671list_a @ Resid @ T3 ) @ ( paths_6439363147398043123list_a @ Resid @ ( cons_P8125392100269589091list_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(1)
thf(fact_554_paths__in__rts_OIde_Oelims_I1_J,axiom,
! [Resid: produc2579390645249093025st_a_a > produc2579390645249093025st_a_a > produc2579390645249093025st_a_a,X2: list_P2210424090985720871st_a_a,Y4: $o] :
( ( paths_4232552623637367430st_a_a @ Resid )
=> ( ( ( paths_822113604939112160st_a_a @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_Pr523822031547952295st_a_a )
=> Y4 )
=> ( ! [T3: produc2579390645249093025st_a_a] :
( ( X2
= ( cons_P2018802349718741079st_a_a @ T3 @ nil_Pr523822031547952295st_a_a ) )
=> ( Y4
= ( ~ ( ide_Pr4006845958993808965st_a_a @ Resid @ T3 ) ) ) )
=> ~ ! [T3: produc2579390645249093025st_a_a,V2: produc2579390645249093025st_a_a,Va: list_P2210424090985720871st_a_a] :
( ( X2
= ( cons_P2018802349718741079st_a_a @ T3 @ ( cons_P2018802349718741079st_a_a @ V2 @ Va ) ) )
=> ( Y4
= ( ~ ( ( ide_Pr4006845958993808965st_a_a @ Resid @ T3 )
& ( paths_822113604939112160st_a_a @ Resid @ ( cons_P2018802349718741079st_a_a @ V2 @ Va ) )
& ( ord_le2808437291371905441st_a_a @ ( target5404720412848966467st_a_a @ Resid @ T3 ) @ ( paths_332773396847195111st_a_a @ Resid @ ( cons_P2018802349718741079st_a_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(1)
thf(fact_555_paths__in__rts_OIde_Oelims_I1_J,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,X2: list_P321204300973800749list_a,Y4: $o] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ( ( paths_5975756276716474598list_a @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_Pr3188421586756112173list_a )
=> Y4 )
=> ( ! [T3: produc9164743771328383783list_a] :
( ( X2
= ( cons_P5184657343811988189list_a @ T3 @ nil_Pr3188421586756112173list_a ) )
=> ( Y4
= ( ~ ( ide_Pr1708403647435852235list_a @ Resid @ T3 ) ) ) )
=> ~ ! [T3: produc9164743771328383783list_a,V2: produc9164743771328383783list_a,Va: list_P321204300973800749list_a] :
( ( X2
= ( cons_P5184657343811988189list_a @ T3 @ ( cons_P5184657343811988189list_a @ V2 @ Va ) ) )
=> ( Y4
= ( ~ ( ( ide_Pr1708403647435852235list_a @ Resid @ T3 )
& ( paths_5975756276716474598list_a @ Resid @ ( cons_P5184657343811988189list_a @ V2 @ Va ) )
& ( ord_le7857023143581076903list_a @ ( target2700282342086602697list_a @ Resid @ T3 ) @ ( paths_6857623006468044397list_a @ Resid @ ( cons_P5184657343811988189list_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(1)
thf(fact_556_paths__in__rts_OIde_Oelims_I1_J,axiom,
! [Resid: nat > nat > nat,X2: list_nat,Y4: $o] :
( ( paths_in_rts_nat @ Resid )
=> ( ( ( paths_in_Ide_nat @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_nat )
=> Y4 )
=> ( ! [T3: nat] :
( ( X2
= ( cons_nat @ T3 @ nil_nat ) )
=> ( Y4
= ( ~ ( ide_nat @ Resid @ T3 ) ) ) )
=> ~ ! [T3: nat,V2: nat,Va: list_nat] :
( ( X2
= ( cons_nat @ T3 @ ( cons_nat @ V2 @ Va ) ) )
=> ( Y4
= ( ~ ( ( ide_nat @ Resid @ T3 )
& ( paths_in_Ide_nat @ Resid @ ( cons_nat @ V2 @ Va ) )
& ( ord_less_eq_set_nat @ ( targets_nat @ Resid @ T3 ) @ ( paths_in_Srcs_nat @ Resid @ ( cons_nat @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(1)
thf(fact_557_paths__in__rts_OIde_Oelims_I1_J,axiom,
! [Resid: a > a > a,X2: list_a,Y4: $o] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Ide_a @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_a )
=> Y4 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y4
= ( ~ ( ide_a @ Resid @ T3 ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y4
= ( ~ ( ( 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_558_paths__in__rts_OIde_Oelims_I2_J,axiom,
! [Resid: produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,X2: list_P321204300973800749list_a] :
( ( paths_893215811548098956list_a @ Resid )
=> ( ( paths_5975756276716474598list_a @ Resid @ X2 )
=> ( ! [T3: produc9164743771328383783list_a] :
( ( X2
= ( cons_P5184657343811988189list_a @ T3 @ nil_Pr3188421586756112173list_a ) )
=> ~ ( ide_Pr1708403647435852235list_a @ Resid @ T3 ) )
=> ~ ! [T3: produc9164743771328383783list_a,V2: produc9164743771328383783list_a,Va: list_P321204300973800749list_a] :
( ( X2
= ( cons_P5184657343811988189list_a @ T3 @ ( cons_P5184657343811988189list_a @ V2 @ Va ) ) )
=> ~ ( ( ide_Pr1708403647435852235list_a @ Resid @ T3 )
& ( paths_5975756276716474598list_a @ Resid @ ( cons_P5184657343811988189list_a @ V2 @ Va ) )
& ( ord_le7857023143581076903list_a @ ( target2700282342086602697list_a @ Resid @ T3 ) @ ( paths_6857623006468044397list_a @ Resid @ ( cons_P5184657343811988189list_a @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(2)
thf(fact_559_paths__in__rts_OIde_Oelims_I2_J,axiom,
! [Resid: nat > nat > nat,X2: list_nat] :
( ( paths_in_rts_nat @ Resid )
=> ( ( paths_in_Ide_nat @ Resid @ X2 )
=> ( ! [T3: nat] :
( ( X2
= ( cons_nat @ T3 @ nil_nat ) )
=> ~ ( ide_nat @ Resid @ T3 ) )
=> ~ ! [T3: nat,V2: nat,Va: list_nat] :
( ( X2
= ( cons_nat @ T3 @ ( cons_nat @ V2 @ Va ) ) )
=> ~ ( ( ide_nat @ Resid @ T3 )
& ( paths_in_Ide_nat @ Resid @ ( cons_nat @ V2 @ Va ) )
& ( ord_less_eq_set_nat @ ( targets_nat @ Resid @ T3 ) @ ( paths_in_Srcs_nat @ Resid @ ( cons_nat @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Ide.elims(2)
thf(fact_560_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_561_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_562_len,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_list_a @ ta ) @ ( size_size_list_a @ ua ) ) @ ( size_size_list_a @ va ) ) @ ( suc @ na ) ).
% len
thf(fact_563__092_060open_062_092_060And_062V_AU_AT_O_A_092_060lbrakk_062T_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_A_092_060noteq_062_A_091_093_059_AV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_A_092_060noteq_062_A_091_093_059_Alength_AT_A_L_Alength_AU_A_L_Alength_AV_A_092_060le_062_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_092_060noteq_062_A_091_093_J_A_061_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060noteq_062_A_091_093_J_A_092_060and_062_A_I_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_092_060noteq_062_A_091_093_A_092_060longrightarrow_062_A_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AT_J_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_AT_J_A_061_A_IV_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_A_092_060_094sup_062_K_092_092_060_094sup_062_K_A_IT_A_092_060_094sup_062_K_092_092_060_094sup_062_K_AU_J_J_092_060close_062,axiom,
! [T4: list_a,U2: list_a,V4: list_a] :
( ( ( paths_in_Resid_a @ resid @ T4 @ U2 )
!= nil_a )
=> ( ( ( paths_in_Resid_a @ resid @ V4 @ 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 @ V4 ) ) @ zero_zero_nat )
=> ( ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V4 @ T4 ) @ ( paths_in_Resid_a @ resid @ U2 @ T4 ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V4 @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) )
!= nil_a ) )
& ( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V4 @ T4 ) @ ( paths_in_Resid_a @ resid @ U2 @ T4 ) )
!= nil_a )
=> ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V4 @ T4 ) @ ( paths_in_Resid_a @ resid @ U2 @ T4 ) )
= ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ V4 @ U2 ) @ ( paths_in_Resid_a @ resid @ T4 @ U2 ) ) ) ) ) ) ) ) ).
% \<open>\<And>V U T. \<lbrakk>T \<^sup>*\\<^sup>* U \<noteq> []; V \<^sup>*\\<^sup>* T \<noteq> []; length T + length U + length V \<le> 0\<rbrakk> \<Longrightarrow> ((V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) \<noteq> []) = ((V \<^sup>*\\<^sup>* U) \<^sup>*\\<^sup>* (T \<^sup>*\\<^sup>* U) \<noteq> []) \<and> ((V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) \<noteq> [] \<longrightarrow> (V \<^sup>*\\<^sup>* T) \<^sup>*\\<^sup>* (U \<^sup>*\\<^sup>* T) = (V \<^sup>*\\<^sup>* U) \<^sup>*\\<^sup>* (T \<^sup>*\\<^sup>* U))\<close>
thf(fact_564_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_565_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_566_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_567_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_568_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_569_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_570_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_571_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_572_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_573_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_574_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_575_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_576_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_577_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_578_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_579_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_580_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_581_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_582_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_583_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y4: nat] :
( ( ( plus_plus_nat @ X2 @ Y4 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_584_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y4: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y4 ) )
= ( ( X2 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_585_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_586_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_587_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_588_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_589_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_590_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_591_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_592_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_593_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_594_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_595_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_596_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_597_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R @ X @ X )
=> ( ! [X: nat,Y3: nat,Z: nat] :
( ( R @ X @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_598_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_599_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_600_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_601_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_602_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_603_Suc__le__D,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_604_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_605_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_606_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_607_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_608_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_609_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_610_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_611_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_612_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_613_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_614_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_615_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_616_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_617_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_618_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_619_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_620_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_621_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_622_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_623_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_624_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_625_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_626_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_627_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_628_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_629_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_630_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_631_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( ( ( plus_plus_nat @ X2 @ Y4 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_632_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y4 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y4 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_633_list_Osize_I4_J,axiom,
! [X21: a,X22: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_634_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y2: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y2 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_635_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y2: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y2 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_636_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_637_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X3: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_638_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_639_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_640_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_641_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_642_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_643_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_644_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_645_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_646_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_647_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_648_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_649_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_650_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_651_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_652_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_653_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_654_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_655_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_656_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_657_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [T5: a] :
( ( member_a @ T5 @ A6 )
=> ( member_a @ T5 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_658_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_659_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_660_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_661_set__eq__subset,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 ) ) ) ) ).
% set_eq_subset
thf(fact_662_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_663_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y4: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y4 ) )
=> ( X2 != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_664_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_665_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_666_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_667_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_668_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_669_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_670_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y2: a,Ys2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_671_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_672_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_673_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_674_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_675_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_676_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_677_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_678_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C4: nat] :
( B4
= ( plus_plus_nat @ A5 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_679_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_680_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_681_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_682_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_683_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_684_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_685_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_686_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_687_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_688_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_689_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_690_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_691_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M5 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_692_length__append__singleton,axiom,
! [Xs: list_a,X2: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_693_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_694_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_695_Arr_Oelims_I1_J,axiom,
! [X2: list_a,Y4: $o] :
( ( ( paths_in_Arr_a @ resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_a )
=> Y4 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y4
= ( ~ ( arr_a @ resid @ T3 ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y4
= ( ~ ( ( 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_696_R_Oprfx__reflexive,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ide_a @ resid @ ( resid @ T @ T ) ) ) ).
% R.prfx_reflexive
thf(fact_697_R_Oide__implies__arr,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( arr_a @ resid @ A ) ) ).
% R.ide_implies_arr
thf(fact_698_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_699_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_700_R_Oarr__resid,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( arr_a @ resid @ ( resid @ T @ U ) ) ) ).
% R.arr_resid
thf(fact_701_R_Oarr__def,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
= ( con_a @ resid @ T @ T ) ) ).
% R.arr_def
thf(fact_702_R_OarrE,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( con_a @ resid @ T @ T ) ) ).
% R.arrE
thf(fact_703_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_704_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_705_R_Onot__arr__null,axiom,
~ ( arr_a @ resid @ ( partial_null_a @ resid ) ) ).
% R.not_arr_null
thf(fact_706_R_OcomposableD_I1_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( arr_a @ resid @ T ) ) ).
% R.composableD(1)
thf(fact_707_R_OcomposableD_I2_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( arr_a @ resid @ U ) ) ).
% R.composableD(2)
thf(fact_708_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_709_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_710_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_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_R_OarrI,axiom,
! [T: a] :
( ( con_a @ resid @ T @ T )
=> ( arr_a @ resid @ T ) ) ).
% R.arrI
thf(fact_719_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_720_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_721_residuation_Oarr_Ocong,axiom,
arr_a = arr_a ).
% residuation.arr.cong
thf(fact_722_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_723_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_724_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ U4 @ NN )
=> ( ( ( sources_a @ Resid @ U )
= ( sources_a @ Resid @ U4 ) )
=> ( ( ( targets_a @ Resid @ U )
= ( targets_a @ Resid @ U4 ) )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U4 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T @ U4 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_725_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_726_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_727_paths__in__rts_OArr_Oelims_I1_J,axiom,
! [Resid: a > a > a,X2: list_a,Y4: $o] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Arr_a @ Resid @ X2 )
= Y4 )
=> ( ( ( X2 = nil_a )
=> Y4 )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( Y4
= ( ~ ( arr_a @ Resid @ T3 ) ) ) )
=> ~ ! [T3: a,V2: a,Va: list_a] :
( ( X2
= ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) )
=> ( Y4
= ( ~ ( ( 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_728_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_729_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_730_length__Cons,axiom,
! [X2: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X2 @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_731_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_732_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_733_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_734_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_735_R_Ocong__respects__seq,axiom,
! [T: a,U: a,T6: a,U4: a] :
( ( seq_a @ resid @ T @ U )
=> ( ( ( ide_a @ resid @ ( resid @ T @ T6 ) )
& ( ide_a @ resid @ ( resid @ T6 @ T ) ) )
=> ( ( ( ide_a @ resid @ ( resid @ U @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U ) ) )
=> ( seq_a @ resid @ T6 @ U4 ) ) ) ) ).
% R.cong_respects_seq
thf(fact_736_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_737_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_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_747_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_748_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_749_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_750_set__ConsD,axiom,
! [Y4: a,X2: a,Xs: list_a] :
( ( member_a @ Y4 @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
=> ( ( Y4 = X2 )
| ( member_a @ Y4 @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_751_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_752_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_753_list_Oset__intros_I2_J,axiom,
! [Y4: a,X22: list_a,X21: a] :
( ( member_a @ Y4 @ ( set_a2 @ X22 ) )
=> ( member_a @ Y4 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_754_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_755_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_756_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_757_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_758_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_759_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 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( X3 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_760_Int__mono,axiom,
! [A2: set_a,C2: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_761_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_762_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_763_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_764_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_765_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_766_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_767_rts_Oseq_Ocong,axiom,
seq_a = seq_a ).
% rts.seq.cong
thf(fact_768_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_769_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_770_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y2: a] :
( ( member_a @ Y2 @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_771_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y2: a] :
( ( member_a @ Y2 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_772_in__set__conv__decomp__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_773_in__set__conv__decomp__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_774_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_775_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_a,X: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_776_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_777_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_a,X: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_778_in__set__conv__decomp,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_779_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_780_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_781_split__list__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_782_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_783_split__list__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_784_split__list,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_785_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_786_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_787_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_le__inf__iff,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y4 @ Z3 ) )
= ( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_801_le__inf__iff,axiom,
! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y4 @ Z3 ) )
= ( ( ord_less_eq_set_a @ X2 @ Y4 )
& ( ord_less_eq_set_a @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_802_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_803_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_804_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_805_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_806_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_807_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_808_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_809_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_810_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_811_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_812_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_813_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_814_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_815_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_816_insert__mono,axiom,
! [C2: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_817_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_818_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_819_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_820_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_821_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_822_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_823_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_824_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_825_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_826_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_827_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_828_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_829_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A5: set_a] :
( ( inf_inf_set_a @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_830_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( inf_inf_nat @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_831_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ( inf_inf_set_a @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_832_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_833_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_834_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_835_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_836_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( A5
= ( inf_inf_nat @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_837_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
( A5
= ( inf_inf_set_a @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_838_inf__greatest,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y4 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_839_inf__greatest,axiom,
! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y4 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_840_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_841_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_842_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_843_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_844_inf__absorb2,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_845_inf__absorb2,axiom,
! [Y4: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( ( inf_inf_set_a @ X2 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_846_inf__absorb1,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( inf_inf_nat @ X2 @ Y4 )
= X2 ) ) ).
% inf_absorb1
thf(fact_847_inf__absorb1,axiom,
! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( inf_inf_set_a @ X2 @ Y4 )
= X2 ) ) ).
% inf_absorb1
thf(fact_848_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_849_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_850_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_851_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_852_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( inf_inf_nat @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_853_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_854_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y4: nat] :
( ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( F @ Y3 @ Z ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y4 )
= ( F @ X2 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_855_inf__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y4: set_a] :
( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( F @ Y3 @ Z ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ Y4 )
= ( F @ X2 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_856_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_857_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_858_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_859_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_860_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_861_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_862_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_863_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_864_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_865_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_866_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_867_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_868_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_869_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_870_inf__le2,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_871_inf__le2,axiom,
! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_872_inf__le1,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y4 ) @ X2 ) ).
% inf_le1
thf(fact_873_inf__le1,axiom,
! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ X2 ) ).
% inf_le1
thf(fact_874_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y4 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_875_inf__sup__ord_I1_J,axiom,
! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_876_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_877_inf__sup__ord_I2_J,axiom,
! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_878_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_879_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_880_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_881_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_882_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_883_R_Otrg__def,axiom,
! [T: a] :
( ( trg_a @ resid @ T )
= ( resid @ T @ T ) ) ).
% R.trg_def
thf(fact_884_R_Ojoin__of__un__upto__cong,axiom,
! [T: a,U: a,V: a,V3: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( join_of_a @ resid @ T @ U @ V3 )
=> ( ( ide_a @ resid @ ( resid @ V @ V3 ) )
& ( ide_a @ resid @ ( resid @ V3 @ V ) ) ) ) ) ).
% R.join_of_un_upto_cong
thf(fact_885_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_886_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_887_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_888_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_889_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_890_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_891_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_892_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_893_R_Ojoinable__def,axiom,
! [T: a,U: a] :
( ( joinable_a @ resid @ T @ U )
= ( ? [X6: a] : ( join_of_a @ resid @ T @ U @ X6 ) ) ) ).
% R.joinable_def
thf(fact_894_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_895_R_Oide__trg,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ide_a @ resid @ ( trg_a @ resid @ T ) ) ) ).
% R.ide_trg
thf(fact_896_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_897_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_898_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_899_residuation_Otrg_Ocong,axiom,
trg_a = trg_a ).
% residuation.trg.cong
thf(fact_900_rts_Ojoin__of_Ocong,axiom,
join_of_a = join_of_a ).
% rts.join_of.cong
thf(fact_901_length__n__lists__elem,axiom,
! [Ys: list_a,N: nat,Xs: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( n_lists_a @ N @ Xs ) ) )
=> ( ( size_size_list_a @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_902_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_903_rts__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a] :
( ( arr_a @ Resid @ T3 )
=> ( ide_a @ Resid @ ( trg_a @ Resid @ T3 ) ) )
=> ( ! [A3: a,T3: a] :
( ( ide_a @ Resid @ A3 )
=> ( ( con_a @ Resid @ T3 @ A3 )
=> ( ( Resid @ T3 @ A3 )
= T3 ) ) )
=> ( ! [A3: a,T3: a] :
( ( ide_a @ Resid @ A3 )
=> ( ( con_a @ Resid @ A3 @ T3 )
=> ( ide_a @ Resid @ ( Resid @ A3 @ 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_904_rts__axioms__def,axiom,
( rts_axioms_a
= ( ^ [Resid2: a > a > a] :
( ! [T5: a] :
( ( arr_a @ Resid2 @ T5 )
=> ( ide_a @ Resid2 @ ( trg_a @ Resid2 @ T5 ) ) )
& ! [A5: a,T5: a] :
( ( ide_a @ Resid2 @ A5 )
=> ( ( con_a @ Resid2 @ T5 @ A5 )
=> ( ( Resid2 @ T5 @ A5 )
= T5 ) ) )
& ! [A5: a,T5: a] :
( ( ide_a @ Resid2 @ A5 )
=> ( ( con_a @ Resid2 @ A5 @ T5 )
=> ( ide_a @ Resid2 @ ( Resid2 @ A5 @ T5 ) ) ) )
& ! [T5: a,U6: a] :
( ( con_a @ Resid2 @ T5 @ U6 )
=> ? [A5: a] :
( ( ide_a @ Resid2 @ A5 )
& ( con_a @ Resid2 @ A5 @ T5 )
& ( con_a @ Resid2 @ A5 @ 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_905_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_906_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_907_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_908_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_909_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_910_le__cases3,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_911_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_912_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_913_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_914_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_915_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_916_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_917_order__antisym,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_918_order__antisym,axiom,
! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_919_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_920_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_921_order__trans,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_922_order__trans,axiom,
! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_923_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( P @ A3 @ B6 ) )
=> ( ! [A3: nat,B6: nat] :
( ( P @ B6 @ A3 )
=> ( P @ A3 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_924_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_925_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_926_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_927_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_928_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_929_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_930_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_931_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_932_order__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order_eq_iff
thf(fact_933_order__eq__iff,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 ) ) ) ) ).
% order_eq_iff
thf(fact_934_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,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_935_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_936_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,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_937_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_938_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,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_939_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,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_940_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_941_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_942_order__eq__refl,axiom,
! [X2: nat,Y4: nat] :
( ( X2 = Y4 )
=> ( ord_less_eq_nat @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_943_order__eq__refl,axiom,
! [X2: set_a,Y4: set_a] :
( ( X2 = Y4 )
=> ( ord_less_eq_set_a @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_944_linorder__linear,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_linear
thf(fact_945_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,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_946_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,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_947_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_948_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_949_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,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_950_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,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_951_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_952_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_953_linorder__le__cases,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_le_cases
thf(fact_954_order__antisym__conv,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_955_order__antisym__conv,axiom,
! [Y4: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_956_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_957_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_958_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_959_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_960_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_961_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_962_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_963_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_964_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_965_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_966_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_967_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_968_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_969_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_970_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_971_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_972_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_973_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_974_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_975_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_976_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_977_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_978_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_979_the__elem__set,axiom,
! [X2: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_980_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_a
= ( ^ [Xs5: list_a] : ( if_nat @ ( Xs5 = nil_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_a @ ( tl_a @ Xs5 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_981_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_982_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_983_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_a
= ( ^ [F2: a > nat,Xs5: list_a] : ( if_nat @ ( Xs5 = nil_a ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F2 @ ( hd_a @ Xs5 ) ) @ ( size_list_a @ F2 @ ( tl_a @ Xs5 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_984_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
( A6
= ( insert_a @ ( the_elem_a @ A6 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_985_Arr_Opelims_I1_J,axiom,
! [X2: list_a,Y4: $o] :
( ( ( paths_in_Arr_a @ resid @ X2 )
= Y4 )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ~ Y4
=> ~ ( accp_list_a @ paths_in_Arr_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y4
= ( 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 ) ) )
=> ( ( Y4
= ( ( 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_986_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_987_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_988_size__list__append,axiom,
! [F: a > nat,Xs: list_a,Ys: list_a] :
( ( size_list_a @ F @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_list_a @ F @ Xs ) @ ( size_list_a @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_989_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_990_Srcs_Opelims,axiom,
! [X2: list_a,Y4: set_a] :
( ( ( paths_in_Srcs_a @ resid @ X2 )
= Y4 )
=> ( ( accp_list_a @ paths_in_Srcs_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y4 = bot_bot_set_a )
=> ~ ( accp_list_a @ paths_in_Srcs_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y4
= ( 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 ) ) )
=> ( ( Y4
= ( sources_a @ resid @ T3 ) )
=> ~ ( accp_list_a @ paths_in_Srcs_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% Srcs.pelims
thf(fact_991_Trgs_Opelims,axiom,
! [X2: list_a,Y4: set_a] :
( ( ( paths_in_Trgs_a @ resid @ X2 )
= Y4 )
=> ( ( accp_list_a @ paths_in_Trgs_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y4 = bot_bot_set_a )
=> ~ ( accp_list_a @ paths_in_Trgs_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y4
= ( 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 ) ) )
=> ( ( Y4
= ( 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_992_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X: a,Y3: a] :
( ( member_a @ X @ A2 )
=> ( ( member_a @ Y3 @ A2 )
=> ( X = Y3 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_993_list_Osize__gen_I1_J,axiom,
! [X2: a > nat] :
( ( size_list_a @ X2 @ nil_a )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_994_size__list__estimation_H,axiom,
! [X2: a,Xs: list_a,Y4: nat,F: a > nat] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ Y4 @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ Y4 @ ( size_list_a @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_995_size__list__pointwise,axiom,
! [Xs: list_a,F: a > nat,G: a > nat] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_nat @ ( size_list_a @ F @ Xs ) @ ( size_list_a @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_996_is__singletonE,axiom,
! [A2: set_a] :
( ( is_singleton_a @ A2 )
=> ~ ! [X: a] :
( A2
!= ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_997_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
? [X3: a] :
( A6
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_998_paths__in__rts_OArr_Opelims_I1_J,axiom,
! [Resid: a > a > a,X2: list_a,Y4: $o] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Arr_a @ Resid @ X2 )
= Y4 )
=> ( ( accp_list_a @ paths_in_Arr_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ~ Y4
=> ~ ( accp_list_a @ paths_in_Arr_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y4
= ( 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 ) ) )
=> ( ( Y4
= ( ( 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_999_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_1000_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_1001_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_1002_Ide_Opelims_I1_J,axiom,
! [X2: list_a,Y4: $o] :
( ( ( paths_in_Ide_a @ resid @ X2 )
= Y4 )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ~ Y4
=> ~ ( accp_list_a @ paths_in_Ide_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y4
= ( 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 ) ) )
=> ( ( Y4
= ( ( 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_1003_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_1004_paths__in__rts_OIde_Opelims_I1_J,axiom,
! [Resid: a > a > a,X2: list_a,Y4: $o] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Ide_a @ Resid @ X2 )
= Y4 )
=> ( ( accp_list_a @ paths_in_Ide_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ~ Y4
=> ~ ( accp_list_a @ paths_in_Ide_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y4
= ( 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 ) ) )
=> ( ( Y4
= ( ( 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_1005_paths__in__rts_OSrcs_Opelims,axiom,
! [Resid: a > a > a,X2: list_a,Y4: set_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Srcs_a @ Resid @ X2 )
= Y4 )
=> ( ( accp_list_a @ paths_in_Srcs_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y4 = bot_bot_set_a )
=> ~ ( accp_list_a @ paths_in_Srcs_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y4
= ( 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 ) ) )
=> ( ( Y4
= ( sources_a @ Resid @ T3 ) )
=> ~ ( accp_list_a @ paths_in_Srcs_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Srcs.pelims
thf(fact_1006_paths__in__rts_OTrgs_Opelims,axiom,
! [Resid: a > a > a,X2: list_a,Y4: set_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( ( paths_in_Trgs_a @ Resid @ X2 )
= Y4 )
=> ( ( accp_list_a @ paths_in_Trgs_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y4 = bot_bot_set_a )
=> ~ ( accp_list_a @ paths_in_Trgs_rel_a @ nil_a ) ) )
=> ( ! [T3: a] :
( ( X2
= ( cons_a @ T3 @ nil_a ) )
=> ( ( Y4
= ( 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 ) ) )
=> ( ( Y4
= ( paths_in_Trgs_a @ Resid @ ( cons_a @ V2 @ Va ) ) )
=> ~ ( accp_list_a @ paths_in_Trgs_rel_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Trgs.pelims
thf(fact_1007_simulation__axioms__def,axiom,
( simula3868467710248865958ms_a_a
= ( ^ [A6: a > a > a,B5: a > a > a,F3: a > a] :
( ! [T5: a] :
( ~ ( arr_a @ A6 @ T5 )
=> ( ( F3 @ T5 )
= ( partial_null_a @ B5 ) ) )
& ! [T5: a,U6: a] :
( ( con_a @ A6 @ T5 @ U6 )
=> ( con_a @ B5 @ ( F3 @ T5 ) @ ( F3 @ U6 ) ) )
& ! [T5: a,U6: a] :
( ( con_a @ A6 @ T5 @ U6 )
=> ( ( F3 @ ( A6 @ T5 @ U6 ) )
= ( B5 @ ( F3 @ T5 ) @ ( F3 @ U6 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_1008_simulation__axioms_Ointro,axiom,
! [A2: a > a > a,F4: a > a,B3: a > a > a] :
( ! [T3: a] :
( ~ ( arr_a @ A2 @ T3 )
=> ( ( F4 @ T3 )
= ( partial_null_a @ B3 ) ) )
=> ( ! [T3: a,U3: a] :
( ( con_a @ A2 @ T3 @ U3 )
=> ( con_a @ B3 @ ( F4 @ T3 ) @ ( F4 @ U3 ) ) )
=> ( ! [T3: a,U3: a] :
( ( con_a @ A2 @ T3 @ U3 )
=> ( ( F4 @ ( A2 @ T3 @ U3 ) )
= ( B3 @ ( F4 @ T3 ) @ ( F4 @ U3 ) ) ) )
=> ( simula3868467710248865958ms_a_a @ A2 @ B3 @ F4 ) ) ) ) ).
% simulation_axioms.intro
thf(fact_1009_coherent__normal__sub__rts__axioms_Ointro,axiom,
! [Resid: a > a > a,NN: set_a] :
( ! [T3: a,U3: a,U7: a] :
( ( arr_a @ Resid @ T3 )
=> ( ( member_a @ U3 @ NN )
=> ( ( member_a @ U7 @ NN )
=> ( ( ( sources_a @ Resid @ U3 )
= ( sources_a @ Resid @ U7 ) )
=> ( ( ( targets_a @ Resid @ U3 )
= ( targets_a @ Resid @ U7 ) )
=> ( ( ( sources_a @ Resid @ T3 )
= ( sources_a @ Resid @ U3 ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T3 @ U3 ) @ ( Resid @ T3 @ U7 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T3 @ U7 ) @ ( Resid @ T3 @ U3 ) ) @ NN ) ) ) ) ) ) ) )
=> ( cohere4894532172567702276ioms_a @ Resid @ NN ) ) ).
% coherent_normal_sub_rts_axioms.intro
thf(fact_1010_coherent__normal__sub__rts__axioms__def,axiom,
( cohere4894532172567702276ioms_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
! [T5: a,U6: a,U8: a] :
( ( arr_a @ Resid2 @ T5 )
=> ( ( member_a @ U6 @ NN2 )
=> ( ( member_a @ U8 @ NN2 )
=> ( ( ( sources_a @ Resid2 @ U6 )
= ( sources_a @ Resid2 @ U8 ) )
=> ( ( ( targets_a @ Resid2 @ U6 )
= ( targets_a @ Resid2 @ U8 ) )
=> ( ( ( sources_a @ Resid2 @ T5 )
= ( sources_a @ Resid2 @ U6 ) )
=> ( ( member_a @ ( Resid2 @ ( Resid2 @ T5 @ U6 ) @ ( Resid2 @ T5 @ U8 ) ) @ NN2 )
& ( member_a @ ( Resid2 @ ( Resid2 @ T5 @ U8 ) @ ( Resid2 @ T5 @ U6 ) ) @ NN2 ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts_axioms_def
thf(fact_1011_coherent__normal__sub__rts_Oaxioms_I2_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( cohere4894532172567702276ioms_a @ Resid @ NN ) ) ).
% coherent_normal_sub_rts.axioms(2)
thf(fact_1012_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_1013_normal__sub__rts_OCong_Ocong,axiom,
normal_sub_Cong_a = normal_sub_Cong_a ).
% normal_sub_rts.Cong.cong
thf(fact_1014_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,T6: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( ( ( sources_a @ Resid @ T6 )
= ( sources_a @ Resid @ U4 ) )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U4 )
=> ( ( con_a @ Resid @ T @ U )
= ( con_a @ Resid @ T6 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_1015_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,U: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U4 )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( ( sources_a @ Resid @ T6 )
= ( sources_a @ Resid @ U4 ) )
=> ( con_a @ Resid @ T6 @ U4 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_1016_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,U: a,U4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U4 )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( ( sources_a @ Resid @ T6 )
= ( sources_a @ Resid @ U4 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ ( Resid @ T @ U ) @ ( Resid @ T6 @ U4 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_1017_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_1018_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_1019_R_Ocomposite__of__unq__upto__cong,axiom,
! [U: a,T: a,V: a,V3: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ( ( composite_of_a @ resid @ U @ T @ V3 )
=> ( ( ide_a @ resid @ ( resid @ V @ V3 ) )
& ( ide_a @ resid @ ( resid @ V3 @ V ) ) ) ) ) ).
% R.composite_of_unq_upto_cong
thf(fact_1020_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_1021_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_1022_R_Ocomposite__of__cancel__left,axiom,
! [T: a,U: a,V: a,U4: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( composite_of_a @ resid @ T @ U4 @ V )
=> ( ( ide_a @ resid @ ( resid @ U @ U4 ) )
& ( ide_a @ resid @ ( resid @ U4 @ U ) ) ) ) ) ).
% R.composite_of_cancel_left
thf(fact_1023_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_1024_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_1025_R_Obounded__imp__con,axiom,
! [T: a,U: a,V: a,T6: a,U4: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( composite_of_a @ resid @ T6 @ U4 @ V )
=> ( con_a @ resid @ T @ T6 ) ) ) ).
% R.bounded_imp_con
thf(fact_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_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_1036_R_Ocomposable__def,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
= ( ? [X6: a] : ( composite_of_a @ resid @ T @ U @ X6 ) ) ) ).
% R.composable_def
thf(fact_1037_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_1038_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_1039_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_1040_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_1041_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_1042_rts_Ocomposite__of_Ocong,axiom,
composite_of_a = composite_of_a ).
% rts.composite_of.cong
thf(fact_1043_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_1044_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_1045_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 ) )
& ! [A5: a] :
( ( ide_a @ Resid2 @ A5 )
=> ( member_a @ A5 @ 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 )
=> ? [X6: a] : ( composite_of_a @ Resid2 @ U6 @ T5 @ X6 ) ) )
& ! [U6: a,T5: a] :
( ( member_a @ U6 @ NN2 )
=> ( ( seq_a @ Resid2 @ T5 @ U6 )
=> ? [X6: a] : ( composite_of_a @ Resid2 @ T5 @ U6 @ X6 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_1046_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_a,Resid: a > a > a] :
( ! [T3: a] :
( ( member_a @ T3 @ NN )
=> ( arr_a @ Resid @ T3 ) )
=> ( ! [A3: a] :
( ( ide_a @ Resid @ A3 )
=> ( member_a @ A3 @ 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_1047_rotate1__hd__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( rotate1_a @ Xs )
= ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_1048_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B6: nat] :
( ( P @ A3 @ B6 )
= ( P @ B6 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B6: nat] :
( ( P @ A3 @ B6 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B6 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1049_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_1050_set__rotate1,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rotate1_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_1051_length__rotate1,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate1
thf(fact_1052_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_1053_rotate1_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X2 @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_1054_R_Oresiduation__axioms,axiom,
residuation_a @ resid ).
% R.residuation_axioms
thf(fact_1055_SuccI,axiom,
! [Kl: list_a,K: a,Kl2: set_list_a] :
( ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_residuation_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( partial_magma_a @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_SuccD,axiom,
! [K: a,Kl2: set_list_a,Kl: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
=> ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_1080_empty__Shift,axiom,
! [Kl2: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl2 )
=> ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_1081_Succ__Shift,axiom,
! [Kl2: set_list_a,K: a,Kl: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) @ Kl )
= ( bNF_Greatest_Succ_a @ Kl2 @ ( cons_a @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_1082_ShiftD,axiom,
! [Kl: list_a,Kl2: set_list_a,K: a] :
( ( member_list_a @ Kl @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) )
=> ( member_list_a @ ( cons_a @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_1083_insert__subsetI,axiom,
! [X2: a,A2: set_a,X5: set_a] :
( ( member_a @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X2 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1084_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X: a] :
~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1085_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1086_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1087_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1088_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1089_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1090_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1091_Resid_Opsimps_I6_J,axiom,
! [T: a,V: a,Va2: list_a,U: a,Vb: a,Vc: list_a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ ( cons_a @ Vb @ Vc ) ) ) )
=> ( ( ( ( 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.psimps(6)
thf(fact_1092_Resid_Opsimps_I1_J,axiom,
! [Uu: list_a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ nil_a @ Uu ) )
=> ( ( paths_in_Resid_a @ resid @ nil_a @ Uu )
= nil_a ) ) ).
% Resid.psimps(1)
thf(fact_1093_Resid_Opsimps_I2_J,axiom,
! [V: a,Va2: list_a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ V @ Va2 ) @ nil_a ) )
=> ( ( paths_in_Resid_a @ resid @ ( cons_a @ V @ Va2 ) @ nil_a )
= nil_a ) ) ).
% Resid.psimps(2)
thf(fact_1094_Resid_Opsimps_I3_J,axiom,
! [T: a,U: a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_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.psimps(3)
thf(fact_1095_Resid_Opsimps_I5_J,axiom,
! [T: a,V: a,Va2: list_a,U: a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ nil_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.psimps(5)
thf(fact_1096_Resid_Opsimps_I4_J,axiom,
! [T: a,U: a,V: a,Va2: list_a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) ) )
=> ( ( ( ( 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.psimps(4)
thf(fact_1097_Resid_Opsimps_I7_J,axiom,
! [T: a,Vb: a,Vc: list_a,U: a,V: a,Va2: list_a] :
( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) ) )
=> ( ( ( ( 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.psimps(7)
thf(fact_1098_paths__in__rts_OResid_Opsimps_I2_J,axiom,
! [Resid: a > a > a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ Resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ V @ Va2 ) @ nil_a ) )
=> ( ( paths_in_Resid_a @ Resid @ ( cons_a @ V @ Va2 ) @ nil_a )
= nil_a ) ) ) ).
% paths_in_rts.Resid.psimps(2)
thf(fact_1099_paths__in__rts_OResid_Opsimps_I1_J,axiom,
! [Resid: a > a > a,Uu: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ Resid ) @ ( produc6837034575241423639list_a @ nil_a @ Uu ) )
=> ( ( paths_in_Resid_a @ Resid @ nil_a @ Uu )
= nil_a ) ) ) ).
% paths_in_rts.Resid.psimps(1)
thf(fact_1100_subrelI,axiom,
! [R2: set_Pr4048851178543822343list_a,S: set_Pr4048851178543822343list_a] :
( ! [X: list_a,Y3: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y3 ) @ R2 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y3 ) @ S ) )
=> ( ord_le7857023143581076903list_a @ R2 @ S ) ) ).
% subrelI
thf(fact_1101_subrelI,axiom,
! [R2: set_Pr8962057229576493569st_a_a,S: set_Pr8962057229576493569st_a_a] :
( ! [X: list_a,Y3: a] :
( ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ X @ Y3 ) @ R2 )
=> ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ X @ Y3 ) @ S ) )
=> ( ord_le2808437291371905441st_a_a @ R2 @ S ) ) ).
% subrelI
thf(fact_1102_subrelI,axiom,
! [R2: set_Pr2070066670564046349list_a,S: set_Pr2070066670564046349list_a] :
( ! [X: a,Y3: list_a] :
( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ X @ Y3 ) @ R2 )
=> ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ X @ Y3 ) @ S ) )
=> ( ord_le5139818769214234029list_a @ R2 @ S ) ) ).
% subrelI
thf(fact_1103_shuffles_Ocases,axiom,
! [X2: produc9164743771328383783list_a] :
( ! [Ys3: list_a] :
( X2
!= ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ( ! [Xs2: list_a] :
( X2
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X: a,Xs2: list_a,Y3: a,Ys3: list_a] :
( X2
!= ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1104_splice_Ocases,axiom,
! [X2: produc9164743771328383783list_a] :
( ! [Ys3: list_a] :
( X2
!= ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ~ ! [X: a,Xs2: list_a,Ys3: list_a] :
( X2
!= ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_1105_paths__in__rts_OResid_Opsimps_I3_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ Resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ nil_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 ) ) ) ) ) ).
% paths_in_rts.Resid.psimps(3)
thf(fact_1106_paths__in__rts_OResidx1_Ocases,axiom,
! [Resid: a > a > a,X2: produc2579390645249093025st_a_a] :
( ( paths_in_rts_a @ Resid )
=> ( ! [U3: a] :
( X2
!= ( produc4781227316648555537st_a_a @ nil_a @ U3 ) )
=> ( ! [T3: a,U3: a] :
( X2
!= ( produc4781227316648555537st_a_a @ ( cons_a @ T3 @ nil_a ) @ U3 ) )
=> ~ ! [T3: a,V2: a,Va: list_a,U3: a] :
( X2
!= ( produc4781227316648555537st_a_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) @ U3 ) ) ) ) ) ).
% paths_in_rts.Residx1.cases
thf(fact_1107_paths__in__rts_OResid1x_Ocases,axiom,
! [Resid: a > a > a,X2: produc8685980395799941037list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ! [T3: a] :
( X2
!= ( produc6670463072477821725list_a @ T3 @ nil_a ) )
=> ( ! [T3: a,U3: a] :
( X2
!= ( produc6670463072477821725list_a @ T3 @ ( cons_a @ U3 @ nil_a ) ) )
=> ~ ! [T3: a,U3: a,V2: a,Va: list_a] :
( X2
!= ( produc6670463072477821725list_a @ T3 @ ( cons_a @ U3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ).
% paths_in_rts.Resid1x.cases
thf(fact_1108_paths__in__rts_OResid_Opsimps_I5_J,axiom,
! [Resid: a > a > a,T: a,V: a,Va2: list_a,U: a] :
( ( paths_in_rts_a @ Resid )
=> ( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ Resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ nil_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 ) ) ) ) ) ).
% paths_in_rts.Resid.psimps(5)
thf(fact_1109_paths__in__rts_OResid_Opsimps_I4_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ Resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T @ nil_a ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) ) )
=> ( ( ( ( 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.psimps(4)
thf(fact_1110_paths__in__rts_OResid_Opsimps_I6_J,axiom,
! [Resid: a > a > a,T: a,V: a,Va2: list_a,U: a,Vb: a,Vc: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ Resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T @ ( cons_a @ V @ Va2 ) ) @ ( cons_a @ U @ ( cons_a @ Vb @ Vc ) ) ) )
=> ( ( ( ( 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.psimps(6)
thf(fact_1111_paths__in__rts_OResid_Opsimps_I7_J,axiom,
! [Resid: a > a > a,T: a,Vb: a,Vc: list_a,U: a,V: a,Va2: list_a] :
( ( paths_in_rts_a @ Resid )
=> ( ( accp_P7377042638478740784list_a @ ( paths_in_Resid_rel_a @ Resid ) @ ( produc6837034575241423639list_a @ ( cons_a @ T @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ U @ ( cons_a @ V @ Va2 ) ) ) )
=> ( ( ( ( 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.psimps(7)
thf(fact_1112_splice_Opinduct,axiom,
! [A0: list_a,A1: list_a,P: list_a > list_a > $o] :
( ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ A0 @ A1 ) )
=> ( ! [Ys3: list_a] :
( ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ( P @ nil_a @ Ys3 ) )
=> ( ! [X: a,Xs2: list_a,Ys3: list_a] :
( ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ Ys3 ) )
=> ( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_1113_shuffles_Opinduct,axiom,
! [A0: list_a,A1: list_a,P: list_a > list_a > $o] :
( ( accp_P7377042638478740784list_a @ shuffles_rel_a @ ( produc6837034575241423639list_a @ A0 @ A1 ) )
=> ( ! [Ys3: list_a] :
( ( accp_P7377042638478740784list_a @ shuffles_rel_a @ ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ( P @ nil_a @ Ys3 ) )
=> ( ! [Xs2: list_a] :
( ( accp_P7377042638478740784list_a @ shuffles_rel_a @ ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ( P @ Xs2 @ nil_a ) )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys3: list_a] :
( ( accp_P7377042638478740784list_a @ shuffles_rel_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( ( P @ Xs2 @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ( P @ ( cons_a @ X @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_1114_Residx1_Ocases,axiom,
! [X2: produc2579390645249093025st_a_a] :
( ! [U3: a] :
( X2
!= ( produc4781227316648555537st_a_a @ nil_a @ U3 ) )
=> ( ! [T3: a,U3: a] :
( X2
!= ( produc4781227316648555537st_a_a @ ( cons_a @ T3 @ nil_a ) @ U3 ) )
=> ~ ! [T3: a,V2: a,Va: list_a,U3: a] :
( X2
!= ( produc4781227316648555537st_a_a @ ( cons_a @ T3 @ ( cons_a @ V2 @ Va ) ) @ U3 ) ) ) ) ).
% Residx1.cases
thf(fact_1115_Resid1x_Ocases,axiom,
! [X2: produc8685980395799941037list_a] :
( ! [T3: a] :
( X2
!= ( produc6670463072477821725list_a @ T3 @ nil_a ) )
=> ( ! [T3: a,U3: a] :
( X2
!= ( produc6670463072477821725list_a @ T3 @ ( cons_a @ U3 @ nil_a ) ) )
=> ~ ! [T3: a,U3: a,V2: a,Va: list_a] :
( X2
!= ( produc6670463072477821725list_a @ T3 @ ( cons_a @ U3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ).
% Resid1x.cases
thf(fact_1116_successively_Ocases,axiom,
! [X2: produc5032551385658279741list_a] :
( ! [P2: a > a > $o] :
( X2
!= ( produc8111569692950616493list_a @ P2 @ nil_a ) )
=> ( ! [P2: a > a > $o,X: a] :
( X2
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X @ nil_a ) ) )
=> ~ ! [P2: a > a > $o,X: a,Y3: a,Xs2: list_a] :
( X2
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1117_sorted__wrt_Ocases,axiom,
! [X2: produc5032551385658279741list_a] :
( ! [P2: a > a > $o] :
( X2
!= ( produc8111569692950616493list_a @ P2 @ nil_a ) )
=> ~ ! [P2: a > a > $o,X: a,Ys3: list_a] :
( X2
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1118_map__tailrec__rev_Ocases,axiom,
! [X2: produc1473018763691903991list_a] :
( ! [F5: a > a,Bs: list_a] :
( X2
!= ( produc8643929849434629545list_a @ F5 @ ( produc6837034575241423639list_a @ nil_a @ Bs ) ) )
=> ~ ! [F5: a > a,A3: a,As2: list_a,Bs: list_a] :
( X2
!= ( produc8643929849434629545list_a @ F5 @ ( produc6837034575241423639list_a @ ( cons_a @ A3 @ As2 ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_1119_Resid1x_Opelims,axiom,
! [X2: a,Xa: list_a,Y4: a] :
( ( ( paths_in_Resid1x_a @ resid @ X2 @ Xa )
= Y4 )
=> ( ( accp_P3213725926765619766list_a @ ( paths_6492648068886854876_rel_a @ resid ) @ ( produc6670463072477821725list_a @ X2 @ Xa ) )
=> ( ( ( Xa = nil_a )
=> ( ( Y4
= ( partial_null_a @ resid ) )
=> ~ ( accp_P3213725926765619766list_a @ ( paths_6492648068886854876_rel_a @ resid ) @ ( produc6670463072477821725list_a @ X2 @ nil_a ) ) ) )
=> ( ! [U3: a] :
( ( Xa
= ( cons_a @ U3 @ nil_a ) )
=> ( ( Y4
= ( resid @ X2 @ U3 ) )
=> ~ ( accp_P3213725926765619766list_a @ ( paths_6492648068886854876_rel_a @ resid ) @ ( produc6670463072477821725list_a @ X2 @ ( cons_a @ U3 @ nil_a ) ) ) ) )
=> ~ ! [U3: a,V2: a,Va: list_a] :
( ( Xa
= ( cons_a @ U3 @ ( cons_a @ V2 @ Va ) ) )
=> ( ( Y4
= ( paths_in_Resid1x_a @ resid @ ( resid @ X2 @ U3 ) @ ( cons_a @ V2 @ Va ) ) )
=> ~ ( accp_P3213725926765619766list_a @ ( paths_6492648068886854876_rel_a @ resid ) @ ( produc6670463072477821725list_a @ X2 @ ( cons_a @ U3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% Resid1x.pelims
thf(fact_1120_splice_Opelims,axiom,
! [X2: list_a,Xa: list_a,Y4: list_a] :
( ( ( splice_a @ X2 @ Xa )
= Y4 )
=> ( ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ X2 @ Xa ) )
=> ( ( ( X2 = nil_a )
=> ( ( Y4 = Xa )
=> ~ ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ nil_a @ Xa ) ) ) )
=> ~ ! [X: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X @ Xs2 ) )
=> ( ( Y4
= ( cons_a @ X @ ( splice_a @ Xa @ Xs2 ) ) )
=> ~ ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_1121_split__Nil__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( splice_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% split_Nil_iff
thf(fact_1122_splice__Nil2,axiom,
! [Xs: list_a] :
( ( splice_a @ Xs @ nil_a )
= Xs ) ).
% splice_Nil2
thf(fact_1123_length__splice,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( splice_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_splice
thf(fact_1124_splice_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( splice_a @ ( cons_a @ X2 @ Xs ) @ Ys )
= ( cons_a @ X2 @ ( splice_a @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_1125_splice_Osimps_I1_J,axiom,
! [Ys: list_a] :
( ( splice_a @ nil_a @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_1126_splice_Oelims,axiom,
! [X2: list_a,Xa: list_a,Y4: list_a] :
( ( ( splice_a @ X2 @ Xa )
= Y4 )
=> ( ( ( X2 = nil_a )
=> ( Y4 != Xa ) )
=> ~ ! [X: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X @ Xs2 ) )
=> ( Y4
!= ( cons_a @ X @ ( splice_a @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_1127_splice_Opsimps_I2_J,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ Ys ) )
=> ( ( splice_a @ ( cons_a @ X2 @ Xs ) @ Ys )
= ( cons_a @ X2 @ ( splice_a @ Ys @ Xs ) ) ) ) ).
% splice.psimps(2)
thf(fact_1128_splice_Opsimps_I1_J,axiom,
! [Ys: list_a] :
( ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ nil_a @ Ys ) )
=> ( ( splice_a @ nil_a @ Ys )
= Ys ) ) ).
% splice.psimps(1)
thf(fact_1129_Cons__in__lex,axiom,
! [X2: list_a,Xs: list_list_a,Y4: list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X2 @ Xs ) @ ( cons_list_a @ Y4 @ Ys ) ) @ ( lex_list_a @ R2 ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ Y4 ) @ R2 )
& ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) )
| ( ( X2 = Y4 )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( lex_list_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_1130_Cons__in__lex,axiom,
! [X2: a,Xs: list_a,Y4: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y4 @ Ys ) ) @ ( lex_a @ R2 ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) )
| ( ( X2 = Y4 )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_1131_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_1132_Nil__notin__lex,axiom,
! [Ys: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ys ) @ ( lex_a @ R2 ) ) ).
% Nil_notin_lex
thf(fact_1133_Nil2__notin__lex,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( lex_a @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_1134_lex__append__leftI,axiom,
! [Ys: list_a,Zs: list_a,R2: set_Product_prod_a_a,Xs: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R2 ) ) ) ).
% lex_append_leftI
thf(fact_1135_lex__append__leftD,axiom,
! [R2: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ! [X: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X ) @ R2 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_1136_lex__append__leftD,axiom,
! [R2: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ! [X: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ X ) @ R2 )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys ) @ ( append_list_a @ Xs @ Zs ) ) @ ( lex_list_a @ R2 ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs ) @ ( lex_list_a @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_1137_lex__append__left__iff,axiom,
! [R2: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ! [X: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X ) @ R2 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R2 ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_1138_lex__append__left__iff,axiom,
! [R2: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ! [X: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ X ) @ R2 )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys ) @ ( append_list_a @ Xs @ Zs ) ) @ ( lex_list_a @ R2 ) )
= ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs ) @ ( lex_list_a @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_1139_lex__append__rightI,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,Vs: list_a,Us2: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Us2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us2 ) @ ( append_a @ Ys @ Vs ) ) @ ( lex_a @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_1140_in__set__product__lists__length,axiom,
! [Xs: list_a,Xss2: list_list_a] :
( ( member_list_a @ Xs @ ( set_list_a2 @ ( product_lists_a @ Xss2 ) ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_1141_bind__simps_I2_J,axiom,
! [X2: a,Xs: list_a,F: a > list_a] :
( ( bind_a_a @ ( cons_a @ X2 @ Xs ) @ F )
= ( append_a @ ( F @ X2 ) @ ( bind_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1142_Gcd__0__iff,axiom,
! [A2: set_nat] :
( ( ( gcd_Gcd_nat @ A2 )
= zero_zero_nat )
= ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% Gcd_0_iff
thf(fact_1143_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_1144_Gcd__empty,axiom,
( ( gcd_Gcd_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Gcd_empty
thf(fact_1145_residuation__def,axiom,
( residuation_a
= ( ^ [Resid2: a > a > a] :
( ( partial_magma_a @ Resid2 )
& ( residuation_axioms_a @ Resid2 ) ) ) ) ).
% residuation_def
thf(fact_1146_residuation_Ointro,axiom,
! [Resid: a > a > a] :
( ( partial_magma_a @ Resid )
=> ( ( residuation_axioms_a @ Resid )
=> ( residuation_a @ Resid ) ) ) ).
% residuation.intro
thf(fact_1147_residuation__axioms__def,axiom,
( residuation_axioms_a
= ( ^ [Resid2: a > a > a] :
( ! [T5: a,U6: a] :
( ( ( Resid2 @ T5 @ U6 )
!= ( partial_null_a @ Resid2 ) )
=> ( ( Resid2 @ U6 @ T5 )
!= ( partial_null_a @ Resid2 ) ) )
& ! [T5: a,U6: a] :
( ( ( Resid2 @ T5 @ U6 )
!= ( partial_null_a @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ T5 @ U6 ) @ ( Resid2 @ T5 @ U6 ) )
!= ( partial_null_a @ Resid2 ) ) )
& ! [V5: a,T5: a,U6: a] :
( ( ( Resid2 @ ( Resid2 @ V5 @ T5 ) @ ( Resid2 @ U6 @ T5 ) )
!= ( partial_null_a @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ V5 @ T5 ) @ ( Resid2 @ U6 @ T5 ) )
= ( Resid2 @ ( Resid2 @ V5 @ U6 ) @ ( Resid2 @ T5 @ U6 ) ) ) ) ) ) ) ).
% residuation_axioms_def
thf(fact_1148_residuation__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a,U3: a] :
( ( ( Resid @ T3 @ U3 )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ U3 @ T3 )
!= ( partial_null_a @ Resid ) ) )
=> ( ! [T3: a,U3: a] :
( ( ( Resid @ T3 @ U3 )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ ( Resid @ T3 @ U3 ) @ ( Resid @ T3 @ U3 ) )
!= ( partial_null_a @ Resid ) ) )
=> ( ! [V2: a,T3: a,U3: a] :
( ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U3 @ T3 ) )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U3 @ T3 ) )
= ( Resid @ ( Resid @ V2 @ U3 ) @ ( Resid @ T3 @ U3 ) ) ) )
=> ( residuation_axioms_a @ Resid ) ) ) ) ).
% residuation_axioms.intro
thf(fact_1149_residuation_Oaxioms_I2_J,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( residuation_axioms_a @ Resid ) ) ).
% residuation.axioms(2)
thf(fact_1150_append__butlast__last__id,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_1151_rts__with__composites__axioms__def,axiom,
( rts_wi2614412583573296275ioms_a
= ( ^ [Resid2: a > a > a] :
! [T5: a,U6: a] :
( ( seq_a @ Resid2 @ T5 @ U6 )
=> ( composable_a @ Resid2 @ T5 @ U6 ) ) ) ) ).
% rts_with_composites_axioms_def
thf(fact_1152_butlast__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_1153_butlast__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_1154_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X2: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X2 @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_1155_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_1156_in__set__butlastD,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
=> ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1157_butlast__tl,axiom,
! [Xs: list_a] :
( ( butlast_a @ ( tl_a @ Xs ) )
= ( tl_a @ ( butlast_a @ Xs ) ) ) ).
% butlast_tl
thf(fact_1158_in__set__butlast__appendI,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
| ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
=> ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1159_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X2: a,Ys: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs )
& ( ( last_a @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_1160_rts__with__composites__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a,U3: a] :
( ( seq_a @ Resid @ T3 @ U3 )
=> ( composable_a @ Resid @ T3 @ U3 ) )
=> ( rts_wi2614412583573296275ioms_a @ Resid ) ) ).
% rts_with_composites_axioms.intro
thf(fact_1161_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_1162_rts__with__joins__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a,U3: a] :
( ( con_a @ Resid @ T3 @ U3 )
=> ( joinable_a @ Resid @ T3 @ U3 ) )
=> ( rts_wi560353115624263628ioms_a @ Resid ) ) ).
% rts_with_joins_axioms.intro
thf(fact_1163_Cons__in__subseqsD,axiom,
! [Y4: a,Ys: list_a,Xs: list_a] :
( ( member_list_a @ ( cons_a @ Y4 @ Ys ) @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) )
=> ( member_list_a @ Ys @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_1164_rts__with__joins__axioms__def,axiom,
( rts_wi560353115624263628ioms_a
= ( ^ [Resid2: a > a > a] :
! [T5: a,U6: a] :
( ( con_a @ Resid2 @ T5 @ U6 )
=> ( joinable_a @ Resid2 @ T5 @ U6 ) ) ) ) ).
% rts_with_joins_axioms_def
thf(fact_1165_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_list_a,X2: list_a,Ys: list_list_a,Y4: list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X2 @ nil_list_a ) ) @ ( append_list_a @ Ys @ ( cons_list_a @ Y4 @ nil_list_a ) ) ) @ ( listrel1_list_a @ R2 ) )
= ( ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R2 ) )
& ( X2 = Y4 ) )
| ( ( Xs = Ys )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ Y4 ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1166_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_a,X2: a,Ys: list_a,Y4: a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) @ ( append_a @ Ys @ ( cons_a @ Y4 @ nil_a ) ) ) @ ( listrel1_a @ R2 ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
& ( X2 = Y4 ) )
| ( ( Xs = Ys )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1167_confluent__rts__axioms__def,axiom,
( conflu3014480972103220363ioms_a
= ( ^ [Resid2: a > a > a] :
! [T5: a,U6: a] :
( ( coinitial_a @ Resid2 @ T5 @ U6 )
=> ( con_a @ Resid2 @ T5 @ U6 ) ) ) ) ).
% confluent_rts_axioms_def
thf(fact_1168_Cons__listrel1__Cons,axiom,
! [X2: list_a,Xs: list_list_a,Y4: list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X2 @ Xs ) @ ( cons_list_a @ Y4 @ Ys ) ) @ ( listrel1_list_a @ R2 ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ Y4 ) @ R2 )
& ( Xs = Ys ) )
| ( ( X2 = Y4 )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1169_Cons__listrel1__Cons,axiom,
! [X2: a,Xs: list_a,Y4: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y4 @ Ys ) ) @ ( listrel1_a @ R2 ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
& ( Xs = Ys ) )
| ( ( X2 = Y4 )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1170_listrel1__eq__len,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_1171_not__listrel1__Nil,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel1_a @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_1172_not__Nil__listrel1,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel1_a @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_1173_listrel1I2,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,X2: a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ X2 @ Ys ) ) @ ( listrel1_a @ R2 ) ) ) ).
% listrel1I2
thf(fact_1174_append__listrel1I,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,Us2: list_a,Vs: list_a] :
( ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
& ( Us2 = Vs ) )
| ( ( Xs = Ys )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us2 @ Vs ) @ ( listrel1_a @ R2 ) ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us2 ) @ ( append_a @ Ys @ Vs ) ) @ ( listrel1_a @ R2 ) ) ) ).
% append_listrel1I
thf(fact_1175_listrel1I1,axiom,
! [X2: a,Y4: a,R2: set_Product_prod_a_a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y4 @ Xs ) ) @ ( listrel1_a @ R2 ) ) ) ).
% listrel1I1
thf(fact_1176_listrel1I1,axiom,
! [X2: list_a,Y4: list_a,R2: set_Pr4048851178543822343list_a,Xs: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ Y4 ) @ R2 )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X2 @ Xs ) @ ( cons_list_a @ Y4 @ Xs ) ) @ ( listrel1_list_a @ R2 ) ) ) ).
% listrel1I1
thf(fact_1177_Cons__listrel1E1,axiom,
! [X2: list_a,Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X2 @ Xs ) @ Ys ) @ ( listrel1_list_a @ R2 ) )
=> ( ! [Y3: list_a] :
( ( Ys
= ( cons_list_a @ Y3 @ Xs ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list_list_a] :
( ( Ys
= ( cons_list_a @ X2 @ Zs2 ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Zs2 ) @ ( listrel1_list_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1178_Cons__listrel1E1,axiom,
! [X2: a,Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ Ys ) @ ( listrel1_a @ R2 ) )
=> ( ! [Y3: a] :
( ( Ys
= ( cons_a @ Y3 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list_a] :
( ( Ys
= ( cons_a @ X2 @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs2 ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1179_Cons__listrel1E2,axiom,
! [Xs: list_list_a,Y4: list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ ( cons_list_a @ Y4 @ Ys ) ) @ ( listrel1_list_a @ R2 ) )
=> ( ! [X: list_a] :
( ( Xs
= ( cons_list_a @ X @ Ys ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y4 ) @ R2 ) )
=> ~ ! [Zs2: list_list_a] :
( ( Xs
= ( cons_list_a @ Y4 @ Zs2 ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Zs2 @ Ys ) @ ( listrel1_list_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1180_Cons__listrel1E2,axiom,
! [Xs: list_a,Y4: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y4 @ Ys ) ) @ ( listrel1_a @ R2 ) )
=> ( ! [X: a] :
( ( Xs
= ( cons_a @ X @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y4 ) @ R2 ) )
=> ~ ! [Zs2: list_a] :
( ( Xs
= ( cons_a @ Y4 @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs2 @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1181_listrel1E,axiom,
! [Xs: list_list_a,Ys: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R2 ) )
=> ~ ! [X: list_a,Y3: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y3 ) @ R2 )
=> ! [Us3: list_list_a,Vs2: list_list_a] :
( ( Xs
= ( append_list_a @ Us3 @ ( cons_list_a @ X @ Vs2 ) ) )
=> ( Ys
!= ( append_list_a @ Us3 @ ( cons_list_a @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1182_listrel1E,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
=> ~ ! [X: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 )
=> ! [Us3: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us3 @ ( cons_a @ X @ Vs2 ) ) )
=> ( Ys
!= ( append_a @ Us3 @ ( cons_a @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1183_listrel1I,axiom,
! [X2: a,Y4: a,R2: set_Product_prod_a_a,Xs: list_a,Us2: list_a,Vs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R2 )
=> ( ( Xs
= ( append_a @ Us2 @ ( cons_a @ X2 @ Vs ) ) )
=> ( ( Ys
= ( append_a @ Us2 @ ( cons_a @ Y4 @ Vs ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1184_listrel1I,axiom,
! [X2: list_a,Y4: list_a,R2: set_Pr4048851178543822343list_a,Xs: list_list_a,Us2: list_list_a,Vs: list_list_a,Ys: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ Y4 ) @ R2 )
=> ( ( Xs
= ( append_list_a @ Us2 @ ( cons_list_a @ X2 @ Vs ) ) )
=> ( ( Ys
= ( append_list_a @ Us2 @ ( cons_list_a @ Y4 @ Vs ) ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1185_confluent__rts__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a,U3: a] :
( ( coinitial_a @ Resid @ T3 @ U3 )
=> ( con_a @ Resid @ T3 @ U3 ) )
=> ( conflu3014480972103220363ioms_a @ Resid ) ) ).
% confluent_rts_axioms.intro
thf(fact_1186_confluent__rts_Oconfluence,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( confluent_rts_a @ Resid )
=> ( ( coinitial_a @ Resid @ T @ U )
=> ( con_a @ Resid @ T @ U ) ) ) ).
% confluent_rts.confluence
thf(fact_1187_distinct__adj__append__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
= ( ( distinct_adj_a @ Xs )
& ( distinct_adj_a @ Ys )
& ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_1188_distinct__adj__Cons__Cons,axiom,
! [X2: a,Y4: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y4 @ Xs ) ) )
= ( ( X2 != Y4 )
& ( distinct_adj_a @ ( cons_a @ Y4 @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_1189_distinct__adj__singleton,axiom,
! [X2: a] : ( distinct_adj_a @ ( cons_a @ X2 @ nil_a ) ) ).
% distinct_adj_singleton
thf(fact_1190_distinct__adj__appendD1,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_1191_distinct__adj__appendD2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
=> ( distinct_adj_a @ Ys ) ) ).
% distinct_adj_appendD2
thf(fact_1192_distinct__adj__Nil,axiom,
distinct_adj_a @ nil_a ).
% distinct_adj_Nil
thf(fact_1193_distinct__adj__ConsD,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ Xs ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_1194_distinct__adj__Cons,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ Xs ) )
= ( ( Xs = nil_a )
| ( ( X2
!= ( hd_a @ Xs ) )
& ( distinct_adj_a @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_1195_R_Oidentities__form__normal__sub__rts,axiom,
normal_sub_rts_a @ resid @ ( collect_a @ ( ide_a @ resid ) ) ).
% R.identities_form_normal_sub_rts
thf(fact_1196_gen__length__def,axiom,
( gen_length_a
= ( ^ [N4: nat,Xs5: list_a] : ( plus_plus_nat @ N4 @ ( size_size_list_a @ Xs5 ) ) ) ) ).
% gen_length_def
thf(fact_1197_normal__sub__rts_Ocomposite__of__normal__arr,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ U @ NN )
=> ( ( composite_of_a @ Resid @ U @ T @ T6 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T6 @ T ) ) ) ) ) ).
% normal_sub_rts.composite_of_normal_arr
thf(fact_1198_normal__sub__rts_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a,V3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( composite_of_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( composite_of_a @ Resid @ U @ ( Resid @ T @ U ) @ V3 )
=> ( ( member_a @ ( Resid @ V @ V3 ) @ NN )
& ( member_a @ ( Resid @ V3 @ V ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.diamond_commutes_upto_Cong\<^sub>0
thf(fact_1199_normal__sub__rts_Ofactor__closed_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( member_a @ V @ NN )
=> ( member_a @ U @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(2)
thf(fact_1200_normal__sub__rts_Ofactor__closed_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( member_a @ V @ NN )
=> ( member_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(1)
thf(fact_1201_normal__sub__rts_OCong_092_060_094sub_0620__iff,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) )
= ( ? [U6: a,U8: a,V5: a,V6: a] :
( ( member_a @ U6 @ NN )
& ( member_a @ U8 @ NN )
& ( member_a @ ( Resid @ V5 @ V6 ) @ NN )
& ( member_a @ ( Resid @ V6 @ V5 ) @ NN )
& ( composite_of_a @ Resid @ T @ U6 @ V5 )
& ( composite_of_a @ Resid @ T6 @ U8 @ V6 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_iff
thf(fact_1202_normal__sub__rts_Ocomposite__closed,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ T @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( member_a @ V @ NN ) ) ) ) ) ).
% normal_sub_rts.composite_closed
thf(fact_1203_normal__sub__rts_Ocomposite__of__arr__normal,axiom,
! [Resid: a > a > a,NN: set_a,Arr: a > $o,T: a,U: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( Arr @ T )
=> ( ( member_a @ U @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ T6 )
=> ( ( member_a @ ( Resid @ T6 @ T ) @ NN )
& ( member_a @ ( Resid @ T @ T6 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.composite_of_arr_normal
thf(fact_1204_normal__sub__rts_OCong_092_060_094sub_0620__cancel__left,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a,U4: a,V3: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( composite_of_a @ Resid @ T @ U4 @ V3 )
=> ( ( ( member_a @ ( Resid @ V @ V3 ) @ NN )
& ( member_a @ ( Resid @ V3 @ V ) @ NN ) )
=> ( ( member_a @ ( Resid @ U @ U4 ) @ NN )
& ( member_a @ ( Resid @ U4 @ U ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_cancel_left
thf(fact_1205_normal__sub__rts_Ocomposite__closed__left,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( seq_a @ Resid @ U @ T )
=> ? [X_12: a] : ( composite_of_a @ Resid @ U @ T @ X_12 ) ) ) ) ).
% normal_sub_rts.composite_closed_left
thf(fact_1206_normal__sub__rts_Ocomposite__closed__right,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( seq_a @ Resid @ T @ U )
=> ? [X_12: a] : ( composite_of_a @ Resid @ T @ U @ X_12 ) ) ) ) ).
% normal_sub_rts.composite_closed_right
thf(fact_1207_normal__sub__rts_Oaxioms_I2_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( normal7698203753654205830ioms_a @ Resid @ NN ) ) ).
% normal_sub_rts.axioms(2)
thf(fact_1208_normal__sub__rts_Oresid__along__elem__preserves__con,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( con_a @ Resid @ T @ T6 )
=> ( ( coinitial_a @ Resid @ T @ U )
=> ( ( member_a @ U @ NN )
=> ( con_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T6 @ U ) ) ) ) ) ) ).
% normal_sub_rts.resid_along_elem_preserves_con
thf(fact_1209_normal__sub__rts_Oelements__are__arr,axiom,
! [Resid: a > a > a,NN: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ T @ NN )
=> ( arr_a @ Resid @ T ) ) ) ).
% normal_sub_rts.elements_are_arr
thf(fact_1210_normal__sub__rts_OCong_092_060_094sub_0620__reflexive,axiom,
! [Resid: a > a > a,NN: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ ( Resid @ T @ T ) @ NN )
& ( member_a @ ( Resid @ T @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_reflexive
thf(fact_1211_normal__sub__rts_Oprfx__closed,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
=> ( member_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.prfx_closed
thf(fact_1212_normal__sub__rts_Oide__closed,axiom,
! [Resid: a > a > a,NN: set_a,A: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ide_a @ Resid @ A )
=> ( member_a @ A @ NN ) ) ) ).
% normal_sub_rts.ide_closed
thf(fact_1213_normal__sub__rts_OCong_092_060_094sub_0620__subst__Con,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,U: a,U4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) )
=> ( ( ( member_a @ ( Resid @ U @ U4 ) @ NN )
& ( member_a @ ( Resid @ U4 @ U ) @ NN ) )
=> ( ( con_a @ Resid @ T @ U )
= ( con_a @ Resid @ T6 @ U4 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_Con
thf(fact_1214_normal__sub__rts_OCong_092_060_094sub_0620__imp__con,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) )
=> ( con_a @ Resid @ T @ T6 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_con
thf(fact_1215_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,U: a,U4: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ U @ U4 ) @ NN )
& ( member_a @ ( Resid @ U4 @ U ) @ NN ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( con_a @ Resid @ T @ U4 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(1)
thf(fact_1216_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,U: a,U4: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ U @ U4 ) @ NN )
& ( member_a @ ( Resid @ U4 @ U ) @ NN ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( member_a @ ( Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U4 @ U ) ) @ ( Resid @ ( Resid @ T @ U4 ) @ ( Resid @ U @ U4 ) ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ ( Resid @ T @ U4 ) @ ( Resid @ U @ U4 ) ) @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U4 @ U ) ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(2)
thf(fact_1217_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( con_a @ Resid @ T6 @ U ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(1)
thf(fact_1218_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( member_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T6 @ U ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(2)
thf(fact_1219_normal__sub__rts_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) )
=> ( ( member_a @ U @ NN )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T6 @ U ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_1220_normal__sub__rts_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) )
=> ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ T6 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_coinitial
thf(fact_1221_normal__sub__rts_Obackward__stable,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ ( Resid @ T @ U ) @ NN )
=> ( member_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.backward_stable
thf(fact_1222_normal__sub__rts_OCong_092_060_094sub_0620__symmetric,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) )
=> ( ( member_a @ ( Resid @ T6 @ T ) @ NN )
& ( member_a @ ( Resid @ T @ T6 ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_symmetric
thf(fact_1223_normal__sub__rts_OCong_092_060_094sub_0620__transitive,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,T8: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) )
=> ( ( ( member_a @ ( Resid @ T6 @ T8 ) @ NN )
& ( member_a @ ( Resid @ T8 @ T6 ) @ NN ) )
=> ( ( member_a @ ( Resid @ T @ T8 ) @ NN )
& ( member_a @ ( Resid @ T8 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_transitive
thf(fact_1224_normal__sub__rts_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T6 @ U ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_1225_coherent__normal__sub__rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( normal_sub_rts_a @ Resid @ NN ) ) ).
% coherent_normal_sub_rts.axioms(1)
thf(fact_1226_normal__sub__rts_Oforward__stable,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( coinitial_a @ Resid @ T @ U )
=> ( member_a @ ( Resid @ U @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.forward_stable
thf(fact_1227_normal__sub__rts_OResid__along__normal__preserves__reflects__con,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( ( con_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T6 @ U ) )
= ( con_a @ Resid @ T @ T6 ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_reflects_con
thf(fact_1228_normal__sub__rts_OCong__imp__arr_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( arr_a @ Resid @ T6 ) ) ) ).
% normal_sub_rts.Cong_imp_arr(2)
thf(fact_1229_normal__sub__rts_OCong__imp__arr_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( arr_a @ Resid @ T ) ) ) ).
% normal_sub_rts.Cong_imp_arr(1)
thf(fact_1230_normal__sub__rts_OCong__reflexive,axiom,
! [Resid: a > a > a,NN: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T ) ) ) ).
% normal_sub_rts.Cong_reflexive
thf(fact_1231_normal__sub__rts_Otargets__are__Cong,axiom,
! [Resid: a > a > a,NN: set_a,B: a,T: a,B2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ Resid @ T ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ B @ B2 ) ) ) ) ).
% normal_sub_rts.targets_are_Cong
thf(fact_1232_normal__sub__rts_Oin__targets__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,B: a,B2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ Resid @ T6 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ B @ B2 ) ) ) ) ) ).
% normal_sub_rts.in_targets_respects_Cong
thf(fact_1233_normal__sub__rts_OCong__closure__props_I3_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ U ) @ NN )
& ( member_a @ ( Resid @ U @ T ) @ NN ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong_closure_props(3)
thf(fact_1234_normal__sub__rts_OCong__closure__props_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ U )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ V )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ V ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(2)
thf(fact_1235_normal__sub__rts_OCong__closure__props_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ U )
=> ( normal_sub_Cong_a @ Resid @ NN @ U @ T ) ) ) ).
% normal_sub_rts.Cong_closure_props(1)
thf(fact_1236_normal__sub__rts_OCongE,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ~ ! [U3: a] :
( ( member_a @ U3 @ NN )
=> ! [U7: a] :
( ( member_a @ U7 @ NN )
=> ~ ( ( member_a @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T6 @ U7 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U7 ) @ ( Resid @ T @ U3 ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.CongE
thf(fact_1237_normal__sub__rts_OCongI,axiom,
! [Resid: a > a > a,NN: set_a,U: a,U4: a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ U4 @ NN )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T6 @ U4 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U4 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 ) ) ) ) ) ).
% normal_sub_rts.CongI
thf(fact_1238_normal__sub__rts_OCong__def,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
= ( ? [U6: a,U8: a] :
( ( member_a @ U6 @ NN )
& ( member_a @ U8 @ NN )
& ( member_a @ ( Resid @ ( Resid @ T @ U6 ) @ ( Resid @ T6 @ U8 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T6 @ U8 ) @ ( Resid @ T @ U6 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong_def
thf(fact_1239_normal__sub__rts_OCong__symmetric,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T6 @ T ) ) ) ).
% normal_sub_rts.Cong_symmetric
thf(fact_1240_normal__sub__rts_OCong__transitive,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T8: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T8 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T8 @ T6 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 ) ) ) ) ).
% normal_sub_rts.Cong_transitive
thf(fact_1241_normal__sub__rts_Onormal__is__Cong__closed,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ T @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( member_a @ T6 @ NN ) ) ) ) ).
% normal_sub_rts.normal_is_Cong_closed
thf(fact_1242_normal__sub__rts_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_implies_Cong
thf(fact_1243_normal__sub__rts_OCong__closure__props_I4_J,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(4)
thf(fact_1244_normal__sub__rts_Osources__are__Cong,axiom,
! [Resid: a > a > a,NN: set_a,A: a,T: a,A4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A4 @ ( sources_a @ Resid @ T ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ A @ A4 ) ) ) ) ).
% normal_sub_rts.sources_are_Cong
thf(fact_1245_normal__sub__rts_Oin__sources__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,A: a,A4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A4 @ ( sources_a @ Resid @ T6 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ A @ A4 ) ) ) ) ) ).
% normal_sub_rts.in_sources_respects_Cong
thf(fact_1246_coherent__normal__sub__rts__def,axiom,
( cohere6072184133013167079_rts_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
( ( normal_sub_rts_a @ Resid2 @ NN2 )
& ( cohere4894532172567702276ioms_a @ Resid2 @ NN2 ) ) ) ) ).
% coherent_normal_sub_rts_def
thf(fact_1247_coherent__normal__sub__rts_Ointro,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( cohere4894532172567702276ioms_a @ Resid @ NN )
=> ( cohere6072184133013167079_rts_a @ Resid @ NN ) ) ) ).
% coherent_normal_sub_rts.intro
thf(fact_1248_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_a @ N @ nil_a )
= N ) ).
% gen_length_code(1)
thf(fact_1249_gen__length__code_I2_J,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( gen_length_a @ N @ ( cons_a @ X2 @ Xs ) )
= ( gen_length_a @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1250_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_1251_remdups__adj__length__ge1,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_1252_shuffles_Opsimps_I1_J,axiom,
! [Ys: list_a] :
( ( accp_P7377042638478740784list_a @ shuffles_rel_a @ ( produc6837034575241423639list_a @ nil_a @ Ys ) )
=> ( ( shuffles_a @ nil_a @ Ys )
= ( insert_list_a @ Ys @ bot_bot_set_list_a ) ) ) ).
% shuffles.psimps(1)
thf(fact_1253_remdups__adj__Nil__iff,axiom,
! [Xs: list_a] :
( ( ( remdups_adj_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% remdups_adj_Nil_iff
thf(fact_1254_Nil__in__shuffles,axiom,
! [Xs: list_a,Ys: list_a] :
( ( member_list_a @ nil_a @ ( shuffles_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_in_shuffles
thf(fact_1255_remdups__adj__set,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( remdups_adj_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_1256_hd__remdups__adj,axiom,
! [Xs: list_a] :
( ( hd_a @ ( remdups_adj_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% hd_remdups_adj
thf(fact_1257_last__remdups__adj,axiom,
! [Xs: list_a] :
( ( last_a @ ( remdups_adj_a @ Xs ) )
= ( last_a @ Xs ) ) ).
% last_remdups_adj
thf(fact_1258_remdups__adj__Cons__alt,axiom,
! [X2: a,Xs: list_a] :
( ( cons_a @ X2 @ ( tl_a @ ( remdups_adj_a @ ( cons_a @ X2 @ Xs ) ) ) )
= ( remdups_adj_a @ ( cons_a @ X2 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_1259_shuffles_Osimps_I1_J,axiom,
! [Ys: list_a] :
( ( shuffles_a @ nil_a @ Ys )
= ( insert_list_a @ Ys @ bot_bot_set_list_a ) ) ).
% shuffles.simps(1)
thf(fact_1260_shuffles_Osimps_I2_J,axiom,
! [Xs: list_a] :
( ( shuffles_a @ Xs @ nil_a )
= ( insert_list_a @ Xs @ bot_bot_set_list_a ) ) ).
% shuffles.simps(2)
thf(fact_1261_remdups__adj_Osimps_I3_J,axiom,
! [X2: a,Y4: a,Xs: list_a] :
( ( ( X2 = Y4 )
=> ( ( remdups_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y4 @ Xs ) ) )
= ( remdups_adj_a @ ( cons_a @ X2 @ Xs ) ) ) )
& ( ( X2 != Y4 )
=> ( ( remdups_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y4 @ Xs ) ) )
= ( cons_a @ X2 @ ( remdups_adj_a @ ( cons_a @ Y4 @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_1262_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_a @ nil_a )
= nil_a ) ).
% remdups_adj.simps(1)
thf(fact_1263_Cons__in__shuffles__rightI,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a,Z3: a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( member_list_a @ ( cons_a @ Z3 @ Zs ) @ ( shuffles_a @ Xs @ ( cons_a @ Z3 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_1264_Cons__in__shuffles__leftI,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a,Z3: a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( member_list_a @ ( cons_a @ Z3 @ Zs ) @ ( shuffles_a @ ( cons_a @ Z3 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_1265_Nil__in__shufflesI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = nil_a )
=> ( ( Ys = nil_a )
=> ( member_list_a @ nil_a @ ( shuffles_a @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_1266_remdups__adj_Oelims,axiom,
! [X2: list_a,Y4: list_a] :
( ( ( remdups_adj_a @ X2 )
= Y4 )
=> ( ( ( X2 = nil_a )
=> ( Y4 != nil_a ) )
=> ( ! [X: a] :
( ( X2
= ( cons_a @ X @ nil_a ) )
=> ( Y4
!= ( cons_a @ X @ nil_a ) ) )
=> ~ ! [X: a,Y3: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ~ ( ( ( X = Y3 )
=> ( Y4
= ( remdups_adj_a @ ( cons_a @ X @ Xs2 ) ) ) )
& ( ( X != Y3 )
=> ( Y4
= ( cons_a @ X @ ( remdups_adj_a @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_1267_remdups__adj_Osimps_I2_J,axiom,
! [X2: a] :
( ( remdups_adj_a @ ( cons_a @ X2 @ nil_a ) )
= ( cons_a @ X2 @ nil_a ) ) ).
% remdups_adj.simps(2)
thf(fact_1268_shufflesE,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_a ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_a ) )
=> ( ! [X: a,Xs3: list_a] :
( ( Xs
= ( cons_a @ X @ Xs3 ) )
=> ! [Z: a,Zs4: list_a] :
( ( Zs
= ( cons_a @ Z @ Zs4 ) )
=> ( ( X = Z )
=> ~ ( member_list_a @ Zs4 @ ( shuffles_a @ Xs3 @ Ys ) ) ) ) )
=> ~ ! [Y3: a,Ys5: list_a] :
( ( Ys
= ( cons_a @ Y3 @ Ys5 ) )
=> ! [Z: a,Zs4: list_a] :
( ( Zs
= ( cons_a @ Z @ Zs4 ) )
=> ( ( Y3 = Z )
=> ~ ( member_list_a @ Zs4 @ ( shuffles_a @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_1269_length__shuffles,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( size_size_list_a @ Zs )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_1270_remdups__adj__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% remdups_adj_length
thf(fact_1271_remdups__adj__append__two,axiom,
! [Xs: list_a,X2: a,Y4: a] :
( ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X2 @ ( cons_a @ Y4 @ nil_a ) ) ) )
= ( append_a @ ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) @ ( if_list_a @ ( X2 = Y4 ) @ nil_a @ ( cons_a @ Y4 @ nil_a ) ) ) ) ).
% remdups_adj_append_two
thf(fact_1272_remdups__adj__append,axiom,
! [Xs_1: list_a,X2: a,Xs_2: list_a] :
( ( remdups_adj_a @ ( append_a @ Xs_1 @ ( cons_a @ X2 @ Xs_2 ) ) )
= ( append_a @ ( remdups_adj_a @ ( append_a @ Xs_1 @ ( cons_a @ X2 @ nil_a ) ) ) @ ( tl_a @ ( remdups_adj_a @ ( cons_a @ X2 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_1273_remdups__adj__append_H,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) )
=> ( ( remdups_adj_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( remdups_adj_a @ Xs ) @ ( remdups_adj_a @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_1274_Cons__in__shuffles__iff,axiom,
! [Z3: a,Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ ( cons_a @ Z3 @ Zs ) @ ( shuffles_a @ Xs @ Ys ) )
= ( ( ( Xs != nil_a )
& ( ( hd_a @ Xs )
= Z3 )
& ( member_list_a @ Zs @ ( shuffles_a @ ( tl_a @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_a )
& ( ( hd_a @ Ys )
= Z3 )
& ( member_list_a @ Zs @ ( shuffles_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_1275_shuffles_Opsimps_I2_J,axiom,
! [Xs: list_a] :
( ( accp_P7377042638478740784list_a @ shuffles_rel_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) )
=> ( ( shuffles_a @ Xs @ nil_a )
= ( insert_list_a @ Xs @ bot_bot_set_list_a ) ) ) ).
% shuffles.psimps(2)
thf(fact_1276_remdups__adj_Opelims,axiom,
! [X2: list_a,Y4: list_a] :
( ( ( remdups_adj_a @ X2 )
= Y4 )
=> ( ( accp_list_a @ remdups_adj_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y4 = nil_a )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ nil_a ) ) )
=> ( ! [X: a] :
( ( X2
= ( cons_a @ X @ nil_a ) )
=> ( ( Y4
= ( cons_a @ X @ nil_a ) )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ ( cons_a @ X @ nil_a ) ) ) )
=> ~ ! [X: a,Y3: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ( ( ( ( X = Y3 )
=> ( Y4
= ( remdups_adj_a @ ( cons_a @ X @ Xs2 ) ) ) )
& ( ( X != Y3 )
=> ( Y4
= ( cons_a @ X @ ( remdups_adj_a @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ ( cons_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y4: nat] :
( ( if_nat @ $false @ X2 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y4: nat] :
( ( if_nat @ $true @ X2 @ Y4 )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y4: list_a] :
( ( if_list_a @ $false @ X2 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y4: list_a] :
( ( if_list_a @ $true @ X2 @ Y4 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ( cons_a @ t2 @ nil_a ) ) @ ( cons_a @ ( resid @ u2 @ t2 ) @ nil_a ) ) @ ( paths_in_Resid_a @ resid @ u @ ( cons_a @ ( resid @ t2 @ u2 ) @ nil_a ) ) )
!= nil_a )
= ( ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ ( paths_in_Resid_a @ resid @ va @ ( cons_a @ u2 @ nil_a ) ) @ ( cons_a @ ( resid @ t2 @ u2 ) @ nil_a ) ) @ ( paths_in_Resid_a @ resid @ u @ ( cons_a @ ( resid @ t2 @ u2 ) @ nil_a ) ) )
!= nil_a ) ) ).
%------------------------------------------------------------------------------