TPTP Problem File: ITP084^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP084^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer KnowledgeKeysSecrets problem prob_119__3292052_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : KnowledgeKeysSecrets/prob_119__3292052_1 [Des21]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 490 ( 194 unt; 138 typ; 0 def)
% Number of atoms : 899 ( 677 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3113 ( 202 ~; 52 |; 122 &;2319 @)
% ( 0 <=>; 418 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 445 ( 445 >; 0 *; 0 +; 0 <<)
% Number of symbols : 120 ( 119 usr; 8 con; 0-3 aty)
% Number of variables : 1113 ( 27 ^; 999 !; 87 ?;1113 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:33:46.381
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__List__Olist_It__List__Olist_It__Secrecy____types__OExpression_J_J,type,
list_l425507653ession: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Secrecy____types__OExpression_J_J,type,
set_li185139947ession: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Secrecy____types__OSecrets_J_J,type,
set_li2007071804ecrets: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Secrecy____types__OKeys_J_J,type,
set_li849033543y_Keys: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Secrecy____types__OKS_J_J,type,
list_list_Secrecy_KS: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Secrecy____types__OKS_J_J,type,
set_list_Secrecy_KS: $tType ).
thf(ty_n_t__List__Olist_It__Secrecy____types__OExpression_J,type,
list_S529945141ession: $tType ).
thf(ty_n_t__Set__Oset_It__Secrecy____types__OExpression_J,type,
set_Se1942955227ession: $tType ).
thf(ty_n_t__List__Olist_It__Secrecy____types__OSecrets_J,type,
list_Secrecy_Secrets: $tType ).
thf(ty_n_t__Set__Oset_It__Secrecy____types__OSecrets_J,type,
set_Secrecy_Secrets: $tType ).
thf(ty_n_t__List__Olist_It__Secrecy____types__OKeys_J,type,
list_Secrecy_Keys: $tType ).
thf(ty_n_t__Set__Oset_It__Secrecy____types__OKeys_J,type,
set_Secrecy_Keys: $tType ).
thf(ty_n_t__List__Olist_It__Secrecy____types__OKS_J,type,
list_Secrecy_KS: $tType ).
thf(ty_n_t__Set__Oset_It__Secrecy____types__OKS_J,type,
set_Secrecy_KS: $tType ).
thf(ty_n_t__Secrecy____types__OExpression,type,
secrecy_Expression: $tType ).
thf(ty_n_t__Secrecy____types__OSecrets,type,
secrecy_Secrets: $tType ).
thf(ty_n_t__Secrecy____types__OspecID,type,
secrecy_specID: $tType ).
thf(ty_n_t__Secrecy____types__OKeys,type,
secrecy_Keys: $tType ).
thf(ty_n_t__Secrecy____types__OKS,type,
secrecy_KS: $tType ).
% Explicit typings (119)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Secrecy____types__OExpression,type,
bNF_Gr275623750ession: set_li185139947ession > secrecy_Expression > set_li185139947ession ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Secrecy____types__OKS,type,
bNF_Gr660123862ecy_KS: set_list_Secrecy_KS > secrecy_KS > set_list_Secrecy_KS ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Secrecy____types__OKeys,type,
bNF_Gr414702370y_Keys: set_li849033543y_Keys > secrecy_Keys > set_li849033543y_Keys ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Secrecy____types__OSecrets,type,
bNF_Gr28627573ecrets: set_li2007071804ecrets > secrecy_Secrets > set_li2007071804ecrets ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Secrecy____types__OExpression,type,
bNF_Gr979629770ession: set_li185139947ession > list_S529945141ession > set_Se1942955227ession ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Secrecy____types__OKS,type,
bNF_Gr577175642ecy_KS: set_list_Secrecy_KS > list_Secrecy_KS > set_Secrecy_KS ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Secrecy____types__OKeys,type,
bNF_Gr344893606y_Keys: set_li849033543y_Keys > list_Secrecy_Keys > set_Secrecy_Keys ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Secrecy____types__OSecrets,type,
bNF_Gr957048817ecrets: set_li2007071804ecrets > list_Secrecy_Secrets > set_Secrecy_Secrets ).
thf(sy_c_CompLocalSecrets_OLocalSecrets,type,
localSecrets: secrecy_specID > set_Secrecy_KS ).
thf(sy_c_Groups_Omonoid_001t__List__Olist_It__Secrecy____types__OExpression_J,type,
monoid929671412ession: ( list_S529945141ession > list_S529945141ession > list_S529945141ession ) > list_S529945141ession > $o ).
thf(sy_c_Groups_Omonoid_001t__List__Olist_It__Secrecy____types__OKS_J,type,
monoid1528146052ecy_KS: ( list_Secrecy_KS > list_Secrecy_KS > list_Secrecy_KS ) > list_Secrecy_KS > $o ).
thf(sy_c_KnowledgeKeysSecrets__Mirabelle__vytwmzrumu_OeoutKnowCorrect,type,
knowle233900195orrect: secrecy_specID > secrecy_KS > $o ).
thf(sy_c_KnowledgeKeysSecrets__Mirabelle__vytwmzrumu_OeoutKnowsECorrect,type,
knowle1268625169orrect: secrecy_specID > secrecy_Expression > $o ).
thf(sy_c_KnowledgeKeysSecrets__Mirabelle__vytwmzrumu_Oknow,type,
knowle945212990e_know: secrecy_specID > secrecy_KS > $o ).
thf(sy_c_KnowledgeKeysSecrets__Mirabelle__vytwmzrumu_Oknows,type,
knowle731562421_knows: secrecy_specID > list_S529945141ession > $o ).
thf(sy_c_ListExtras_Odisjoint_001t__Secrecy____types__OExpression,type,
disjoi1945222607ession: list_S529945141ession > list_S529945141ession > $o ).
thf(sy_c_ListExtras_OmemS_001t__Secrecy____types__OExpression,type,
memS_S2026082737ession: secrecy_Expression > list_S529945141ession > $o ).
thf(sy_c_ListExtras_Omem_001t__Secrecy____types__OExpression,type,
mem_Se1967555118ession: secrecy_Expression > list_S529945141ession > $o ).
thf(sy_c_ListExtras_Omem_001t__Secrecy____types__OKS,type,
mem_Secrecy_KS: secrecy_KS > list_Secrecy_KS > $o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Secrecy____types__OExpression_J,type,
append1105297242ession: list_l425507653ession > list_l425507653ession > list_l425507653ession ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Secrecy____types__OKS_J,type,
append1376766698ecy_KS: list_list_Secrecy_KS > list_list_Secrecy_KS > list_list_Secrecy_KS ).
thf(sy_c_List_Oappend_001t__Secrecy____types__OExpression,type,
append2089414218ession: list_S529945141ession > list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Oappend_001t__Secrecy____types__OKS,type,
append_Secrecy_KS: list_Secrecy_KS > list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Oappend_001t__Secrecy____types__OKeys,type,
append_Secrecy_Keys: list_Secrecy_Keys > list_Secrecy_Keys > list_Secrecy_Keys ).
thf(sy_c_List_Oappend_001t__Secrecy____types__OSecrets,type,
append186372209ecrets: list_Secrecy_Secrets > list_Secrecy_Secrets > list_Secrecy_Secrets ).
thf(sy_c_List_Obind_001t__Secrecy____types__OExpression_001t__Secrecy____types__OExpression,type,
bind_S815874003ession: list_S529945141ession > ( secrecy_Expression > list_S529945141ession ) > list_S529945141ession ).
thf(sy_c_List_Obind_001t__Secrecy____types__OExpression_001t__Secrecy____types__OKS,type,
bind_S1823644515ecy_KS: list_S529945141ession > ( secrecy_Expression > list_Secrecy_KS ) > list_Secrecy_KS ).
thf(sy_c_List_Obind_001t__Secrecy____types__OKS_001t__Secrecy____types__OExpression,type,
bind_S711229283ession: list_Secrecy_KS > ( secrecy_KS > list_S529945141ession ) > list_S529945141ession ).
thf(sy_c_List_Obind_001t__Secrecy____types__OKS_001t__Secrecy____types__OKS,type,
bind_S645462771ecy_KS: list_Secrecy_KS > ( secrecy_KS > list_Secrecy_KS ) > list_Secrecy_KS ).
thf(sy_c_List_Obutlast_001t__Secrecy____types__OExpression,type,
butlas171474643ession: list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Obutlast_001t__Secrecy____types__OKS,type,
butlast_Secrecy_KS: list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Oconcat_001t__Secrecy____types__OExpression,type,
concat388015140ession: list_l425507653ession > list_S529945141ession ).
thf(sy_c_List_Oconcat_001t__Secrecy____types__OKS,type,
concat_Secrecy_KS: list_list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Odistinct__adj_001t__Secrecy____types__OExpression,type,
distin1076152135ession: list_S529945141ession > $o ).
thf(sy_c_List_Odistinct__adj_001t__Secrecy____types__OKS,type,
distin1495927511ecy_KS: list_Secrecy_KS > $o ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Secrecy____types__OExpression_J_001t__List__Olist_It__Secrecy____types__OExpression_J,type,
foldr_1531709715ession: ( list_S529945141ession > list_S529945141ession > list_S529945141ession ) > list_l425507653ession > list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Secrecy____types__OKS_J_001t__List__Olist_It__Secrecy____types__OKS_J,type,
foldr_1711005747ecy_KS: ( list_Secrecy_KS > list_Secrecy_KS > list_Secrecy_KS ) > list_list_Secrecy_KS > list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Oinsert_001t__Secrecy____types__OExpression,type,
insert1173886601ession: secrecy_Expression > list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Oinsert_001t__Secrecy____types__OKS,type,
insert_Secrecy_KS: secrecy_KS > list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Olast_001t__Secrecy____types__OExpression,type,
last_S1532390982ession: list_S529945141ession > secrecy_Expression ).
thf(sy_c_List_Olast_001t__Secrecy____types__OKS,type,
last_Secrecy_KS: list_Secrecy_KS > secrecy_KS ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Secrecy____types__OExpression_J,type,
cons_l533358079ession: list_S529945141ession > list_l425507653ession > list_l425507653ession ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Secrecy____types__OKS_J,type,
cons_list_Secrecy_KS: list_Secrecy_KS > list_list_Secrecy_KS > list_list_Secrecy_KS ).
thf(sy_c_List_Olist_OCons_001t__Secrecy____types__OExpression,type,
cons_S112077679ession: secrecy_Expression > list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Olist_OCons_001t__Secrecy____types__OKS,type,
cons_Secrecy_KS: secrecy_KS > list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Olist_OCons_001t__Secrecy____types__OKeys,type,
cons_Secrecy_Keys: secrecy_Keys > list_Secrecy_Keys > list_Secrecy_Keys ).
thf(sy_c_List_Olist_OCons_001t__Secrecy____types__OSecrets,type,
cons_Secrecy_Secrets: secrecy_Secrets > list_Secrecy_Secrets > list_Secrecy_Secrets ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Secrecy____types__OExpression_J,type,
nil_li311616943ession: list_l425507653ession ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Secrecy____types__OKS_J,type,
nil_list_Secrecy_KS: list_list_Secrecy_KS ).
thf(sy_c_List_Olist_ONil_001t__Secrecy____types__OExpression,type,
nil_Se901661471ession: list_S529945141ession ).
thf(sy_c_List_Olist_ONil_001t__Secrecy____types__OKS,type,
nil_Secrecy_KS: list_Secrecy_KS ).
thf(sy_c_List_Olist_ONil_001t__Secrecy____types__OKeys,type,
nil_Secrecy_Keys: list_Secrecy_Keys ).
thf(sy_c_List_Olist_ONil_001t__Secrecy____types__OSecrets,type,
nil_Secrecy_Secrets: list_Secrecy_Secrets ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__Secrecy____types__OExpression_J,type,
hd_lis1623417098ession: list_l425507653ession > list_S529945141ession ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__Secrecy____types__OKS_J,type,
hd_list_Secrecy_KS: list_list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Olist_Ohd_001t__Secrecy____types__OExpression,type,
hd_Sec357328890ession: list_S529945141ession > secrecy_Expression ).
thf(sy_c_List_Olist_Ohd_001t__Secrecy____types__OKS,type,
hd_Secrecy_KS: list_Secrecy_KS > secrecy_KS ).
thf(sy_c_List_Olist_Otl_001t__Secrecy____types__OExpression,type,
tl_Sec1905266422ession: list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Olist_Otl_001t__Secrecy____types__OKS,type,
tl_Secrecy_KS: list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Olist__ex1_001t__Secrecy____types__OExpression,type,
list_e1486860828ession: ( secrecy_Expression > $o ) > list_S529945141ession > $o ).
thf(sy_c_List_Olist__ex1_001t__Secrecy____types__OKS,type,
list_ex1_Secrecy_KS: ( secrecy_KS > $o ) > list_Secrecy_KS > $o ).
thf(sy_c_List_Olistrelp_001t__Secrecy____types__OExpression_001t__Secrecy____types__OExpression,type,
listre84909707ession: ( secrecy_Expression > secrecy_Expression > $o ) > list_S529945141ession > list_S529945141ession > $o ).
thf(sy_c_List_Olistrelp_001t__Secrecy____types__OExpression_001t__Secrecy____types__OKS,type,
listre565996059ecy_KS: ( secrecy_Expression > secrecy_KS > $o ) > list_S529945141ession > list_Secrecy_KS > $o ).
thf(sy_c_List_Olistrelp_001t__Secrecy____types__OKS_001t__Secrecy____types__OExpression,type,
listre1601064475ession: ( secrecy_KS > secrecy_Expression > $o ) > list_Secrecy_KS > list_S529945141ession > $o ).
thf(sy_c_List_Olistrelp_001t__Secrecy____types__OKS_001t__Secrecy____types__OKS,type,
listre788412843ecy_KS: ( secrecy_KS > secrecy_KS > $o ) > list_Secrecy_KS > list_Secrecy_KS > $o ).
thf(sy_c_List_Omap__tailrec__rev_001t__Secrecy____types__OExpression_001t__Secrecy____types__OExpression,type,
map_ta1624757057ession: ( secrecy_Expression > secrecy_Expression ) > list_S529945141ession > list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Omap__tailrec__rev_001t__Secrecy____types__OKS_001t__Secrecy____types__OExpression,type,
map_ta623472337ession: ( secrecy_KS > secrecy_Expression ) > list_Secrecy_KS > list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Omaps_001t__Secrecy____types__OExpression_001t__Secrecy____types__OExpression,type,
maps_S599402637ession: ( secrecy_Expression > list_S529945141ession ) > list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Omaps_001t__Secrecy____types__OExpression_001t__Secrecy____types__OKS,type,
maps_S399472669ecy_KS: ( secrecy_Expression > list_Secrecy_KS ) > list_S529945141ession > list_Secrecy_KS ).
thf(sy_c_List_Omaps_001t__Secrecy____types__OKS_001t__Secrecy____types__OExpression,type,
maps_S1434541085ession: ( secrecy_KS > list_S529945141ession ) > list_Secrecy_KS > list_S529945141ession ).
thf(sy_c_List_Omaps_001t__Secrecy____types__OKS_001t__Secrecy____types__OKS,type,
maps_S377605037ecy_KS: ( secrecy_KS > list_Secrecy_KS ) > list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Omember_001t__Secrecy____types__OExpression,type,
member2020590922ession: list_S529945141ession > secrecy_Expression > $o ).
thf(sy_c_List_Omember_001t__Secrecy____types__OKS,type,
member_Secrecy_KS: list_Secrecy_KS > secrecy_KS > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001t__Secrecy____types__OExpression,type,
lexord1123423577ession: ( secrecy_Expression > secrecy_Expression > $o ) > list_S529945141ession > list_S529945141ession > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001t__Secrecy____types__OKS,type,
lexord948960489ecy_KS: ( secrecy_KS > secrecy_KS > $o ) > list_Secrecy_KS > list_Secrecy_KS > $o ).
thf(sy_c_List_Oproduct__lists_001t__Secrecy____types__OExpression,type,
produc1450295286ession: list_l425507653ession > list_l425507653ession ).
thf(sy_c_List_Oproduct__lists_001t__Secrecy____types__OKS,type,
produc713695622ecy_KS: list_list_Secrecy_KS > list_list_Secrecy_KS ).
thf(sy_c_List_Orev_001t__Secrecy____types__OExpression,type,
rev_Se1073238207ession: list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Orev_001t__Secrecy____types__OKS,type,
rev_Secrecy_KS: list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Orotate1_001t__Secrecy____types__OExpression,type,
rotate1237234450ession: list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Orotate1_001t__Secrecy____types__OKS,type,
rotate1_Secrecy_KS: list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Oshuffles_001t__Secrecy____types__OExpression,type,
shuffl552897610ession: list_S529945141ession > list_S529945141ession > set_li185139947ession ).
thf(sy_c_List_Oshuffles_001t__Secrecy____types__OKS,type,
shuffles_Secrecy_KS: list_Secrecy_KS > list_Secrecy_KS > set_list_Secrecy_KS ).
thf(sy_c_List_Osplice_001t__Secrecy____types__OExpression,type,
splice1390717708ession: list_S529945141ession > list_S529945141ession > list_S529945141ession ).
thf(sy_c_List_Osplice_001t__Secrecy____types__OKS,type,
splice_Secrecy_KS: list_Secrecy_KS > list_Secrecy_KS > list_Secrecy_KS ).
thf(sy_c_List_Osubseqs_001t__Secrecy____types__OExpression,type,
subseq348280080ession: list_S529945141ession > list_l425507653ession ).
thf(sy_c_List_Osubseqs_001t__Secrecy____types__OKS,type,
subseqs_Secrecy_KS: list_Secrecy_KS > list_list_Secrecy_KS ).
thf(sy_c_List_Osuccessively_001t__Secrecy____types__OExpression,type,
succes1388047762ession: ( secrecy_Expression > secrecy_Expression > $o ) > list_S529945141ession > $o ).
thf(sy_c_List_Osuccessively_001t__Secrecy____types__OKS,type,
succes590251810ecy_KS: ( secrecy_KS > secrecy_KS > $o ) > list_Secrecy_KS > $o ).
thf(sy_c_Secrecy_ODecr,type,
decr: secrecy_Keys > list_S529945141ession > list_S529945141ession ).
thf(sy_c_Secrecy_OEnc,type,
enc: secrecy_Keys > list_S529945141ession > list_S529945141ession ).
thf(sy_c_Secrecy_OEncrDecrKeys,type,
encrDecrKeys: secrecy_Keys > secrecy_Keys > $o ).
thf(sy_c_Secrecy_OExt,type,
ext: secrecy_Keys > list_S529945141ession > list_S529945141ession ).
thf(sy_c_Secrecy_OSign,type,
sign: secrecy_Keys > list_S529945141ession > list_S529945141ession ).
thf(sy_c_Secrecy_Oeout,type,
eout: secrecy_specID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_Oine,type,
ine: secrecy_specID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_OnotSpecKeysSecretsExpr,type,
notSpe1586871396tsExpr: secrecy_specID > list_S529945141ession > $o ).
thf(sy_c_Secrecy_OspecKeys,type,
specKeys: secrecy_specID > set_Secrecy_Keys ).
thf(sy_c_Secrecy_OspecKeysSecrets,type,
specKeysSecrets: secrecy_specID > set_Secrecy_KS ).
thf(sy_c_Secrecy_OspecSecrets,type,
specSecrets: secrecy_specID > set_Secrecy_Secrets ).
thf(sy_c_Secrecy__types_OExpression2KSL,type,
secrec1371625429on2KSL: list_S529945141ession > list_Secrecy_KS ).
thf(sy_c_Secrecy__types_OExpression_OkE,type,
secrecy_kE: secrecy_Keys > secrecy_Expression ).
thf(sy_c_Secrecy__types_OExpression_OsE,type,
secrecy_sE: secrecy_Secrets > secrecy_Expression ).
thf(sy_c_Secrecy__types_OKS2Expression,type,
secrec1016742679ession: secrecy_KS > secrecy_Expression ).
thf(sy_c_Secrecy__types_OKS_OkKS,type,
secrecy_kKS: secrecy_Keys > secrecy_KS ).
thf(sy_c_Secrecy__types_OKS_OsKS,type,
secrecy_sKS: secrecy_Secrets > secrecy_KS ).
thf(sy_c_Set_OCollect_001t__Secrecy____types__OKS,type,
collect_Secrecy_KS: ( secrecy_KS > $o ) > set_Secrecy_KS ).
thf(sy_c_Set_OCollect_001t__Secrecy____types__OKeys,type,
collect_Secrecy_Keys: ( secrecy_Keys > $o ) > set_Secrecy_Keys ).
thf(sy_c_Set_OCollect_001t__Secrecy____types__OSecrets,type,
collec836099393ecrets: ( secrecy_Secrets > $o ) > set_Secrecy_Secrets ).
thf(sy_c_member_001t__List__Olist_It__Secrecy____types__OExpression_J,type,
member249235532ession: list_S529945141ession > set_li185139947ession > $o ).
thf(sy_c_member_001t__List__Olist_It__Secrecy____types__OKS_J,type,
member193705948ecy_KS: list_Secrecy_KS > set_list_Secrecy_KS > $o ).
thf(sy_c_member_001t__List__Olist_It__Secrecy____types__OKeys_J,type,
member341114536y_Keys: list_Secrecy_Keys > set_li849033543y_Keys > $o ).
thf(sy_c_member_001t__List__Olist_It__Secrecy____types__OSecrets_J,type,
member1138497285ecrets: list_Secrecy_Secrets > set_li2007071804ecrets > $o ).
thf(sy_c_member_001t__Secrecy____types__OExpression,type,
member279582780ession: secrecy_Expression > set_Se1942955227ession > $o ).
thf(sy_c_member_001t__Secrecy____types__OKS,type,
member_Secrecy_KS2: secrecy_KS > set_Secrecy_KS > $o ).
thf(sy_c_member_001t__Secrecy____types__OKeys,type,
member_Secrecy_Keys: secrecy_Keys > set_Secrecy_Keys > $o ).
thf(sy_c_member_001t__Secrecy____types__OSecrets,type,
member2138796287ecrets: secrecy_Secrets > set_Secrecy_Secrets > $o ).
thf(sy_v_A,type,
a: secrecy_specID ).
thf(sy_v_m,type,
m: secrecy_Keys ).
% Relevant facts (351)
thf(fact_0_assms,axiom,
knowle731562421_knows @ a @ ( cons_S112077679ession @ ( secrecy_kE @ m ) @ nil_Se901661471ession ) ).
% assms
thf(fact_1_KS_Oinject_I1_J,axiom,
! [X1: secrecy_Keys,Y1: secrecy_Keys] :
( ( ( secrecy_kKS @ X1 )
= ( secrecy_kKS @ Y1 ) )
= ( X1 = Y1 ) ) ).
% KS.inject(1)
thf(fact_2_knows4,axiom,
! [IncrDecrKeys: secrecy_Keys > secrecy_Keys > $o,K1: secrecy_Keys,K2: secrecy_Keys,A: secrecy_specID,E: list_S529945141ession] :
( ( ( IncrDecrKeys @ K1 @ K2 )
& ( knowle945212990e_know @ A @ ( secrecy_kKS @ K2 ) )
& ( knowle731562421_knows @ A @ ( enc @ K1 @ E ) ) )
=> ( knowle731562421_knows @ A @ E ) ) ).
% knows4
thf(fact_3_knows5,axiom,
! [IncrDecrKeys: secrecy_Keys > secrecy_Keys > $o,K1: secrecy_Keys,K2: secrecy_Keys,A: secrecy_specID,E: list_S529945141ession] :
( ( ( IncrDecrKeys @ K1 @ K2 )
& ( knowle945212990e_know @ A @ ( secrecy_kKS @ K1 ) )
& ( knowle731562421_knows @ A @ ( sign @ K2 @ E ) ) )
=> ( knowle731562421_knows @ A @ E ) ) ).
% knows5
thf(fact_4_knows6,axiom,
! [A: secrecy_specID,K: secrecy_Keys,E1: list_S529945141ession] :
( ( ( knowle945212990e_know @ A @ ( secrecy_kKS @ K ) )
& ( knowle731562421_knows @ A @ E1 ) )
=> ( knowle731562421_knows @ A @ ( enc @ K @ E1 ) ) ) ).
% knows6
thf(fact_5_knows7,axiom,
! [A: secrecy_specID,K: secrecy_Keys,E1: list_S529945141ession] :
( ( ( knowle945212990e_know @ A @ ( secrecy_kKS @ K ) )
& ( knowle731562421_knows @ A @ E1 ) )
=> ( knowle731562421_knows @ A @ ( sign @ K @ E1 ) ) ) ).
% knows7
thf(fact_6_know2knows__k,axiom,
! [A: secrecy_specID,M: secrecy_Keys] :
( ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) )
=> ( knowle731562421_knows @ A @ ( cons_S112077679ession @ ( secrecy_kE @ M ) @ nil_Se901661471ession ) ) ) ).
% know2knows_k
thf(fact_7_know1k,axiom,
! [C: secrecy_specID,M1: secrecy_Keys] :
( ( knowle731562421_knows @ C @ ( cons_S112077679ession @ ( secrec1016742679ession @ ( secrecy_kKS @ M1 ) ) @ nil_Se901661471ession ) )
= ( knowle945212990e_know @ C @ ( secrecy_kKS @ M1 ) ) ) ).
% know1k
thf(fact_8_eoutKnowCorrect__L1k,axiom,
! [C: secrecy_specID,M: secrecy_Keys] :
( ( knowle233900195orrect @ C @ ( secrecy_kKS @ M ) )
=> ( ( eout @ C @ ( secrecy_kE @ M ) )
=> ( ( member_Secrecy_Keys @ M @ ( specKeys @ C ) )
| ( knowle945212990e_know @ C @ ( secrecy_kKS @ M ) ) ) ) ) ).
% eoutKnowCorrect_L1k
thf(fact_9_eout__know__k,axiom,
! [C: secrecy_specID,M: secrecy_Keys] :
( ( knowle233900195orrect @ C @ ( secrecy_kKS @ M ) )
= ( ( eout @ C @ ( secrecy_kE @ M ) )
= ( ( member_Secrecy_Keys @ M @ ( specKeys @ C ) )
| ( knowle945212990e_know @ C @ ( secrecy_kKS @ M ) ) ) ) ) ).
% eout_know_k
thf(fact_10_know_Osimps_I1_J,axiom,
! [A: secrecy_specID,M: secrecy_Keys] :
( ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) )
= ( ( ine @ A @ ( secrecy_kE @ M ) )
| ( member_Secrecy_KS2 @ ( secrecy_kKS @ M ) @ ( localSecrets @ A ) ) ) ) ).
% know.simps(1)
thf(fact_11_Expression_Oinject_I1_J,axiom,
! [X1: secrecy_Keys,Y1: secrecy_Keys] :
( ( ( secrecy_kE @ X1 )
= ( secrecy_kE @ Y1 ) )
= ( X1 = Y1 ) ) ).
% Expression.inject(1)
thf(fact_12_knows__emptyexpression,axiom,
! [C: secrecy_specID] : ( knowle731562421_knows @ C @ nil_Se901661471ession ) ).
% knows_emptyexpression
thf(fact_13_KS2Expression_Osimps_I1_J,axiom,
! [M: secrecy_Keys] :
( ( secrec1016742679ession @ ( secrecy_kKS @ M ) )
= ( secrecy_kE @ M ) ) ).
% KS2Expression.simps(1)
thf(fact_14_list_Oinject,axiom,
! [X21: secrecy_KS,X22: list_Secrecy_KS,Y21: secrecy_KS,Y22: list_Secrecy_KS] :
( ( ( cons_Secrecy_KS @ X21 @ X22 )
= ( cons_Secrecy_KS @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_15_list_Oinject,axiom,
! [X21: list_S529945141ession,X22: list_l425507653ession,Y21: list_S529945141ession,Y22: list_l425507653ession] :
( ( ( cons_l533358079ession @ X21 @ X22 )
= ( cons_l533358079ession @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_16_list_Oinject,axiom,
! [X21: list_Secrecy_KS,X22: list_list_Secrecy_KS,Y21: list_Secrecy_KS,Y22: list_list_Secrecy_KS] :
( ( ( cons_list_Secrecy_KS @ X21 @ X22 )
= ( cons_list_Secrecy_KS @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_17_list_Oinject,axiom,
! [X21: secrecy_Secrets,X22: list_Secrecy_Secrets,Y21: secrecy_Secrets,Y22: list_Secrecy_Secrets] :
( ( ( cons_Secrecy_Secrets @ X21 @ X22 )
= ( cons_Secrecy_Secrets @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_18_list_Oinject,axiom,
! [X21: secrecy_Keys,X22: list_Secrecy_Keys,Y21: secrecy_Keys,Y22: list_Secrecy_Keys] :
( ( ( cons_Secrecy_Keys @ X21 @ X22 )
= ( cons_Secrecy_Keys @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_19_list_Oinject,axiom,
! [X21: secrecy_Expression,X22: list_S529945141ession,Y21: secrecy_Expression,Y22: list_S529945141ession] :
( ( ( cons_S112077679ession @ X21 @ X22 )
= ( cons_S112077679ession @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_20_know1s,axiom,
! [C: secrecy_specID,M2: secrecy_Secrets] :
( ( knowle731562421_knows @ C @ ( cons_S112077679ession @ ( secrec1016742679ession @ ( secrecy_sKS @ M2 ) ) @ nil_Se901661471ession ) )
= ( knowle945212990e_know @ C @ ( secrecy_sKS @ M2 ) ) ) ).
% know1s
thf(fact_21_list_Odistinct_I1_J,axiom,
! [X21: list_S529945141ession,X22: list_l425507653ession] :
( nil_li311616943ession
!= ( cons_l533358079ession @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_22_list_Odistinct_I1_J,axiom,
! [X21: list_Secrecy_KS,X22: list_list_Secrecy_KS] :
( nil_list_Secrecy_KS
!= ( cons_list_Secrecy_KS @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_23_list_Odistinct_I1_J,axiom,
! [X21: secrecy_Secrets,X22: list_Secrecy_Secrets] :
( nil_Secrecy_Secrets
!= ( cons_Secrecy_Secrets @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_24_list_Odistinct_I1_J,axiom,
! [X21: secrecy_Keys,X22: list_Secrecy_Keys] :
( nil_Secrecy_Keys
!= ( cons_Secrecy_Keys @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_25_list_Odistinct_I1_J,axiom,
! [X21: secrecy_KS,X22: list_Secrecy_KS] :
( nil_Secrecy_KS
!= ( cons_Secrecy_KS @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_26_list_Odistinct_I1_J,axiom,
! [X21: secrecy_Expression,X22: list_S529945141ession] :
( nil_Se901661471ession
!= ( cons_S112077679ession @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_27_list_OdiscI,axiom,
! [List: list_l425507653ession,X21: list_S529945141ession,X22: list_l425507653ession] :
( ( List
= ( cons_l533358079ession @ X21 @ X22 ) )
=> ( List != nil_li311616943ession ) ) ).
% list.discI
thf(fact_28_list_OdiscI,axiom,
! [List: list_list_Secrecy_KS,X21: list_Secrecy_KS,X22: list_list_Secrecy_KS] :
( ( List
= ( cons_list_Secrecy_KS @ X21 @ X22 ) )
=> ( List != nil_list_Secrecy_KS ) ) ).
% list.discI
thf(fact_29_list_OdiscI,axiom,
! [List: list_Secrecy_Secrets,X21: secrecy_Secrets,X22: list_Secrecy_Secrets] :
( ( List
= ( cons_Secrecy_Secrets @ X21 @ X22 ) )
=> ( List != nil_Secrecy_Secrets ) ) ).
% list.discI
thf(fact_30_list_OdiscI,axiom,
! [List: list_Secrecy_Keys,X21: secrecy_Keys,X22: list_Secrecy_Keys] :
( ( List
= ( cons_Secrecy_Keys @ X21 @ X22 ) )
=> ( List != nil_Secrecy_Keys ) ) ).
% list.discI
thf(fact_31_list_OdiscI,axiom,
! [List: list_Secrecy_KS,X21: secrecy_KS,X22: list_Secrecy_KS] :
( ( List
= ( cons_Secrecy_KS @ X21 @ X22 ) )
=> ( List != nil_Secrecy_KS ) ) ).
% list.discI
thf(fact_32_list_OdiscI,axiom,
! [List: list_S529945141ession,X21: secrecy_Expression,X22: list_S529945141ession] :
( ( List
= ( cons_S112077679ession @ X21 @ X22 ) )
=> ( List != nil_Se901661471ession ) ) ).
% list.discI
thf(fact_33_list_Oexhaust,axiom,
! [Y: list_l425507653ession] :
( ( Y != nil_li311616943ession )
=> ~ ! [X212: list_S529945141ession,X222: list_l425507653ession] :
( Y
!= ( cons_l533358079ession @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_34_list_Oexhaust,axiom,
! [Y: list_list_Secrecy_KS] :
( ( Y != nil_list_Secrecy_KS )
=> ~ ! [X212: list_Secrecy_KS,X222: list_list_Secrecy_KS] :
( Y
!= ( cons_list_Secrecy_KS @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_35_list_Oexhaust,axiom,
! [Y: list_Secrecy_Secrets] :
( ( Y != nil_Secrecy_Secrets )
=> ~ ! [X212: secrecy_Secrets,X222: list_Secrecy_Secrets] :
( Y
!= ( cons_Secrecy_Secrets @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_36_list_Oexhaust,axiom,
! [Y: list_Secrecy_Keys] :
( ( Y != nil_Secrecy_Keys )
=> ~ ! [X212: secrecy_Keys,X222: list_Secrecy_Keys] :
( Y
!= ( cons_Secrecy_Keys @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_37_list_Oexhaust,axiom,
! [Y: list_Secrecy_KS] :
( ( Y != nil_Secrecy_KS )
=> ~ ! [X212: secrecy_KS,X222: list_Secrecy_KS] :
( Y
!= ( cons_Secrecy_KS @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_38_list_Oexhaust,axiom,
! [Y: list_S529945141ession] :
( ( Y != nil_Se901661471ession )
=> ~ ! [X212: secrecy_Expression,X222: list_S529945141ession] :
( Y
!= ( cons_S112077679ession @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_39_list_Oinducts,axiom,
! [P: list_l425507653ession > $o,List: list_l425507653ession] :
( ( P @ nil_li311616943ession )
=> ( ! [X12: list_S529945141ession,X2: list_l425507653ession] :
( ( P @ X2 )
=> ( P @ ( cons_l533358079ession @ X12 @ X2 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_40_list_Oinducts,axiom,
! [P: list_list_Secrecy_KS > $o,List: list_list_Secrecy_KS] :
( ( P @ nil_list_Secrecy_KS )
=> ( ! [X12: list_Secrecy_KS,X2: list_list_Secrecy_KS] :
( ( P @ X2 )
=> ( P @ ( cons_list_Secrecy_KS @ X12 @ X2 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_41_list_Oinducts,axiom,
! [P: list_Secrecy_Secrets > $o,List: list_Secrecy_Secrets] :
( ( P @ nil_Secrecy_Secrets )
=> ( ! [X12: secrecy_Secrets,X2: list_Secrecy_Secrets] :
( ( P @ X2 )
=> ( P @ ( cons_Secrecy_Secrets @ X12 @ X2 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_42_list_Oinducts,axiom,
! [P: list_Secrecy_Keys > $o,List: list_Secrecy_Keys] :
( ( P @ nil_Secrecy_Keys )
=> ( ! [X12: secrecy_Keys,X2: list_Secrecy_Keys] :
( ( P @ X2 )
=> ( P @ ( cons_Secrecy_Keys @ X12 @ X2 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_43_list_Oinducts,axiom,
! [P: list_Secrecy_KS > $o,List: list_Secrecy_KS] :
( ( P @ nil_Secrecy_KS )
=> ( ! [X12: secrecy_KS,X2: list_Secrecy_KS] :
( ( P @ X2 )
=> ( P @ ( cons_Secrecy_KS @ X12 @ X2 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_44_list_Oinducts,axiom,
! [P: list_S529945141ession > $o,List: list_S529945141ession] :
( ( P @ nil_Se901661471ession )
=> ( ! [X12: secrecy_Expression,X2: list_S529945141ession] :
( ( P @ X2 )
=> ( P @ ( cons_S112077679ession @ X12 @ X2 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_45_neq__Nil__conv,axiom,
! [Xs: list_l425507653ession] :
( ( Xs != nil_li311616943ession )
= ( ? [Y2: list_S529945141ession,Ys: list_l425507653ession] :
( Xs
= ( cons_l533358079ession @ Y2 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_46_neq__Nil__conv,axiom,
! [Xs: list_list_Secrecy_KS] :
( ( Xs != nil_list_Secrecy_KS )
= ( ? [Y2: list_Secrecy_KS,Ys: list_list_Secrecy_KS] :
( Xs
= ( cons_list_Secrecy_KS @ Y2 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_47_neq__Nil__conv,axiom,
! [Xs: list_Secrecy_Secrets] :
( ( Xs != nil_Secrecy_Secrets )
= ( ? [Y2: secrecy_Secrets,Ys: list_Secrecy_Secrets] :
( Xs
= ( cons_Secrecy_Secrets @ Y2 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_48_neq__Nil__conv,axiom,
! [Xs: list_Secrecy_Keys] :
( ( Xs != nil_Secrecy_Keys )
= ( ? [Y2: secrecy_Keys,Ys: list_Secrecy_Keys] :
( Xs
= ( cons_Secrecy_Keys @ Y2 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_49_neq__Nil__conv,axiom,
! [Xs: list_Secrecy_KS] :
( ( Xs != nil_Secrecy_KS )
= ( ? [Y2: secrecy_KS,Ys: list_Secrecy_KS] :
( Xs
= ( cons_Secrecy_KS @ Y2 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_50_neq__Nil__conv,axiom,
! [Xs: list_S529945141ession] :
( ( Xs != nil_Se901661471ession )
= ( ? [Y2: secrecy_Expression,Ys: list_S529945141ession] :
( Xs
= ( cons_S112077679ession @ Y2 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_51_list__induct2_H,axiom,
! [P: list_Secrecy_Keys > list_Secrecy_KS > $o,Xs: list_Secrecy_Keys,Ys2: list_Secrecy_KS] :
( ( P @ nil_Secrecy_Keys @ nil_Secrecy_KS )
=> ( ! [X: secrecy_Keys,Xs2: list_Secrecy_Keys] : ( P @ ( cons_Secrecy_Keys @ X @ Xs2 ) @ nil_Secrecy_KS )
=> ( ! [Y3: secrecy_KS,Ys3: list_Secrecy_KS] : ( P @ nil_Secrecy_Keys @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) )
=> ( ! [X: secrecy_Keys,Xs2: list_Secrecy_Keys,Y3: secrecy_KS,Ys3: list_Secrecy_KS] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_Secrecy_Keys @ X @ Xs2 ) @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_52_list__induct2_H,axiom,
! [P: list_Secrecy_Keys > list_l425507653ession > $o,Xs: list_Secrecy_Keys,Ys2: list_l425507653ession] :
( ( P @ nil_Secrecy_Keys @ nil_li311616943ession )
=> ( ! [X: secrecy_Keys,Xs2: list_Secrecy_Keys] : ( P @ ( cons_Secrecy_Keys @ X @ Xs2 ) @ nil_li311616943ession )
=> ( ! [Y3: list_S529945141ession,Ys3: list_l425507653ession] : ( P @ nil_Secrecy_Keys @ ( cons_l533358079ession @ Y3 @ Ys3 ) )
=> ( ! [X: secrecy_Keys,Xs2: list_Secrecy_Keys,Y3: list_S529945141ession,Ys3: list_l425507653ession] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_Secrecy_Keys @ X @ Xs2 ) @ ( cons_l533358079ession @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_53_list__induct2_H,axiom,
! [P: list_Secrecy_Keys > list_list_Secrecy_KS > $o,Xs: list_Secrecy_Keys,Ys2: list_list_Secrecy_KS] :
( ( P @ nil_Secrecy_Keys @ nil_list_Secrecy_KS )
=> ( ! [X: secrecy_Keys,Xs2: list_Secrecy_Keys] : ( P @ ( cons_Secrecy_Keys @ X @ Xs2 ) @ nil_list_Secrecy_KS )
=> ( ! [Y3: list_Secrecy_KS,Ys3: list_list_Secrecy_KS] : ( P @ nil_Secrecy_Keys @ ( cons_list_Secrecy_KS @ Y3 @ Ys3 ) )
=> ( ! [X: secrecy_Keys,Xs2: list_Secrecy_Keys,Y3: list_Secrecy_KS,Ys3: list_list_Secrecy_KS] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_Secrecy_Keys @ X @ Xs2 ) @ ( cons_list_Secrecy_KS @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_54_list__induct2_H,axiom,
! [P: list_Secrecy_Keys > list_Secrecy_Secrets > $o,Xs: list_Secrecy_Keys,Ys2: list_Secrecy_Secrets] :
( ( P @ nil_Secrecy_Keys @ nil_Secrecy_Secrets )
=> ( ! [X: secrecy_Keys,Xs2: list_Secrecy_Keys] : ( P @ ( cons_Secrecy_Keys @ X @ Xs2 ) @ nil_Secrecy_Secrets )
=> ( ! [Y3: secrecy_Secrets,Ys3: list_Secrecy_Secrets] : ( P @ nil_Secrecy_Keys @ ( cons_Secrecy_Secrets @ Y3 @ Ys3 ) )
=> ( ! [X: secrecy_Keys,Xs2: list_Secrecy_Keys,Y3: secrecy_Secrets,Ys3: list_Secrecy_Secrets] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_Secrecy_Keys @ X @ Xs2 ) @ ( cons_Secrecy_Secrets @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_55_list__induct2_H,axiom,
! [P: list_Secrecy_Keys > list_Secrecy_Keys > $o,Xs: list_Secrecy_Keys,Ys2: list_Secrecy_Keys] :
( ( P @ nil_Secrecy_Keys @ nil_Secrecy_Keys )
=> ( ! [X: secrecy_Keys,Xs2: list_Secrecy_Keys] : ( P @ ( cons_Secrecy_Keys @ X @ Xs2 ) @ nil_Secrecy_Keys )
=> ( ! [Y3: secrecy_Keys,Ys3: list_Secrecy_Keys] : ( P @ nil_Secrecy_Keys @ ( cons_Secrecy_Keys @ Y3 @ Ys3 ) )
=> ( ! [X: secrecy_Keys,Xs2: list_Secrecy_Keys,Y3: secrecy_Keys,Ys3: list_Secrecy_Keys] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_Secrecy_Keys @ X @ Xs2 ) @ ( cons_Secrecy_Keys @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_56_list__induct2_H,axiom,
! [P: list_Secrecy_KS > list_Secrecy_KS > $o,Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( P @ nil_Secrecy_KS @ nil_Secrecy_KS )
=> ( ! [X: secrecy_KS,Xs2: list_Secrecy_KS] : ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) @ nil_Secrecy_KS )
=> ( ! [Y3: secrecy_KS,Ys3: list_Secrecy_KS] : ( P @ nil_Secrecy_KS @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) )
=> ( ! [X: secrecy_KS,Xs2: list_Secrecy_KS,Y3: secrecy_KS,Ys3: list_Secrecy_KS] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_57_list__induct2_H,axiom,
! [P: list_Secrecy_KS > list_S529945141ession > $o,Xs: list_Secrecy_KS,Ys2: list_S529945141ession] :
( ( P @ nil_Secrecy_KS @ nil_Se901661471ession )
=> ( ! [X: secrecy_KS,Xs2: list_Secrecy_KS] : ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) @ nil_Se901661471ession )
=> ( ! [Y3: secrecy_Expression,Ys3: list_S529945141ession] : ( P @ nil_Secrecy_KS @ ( cons_S112077679ession @ Y3 @ Ys3 ) )
=> ( ! [X: secrecy_KS,Xs2: list_Secrecy_KS,Y3: secrecy_Expression,Ys3: list_S529945141ession] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) @ ( cons_S112077679ession @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_58_list__induct2_H,axiom,
! [P: list_S529945141ession > list_Secrecy_KS > $o,Xs: list_S529945141ession,Ys2: list_Secrecy_KS] :
( ( P @ nil_Se901661471ession @ nil_Secrecy_KS )
=> ( ! [X: secrecy_Expression,Xs2: list_S529945141ession] : ( P @ ( cons_S112077679ession @ X @ Xs2 ) @ nil_Secrecy_KS )
=> ( ! [Y3: secrecy_KS,Ys3: list_Secrecy_KS] : ( P @ nil_Se901661471ession @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) )
=> ( ! [X: secrecy_Expression,Xs2: list_S529945141ession,Y3: secrecy_KS,Ys3: list_Secrecy_KS] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_S112077679ession @ X @ Xs2 ) @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_59_list__induct2_H,axiom,
! [P: list_S529945141ession > list_S529945141ession > $o,Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( P @ nil_Se901661471ession @ nil_Se901661471ession )
=> ( ! [X: secrecy_Expression,Xs2: list_S529945141ession] : ( P @ ( cons_S112077679ession @ X @ Xs2 ) @ nil_Se901661471ession )
=> ( ! [Y3: secrecy_Expression,Ys3: list_S529945141ession] : ( P @ nil_Se901661471ession @ ( cons_S112077679ession @ Y3 @ Ys3 ) )
=> ( ! [X: secrecy_Expression,Xs2: list_S529945141ession,Y3: secrecy_Expression,Ys3: list_S529945141ession] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_S112077679ession @ X @ Xs2 ) @ ( cons_S112077679ession @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_60_splice_Oinduct,axiom,
! [P: list_Secrecy_KS > list_Secrecy_KS > $o,A0: list_Secrecy_KS,A1: list_Secrecy_KS] :
( ! [X_1: list_Secrecy_KS] : ( P @ nil_Secrecy_KS @ X_1 )
=> ( ! [X: secrecy_KS,Xs2: list_Secrecy_KS,Ys3: list_Secrecy_KS] :
( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_61_splice_Oinduct,axiom,
! [P: list_S529945141ession > list_S529945141ession > $o,A0: list_S529945141ession,A1: list_S529945141ession] :
( ! [X_1: list_S529945141ession] : ( P @ nil_Se901661471ession @ X_1 )
=> ( ! [X: secrecy_Expression,Xs2: list_S529945141ession,Ys3: list_S529945141ession] :
( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons_S112077679ession @ X @ Xs2 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_62_induct__list012,axiom,
! [P: list_Secrecy_KS > $o,Xs: list_Secrecy_KS] :
( ( P @ nil_Secrecy_KS )
=> ( ! [X: secrecy_KS] : ( P @ ( cons_Secrecy_KS @ X @ nil_Secrecy_KS ) )
=> ( ! [X: secrecy_KS,Y3: secrecy_KS,Zs: list_Secrecy_KS] :
( ( P @ Zs )
=> ( ( P @ ( cons_Secrecy_KS @ Y3 @ Zs ) )
=> ( P @ ( cons_Secrecy_KS @ X @ ( cons_Secrecy_KS @ Y3 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_63_induct__list012,axiom,
! [P: list_S529945141ession > $o,Xs: list_S529945141ession] :
( ( P @ nil_Se901661471ession )
=> ( ! [X: secrecy_Expression] : ( P @ ( cons_S112077679ession @ X @ nil_Se901661471ession ) )
=> ( ! [X: secrecy_Expression,Y3: secrecy_Expression,Zs: list_S529945141ession] :
( ( P @ Zs )
=> ( ( P @ ( cons_S112077679ession @ Y3 @ Zs ) )
=> ( P @ ( cons_S112077679ession @ X @ ( cons_S112077679ession @ Y3 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_64_KS_Oinject_I2_J,axiom,
! [X23: secrecy_Secrets,Y23: secrecy_Secrets] :
( ( ( secrecy_sKS @ X23 )
= ( secrecy_sKS @ Y23 ) )
= ( X23 = Y23 ) ) ).
% KS.inject(2)
thf(fact_65_KS_Odistinct_I1_J,axiom,
! [X1: secrecy_Keys,X23: secrecy_Secrets] :
( ( secrecy_kKS @ X1 )
!= ( secrecy_sKS @ X23 ) ) ).
% KS.distinct(1)
thf(fact_66_KS_Oinduct,axiom,
! [P: secrecy_KS > $o,KS: secrecy_KS] :
( ! [X: secrecy_Keys] : ( P @ ( secrecy_kKS @ X ) )
=> ( ! [X: secrecy_Secrets] : ( P @ ( secrecy_sKS @ X ) )
=> ( P @ KS ) ) ) ).
% KS.induct
thf(fact_67_KS_Oexhaust,axiom,
! [Y: secrecy_KS] :
( ! [X12: secrecy_Keys] :
( Y
!= ( secrecy_kKS @ X12 ) )
=> ~ ! [X2: secrecy_Secrets] :
( Y
!= ( secrecy_sKS @ X2 ) ) ) ).
% KS.exhaust
thf(fact_68_not__Cons__self2,axiom,
! [X3: secrecy_Expression,Xs: list_S529945141ession] :
( ( cons_S112077679ession @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_69_map__tailrec__rev_Oinduct,axiom,
! [P: ( secrecy_KS > secrecy_Expression ) > list_Secrecy_KS > list_S529945141ession > $o,A0: secrecy_KS > secrecy_Expression,A1: list_Secrecy_KS,A2: list_S529945141ession] :
( ! [F: secrecy_KS > secrecy_Expression,X_1: list_S529945141ession] : ( P @ F @ nil_Secrecy_KS @ X_1 )
=> ( ! [F: secrecy_KS > secrecy_Expression,A3: secrecy_KS,As: list_Secrecy_KS,Bs: list_S529945141ession] :
( ( P @ F @ As @ ( cons_S112077679ession @ ( F @ A3 ) @ Bs ) )
=> ( P @ F @ ( cons_Secrecy_KS @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_70_map__tailrec__rev_Oinduct,axiom,
! [P: ( secrecy_Expression > secrecy_Expression ) > list_S529945141ession > list_S529945141ession > $o,A0: secrecy_Expression > secrecy_Expression,A1: list_S529945141ession,A2: list_S529945141ession] :
( ! [F: secrecy_Expression > secrecy_Expression,X_1: list_S529945141ession] : ( P @ F @ nil_Se901661471ession @ X_1 )
=> ( ! [F: secrecy_Expression > secrecy_Expression,A3: secrecy_Expression,As: list_S529945141ession,Bs: list_S529945141ession] :
( ( P @ F @ As @ ( cons_S112077679ession @ ( F @ A3 ) @ Bs ) )
=> ( P @ F @ ( cons_S112077679ession @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_71_list__nonempty__induct,axiom,
! [Xs: list_Secrecy_KS,P: list_Secrecy_KS > $o] :
( ( Xs != nil_Secrecy_KS )
=> ( ! [X: secrecy_KS] : ( P @ ( cons_Secrecy_KS @ X @ nil_Secrecy_KS ) )
=> ( ! [X: secrecy_KS,Xs2: list_Secrecy_KS] :
( ( Xs2 != nil_Secrecy_KS )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_72_list__nonempty__induct,axiom,
! [Xs: list_S529945141ession,P: list_S529945141ession > $o] :
( ( Xs != nil_Se901661471ession )
=> ( ! [X: secrecy_Expression] : ( P @ ( cons_S112077679ession @ X @ nil_Se901661471ession ) )
=> ( ! [X: secrecy_Expression,Xs2: list_S529945141ession] :
( ( Xs2 != nil_Se901661471ession )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_S112077679ession @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_73_successively_Oinduct,axiom,
! [P: ( secrecy_KS > secrecy_KS > $o ) > list_Secrecy_KS > $o,A0: secrecy_KS > secrecy_KS > $o,A1: list_Secrecy_KS] :
( ! [P2: secrecy_KS > secrecy_KS > $o] : ( P @ P2 @ nil_Secrecy_KS )
=> ( ! [P2: secrecy_KS > secrecy_KS > $o,X: secrecy_KS] : ( P @ P2 @ ( cons_Secrecy_KS @ X @ nil_Secrecy_KS ) )
=> ( ! [P2: secrecy_KS > secrecy_KS > $o,X: secrecy_KS,Y3: secrecy_KS,Xs2: list_Secrecy_KS] :
( ( P @ P2 @ ( cons_Secrecy_KS @ Y3 @ Xs2 ) )
=> ( P @ P2 @ ( cons_Secrecy_KS @ X @ ( cons_Secrecy_KS @ Y3 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_74_successively_Oinduct,axiom,
! [P: ( secrecy_Expression > secrecy_Expression > $o ) > list_S529945141ession > $o,A0: secrecy_Expression > secrecy_Expression > $o,A1: list_S529945141ession] :
( ! [P2: secrecy_Expression > secrecy_Expression > $o] : ( P @ P2 @ nil_Se901661471ession )
=> ( ! [P2: secrecy_Expression > secrecy_Expression > $o,X: secrecy_Expression] : ( P @ P2 @ ( cons_S112077679ession @ X @ nil_Se901661471ession ) )
=> ( ! [P2: secrecy_Expression > secrecy_Expression > $o,X: secrecy_Expression,Y3: secrecy_Expression,Xs2: list_S529945141ession] :
( ( P @ P2 @ ( cons_S112077679ession @ Y3 @ Xs2 ) )
=> ( P @ P2 @ ( cons_S112077679ession @ X @ ( cons_S112077679ession @ Y3 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_75_remdups__adj_Oinduct,axiom,
! [P: list_Secrecy_KS > $o,A0: list_Secrecy_KS] :
( ( P @ nil_Secrecy_KS )
=> ( ! [X: secrecy_KS] : ( P @ ( cons_Secrecy_KS @ X @ nil_Secrecy_KS ) )
=> ( ! [X: secrecy_KS,Y3: secrecy_KS,Xs2: list_Secrecy_KS] :
( ( ( X = Y3 )
=> ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) ) )
=> ( ( ( X != Y3 )
=> ( P @ ( cons_Secrecy_KS @ Y3 @ Xs2 ) ) )
=> ( P @ ( cons_Secrecy_KS @ X @ ( cons_Secrecy_KS @ Y3 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_76_remdups__adj_Oinduct,axiom,
! [P: list_S529945141ession > $o,A0: list_S529945141ession] :
( ( P @ nil_Se901661471ession )
=> ( ! [X: secrecy_Expression] : ( P @ ( cons_S112077679ession @ X @ nil_Se901661471ession ) )
=> ( ! [X: secrecy_Expression,Y3: secrecy_Expression,Xs2: list_S529945141ession] :
( ( ( X = Y3 )
=> ( P @ ( cons_S112077679ession @ X @ Xs2 ) ) )
=> ( ( ( X != Y3 )
=> ( P @ ( cons_S112077679ession @ Y3 @ Xs2 ) ) )
=> ( P @ ( cons_S112077679ession @ X @ ( cons_S112077679ession @ Y3 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_77_sorted__wrt_Oinduct,axiom,
! [P: ( secrecy_KS > secrecy_KS > $o ) > list_Secrecy_KS > $o,A0: secrecy_KS > secrecy_KS > $o,A1: list_Secrecy_KS] :
( ! [P2: secrecy_KS > secrecy_KS > $o] : ( P @ P2 @ nil_Secrecy_KS )
=> ( ! [P2: secrecy_KS > secrecy_KS > $o,X: secrecy_KS,Ys3: list_Secrecy_KS] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons_Secrecy_KS @ X @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_78_sorted__wrt_Oinduct,axiom,
! [P: ( secrecy_Expression > secrecy_Expression > $o ) > list_S529945141ession > $o,A0: secrecy_Expression > secrecy_Expression > $o,A1: list_S529945141ession] :
( ! [P2: secrecy_Expression > secrecy_Expression > $o] : ( P @ P2 @ nil_Se901661471ession )
=> ( ! [P2: secrecy_Expression > secrecy_Expression > $o,X: secrecy_Expression,Ys3: list_S529945141ession] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons_S112077679ession @ X @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_79_remdups__adj_Ocases,axiom,
! [X3: list_Secrecy_KS] :
( ( X3 != nil_Secrecy_KS )
=> ( ! [X: secrecy_KS] :
( X3
!= ( cons_Secrecy_KS @ X @ nil_Secrecy_KS ) )
=> ~ ! [X: secrecy_KS,Y3: secrecy_KS,Xs2: list_Secrecy_KS] :
( X3
!= ( cons_Secrecy_KS @ X @ ( cons_Secrecy_KS @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_80_remdups__adj_Ocases,axiom,
! [X3: list_S529945141ession] :
( ( X3 != nil_Se901661471ession )
=> ( ! [X: secrecy_Expression] :
( X3
!= ( cons_S112077679ession @ X @ nil_Se901661471ession ) )
=> ~ ! [X: secrecy_Expression,Y3: secrecy_Expression,Xs2: list_S529945141ession] :
( X3
!= ( cons_S112077679ession @ X @ ( cons_S112077679ession @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_81_transpose_Ocases,axiom,
! [X3: list_list_Secrecy_KS] :
( ( X3 != nil_list_Secrecy_KS )
=> ( ! [Xss: list_list_Secrecy_KS] :
( X3
!= ( cons_list_Secrecy_KS @ nil_Secrecy_KS @ Xss ) )
=> ~ ! [X: secrecy_KS,Xs2: list_Secrecy_KS,Xss: list_list_Secrecy_KS] :
( X3
!= ( cons_list_Secrecy_KS @ ( cons_Secrecy_KS @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_82_transpose_Ocases,axiom,
! [X3: list_l425507653ession] :
( ( X3 != nil_li311616943ession )
=> ( ! [Xss: list_l425507653ession] :
( X3
!= ( cons_l533358079ession @ nil_Se901661471ession @ Xss ) )
=> ~ ! [X: secrecy_Expression,Xs2: list_S529945141ession,Xss: list_l425507653ession] :
( X3
!= ( cons_l533358079ession @ ( cons_S112077679ession @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_83_mem__Collect__eq,axiom,
! [A4: secrecy_Keys,P: secrecy_Keys > $o] :
( ( member_Secrecy_Keys @ A4 @ ( collect_Secrecy_Keys @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A4: secrecy_KS,P: secrecy_KS > $o] :
( ( member_Secrecy_KS2 @ A4 @ ( collect_Secrecy_KS @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
! [A4: secrecy_Secrets,P: secrecy_Secrets > $o] :
( ( member2138796287ecrets @ A4 @ ( collec836099393ecrets @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A: set_Secrecy_Keys] :
( ( collect_Secrecy_Keys
@ ^ [X4: secrecy_Keys] : ( member_Secrecy_Keys @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A: set_Secrecy_KS] :
( ( collect_Secrecy_KS
@ ^ [X4: secrecy_KS] : ( member_Secrecy_KS2 @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A: set_Secrecy_Secrets] :
( ( collec836099393ecrets
@ ^ [X4: secrecy_Secrets] : ( member2138796287ecrets @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_89_shuffles_Oinduct,axiom,
! [P: list_Secrecy_KS > list_Secrecy_KS > $o,A0: list_Secrecy_KS,A1: list_Secrecy_KS] :
( ! [X_1: list_Secrecy_KS] : ( P @ nil_Secrecy_KS @ X_1 )
=> ( ! [Xs2: list_Secrecy_KS] : ( P @ Xs2 @ nil_Secrecy_KS )
=> ( ! [X: secrecy_KS,Xs2: list_Secrecy_KS,Y3: secrecy_KS,Ys3: list_Secrecy_KS] :
( ( P @ Xs2 @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) )
=> ( ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_90_shuffles_Oinduct,axiom,
! [P: list_S529945141ession > list_S529945141ession > $o,A0: list_S529945141ession,A1: list_S529945141ession] :
( ! [X_1: list_S529945141ession] : ( P @ nil_Se901661471ession @ X_1 )
=> ( ! [Xs2: list_S529945141ession] : ( P @ Xs2 @ nil_Se901661471ession )
=> ( ! [X: secrecy_Expression,Xs2: list_S529945141ession,Y3: secrecy_Expression,Ys3: list_S529945141ession] :
( ( P @ Xs2 @ ( cons_S112077679ession @ Y3 @ Ys3 ) )
=> ( ( P @ ( cons_S112077679ession @ X @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_S112077679ession @ X @ Xs2 ) @ ( cons_S112077679ession @ Y3 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_91_know_Osimps_I2_J,axiom,
! [A: secrecy_specID,M: secrecy_Secrets] :
( ( knowle945212990e_know @ A @ ( secrecy_sKS @ M ) )
= ( ( ine @ A @ ( secrecy_sE @ M ) )
| ( member_Secrecy_KS2 @ ( secrecy_sKS @ M ) @ ( localSecrets @ A ) ) ) ) ).
% know.simps(2)
thf(fact_92_Expression2KSL_Osimps_I1_J,axiom,
( ( secrec1371625429on2KSL @ nil_Se901661471ession )
= nil_Secrecy_KS ) ).
% Expression2KSL.simps(1)
thf(fact_93_insert__Nil,axiom,
! [X3: secrecy_KS] :
( ( insert_Secrecy_KS @ X3 @ nil_Secrecy_KS )
= ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS ) ) ).
% insert_Nil
thf(fact_94_insert__Nil,axiom,
! [X3: secrecy_Expression] :
( ( insert1173886601ession @ X3 @ nil_Se901661471ession )
= ( cons_S112077679ession @ X3 @ nil_Se901661471ession ) ) ).
% insert_Nil
thf(fact_95_list__ex1__simps_I1_J,axiom,
! [P: secrecy_Expression > $o] :
~ ( list_e1486860828ession @ P @ nil_Se901661471ession ) ).
% list_ex1_simps(1)
thf(fact_96_list__ex1__simps_I1_J,axiom,
! [P: secrecy_KS > $o] :
~ ( list_ex1_Secrecy_KS @ P @ nil_Secrecy_KS ) ).
% list_ex1_simps(1)
thf(fact_97_eoutKnowsECorrect__def,axiom,
( knowle1268625169orrect
= ( ^ [C2: secrecy_specID,E2: secrecy_Expression] :
( ( eout @ C2 @ E2 )
= ( ? [K3: secrecy_Keys] :
( ( E2
= ( secrecy_kE @ K3 ) )
& ( member_Secrecy_Keys @ K3 @ ( specKeys @ C2 ) ) )
| ? [S: secrecy_Secrets] :
( ( E2
= ( secrecy_sE @ S ) )
& ( member2138796287ecrets @ S @ ( specSecrets @ C2 ) ) )
| ( knowle731562421_knows @ C2 @ ( cons_S112077679ession @ E2 @ nil_Se901661471ession ) ) ) ) ) ) ).
% eoutKnowsECorrect_def
thf(fact_98_eoutKnowsECorrect__L1,axiom,
! [C: secrecy_specID,E: secrecy_Expression] :
( ( knowle1268625169orrect @ C @ E )
=> ( ( eout @ C @ E )
=> ( ? [K4: secrecy_Keys] :
( ( E
= ( secrecy_kE @ K4 ) )
& ( member_Secrecy_Keys @ K4 @ ( specKeys @ C ) ) )
| ? [S2: secrecy_Secrets] :
( ( E
= ( secrecy_sE @ S2 ) )
& ( member2138796287ecrets @ S2 @ ( specSecrets @ C ) ) )
| ( knowle731562421_knows @ C @ ( cons_S112077679ession @ E @ nil_Se901661471ession ) ) ) ) ) ).
% eoutKnowsECorrect_L1
thf(fact_99_map__tailrec__rev_Oelims,axiom,
! [X3: secrecy_KS > secrecy_Expression,Xa: list_Secrecy_KS,Xb: list_S529945141ession,Y: list_S529945141ession] :
( ( ( map_ta623472337ession @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Secrecy_KS )
=> ( Y != Xb ) )
=> ~ ! [A3: secrecy_KS,As: list_Secrecy_KS] :
( ( Xa
= ( cons_Secrecy_KS @ A3 @ As ) )
=> ( Y
!= ( map_ta623472337ession @ X3 @ As @ ( cons_S112077679ession @ ( X3 @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_100_map__tailrec__rev_Oelims,axiom,
! [X3: secrecy_Expression > secrecy_Expression,Xa: list_S529945141ession,Xb: list_S529945141ession,Y: list_S529945141ession] :
( ( ( map_ta1624757057ession @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Se901661471ession )
=> ( Y != Xb ) )
=> ~ ! [A3: secrecy_Expression,As: list_S529945141ession] :
( ( Xa
= ( cons_S112077679ession @ A3 @ As ) )
=> ( Y
!= ( map_ta1624757057ession @ X3 @ As @ ( cons_S112077679ession @ ( X3 @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_101_notSpecKeysSecretsExpr__L1,axiom,
! [P: secrecy_specID,A4: secrecy_Expression,L: list_S529945141ession] :
( ( notSpe1586871396tsExpr @ P @ ( cons_S112077679ession @ A4 @ L ) )
=> ( notSpe1586871396tsExpr @ P @ ( cons_S112077679ession @ A4 @ nil_Se901661471ession ) ) ) ).
% notSpecKeysSecretsExpr_L1
thf(fact_102_Expression_Oinject_I2_J,axiom,
! [X23: secrecy_Secrets,Y23: secrecy_Secrets] :
( ( ( secrecy_sE @ X23 )
= ( secrecy_sE @ Y23 ) )
= ( X23 = Y23 ) ) ).
% Expression.inject(2)
thf(fact_103_Expression_Odistinct_I1_J,axiom,
! [X1: secrecy_Keys,X23: secrecy_Secrets] :
( ( secrecy_kE @ X1 )
!= ( secrecy_sE @ X23 ) ) ).
% Expression.distinct(1)
thf(fact_104_notSpecKeysSecretsExpr__L2,axiom,
! [P: secrecy_specID,A4: secrecy_Expression,L: list_S529945141ession] :
( ( notSpe1586871396tsExpr @ P @ ( cons_S112077679ession @ A4 @ L ) )
=> ( notSpe1586871396tsExpr @ P @ L ) ) ).
% notSpecKeysSecretsExpr_L2
thf(fact_105_map__tailrec__rev_Osimps_I2_J,axiom,
! [F2: secrecy_Expression > secrecy_Expression,A4: secrecy_Expression,As2: list_S529945141ession,Bs2: list_S529945141ession] :
( ( map_ta1624757057ession @ F2 @ ( cons_S112077679ession @ A4 @ As2 ) @ Bs2 )
= ( map_ta1624757057ession @ F2 @ As2 @ ( cons_S112077679ession @ ( F2 @ A4 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_106_KS2Expression_Osimps_I2_J,axiom,
! [M: secrecy_Secrets] :
( ( secrec1016742679ession @ ( secrecy_sKS @ M ) )
= ( secrecy_sE @ M ) ) ).
% KS2Expression.simps(2)
thf(fact_107_eoutKnowCorrect__L1s,axiom,
! [C: secrecy_specID,M: secrecy_Secrets] :
( ( knowle233900195orrect @ C @ ( secrecy_sKS @ M ) )
=> ( ( eout @ C @ ( secrecy_sE @ M ) )
=> ( ( member2138796287ecrets @ M @ ( specSecrets @ C ) )
| ( knowle945212990e_know @ C @ ( secrecy_sKS @ M ) ) ) ) ) ).
% eoutKnowCorrect_L1s
thf(fact_108_eout__know__s,axiom,
! [C: secrecy_specID,M: secrecy_Secrets] :
( ( knowle233900195orrect @ C @ ( secrecy_sKS @ M ) )
= ( ( eout @ C @ ( secrecy_sE @ M ) )
= ( ( member2138796287ecrets @ M @ ( specSecrets @ C ) )
| ( knowle945212990e_know @ C @ ( secrecy_sKS @ M ) ) ) ) ) ).
% eout_know_s
thf(fact_109_product__lists_Osimps_I1_J,axiom,
( ( produc1450295286ession @ nil_li311616943ession )
= ( cons_l533358079ession @ nil_Se901661471ession @ nil_li311616943ession ) ) ).
% product_lists.simps(1)
thf(fact_110_product__lists_Osimps_I1_J,axiom,
( ( produc713695622ecy_KS @ nil_list_Secrecy_KS )
= ( cons_list_Secrecy_KS @ nil_Secrecy_KS @ nil_list_Secrecy_KS ) ) ).
% product_lists.simps(1)
thf(fact_111_subseqs_Osimps_I1_J,axiom,
( ( subseq348280080ession @ nil_Se901661471ession )
= ( cons_l533358079ession @ nil_Se901661471ession @ nil_li311616943ession ) ) ).
% subseqs.simps(1)
thf(fact_112_subseqs_Osimps_I1_J,axiom,
( ( subseqs_Secrecy_KS @ nil_Secrecy_KS )
= ( cons_list_Secrecy_KS @ nil_Secrecy_KS @ nil_list_Secrecy_KS ) ) ).
% subseqs.simps(1)
thf(fact_113_DecrEnc,axiom,
! [K12: secrecy_Keys,K22: secrecy_Keys,E3: list_S529945141ession] :
( ( encrDecrKeys @ K12 @ K22 )
=> ( ( decr @ K22 @ ( enc @ K12 @ E3 ) )
= E3 ) ) ).
% DecrEnc
thf(fact_114_ExtSign,axiom,
! [K12: secrecy_Keys,K22: secrecy_Keys,E3: list_S529945141ession] :
( ( encrDecrKeys @ K12 @ K22 )
=> ( ( ext @ K12 @ ( sign @ K22 @ E3 ) )
= E3 ) ) ).
% ExtSign
thf(fact_115_notSpecKeysSecretsExpr__def,axiom,
( notSpe1586871396tsExpr
= ( ^ [P3: secrecy_specID,E2: list_S529945141ession] :
( ! [X4: secrecy_Keys] :
( ( mem_Se1967555118ession @ ( secrecy_kE @ X4 ) @ E2 )
=> ~ ( member_Secrecy_KS2 @ ( secrecy_kKS @ X4 ) @ ( specKeysSecrets @ P3 ) ) )
& ! [Y2: secrecy_Secrets] :
( ( mem_Se1967555118ession @ ( secrecy_sE @ Y2 ) @ E2 )
=> ~ ( member_Secrecy_KS2 @ ( secrecy_sKS @ Y2 ) @ ( specKeysSecrets @ P3 ) ) ) ) ) ) ).
% notSpecKeysSecretsExpr_def
thf(fact_116_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: secrecy_KS > secrecy_KS > $o,X3: secrecy_KS,Xs: list_Secrecy_KS] :
~ ( lexord948960489ecy_KS @ Less @ ( cons_Secrecy_KS @ X3 @ Xs ) @ nil_Secrecy_KS ) ).
% ord.lexordp_eq_simps(3)
thf(fact_117_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: secrecy_Expression > secrecy_Expression > $o,X3: secrecy_Expression,Xs: list_S529945141ession] :
~ ( lexord1123423577ession @ Less @ ( cons_S112077679ession @ X3 @ Xs ) @ nil_Se901661471ession ) ).
% ord.lexordp_eq_simps(3)
thf(fact_118_bind__simps_I1_J,axiom,
! [F2: secrecy_Expression > list_S529945141ession] :
( ( bind_S815874003ession @ nil_Se901661471ession @ F2 )
= nil_Se901661471ession ) ).
% bind_simps(1)
thf(fact_119_bind__simps_I1_J,axiom,
! [F2: secrecy_Expression > list_Secrecy_KS] :
( ( bind_S1823644515ecy_KS @ nil_Se901661471ession @ F2 )
= nil_Secrecy_KS ) ).
% bind_simps(1)
thf(fact_120_bind__simps_I1_J,axiom,
! [F2: secrecy_KS > list_S529945141ession] :
( ( bind_S711229283ession @ nil_Secrecy_KS @ F2 )
= nil_Se901661471ession ) ).
% bind_simps(1)
thf(fact_121_bind__simps_I1_J,axiom,
! [F2: secrecy_KS > list_Secrecy_KS] :
( ( bind_S645462771ecy_KS @ nil_Secrecy_KS @ F2 )
= nil_Secrecy_KS ) ).
% bind_simps(1)
thf(fact_122_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: secrecy_Expression > secrecy_Expression > $o,X3: secrecy_Expression,Xs: list_S529945141ession,Y: secrecy_Expression,Ys2: list_S529945141ession] :
( ( lexord1123423577ession @ Less @ ( cons_S112077679ession @ X3 @ Xs ) @ ( cons_S112077679ession @ Y @ Ys2 ) )
= ( ( Less @ X3 @ Y )
| ( ~ ( Less @ Y @ X3 )
& ( lexord1123423577ession @ Less @ Xs @ Ys2 ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_123_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: secrecy_Expression > secrecy_Expression > $o,Ys2: list_S529945141ession] : ( lexord1123423577ession @ Less @ nil_Se901661471ession @ Ys2 ) ).
% ord.lexordp_eq_simps(1)
thf(fact_124_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: secrecy_KS > secrecy_KS > $o,Ys2: list_Secrecy_KS] : ( lexord948960489ecy_KS @ Less @ nil_Secrecy_KS @ Ys2 ) ).
% ord.lexordp_eq_simps(1)
thf(fact_125_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: secrecy_Expression > secrecy_Expression > $o,Xs: list_S529945141ession] :
( ( lexord1123423577ession @ Less @ Xs @ nil_Se901661471ession )
= ( Xs = nil_Se901661471ession ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_126_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: secrecy_KS > secrecy_KS > $o,Xs: list_Secrecy_KS] :
( ( lexord948960489ecy_KS @ Less @ Xs @ nil_Secrecy_KS )
= ( Xs = nil_Secrecy_KS ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_127_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: secrecy_Expression > secrecy_Expression > $o,X3: secrecy_Expression,Y: secrecy_Expression,Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ~ ( Less @ X3 @ Y )
=> ( ~ ( Less @ Y @ X3 )
=> ( ( lexord1123423577ession @ Less @ Xs @ Ys2 )
=> ( lexord1123423577ession @ Less @ ( cons_S112077679ession @ X3 @ Xs ) @ ( cons_S112077679ession @ Y @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_128_ord_Olexordp__eq_OCons,axiom,
! [Less: secrecy_Expression > secrecy_Expression > $o,X3: secrecy_Expression,Y: secrecy_Expression,Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( Less @ X3 @ Y )
=> ( lexord1123423577ession @ Less @ ( cons_S112077679ession @ X3 @ Xs ) @ ( cons_S112077679ession @ Y @ Ys2 ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_129_ord_Olexordp__eq_ONil,axiom,
! [Less: secrecy_Expression > secrecy_Expression > $o,Ys2: list_S529945141ession] : ( lexord1123423577ession @ Less @ nil_Se901661471ession @ Ys2 ) ).
% ord.lexordp_eq.Nil
thf(fact_130_ord_Olexordp__eq_ONil,axiom,
! [Less: secrecy_KS > secrecy_KS > $o,Ys2: list_Secrecy_KS] : ( lexord948960489ecy_KS @ Less @ nil_Secrecy_KS @ Ys2 ) ).
% ord.lexordp_eq.Nil
thf(fact_131_ord_Olexordp__eq_Oinducts,axiom,
! [Less: secrecy_KS > secrecy_KS > $o,X1: list_Secrecy_KS,X23: list_Secrecy_KS,P: list_Secrecy_KS > list_Secrecy_KS > $o] :
( ( lexord948960489ecy_KS @ Less @ X1 @ X23 )
=> ( ! [X_1: list_Secrecy_KS] : ( P @ nil_Secrecy_KS @ X_1 )
=> ( ! [X: secrecy_KS,Y3: secrecy_KS,Xs2: list_Secrecy_KS,Ys3: list_Secrecy_KS] :
( ( Less @ X @ Y3 )
=> ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) ) )
=> ( ! [X: secrecy_KS,Y3: secrecy_KS,Xs2: list_Secrecy_KS,Ys3: list_Secrecy_KS] :
( ~ ( Less @ X @ Y3 )
=> ( ~ ( Less @ Y3 @ X )
=> ( ( lexord948960489ecy_KS @ Less @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_132_ord_Olexordp__eq_Oinducts,axiom,
! [Less: secrecy_Expression > secrecy_Expression > $o,X1: list_S529945141ession,X23: list_S529945141ession,P: list_S529945141ession > list_S529945141ession > $o] :
( ( lexord1123423577ession @ Less @ X1 @ X23 )
=> ( ! [X_1: list_S529945141ession] : ( P @ nil_Se901661471ession @ X_1 )
=> ( ! [X: secrecy_Expression,Y3: secrecy_Expression,Xs2: list_S529945141ession,Ys3: list_S529945141ession] :
( ( Less @ X @ Y3 )
=> ( P @ ( cons_S112077679ession @ X @ Xs2 ) @ ( cons_S112077679ession @ Y3 @ Ys3 ) ) )
=> ( ! [X: secrecy_Expression,Y3: secrecy_Expression,Xs2: list_S529945141ession,Ys3: list_S529945141ession] :
( ~ ( Less @ X @ Y3 )
=> ( ~ ( Less @ Y3 @ X )
=> ( ( lexord1123423577ession @ Less @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_S112077679ession @ X @ Xs2 ) @ ( cons_S112077679ession @ Y3 @ Ys3 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_133_ord_Olexordp__eq_Osimps,axiom,
( lexord948960489ecy_KS
= ( ^ [Less2: secrecy_KS > secrecy_KS > $o,A12: list_Secrecy_KS,A22: list_Secrecy_KS] :
( ? [Ys: list_Secrecy_KS] :
( ( A12 = nil_Secrecy_KS )
& ( A22 = Ys ) )
| ? [X4: secrecy_KS,Y2: secrecy_KS,Xs3: list_Secrecy_KS,Ys: list_Secrecy_KS] :
( ( A12
= ( cons_Secrecy_KS @ X4 @ Xs3 ) )
& ( A22
= ( cons_Secrecy_KS @ Y2 @ Ys ) )
& ( Less2 @ X4 @ Y2 ) )
| ? [X4: secrecy_KS,Y2: secrecy_KS,Xs3: list_Secrecy_KS,Ys: list_Secrecy_KS] :
( ( A12
= ( cons_Secrecy_KS @ X4 @ Xs3 ) )
& ( A22
= ( cons_Secrecy_KS @ Y2 @ Ys ) )
& ~ ( Less2 @ X4 @ Y2 )
& ~ ( Less2 @ Y2 @ X4 )
& ( lexord948960489ecy_KS @ Less2 @ Xs3 @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_134_ord_Olexordp__eq_Osimps,axiom,
( lexord1123423577ession
= ( ^ [Less2: secrecy_Expression > secrecy_Expression > $o,A12: list_S529945141ession,A22: list_S529945141ession] :
( ? [Ys: list_S529945141ession] :
( ( A12 = nil_Se901661471ession )
& ( A22 = Ys ) )
| ? [X4: secrecy_Expression,Y2: secrecy_Expression,Xs3: list_S529945141ession,Ys: list_S529945141ession] :
( ( A12
= ( cons_S112077679ession @ X4 @ Xs3 ) )
& ( A22
= ( cons_S112077679ession @ Y2 @ Ys ) )
& ( Less2 @ X4 @ Y2 ) )
| ? [X4: secrecy_Expression,Y2: secrecy_Expression,Xs3: list_S529945141ession,Ys: list_S529945141ession] :
( ( A12
= ( cons_S112077679ession @ X4 @ Xs3 ) )
& ( A22
= ( cons_S112077679ession @ Y2 @ Ys ) )
& ~ ( Less2 @ X4 @ Y2 )
& ~ ( Less2 @ Y2 @ X4 )
& ( lexord1123423577ession @ Less2 @ Xs3 @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_135_ord_Olexordp__eq_Ocases,axiom,
! [Less: secrecy_KS > secrecy_KS > $o,A1: list_Secrecy_KS,A2: list_Secrecy_KS] :
( ( lexord948960489ecy_KS @ Less @ A1 @ A2 )
=> ( ( A1 != nil_Secrecy_KS )
=> ( ! [X: secrecy_KS] :
( ? [Xs2: list_Secrecy_KS] :
( A1
= ( cons_Secrecy_KS @ X @ Xs2 ) )
=> ! [Y3: secrecy_KS] :
( ? [Ys3: list_Secrecy_KS] :
( A2
= ( cons_Secrecy_KS @ Y3 @ Ys3 ) )
=> ~ ( Less @ X @ Y3 ) ) )
=> ~ ! [X: secrecy_KS,Y3: secrecy_KS,Xs2: list_Secrecy_KS] :
( ( A1
= ( cons_Secrecy_KS @ X @ Xs2 ) )
=> ! [Ys3: list_Secrecy_KS] :
( ( A2
= ( cons_Secrecy_KS @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X @ Y3 )
=> ( ~ ( Less @ Y3 @ X )
=> ~ ( lexord948960489ecy_KS @ Less @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_136_ord_Olexordp__eq_Ocases,axiom,
! [Less: secrecy_Expression > secrecy_Expression > $o,A1: list_S529945141ession,A2: list_S529945141ession] :
( ( lexord1123423577ession @ Less @ A1 @ A2 )
=> ( ( A1 != nil_Se901661471ession )
=> ( ! [X: secrecy_Expression] :
( ? [Xs2: list_S529945141ession] :
( A1
= ( cons_S112077679ession @ X @ Xs2 ) )
=> ! [Y3: secrecy_Expression] :
( ? [Ys3: list_S529945141ession] :
( A2
= ( cons_S112077679ession @ Y3 @ Ys3 ) )
=> ~ ( Less @ X @ Y3 ) ) )
=> ~ ! [X: secrecy_Expression,Y3: secrecy_Expression,Xs2: list_S529945141ession] :
( ( A1
= ( cons_S112077679ession @ X @ Xs2 ) )
=> ! [Ys3: list_S529945141ession] :
( ( A2
= ( cons_S112077679ession @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X @ Y3 )
=> ( ~ ( Less @ Y3 @ X )
=> ~ ( lexord1123423577ession @ Less @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_137_mem_Osimps_I1_J,axiom,
! [X3: secrecy_KS] :
~ ( mem_Secrecy_KS @ X3 @ nil_Secrecy_KS ) ).
% mem.simps(1)
thf(fact_138_mem_Osimps_I1_J,axiom,
! [X3: secrecy_Expression] :
~ ( mem_Se1967555118ession @ X3 @ nil_Se901661471ession ) ).
% mem.simps(1)
thf(fact_139_mem_Osimps_I2_J,axiom,
! [X3: secrecy_Expression,Y: secrecy_Expression,L: list_S529945141ession] :
( ( mem_Se1967555118ession @ X3 @ ( cons_S112077679ession @ Y @ L ) )
= ( ( X3 = Y )
| ( mem_Se1967555118ession @ X3 @ L ) ) ) ).
% mem.simps(2)
thf(fact_140_listrelp_Oinducts,axiom,
! [R: secrecy_KS > secrecy_KS > $o,X1: list_Secrecy_KS,X23: list_Secrecy_KS,P: list_Secrecy_KS > list_Secrecy_KS > $o] :
( ( listre788412843ecy_KS @ R @ X1 @ X23 )
=> ( ( P @ nil_Secrecy_KS @ nil_Secrecy_KS )
=> ( ! [X: secrecy_KS,Y3: secrecy_KS,Xs2: list_Secrecy_KS,Ys3: list_Secrecy_KS] :
( ( R @ X @ Y3 )
=> ( ( listre788412843ecy_KS @ R @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ).
% listrelp.inducts
thf(fact_141_listrelp_Oinducts,axiom,
! [R: secrecy_KS > secrecy_Expression > $o,X1: list_Secrecy_KS,X23: list_S529945141ession,P: list_Secrecy_KS > list_S529945141ession > $o] :
( ( listre1601064475ession @ R @ X1 @ X23 )
=> ( ( P @ nil_Secrecy_KS @ nil_Se901661471ession )
=> ( ! [X: secrecy_KS,Y3: secrecy_Expression,Xs2: list_Secrecy_KS,Ys3: list_S529945141ession] :
( ( R @ X @ Y3 )
=> ( ( listre1601064475ession @ R @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_Secrecy_KS @ X @ Xs2 ) @ ( cons_S112077679ession @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ).
% listrelp.inducts
thf(fact_142_listrelp_Oinducts,axiom,
! [R: secrecy_Expression > secrecy_KS > $o,X1: list_S529945141ession,X23: list_Secrecy_KS,P: list_S529945141ession > list_Secrecy_KS > $o] :
( ( listre565996059ecy_KS @ R @ X1 @ X23 )
=> ( ( P @ nil_Se901661471ession @ nil_Secrecy_KS )
=> ( ! [X: secrecy_Expression,Y3: secrecy_KS,Xs2: list_S529945141ession,Ys3: list_Secrecy_KS] :
( ( R @ X @ Y3 )
=> ( ( listre565996059ecy_KS @ R @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_S112077679ession @ X @ Xs2 ) @ ( cons_Secrecy_KS @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ).
% listrelp.inducts
thf(fact_143_listrelp_Oinducts,axiom,
! [R: secrecy_Expression > secrecy_Expression > $o,X1: list_S529945141ession,X23: list_S529945141ession,P: list_S529945141ession > list_S529945141ession > $o] :
( ( listre84909707ession @ R @ X1 @ X23 )
=> ( ( P @ nil_Se901661471ession @ nil_Se901661471ession )
=> ( ! [X: secrecy_Expression,Y3: secrecy_Expression,Xs2: list_S529945141ession,Ys3: list_S529945141ession] :
( ( R @ X @ Y3 )
=> ( ( listre84909707ession @ R @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_S112077679ession @ X @ Xs2 ) @ ( cons_S112077679ession @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ).
% listrelp.inducts
thf(fact_144_listrelp_Osimps,axiom,
( listre788412843ecy_KS
= ( ^ [R2: secrecy_KS > secrecy_KS > $o,A12: list_Secrecy_KS,A22: list_Secrecy_KS] :
( ( ( A12 = nil_Secrecy_KS )
& ( A22 = nil_Secrecy_KS ) )
| ? [X4: secrecy_KS,Y2: secrecy_KS,Xs3: list_Secrecy_KS,Ys: list_Secrecy_KS] :
( ( A12
= ( cons_Secrecy_KS @ X4 @ Xs3 ) )
& ( A22
= ( cons_Secrecy_KS @ Y2 @ Ys ) )
& ( R2 @ X4 @ Y2 )
& ( listre788412843ecy_KS @ R2 @ Xs3 @ Ys ) ) ) ) ) ).
% listrelp.simps
thf(fact_145_listrelp_Osimps,axiom,
( listre1601064475ession
= ( ^ [R2: secrecy_KS > secrecy_Expression > $o,A12: list_Secrecy_KS,A22: list_S529945141ession] :
( ( ( A12 = nil_Secrecy_KS )
& ( A22 = nil_Se901661471ession ) )
| ? [X4: secrecy_KS,Y2: secrecy_Expression,Xs3: list_Secrecy_KS,Ys: list_S529945141ession] :
( ( A12
= ( cons_Secrecy_KS @ X4 @ Xs3 ) )
& ( A22
= ( cons_S112077679ession @ Y2 @ Ys ) )
& ( R2 @ X4 @ Y2 )
& ( listre1601064475ession @ R2 @ Xs3 @ Ys ) ) ) ) ) ).
% listrelp.simps
thf(fact_146_listrelp_Osimps,axiom,
( listre565996059ecy_KS
= ( ^ [R2: secrecy_Expression > secrecy_KS > $o,A12: list_S529945141ession,A22: list_Secrecy_KS] :
( ( ( A12 = nil_Se901661471ession )
& ( A22 = nil_Secrecy_KS ) )
| ? [X4: secrecy_Expression,Y2: secrecy_KS,Xs3: list_S529945141ession,Ys: list_Secrecy_KS] :
( ( A12
= ( cons_S112077679ession @ X4 @ Xs3 ) )
& ( A22
= ( cons_Secrecy_KS @ Y2 @ Ys ) )
& ( R2 @ X4 @ Y2 )
& ( listre565996059ecy_KS @ R2 @ Xs3 @ Ys ) ) ) ) ) ).
% listrelp.simps
thf(fact_147_listrelp_Osimps,axiom,
( listre84909707ession
= ( ^ [R2: secrecy_Expression > secrecy_Expression > $o,A12: list_S529945141ession,A22: list_S529945141ession] :
( ( ( A12 = nil_Se901661471ession )
& ( A22 = nil_Se901661471ession ) )
| ? [X4: secrecy_Expression,Y2: secrecy_Expression,Xs3: list_S529945141ession,Ys: list_S529945141ession] :
( ( A12
= ( cons_S112077679ession @ X4 @ Xs3 ) )
& ( A22
= ( cons_S112077679ession @ Y2 @ Ys ) )
& ( R2 @ X4 @ Y2 )
& ( listre84909707ession @ R2 @ Xs3 @ Ys ) ) ) ) ) ).
% listrelp.simps
thf(fact_148_listrelp_Ocases,axiom,
! [R: secrecy_KS > secrecy_KS > $o,A1: list_Secrecy_KS,A2: list_Secrecy_KS] :
( ( listre788412843ecy_KS @ R @ A1 @ A2 )
=> ( ( ( A1 = nil_Secrecy_KS )
=> ( A2 != nil_Secrecy_KS ) )
=> ~ ! [X: secrecy_KS,Y3: secrecy_KS,Xs2: list_Secrecy_KS] :
( ( A1
= ( cons_Secrecy_KS @ X @ Xs2 ) )
=> ! [Ys3: list_Secrecy_KS] :
( ( A2
= ( cons_Secrecy_KS @ Y3 @ Ys3 ) )
=> ( ( R @ X @ Y3 )
=> ~ ( listre788412843ecy_KS @ R @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_149_listrelp_Ocases,axiom,
! [R: secrecy_KS > secrecy_Expression > $o,A1: list_Secrecy_KS,A2: list_S529945141ession] :
( ( listre1601064475ession @ R @ A1 @ A2 )
=> ( ( ( A1 = nil_Secrecy_KS )
=> ( A2 != nil_Se901661471ession ) )
=> ~ ! [X: secrecy_KS,Y3: secrecy_Expression,Xs2: list_Secrecy_KS] :
( ( A1
= ( cons_Secrecy_KS @ X @ Xs2 ) )
=> ! [Ys3: list_S529945141ession] :
( ( A2
= ( cons_S112077679ession @ Y3 @ Ys3 ) )
=> ( ( R @ X @ Y3 )
=> ~ ( listre1601064475ession @ R @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_150_listrelp_Ocases,axiom,
! [R: secrecy_Expression > secrecy_KS > $o,A1: list_S529945141ession,A2: list_Secrecy_KS] :
( ( listre565996059ecy_KS @ R @ A1 @ A2 )
=> ( ( ( A1 = nil_Se901661471ession )
=> ( A2 != nil_Secrecy_KS ) )
=> ~ ! [X: secrecy_Expression,Y3: secrecy_KS,Xs2: list_S529945141ession] :
( ( A1
= ( cons_S112077679ession @ X @ Xs2 ) )
=> ! [Ys3: list_Secrecy_KS] :
( ( A2
= ( cons_Secrecy_KS @ Y3 @ Ys3 ) )
=> ( ( R @ X @ Y3 )
=> ~ ( listre565996059ecy_KS @ R @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_151_listrelp_Ocases,axiom,
! [R: secrecy_Expression > secrecy_Expression > $o,A1: list_S529945141ession,A2: list_S529945141ession] :
( ( listre84909707ession @ R @ A1 @ A2 )
=> ( ( ( A1 = nil_Se901661471ession )
=> ( A2 != nil_Se901661471ession ) )
=> ~ ! [X: secrecy_Expression,Y3: secrecy_Expression,Xs2: list_S529945141ession] :
( ( A1
= ( cons_S112077679ession @ X @ Xs2 ) )
=> ! [Ys3: list_S529945141ession] :
( ( A2
= ( cons_S112077679ession @ Y3 @ Ys3 ) )
=> ( ( R @ X @ Y3 )
=> ~ ( listre84909707ession @ R @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_152_listrelp_OCons,axiom,
! [R: secrecy_Expression > secrecy_Expression > $o,X3: secrecy_Expression,Y: secrecy_Expression,Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( R @ X3 @ Y )
=> ( ( listre84909707ession @ R @ Xs @ Ys2 )
=> ( listre84909707ession @ R @ ( cons_S112077679ession @ X3 @ Xs ) @ ( cons_S112077679ession @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_153_listrelp_ONil,axiom,
! [R: secrecy_Expression > secrecy_Expression > $o] : ( listre84909707ession @ R @ nil_Se901661471ession @ nil_Se901661471ession ) ).
% listrelp.Nil
thf(fact_154_listrelp_ONil,axiom,
! [R: secrecy_Expression > secrecy_KS > $o] : ( listre565996059ecy_KS @ R @ nil_Se901661471ession @ nil_Secrecy_KS ) ).
% listrelp.Nil
thf(fact_155_listrelp_ONil,axiom,
! [R: secrecy_KS > secrecy_Expression > $o] : ( listre1601064475ession @ R @ nil_Secrecy_KS @ nil_Se901661471ession ) ).
% listrelp.Nil
thf(fact_156_listrelp_ONil,axiom,
! [R: secrecy_KS > secrecy_KS > $o] : ( listre788412843ecy_KS @ R @ nil_Secrecy_KS @ nil_Secrecy_KS ) ).
% listrelp.Nil
thf(fact_157_mem__memS__eq,axiom,
mem_Se1967555118ession = memS_S2026082737ession ).
% mem_memS_eq
thf(fact_158_mem__notdisjoint,axiom,
! [X3: secrecy_Expression,L1: list_S529945141ession,L2: list_S529945141ession] :
( ( mem_Se1967555118ession @ X3 @ L1 )
=> ( ( mem_Se1967555118ession @ X3 @ L2 )
=> ~ ( disjoi1945222607ession @ L1 @ L2 ) ) ) ).
% mem_notdisjoint
thf(fact_159_member__rec_I2_J,axiom,
! [Y: secrecy_Expression] :
~ ( member2020590922ession @ nil_Se901661471ession @ Y ) ).
% member_rec(2)
thf(fact_160_member__rec_I2_J,axiom,
! [Y: secrecy_KS] :
~ ( member_Secrecy_KS @ nil_Secrecy_KS @ Y ) ).
% member_rec(2)
thf(fact_161_member__rec_I1_J,axiom,
! [X3: secrecy_Expression,Xs: list_S529945141ession,Y: secrecy_Expression] :
( ( member2020590922ession @ ( cons_S112077679ession @ X3 @ Xs ) @ Y )
= ( ( X3 = Y )
| ( member2020590922ession @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_162_splice_Oelims,axiom,
! [X3: list_Secrecy_KS,Xa: list_Secrecy_KS,Y: list_Secrecy_KS] :
( ( ( splice_Secrecy_KS @ X3 @ Xa )
= Y )
=> ( ( ( X3 = nil_Secrecy_KS )
=> ( Y != Xa ) )
=> ~ ! [X: secrecy_KS,Xs2: list_Secrecy_KS] :
( ( X3
= ( cons_Secrecy_KS @ X @ Xs2 ) )
=> ( Y
!= ( cons_Secrecy_KS @ X @ ( splice_Secrecy_KS @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_163_splice_Oelims,axiom,
! [X3: list_S529945141ession,Xa: list_S529945141ession,Y: list_S529945141ession] :
( ( ( splice1390717708ession @ X3 @ Xa )
= Y )
=> ( ( ( X3 = nil_Se901661471ession )
=> ( Y != Xa ) )
=> ~ ! [X: secrecy_Expression,Xs2: list_S529945141ession] :
( ( X3
= ( cons_S112077679ession @ X @ Xs2 ) )
=> ( Y
!= ( cons_S112077679ession @ X @ ( splice1390717708ession @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_164_append_Oassoc,axiom,
! [A4: list_S529945141ession,B: list_S529945141ession,C3: list_S529945141ession] :
( ( append2089414218ession @ ( append2089414218ession @ A4 @ B ) @ C3 )
= ( append2089414218ession @ A4 @ ( append2089414218ession @ B @ C3 ) ) ) ).
% append.assoc
thf(fact_165_append__assoc,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession,Zs2: list_S529945141ession] :
( ( append2089414218ession @ ( append2089414218ession @ Xs @ Ys2 ) @ Zs2 )
= ( append2089414218ession @ Xs @ ( append2089414218ession @ Ys2 @ Zs2 ) ) ) ).
% append_assoc
thf(fact_166_append__same__eq,axiom,
! [Ys2: list_S529945141ession,Xs: list_S529945141ession,Zs2: list_S529945141ession] :
( ( ( append2089414218ession @ Ys2 @ Xs )
= ( append2089414218ession @ Zs2 @ Xs ) )
= ( Ys2 = Zs2 ) ) ).
% append_same_eq
thf(fact_167_same__append__eq,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession,Zs2: list_S529945141ession] :
( ( ( append2089414218ession @ Xs @ Ys2 )
= ( append2089414218ession @ Xs @ Zs2 ) )
= ( Ys2 = Zs2 ) ) ).
% same_append_eq
thf(fact_168_append_Oright__neutral,axiom,
! [A4: list_S529945141ession] :
( ( append2089414218ession @ A4 @ nil_Se901661471ession )
= A4 ) ).
% append.right_neutral
thf(fact_169_append_Oright__neutral,axiom,
! [A4: list_Secrecy_KS] :
( ( append_Secrecy_KS @ A4 @ nil_Secrecy_KS )
= A4 ) ).
% append.right_neutral
thf(fact_170_append__is__Nil__conv,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( ( append2089414218ession @ Xs @ Ys2 )
= nil_Se901661471ession )
= ( ( Xs = nil_Se901661471ession )
& ( Ys2 = nil_Se901661471ession ) ) ) ).
% append_is_Nil_conv
thf(fact_171_append__is__Nil__conv,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( ( append_Secrecy_KS @ Xs @ Ys2 )
= nil_Secrecy_KS )
= ( ( Xs = nil_Secrecy_KS )
& ( Ys2 = nil_Secrecy_KS ) ) ) ).
% append_is_Nil_conv
thf(fact_172_Nil__is__append__conv,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( nil_Se901661471ession
= ( append2089414218ession @ Xs @ Ys2 ) )
= ( ( Xs = nil_Se901661471ession )
& ( Ys2 = nil_Se901661471ession ) ) ) ).
% Nil_is_append_conv
thf(fact_173_Nil__is__append__conv,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( nil_Secrecy_KS
= ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( ( Xs = nil_Secrecy_KS )
& ( Ys2 = nil_Secrecy_KS ) ) ) ).
% Nil_is_append_conv
thf(fact_174_self__append__conv2,axiom,
! [Ys2: list_S529945141ession,Xs: list_S529945141ession] :
( ( Ys2
= ( append2089414218ession @ Xs @ Ys2 ) )
= ( Xs = nil_Se901661471ession ) ) ).
% self_append_conv2
thf(fact_175_self__append__conv2,axiom,
! [Ys2: list_Secrecy_KS,Xs: list_Secrecy_KS] :
( ( Ys2
= ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( Xs = nil_Secrecy_KS ) ) ).
% self_append_conv2
thf(fact_176_append__self__conv2,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( ( append2089414218ession @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_Se901661471ession ) ) ).
% append_self_conv2
thf(fact_177_append__self__conv2,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( ( append_Secrecy_KS @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_Secrecy_KS ) ) ).
% append_self_conv2
thf(fact_178_self__append__conv,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( Xs
= ( append2089414218ession @ Xs @ Ys2 ) )
= ( Ys2 = nil_Se901661471ession ) ) ).
% self_append_conv
thf(fact_179_self__append__conv,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( Xs
= ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( Ys2 = nil_Secrecy_KS ) ) ).
% self_append_conv
thf(fact_180_append__self__conv,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( ( append2089414218ession @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_Se901661471ession ) ) ).
% append_self_conv
thf(fact_181_append__self__conv,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( ( append_Secrecy_KS @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_Secrecy_KS ) ) ).
% append_self_conv
thf(fact_182_append__Nil2,axiom,
! [Xs: list_S529945141ession] :
( ( append2089414218ession @ Xs @ nil_Se901661471ession )
= Xs ) ).
% append_Nil2
thf(fact_183_append__Nil2,axiom,
! [Xs: list_Secrecy_KS] :
( ( append_Secrecy_KS @ Xs @ nil_Secrecy_KS )
= Xs ) ).
% append_Nil2
thf(fact_184_split__Nil__iff,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( ( splice1390717708ession @ Xs @ Ys2 )
= nil_Se901661471ession )
= ( ( Xs = nil_Se901661471ession )
& ( Ys2 = nil_Se901661471ession ) ) ) ).
% split_Nil_iff
thf(fact_185_split__Nil__iff,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( ( splice_Secrecy_KS @ Xs @ Ys2 )
= nil_Secrecy_KS )
= ( ( Xs = nil_Secrecy_KS )
& ( Ys2 = nil_Secrecy_KS ) ) ) ).
% split_Nil_iff
thf(fact_186_splice__Nil2,axiom,
! [Xs: list_S529945141ession] :
( ( splice1390717708ession @ Xs @ nil_Se901661471ession )
= Xs ) ).
% splice_Nil2
thf(fact_187_splice__Nil2,axiom,
! [Xs: list_Secrecy_KS] :
( ( splice_Secrecy_KS @ Xs @ nil_Secrecy_KS )
= Xs ) ).
% splice_Nil2
thf(fact_188_append1__eq__conv,axiom,
! [Xs: list_Secrecy_KS,X3: secrecy_KS,Ys2: list_Secrecy_KS,Y: secrecy_KS] :
( ( ( append_Secrecy_KS @ Xs @ ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS ) )
= ( append_Secrecy_KS @ Ys2 @ ( cons_Secrecy_KS @ Y @ nil_Secrecy_KS ) ) )
= ( ( Xs = Ys2 )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_189_append1__eq__conv,axiom,
! [Xs: list_S529945141ession,X3: secrecy_Expression,Ys2: list_S529945141ession,Y: secrecy_Expression] :
( ( ( append2089414218ession @ Xs @ ( cons_S112077679ession @ X3 @ nil_Se901661471ession ) )
= ( append2089414218ession @ Ys2 @ ( cons_S112077679ession @ Y @ nil_Se901661471ession ) ) )
= ( ( Xs = Ys2 )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_190_bind__simps_I2_J,axiom,
! [X3: secrecy_Expression,Xs: list_S529945141ession,F2: secrecy_Expression > list_S529945141ession] :
( ( bind_S815874003ession @ ( cons_S112077679ession @ X3 @ Xs ) @ F2 )
= ( append2089414218ession @ ( F2 @ X3 ) @ ( bind_S815874003ession @ Xs @ F2 ) ) ) ).
% bind_simps(2)
thf(fact_191_append__eq__appendI,axiom,
! [Xs: list_S529945141ession,Xs1: list_S529945141ession,Zs2: list_S529945141ession,Ys2: list_S529945141ession,Us: list_S529945141ession] :
( ( ( append2089414218ession @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys2
= ( append2089414218ession @ Xs1 @ Us ) )
=> ( ( append2089414218ession @ Xs @ Ys2 )
= ( append2089414218ession @ Zs2 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_192_append__eq__append__conv2,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession,Zs2: list_S529945141ession,Ts: list_S529945141ession] :
( ( ( append2089414218ession @ Xs @ Ys2 )
= ( append2089414218ession @ Zs2 @ Ts ) )
= ( ? [Us2: list_S529945141ession] :
( ( ( Xs
= ( append2089414218ession @ Zs2 @ Us2 ) )
& ( ( append2089414218ession @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append2089414218ession @ Xs @ Us2 )
= Zs2 )
& ( Ys2
= ( append2089414218ession @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_193_eq__Nil__appendI,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( Xs = Ys2 )
=> ( Xs
= ( append2089414218ession @ nil_Se901661471ession @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_194_eq__Nil__appendI,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_Secrecy_KS @ nil_Secrecy_KS @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_195_append__Nil,axiom,
! [Ys2: list_S529945141ession] :
( ( append2089414218ession @ nil_Se901661471ession @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_196_append__Nil,axiom,
! [Ys2: list_Secrecy_KS] :
( ( append_Secrecy_KS @ nil_Secrecy_KS @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_197_append_Oleft__neutral,axiom,
! [A4: list_S529945141ession] :
( ( append2089414218ession @ nil_Se901661471ession @ A4 )
= A4 ) ).
% append.left_neutral
thf(fact_198_append_Oleft__neutral,axiom,
! [A4: list_Secrecy_KS] :
( ( append_Secrecy_KS @ nil_Secrecy_KS @ A4 )
= A4 ) ).
% append.left_neutral
thf(fact_199_Cons__eq__appendI,axiom,
! [X3: secrecy_Expression,Xs1: list_S529945141ession,Ys2: list_S529945141ession,Xs: list_S529945141ession,Zs2: list_S529945141ession] :
( ( ( cons_S112077679ession @ X3 @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append2089414218ession @ Xs1 @ Zs2 ) )
=> ( ( cons_S112077679ession @ X3 @ Xs )
= ( append2089414218ession @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_200_append__Cons,axiom,
! [X3: secrecy_Expression,Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( append2089414218ession @ ( cons_S112077679ession @ X3 @ Xs ) @ Ys2 )
= ( cons_S112077679ession @ X3 @ ( append2089414218ession @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_201_ord_Olexordp__eq__pref,axiom,
! [Less: secrecy_Expression > secrecy_Expression > $o,U: list_S529945141ession,V: list_S529945141ession] : ( lexord1123423577ession @ Less @ U @ ( append2089414218ession @ U @ V ) ) ).
% ord.lexordp_eq_pref
thf(fact_202_knows2b,axiom,
! [A: secrecy_specID,E: list_S529945141ession,E1: list_S529945141ession] :
( ( knowle731562421_knows @ A @ ( append2089414218ession @ E @ E1 ) )
=> ( knowle731562421_knows @ A @ E ) ) ).
% knows2b
thf(fact_203_knows2a,axiom,
! [A: secrecy_specID,E1: list_S529945141ession,E: list_S529945141ession] :
( ( knowle731562421_knows @ A @ ( append2089414218ession @ E1 @ E ) )
=> ( knowle731562421_knows @ A @ E ) ) ).
% knows2a
thf(fact_204_knows3,axiom,
! [A: secrecy_specID,E1: list_S529945141ession,E22: list_S529945141ession] :
( ( ( knowle731562421_knows @ A @ E1 )
& ( knowle731562421_knows @ A @ E22 ) )
=> ( knowle731562421_knows @ A @ ( append2089414218ession @ E1 @ E22 ) ) ) ).
% knows3
thf(fact_205_splice_Osimps_I2_J,axiom,
! [X3: secrecy_Expression,Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( splice1390717708ession @ ( cons_S112077679ession @ X3 @ Xs ) @ Ys2 )
= ( cons_S112077679ession @ X3 @ ( splice1390717708ession @ Ys2 @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_206_splice_Osimps_I1_J,axiom,
! [Ys2: list_S529945141ession] :
( ( splice1390717708ession @ nil_Se901661471ession @ Ys2 )
= Ys2 ) ).
% splice.simps(1)
thf(fact_207_splice_Osimps_I1_J,axiom,
! [Ys2: list_Secrecy_KS] :
( ( splice_Secrecy_KS @ nil_Secrecy_KS @ Ys2 )
= Ys2 ) ).
% splice.simps(1)
thf(fact_208_rev__nonempty__induct,axiom,
! [Xs: list_Secrecy_KS,P: list_Secrecy_KS > $o] :
( ( Xs != nil_Secrecy_KS )
=> ( ! [X: secrecy_KS] : ( P @ ( cons_Secrecy_KS @ X @ nil_Secrecy_KS ) )
=> ( ! [X: secrecy_KS,Xs2: list_Secrecy_KS] :
( ( Xs2 != nil_Secrecy_KS )
=> ( ( P @ Xs2 )
=> ( P @ ( append_Secrecy_KS @ Xs2 @ ( cons_Secrecy_KS @ X @ nil_Secrecy_KS ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_209_rev__nonempty__induct,axiom,
! [Xs: list_S529945141ession,P: list_S529945141ession > $o] :
( ( Xs != nil_Se901661471ession )
=> ( ! [X: secrecy_Expression] : ( P @ ( cons_S112077679ession @ X @ nil_Se901661471ession ) )
=> ( ! [X: secrecy_Expression,Xs2: list_S529945141ession] :
( ( Xs2 != nil_Se901661471ession )
=> ( ( P @ Xs2 )
=> ( P @ ( append2089414218ession @ Xs2 @ ( cons_S112077679ession @ X @ nil_Se901661471ession ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_210_append__eq__Cons__conv,axiom,
! [Ys2: list_Secrecy_KS,Zs2: list_Secrecy_KS,X3: secrecy_KS,Xs: list_Secrecy_KS] :
( ( ( append_Secrecy_KS @ Ys2 @ Zs2 )
= ( cons_Secrecy_KS @ X3 @ Xs ) )
= ( ( ( Ys2 = nil_Secrecy_KS )
& ( Zs2
= ( cons_Secrecy_KS @ X3 @ Xs ) ) )
| ? [Ys4: list_Secrecy_KS] :
( ( Ys2
= ( cons_Secrecy_KS @ X3 @ Ys4 ) )
& ( ( append_Secrecy_KS @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_211_append__eq__Cons__conv,axiom,
! [Ys2: list_S529945141ession,Zs2: list_S529945141ession,X3: secrecy_Expression,Xs: list_S529945141ession] :
( ( ( append2089414218ession @ Ys2 @ Zs2 )
= ( cons_S112077679ession @ X3 @ Xs ) )
= ( ( ( Ys2 = nil_Se901661471ession )
& ( Zs2
= ( cons_S112077679ession @ X3 @ Xs ) ) )
| ? [Ys4: list_S529945141ession] :
( ( Ys2
= ( cons_S112077679ession @ X3 @ Ys4 ) )
& ( ( append2089414218ession @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_212_Cons__eq__append__conv,axiom,
! [X3: secrecy_KS,Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS,Zs2: list_Secrecy_KS] :
( ( ( cons_Secrecy_KS @ X3 @ Xs )
= ( append_Secrecy_KS @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_Secrecy_KS )
& ( ( cons_Secrecy_KS @ X3 @ Xs )
= Zs2 ) )
| ? [Ys4: list_Secrecy_KS] :
( ( ( cons_Secrecy_KS @ X3 @ Ys4 )
= Ys2 )
& ( Xs
= ( append_Secrecy_KS @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_213_Cons__eq__append__conv,axiom,
! [X3: secrecy_Expression,Xs: list_S529945141ession,Ys2: list_S529945141ession,Zs2: list_S529945141ession] :
( ( ( cons_S112077679ession @ X3 @ Xs )
= ( append2089414218ession @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_Se901661471ession )
& ( ( cons_S112077679ession @ X3 @ Xs )
= Zs2 ) )
| ? [Ys4: list_S529945141ession] :
( ( ( cons_S112077679ession @ X3 @ Ys4 )
= Ys2 )
& ( Xs
= ( append2089414218ession @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_214_rev__exhaust,axiom,
! [Xs: list_Secrecy_KS] :
( ( Xs != nil_Secrecy_KS )
=> ~ ! [Ys3: list_Secrecy_KS,Y3: secrecy_KS] :
( Xs
!= ( append_Secrecy_KS @ Ys3 @ ( cons_Secrecy_KS @ Y3 @ nil_Secrecy_KS ) ) ) ) ).
% rev_exhaust
thf(fact_215_rev__exhaust,axiom,
! [Xs: list_S529945141ession] :
( ( Xs != nil_Se901661471ession )
=> ~ ! [Ys3: list_S529945141ession,Y3: secrecy_Expression] :
( Xs
!= ( append2089414218ession @ Ys3 @ ( cons_S112077679ession @ Y3 @ nil_Se901661471ession ) ) ) ) ).
% rev_exhaust
thf(fact_216_rev__induct,axiom,
! [P: list_Secrecy_KS > $o,Xs: list_Secrecy_KS] :
( ( P @ nil_Secrecy_KS )
=> ( ! [X: secrecy_KS,Xs2: list_Secrecy_KS] :
( ( P @ Xs2 )
=> ( P @ ( append_Secrecy_KS @ Xs2 @ ( cons_Secrecy_KS @ X @ nil_Secrecy_KS ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_217_rev__induct,axiom,
! [P: list_S529945141ession > $o,Xs: list_S529945141ession] :
( ( P @ nil_Se901661471ession )
=> ( ! [X: secrecy_Expression,Xs2: list_S529945141ession] :
( ( P @ Xs2 )
=> ( P @ ( append2089414218ession @ Xs2 @ ( cons_S112077679ession @ X @ nil_Se901661471ession ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_218_maps__simps_I1_J,axiom,
! [F2: secrecy_Expression > list_S529945141ession,X3: secrecy_Expression,Xs: list_S529945141ession] :
( ( maps_S599402637ession @ F2 @ ( cons_S112077679ession @ X3 @ Xs ) )
= ( append2089414218ession @ ( F2 @ X3 ) @ ( maps_S599402637ession @ F2 @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_219_concat__eq__append__conv,axiom,
! [Xss2: list_l425507653ession,Ys2: list_S529945141ession,Zs2: list_S529945141ession] :
( ( ( concat388015140ession @ Xss2 )
= ( append2089414218ession @ Ys2 @ Zs2 ) )
= ( ( ( Xss2 = nil_li311616943ession )
=> ( ( Ys2 = nil_Se901661471ession )
& ( Zs2 = nil_Se901661471ession ) ) )
& ( ( Xss2 != nil_li311616943ession )
=> ? [Xss1: list_l425507653ession,Xs3: list_S529945141ession,Xs4: list_S529945141ession,Xss22: list_l425507653ession] :
( ( Xss2
= ( append1105297242ession @ Xss1 @ ( cons_l533358079ession @ ( append2089414218ession @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys2
= ( append2089414218ession @ ( concat388015140ession @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append2089414218ession @ Xs4 @ ( concat388015140ession @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_220_concat__eq__append__conv,axiom,
! [Xss2: list_list_Secrecy_KS,Ys2: list_Secrecy_KS,Zs2: list_Secrecy_KS] :
( ( ( concat_Secrecy_KS @ Xss2 )
= ( append_Secrecy_KS @ Ys2 @ Zs2 ) )
= ( ( ( Xss2 = nil_list_Secrecy_KS )
=> ( ( Ys2 = nil_Secrecy_KS )
& ( Zs2 = nil_Secrecy_KS ) ) )
& ( ( Xss2 != nil_list_Secrecy_KS )
=> ? [Xss1: list_list_Secrecy_KS,Xs3: list_Secrecy_KS,Xs4: list_Secrecy_KS,Xss22: list_list_Secrecy_KS] :
( ( Xss2
= ( append1376766698ecy_KS @ Xss1 @ ( cons_list_Secrecy_KS @ ( append_Secrecy_KS @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys2
= ( append_Secrecy_KS @ ( concat_Secrecy_KS @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append_Secrecy_KS @ Xs4 @ ( concat_Secrecy_KS @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_221_rotate1_Osimps_I2_J,axiom,
! [X3: secrecy_KS,Xs: list_Secrecy_KS] :
( ( rotate1_Secrecy_KS @ ( cons_Secrecy_KS @ X3 @ Xs ) )
= ( append_Secrecy_KS @ Xs @ ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS ) ) ) ).
% rotate1.simps(2)
thf(fact_222_rotate1_Osimps_I2_J,axiom,
! [X3: secrecy_Expression,Xs: list_S529945141ession] :
( ( rotate1237234450ession @ ( cons_S112077679ession @ X3 @ Xs ) )
= ( append2089414218ession @ Xs @ ( cons_S112077679ession @ X3 @ nil_Se901661471ession ) ) ) ).
% rotate1.simps(2)
thf(fact_223_butlast__snoc,axiom,
! [Xs: list_Secrecy_KS,X3: secrecy_KS] :
( ( butlast_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_224_butlast__snoc,axiom,
! [Xs: list_S529945141ession,X3: secrecy_Expression] :
( ( butlas171474643ession @ ( append2089414218ession @ Xs @ ( cons_S112077679ession @ X3 @ nil_Se901661471ession ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_225_rotate1__is__Nil__conv,axiom,
! [Xs: list_S529945141ession] :
( ( ( rotate1237234450ession @ Xs )
= nil_Se901661471ession )
= ( Xs = nil_Se901661471ession ) ) ).
% rotate1_is_Nil_conv
thf(fact_226_rotate1__is__Nil__conv,axiom,
! [Xs: list_Secrecy_KS] :
( ( ( rotate1_Secrecy_KS @ Xs )
= nil_Secrecy_KS )
= ( Xs = nil_Secrecy_KS ) ) ).
% rotate1_is_Nil_conv
thf(fact_227_concat__append,axiom,
! [Xs: list_l425507653ession,Ys2: list_l425507653ession] :
( ( concat388015140ession @ ( append1105297242ession @ Xs @ Ys2 ) )
= ( append2089414218ession @ ( concat388015140ession @ Xs ) @ ( concat388015140ession @ Ys2 ) ) ) ).
% concat_append
thf(fact_228_butlast_Osimps_I1_J,axiom,
( ( butlas171474643ession @ nil_Se901661471ession )
= nil_Se901661471ession ) ).
% butlast.simps(1)
thf(fact_229_butlast_Osimps_I1_J,axiom,
( ( butlast_Secrecy_KS @ nil_Secrecy_KS )
= nil_Secrecy_KS ) ).
% butlast.simps(1)
thf(fact_230_rotate1_Osimps_I1_J,axiom,
( ( rotate1237234450ession @ nil_Se901661471ession )
= nil_Se901661471ession ) ).
% rotate1.simps(1)
thf(fact_231_rotate1_Osimps_I1_J,axiom,
( ( rotate1_Secrecy_KS @ nil_Secrecy_KS )
= nil_Secrecy_KS ) ).
% rotate1.simps(1)
thf(fact_232_butlast_Osimps_I2_J,axiom,
! [Xs: list_Secrecy_KS,X3: secrecy_KS] :
( ( ( Xs = nil_Secrecy_KS )
=> ( ( butlast_Secrecy_KS @ ( cons_Secrecy_KS @ X3 @ Xs ) )
= nil_Secrecy_KS ) )
& ( ( Xs != nil_Secrecy_KS )
=> ( ( butlast_Secrecy_KS @ ( cons_Secrecy_KS @ X3 @ Xs ) )
= ( cons_Secrecy_KS @ X3 @ ( butlast_Secrecy_KS @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_233_butlast_Osimps_I2_J,axiom,
! [Xs: list_S529945141ession,X3: secrecy_Expression] :
( ( ( Xs = nil_Se901661471ession )
=> ( ( butlas171474643ession @ ( cons_S112077679ession @ X3 @ Xs ) )
= nil_Se901661471ession ) )
& ( ( Xs != nil_Se901661471ession )
=> ( ( butlas171474643ession @ ( cons_S112077679ession @ X3 @ Xs ) )
= ( cons_S112077679ession @ X3 @ ( butlas171474643ession @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_234_concat_Osimps_I1_J,axiom,
( ( concat388015140ession @ nil_li311616943ession )
= nil_Se901661471ession ) ).
% concat.simps(1)
thf(fact_235_concat_Osimps_I1_J,axiom,
( ( concat_Secrecy_KS @ nil_list_Secrecy_KS )
= nil_Secrecy_KS ) ).
% concat.simps(1)
thf(fact_236_butlast__append,axiom,
! [Ys2: list_S529945141ession,Xs: list_S529945141ession] :
( ( ( Ys2 = nil_Se901661471ession )
=> ( ( butlas171474643ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( butlas171474643ession @ Xs ) ) )
& ( ( Ys2 != nil_Se901661471ession )
=> ( ( butlas171474643ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( append2089414218ession @ Xs @ ( butlas171474643ession @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_237_butlast__append,axiom,
! [Ys2: list_Secrecy_KS,Xs: list_Secrecy_KS] :
( ( ( Ys2 = nil_Secrecy_KS )
=> ( ( butlast_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( butlast_Secrecy_KS @ Xs ) ) )
& ( ( Ys2 != nil_Secrecy_KS )
=> ( ( butlast_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( append_Secrecy_KS @ Xs @ ( butlast_Secrecy_KS @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_238_concat_Osimps_I2_J,axiom,
! [X3: list_S529945141ession,Xs: list_l425507653ession] :
( ( concat388015140ession @ ( cons_l533358079ession @ X3 @ Xs ) )
= ( append2089414218ession @ X3 @ ( concat388015140ession @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_239_maps__simps_I2_J,axiom,
! [F2: secrecy_Expression > list_S529945141ession] :
( ( maps_S599402637ession @ F2 @ nil_Se901661471ession )
= nil_Se901661471ession ) ).
% maps_simps(2)
thf(fact_240_maps__simps_I2_J,axiom,
! [F2: secrecy_Expression > list_Secrecy_KS] :
( ( maps_S399472669ecy_KS @ F2 @ nil_Se901661471ession )
= nil_Secrecy_KS ) ).
% maps_simps(2)
thf(fact_241_maps__simps_I2_J,axiom,
! [F2: secrecy_KS > list_S529945141ession] :
( ( maps_S1434541085ession @ F2 @ nil_Secrecy_KS )
= nil_Se901661471ession ) ).
% maps_simps(2)
thf(fact_242_maps__simps_I2_J,axiom,
! [F2: secrecy_KS > list_Secrecy_KS] :
( ( maps_S377605037ecy_KS @ F2 @ nil_Secrecy_KS )
= nil_Secrecy_KS ) ).
% maps_simps(2)
thf(fact_243_concat__eq__appendD,axiom,
! [Xss2: list_l425507653ession,Ys2: list_S529945141ession,Zs2: list_S529945141ession] :
( ( ( concat388015140ession @ Xss2 )
= ( append2089414218ession @ Ys2 @ Zs2 ) )
=> ( ( Xss2 != nil_li311616943ession )
=> ? [Xss12: list_l425507653ession,Xs2: list_S529945141ession,Xs5: list_S529945141ession,Xss23: list_l425507653ession] :
( ( Xss2
= ( append1105297242ession @ Xss12 @ ( cons_l533358079ession @ ( append2089414218ession @ Xs2 @ Xs5 ) @ Xss23 ) ) )
& ( Ys2
= ( append2089414218ession @ ( concat388015140ession @ Xss12 ) @ Xs2 ) )
& ( Zs2
= ( append2089414218ession @ Xs5 @ ( concat388015140ession @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_244_append__butlast__last__id,axiom,
! [Xs: list_Secrecy_KS] :
( ( Xs != nil_Secrecy_KS )
=> ( ( append_Secrecy_KS @ ( butlast_Secrecy_KS @ Xs ) @ ( cons_Secrecy_KS @ ( last_Secrecy_KS @ Xs ) @ nil_Secrecy_KS ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_245_append__butlast__last__id,axiom,
! [Xs: list_S529945141ession] :
( ( Xs != nil_Se901661471ession )
=> ( ( append2089414218ession @ ( butlas171474643ession @ Xs ) @ ( cons_S112077679ession @ ( last_S1532390982ession @ Xs ) @ nil_Se901661471ession ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_246_snoc__eq__iff__butlast,axiom,
! [Xs: list_Secrecy_KS,X3: secrecy_KS,Ys2: list_Secrecy_KS] :
( ( ( append_Secrecy_KS @ Xs @ ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS ) )
= Ys2 )
= ( ( Ys2 != nil_Secrecy_KS )
& ( ( butlast_Secrecy_KS @ Ys2 )
= Xs )
& ( ( last_Secrecy_KS @ Ys2 )
= X3 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_247_snoc__eq__iff__butlast,axiom,
! [Xs: list_S529945141ession,X3: secrecy_Expression,Ys2: list_S529945141ession] :
( ( ( append2089414218ession @ Xs @ ( cons_S112077679ession @ X3 @ nil_Se901661471ession ) )
= Ys2 )
= ( ( Ys2 != nil_Se901661471ession )
& ( ( butlas171474643ession @ Ys2 )
= Xs )
& ( ( last_S1532390982ession @ Ys2 )
= X3 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_248_last__snoc,axiom,
! [Xs: list_Secrecy_KS,X3: secrecy_KS] :
( ( last_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS ) ) )
= X3 ) ).
% last_snoc
thf(fact_249_last__snoc,axiom,
! [Xs: list_S529945141ession,X3: secrecy_Expression] :
( ( last_S1532390982ession @ ( append2089414218ession @ Xs @ ( cons_S112077679ession @ X3 @ nil_Se901661471ession ) ) )
= X3 ) ).
% last_snoc
thf(fact_250_last__appendR,axiom,
! [Ys2: list_S529945141ession,Xs: list_S529945141ession] :
( ( Ys2 != nil_Se901661471ession )
=> ( ( last_S1532390982ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( last_S1532390982ession @ Ys2 ) ) ) ).
% last_appendR
thf(fact_251_last__appendR,axiom,
! [Ys2: list_Secrecy_KS,Xs: list_Secrecy_KS] :
( ( Ys2 != nil_Secrecy_KS )
=> ( ( last_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( last_Secrecy_KS @ Ys2 ) ) ) ).
% last_appendR
thf(fact_252_last__appendL,axiom,
! [Ys2: list_S529945141ession,Xs: list_S529945141ession] :
( ( Ys2 = nil_Se901661471ession )
=> ( ( last_S1532390982ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( last_S1532390982ession @ Xs ) ) ) ).
% last_appendL
thf(fact_253_last__appendL,axiom,
! [Ys2: list_Secrecy_KS,Xs: list_Secrecy_KS] :
( ( Ys2 = nil_Secrecy_KS )
=> ( ( last_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( last_Secrecy_KS @ Xs ) ) ) ).
% last_appendL
thf(fact_254_last__ConsR,axiom,
! [Xs: list_Secrecy_KS,X3: secrecy_KS] :
( ( Xs != nil_Secrecy_KS )
=> ( ( last_Secrecy_KS @ ( cons_Secrecy_KS @ X3 @ Xs ) )
= ( last_Secrecy_KS @ Xs ) ) ) ).
% last_ConsR
thf(fact_255_last__ConsR,axiom,
! [Xs: list_S529945141ession,X3: secrecy_Expression] :
( ( Xs != nil_Se901661471ession )
=> ( ( last_S1532390982ession @ ( cons_S112077679ession @ X3 @ Xs ) )
= ( last_S1532390982ession @ Xs ) ) ) ).
% last_ConsR
thf(fact_256_last__ConsL,axiom,
! [Xs: list_Secrecy_KS,X3: secrecy_KS] :
( ( Xs = nil_Secrecy_KS )
=> ( ( last_Secrecy_KS @ ( cons_Secrecy_KS @ X3 @ Xs ) )
= X3 ) ) ).
% last_ConsL
thf(fact_257_last__ConsL,axiom,
! [Xs: list_S529945141ession,X3: secrecy_Expression] :
( ( Xs = nil_Se901661471ession )
=> ( ( last_S1532390982ession @ ( cons_S112077679ession @ X3 @ Xs ) )
= X3 ) ) ).
% last_ConsL
thf(fact_258_last_Osimps,axiom,
! [Xs: list_Secrecy_KS,X3: secrecy_KS] :
( ( ( Xs = nil_Secrecy_KS )
=> ( ( last_Secrecy_KS @ ( cons_Secrecy_KS @ X3 @ Xs ) )
= X3 ) )
& ( ( Xs != nil_Secrecy_KS )
=> ( ( last_Secrecy_KS @ ( cons_Secrecy_KS @ X3 @ Xs ) )
= ( last_Secrecy_KS @ Xs ) ) ) ) ).
% last.simps
thf(fact_259_last_Osimps,axiom,
! [Xs: list_S529945141ession,X3: secrecy_Expression] :
( ( ( Xs = nil_Se901661471ession )
=> ( ( last_S1532390982ession @ ( cons_S112077679ession @ X3 @ Xs ) )
= X3 ) )
& ( ( Xs != nil_Se901661471ession )
=> ( ( last_S1532390982ession @ ( cons_S112077679ession @ X3 @ Xs ) )
= ( last_S1532390982ession @ Xs ) ) ) ) ).
% last.simps
thf(fact_260_last__append,axiom,
! [Ys2: list_S529945141ession,Xs: list_S529945141ession] :
( ( ( Ys2 = nil_Se901661471ession )
=> ( ( last_S1532390982ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( last_S1532390982ession @ Xs ) ) )
& ( ( Ys2 != nil_Se901661471ession )
=> ( ( last_S1532390982ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( last_S1532390982ession @ Ys2 ) ) ) ) ).
% last_append
thf(fact_261_last__append,axiom,
! [Ys2: list_Secrecy_KS,Xs: list_Secrecy_KS] :
( ( ( Ys2 = nil_Secrecy_KS )
=> ( ( last_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( last_Secrecy_KS @ Xs ) ) )
& ( ( Ys2 != nil_Secrecy_KS )
=> ( ( last_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( last_Secrecy_KS @ Ys2 ) ) ) ) ).
% last_append
thf(fact_262_longest__common__suffix,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
? [Ss: list_S529945141ession,Xs5: list_S529945141ession,Ys5: list_S529945141ession] :
( ( Xs
= ( append2089414218ession @ Xs5 @ Ss ) )
& ( Ys2
= ( append2089414218ession @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_Se901661471ession )
| ( Ys5 = nil_Se901661471ession )
| ( ( last_S1532390982ession @ Xs5 )
!= ( last_S1532390982ession @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_263_longest__common__suffix,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
? [Ss: list_Secrecy_KS,Xs5: list_Secrecy_KS,Ys5: list_Secrecy_KS] :
( ( Xs
= ( append_Secrecy_KS @ Xs5 @ Ss ) )
& ( Ys2
= ( append_Secrecy_KS @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_Secrecy_KS )
| ( Ys5 = nil_Secrecy_KS )
| ( ( last_Secrecy_KS @ Xs5 )
!= ( last_Secrecy_KS @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_264_concat__conv__foldr,axiom,
( concat388015140ession
= ( ^ [Xss3: list_l425507653ession] : ( foldr_1531709715ession @ append2089414218ession @ Xss3 @ nil_Se901661471ession ) ) ) ).
% concat_conv_foldr
thf(fact_265_concat__conv__foldr,axiom,
( concat_Secrecy_KS
= ( ^ [Xss3: list_list_Secrecy_KS] : ( foldr_1711005747ecy_KS @ append_Secrecy_KS @ Xss3 @ nil_Secrecy_KS ) ) ) ).
% concat_conv_foldr
thf(fact_266_SuccD,axiom,
! [K: secrecy_Keys,Kl: set_li849033543y_Keys,Kl2: list_Secrecy_Keys] :
( ( member_Secrecy_Keys @ K @ ( bNF_Gr344893606y_Keys @ Kl @ Kl2 ) )
=> ( member341114536y_Keys @ ( append_Secrecy_Keys @ Kl2 @ ( cons_Secrecy_Keys @ K @ nil_Secrecy_Keys ) ) @ Kl ) ) ).
% SuccD
thf(fact_267_SuccD,axiom,
! [K: secrecy_Secrets,Kl: set_li2007071804ecrets,Kl2: list_Secrecy_Secrets] :
( ( member2138796287ecrets @ K @ ( bNF_Gr957048817ecrets @ Kl @ Kl2 ) )
=> ( member1138497285ecrets @ ( append186372209ecrets @ Kl2 @ ( cons_Secrecy_Secrets @ K @ nil_Secrecy_Secrets ) ) @ Kl ) ) ).
% SuccD
thf(fact_268_SuccD,axiom,
! [K: secrecy_KS,Kl: set_list_Secrecy_KS,Kl2: list_Secrecy_KS] :
( ( member_Secrecy_KS2 @ K @ ( bNF_Gr577175642ecy_KS @ Kl @ Kl2 ) )
=> ( member193705948ecy_KS @ ( append_Secrecy_KS @ Kl2 @ ( cons_Secrecy_KS @ K @ nil_Secrecy_KS ) ) @ Kl ) ) ).
% SuccD
thf(fact_269_SuccD,axiom,
! [K: secrecy_Expression,Kl: set_li185139947ession,Kl2: list_S529945141ession] :
( ( member279582780ession @ K @ ( bNF_Gr979629770ession @ Kl @ Kl2 ) )
=> ( member249235532ession @ ( append2089414218ession @ Kl2 @ ( cons_S112077679ession @ K @ nil_Se901661471ession ) ) @ Kl ) ) ).
% SuccD
thf(fact_270_SuccI,axiom,
! [Kl2: list_Secrecy_Keys,K: secrecy_Keys,Kl: set_li849033543y_Keys] :
( ( member341114536y_Keys @ ( append_Secrecy_Keys @ Kl2 @ ( cons_Secrecy_Keys @ K @ nil_Secrecy_Keys ) ) @ Kl )
=> ( member_Secrecy_Keys @ K @ ( bNF_Gr344893606y_Keys @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_271_SuccI,axiom,
! [Kl2: list_Secrecy_Secrets,K: secrecy_Secrets,Kl: set_li2007071804ecrets] :
( ( member1138497285ecrets @ ( append186372209ecrets @ Kl2 @ ( cons_Secrecy_Secrets @ K @ nil_Secrecy_Secrets ) ) @ Kl )
=> ( member2138796287ecrets @ K @ ( bNF_Gr957048817ecrets @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_272_SuccI,axiom,
! [Kl2: list_Secrecy_KS,K: secrecy_KS,Kl: set_list_Secrecy_KS] :
( ( member193705948ecy_KS @ ( append_Secrecy_KS @ Kl2 @ ( cons_Secrecy_KS @ K @ nil_Secrecy_KS ) ) @ Kl )
=> ( member_Secrecy_KS2 @ K @ ( bNF_Gr577175642ecy_KS @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_273_SuccI,axiom,
! [Kl2: list_S529945141ession,K: secrecy_Expression,Kl: set_li185139947ession] :
( ( member249235532ession @ ( append2089414218ession @ Kl2 @ ( cons_S112077679ession @ K @ nil_Se901661471ession ) ) @ Kl )
=> ( member279582780ession @ K @ ( bNF_Gr979629770ession @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_274_empty__Shift,axiom,
! [Kl: set_li849033543y_Keys,K: secrecy_Keys] :
( ( member341114536y_Keys @ nil_Secrecy_Keys @ Kl )
=> ( ( member_Secrecy_Keys @ K @ ( bNF_Gr344893606y_Keys @ Kl @ nil_Secrecy_Keys ) )
=> ( member341114536y_Keys @ nil_Secrecy_Keys @ ( bNF_Gr414702370y_Keys @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_275_empty__Shift,axiom,
! [Kl: set_li2007071804ecrets,K: secrecy_Secrets] :
( ( member1138497285ecrets @ nil_Secrecy_Secrets @ Kl )
=> ( ( member2138796287ecrets @ K @ ( bNF_Gr957048817ecrets @ Kl @ nil_Secrecy_Secrets ) )
=> ( member1138497285ecrets @ nil_Secrecy_Secrets @ ( bNF_Gr28627573ecrets @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_276_empty__Shift,axiom,
! [Kl: set_li185139947ession,K: secrecy_Expression] :
( ( member249235532ession @ nil_Se901661471ession @ Kl )
=> ( ( member279582780ession @ K @ ( bNF_Gr979629770ession @ Kl @ nil_Se901661471ession ) )
=> ( member249235532ession @ nil_Se901661471ession @ ( bNF_Gr275623750ession @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_277_empty__Shift,axiom,
! [Kl: set_list_Secrecy_KS,K: secrecy_KS] :
( ( member193705948ecy_KS @ nil_Secrecy_KS @ Kl )
=> ( ( member_Secrecy_KS2 @ K @ ( bNF_Gr577175642ecy_KS @ Kl @ nil_Secrecy_KS ) )
=> ( member193705948ecy_KS @ nil_Secrecy_KS @ ( bNF_Gr660123862ecy_KS @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_278_ShiftD,axiom,
! [Kl2: list_S529945141ession,Kl: set_li185139947ession,K: secrecy_Expression] :
( ( member249235532ession @ Kl2 @ ( bNF_Gr275623750ession @ Kl @ K ) )
=> ( member249235532ession @ ( cons_S112077679ession @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_279_Succ__Shift,axiom,
! [Kl: set_li185139947ession,K: secrecy_Expression,Kl2: list_S529945141ession] :
( ( bNF_Gr979629770ession @ ( bNF_Gr275623750ession @ Kl @ K ) @ Kl2 )
= ( bNF_Gr979629770ession @ Kl @ ( cons_S112077679ession @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_280_rev__eq__Cons__iff,axiom,
! [Xs: list_Secrecy_KS,Y: secrecy_KS,Ys2: list_Secrecy_KS] :
( ( ( rev_Secrecy_KS @ Xs )
= ( cons_Secrecy_KS @ Y @ Ys2 ) )
= ( Xs
= ( append_Secrecy_KS @ ( rev_Secrecy_KS @ Ys2 ) @ ( cons_Secrecy_KS @ Y @ nil_Secrecy_KS ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_281_rev__eq__Cons__iff,axiom,
! [Xs: list_S529945141ession,Y: secrecy_Expression,Ys2: list_S529945141ession] :
( ( ( rev_Se1073238207ession @ Xs )
= ( cons_S112077679ession @ Y @ Ys2 ) )
= ( Xs
= ( append2089414218ession @ ( rev_Se1073238207ession @ Ys2 ) @ ( cons_S112077679ession @ Y @ nil_Se901661471ession ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_282_append_Omonoid__axioms,axiom,
monoid929671412ession @ append2089414218ession @ nil_Se901661471ession ).
% append.monoid_axioms
thf(fact_283_append_Omonoid__axioms,axiom,
monoid1528146052ecy_KS @ append_Secrecy_KS @ nil_Secrecy_KS ).
% append.monoid_axioms
thf(fact_284_rev__is__Nil__conv,axiom,
! [Xs: list_S529945141ession] :
( ( ( rev_Se1073238207ession @ Xs )
= nil_Se901661471ession )
= ( Xs = nil_Se901661471ession ) ) ).
% rev_is_Nil_conv
thf(fact_285_rev__is__Nil__conv,axiom,
! [Xs: list_Secrecy_KS] :
( ( ( rev_Secrecy_KS @ Xs )
= nil_Secrecy_KS )
= ( Xs = nil_Secrecy_KS ) ) ).
% rev_is_Nil_conv
thf(fact_286_Nil__is__rev__conv,axiom,
! [Xs: list_S529945141ession] :
( ( nil_Se901661471ession
= ( rev_Se1073238207ession @ Xs ) )
= ( Xs = nil_Se901661471ession ) ) ).
% Nil_is_rev_conv
thf(fact_287_Nil__is__rev__conv,axiom,
! [Xs: list_Secrecy_KS] :
( ( nil_Secrecy_KS
= ( rev_Secrecy_KS @ Xs ) )
= ( Xs = nil_Secrecy_KS ) ) ).
% Nil_is_rev_conv
thf(fact_288_rev__append,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( rev_Se1073238207ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( append2089414218ession @ ( rev_Se1073238207ession @ Ys2 ) @ ( rev_Se1073238207ession @ Xs ) ) ) ).
% rev_append
thf(fact_289_singleton__rev__conv,axiom,
! [X3: secrecy_KS,Xs: list_Secrecy_KS] :
( ( ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS )
= ( rev_Secrecy_KS @ Xs ) )
= ( Xs
= ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS ) ) ) ).
% singleton_rev_conv
thf(fact_290_singleton__rev__conv,axiom,
! [X3: secrecy_Expression,Xs: list_S529945141ession] :
( ( ( cons_S112077679ession @ X3 @ nil_Se901661471ession )
= ( rev_Se1073238207ession @ Xs ) )
= ( Xs
= ( cons_S112077679ession @ X3 @ nil_Se901661471ession ) ) ) ).
% singleton_rev_conv
thf(fact_291_rev__singleton__conv,axiom,
! [Xs: list_Secrecy_KS,X3: secrecy_KS] :
( ( ( rev_Secrecy_KS @ Xs )
= ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS ) )
= ( Xs
= ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS ) ) ) ).
% rev_singleton_conv
thf(fact_292_rev__singleton__conv,axiom,
! [Xs: list_S529945141ession,X3: secrecy_Expression] :
( ( ( rev_Se1073238207ession @ Xs )
= ( cons_S112077679ession @ X3 @ nil_Se901661471ession ) )
= ( Xs
= ( cons_S112077679ession @ X3 @ nil_Se901661471ession ) ) ) ).
% rev_singleton_conv
thf(fact_293_rev_Osimps_I1_J,axiom,
( ( rev_Se1073238207ession @ nil_Se901661471ession )
= nil_Se901661471ession ) ).
% rev.simps(1)
thf(fact_294_rev_Osimps_I1_J,axiom,
( ( rev_Secrecy_KS @ nil_Secrecy_KS )
= nil_Secrecy_KS ) ).
% rev.simps(1)
thf(fact_295_rev_Osimps_I2_J,axiom,
! [X3: secrecy_KS,Xs: list_Secrecy_KS] :
( ( rev_Secrecy_KS @ ( cons_Secrecy_KS @ X3 @ Xs ) )
= ( append_Secrecy_KS @ ( rev_Secrecy_KS @ Xs ) @ ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS ) ) ) ).
% rev.simps(2)
thf(fact_296_rev_Osimps_I2_J,axiom,
! [X3: secrecy_Expression,Xs: list_S529945141ession] :
( ( rev_Se1073238207ession @ ( cons_S112077679ession @ X3 @ Xs ) )
= ( append2089414218ession @ ( rev_Se1073238207ession @ Xs ) @ ( cons_S112077679ession @ X3 @ nil_Se901661471ession ) ) ) ).
% rev.simps(2)
thf(fact_297_hd__rev,axiom,
! [Xs: list_S529945141ession] :
( ( Xs != nil_Se901661471ession )
=> ( ( hd_Sec357328890ession @ ( rev_Se1073238207ession @ Xs ) )
= ( last_S1532390982ession @ Xs ) ) ) ).
% hd_rev
thf(fact_298_hd__rev,axiom,
! [Xs: list_Secrecy_KS] :
( ( Xs != nil_Secrecy_KS )
=> ( ( hd_Secrecy_KS @ ( rev_Secrecy_KS @ Xs ) )
= ( last_Secrecy_KS @ Xs ) ) ) ).
% hd_rev
thf(fact_299_hd__append2,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( Xs != nil_Se901661471ession )
=> ( ( hd_Sec357328890ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( hd_Sec357328890ession @ Xs ) ) ) ).
% hd_append2
thf(fact_300_hd__append2,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( Xs != nil_Secrecy_KS )
=> ( ( hd_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( hd_Secrecy_KS @ Xs ) ) ) ).
% hd_append2
thf(fact_301_hd__concat,axiom,
! [Xs: list_l425507653ession] :
( ( Xs != nil_li311616943ession )
=> ( ( ( hd_lis1623417098ession @ Xs )
!= nil_Se901661471ession )
=> ( ( hd_Sec357328890ession @ ( concat388015140ession @ Xs ) )
= ( hd_Sec357328890ession @ ( hd_lis1623417098ession @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_302_hd__concat,axiom,
! [Xs: list_list_Secrecy_KS] :
( ( Xs != nil_list_Secrecy_KS )
=> ( ( ( hd_list_Secrecy_KS @ Xs )
!= nil_Secrecy_KS )
=> ( ( hd_Secrecy_KS @ ( concat_Secrecy_KS @ Xs ) )
= ( hd_Secrecy_KS @ ( hd_list_Secrecy_KS @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_303_list_Osel_I1_J,axiom,
! [X21: secrecy_Expression,X22: list_S529945141ession] :
( ( hd_Sec357328890ession @ ( cons_S112077679ession @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_304_longest__common__prefix,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
? [Ps: list_S529945141ession,Xs5: list_S529945141ession,Ys5: list_S529945141ession] :
( ( Xs
= ( append2089414218ession @ Ps @ Xs5 ) )
& ( Ys2
= ( append2089414218ession @ Ps @ Ys5 ) )
& ( ( Xs5 = nil_Se901661471ession )
| ( Ys5 = nil_Se901661471ession )
| ( ( hd_Sec357328890ession @ Xs5 )
!= ( hd_Sec357328890ession @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_305_longest__common__prefix,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
? [Ps: list_Secrecy_KS,Xs5: list_Secrecy_KS,Ys5: list_Secrecy_KS] :
( ( Xs
= ( append_Secrecy_KS @ Ps @ Xs5 ) )
& ( Ys2
= ( append_Secrecy_KS @ Ps @ Ys5 ) )
& ( ( Xs5 = nil_Secrecy_KS )
| ( Ys5 = nil_Secrecy_KS )
| ( ( hd_Secrecy_KS @ Xs5 )
!= ( hd_Secrecy_KS @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_306_hd__append,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( ( Xs = nil_Se901661471ession )
=> ( ( hd_Sec357328890ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( hd_Sec357328890ession @ Ys2 ) ) )
& ( ( Xs != nil_Se901661471ession )
=> ( ( hd_Sec357328890ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( hd_Sec357328890ession @ Xs ) ) ) ) ).
% hd_append
thf(fact_307_hd__append,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( ( Xs = nil_Secrecy_KS )
=> ( ( hd_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( hd_Secrecy_KS @ Ys2 ) ) )
& ( ( Xs != nil_Secrecy_KS )
=> ( ( hd_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( hd_Secrecy_KS @ Xs ) ) ) ) ).
% hd_append
thf(fact_308_last__rev,axiom,
! [Xs: list_S529945141ession] :
( ( Xs != nil_Se901661471ession )
=> ( ( last_S1532390982ession @ ( rev_Se1073238207ession @ Xs ) )
= ( hd_Sec357328890ession @ Xs ) ) ) ).
% last_rev
thf(fact_309_last__rev,axiom,
! [Xs: list_Secrecy_KS] :
( ( Xs != nil_Secrecy_KS )
=> ( ( last_Secrecy_KS @ ( rev_Secrecy_KS @ Xs ) )
= ( hd_Secrecy_KS @ Xs ) ) ) ).
% last_rev
thf(fact_310_rotate1__hd__tl,axiom,
! [Xs: list_Secrecy_KS] :
( ( Xs != nil_Secrecy_KS )
=> ( ( rotate1_Secrecy_KS @ Xs )
= ( append_Secrecy_KS @ ( tl_Secrecy_KS @ Xs ) @ ( cons_Secrecy_KS @ ( hd_Secrecy_KS @ Xs ) @ nil_Secrecy_KS ) ) ) ) ).
% rotate1_hd_tl
thf(fact_311_rotate1__hd__tl,axiom,
! [Xs: list_S529945141ession] :
( ( Xs != nil_Se901661471ession )
=> ( ( rotate1237234450ession @ Xs )
= ( append2089414218ession @ ( tl_Sec1905266422ession @ Xs ) @ ( cons_S112077679ession @ ( hd_Sec357328890ession @ Xs ) @ nil_Se901661471ession ) ) ) ) ).
% rotate1_hd_tl
thf(fact_312_distinct__adj__append__iff,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( distin1076152135ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( ( distin1076152135ession @ Xs )
& ( distin1076152135ession @ Ys2 )
& ( ( Xs = nil_Se901661471ession )
| ( Ys2 = nil_Se901661471ession )
| ( ( last_S1532390982ession @ Xs )
!= ( hd_Sec357328890ession @ Ys2 ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_313_distinct__adj__append__iff,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( distin1495927511ecy_KS @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( ( distin1495927511ecy_KS @ Xs )
& ( distin1495927511ecy_KS @ Ys2 )
& ( ( Xs = nil_Secrecy_KS )
| ( Ys2 = nil_Secrecy_KS )
| ( ( last_Secrecy_KS @ Xs )
!= ( hd_Secrecy_KS @ Ys2 ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_314_distinct__adj__Cons__Cons,axiom,
! [X3: secrecy_Expression,Y: secrecy_Expression,Xs: list_S529945141ession] :
( ( distin1076152135ession @ ( cons_S112077679ession @ X3 @ ( cons_S112077679ession @ Y @ Xs ) ) )
= ( ( X3 != Y )
& ( distin1076152135ession @ ( cons_S112077679ession @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_315_tl__append2,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( Xs != nil_Se901661471ession )
=> ( ( tl_Sec1905266422ession @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( append2089414218ession @ ( tl_Sec1905266422ession @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_316_tl__append2,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( Xs != nil_Secrecy_KS )
=> ( ( tl_Secrecy_KS @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( append_Secrecy_KS @ ( tl_Secrecy_KS @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_317_list_Ocollapse,axiom,
! [List: list_Secrecy_KS] :
( ( List != nil_Secrecy_KS )
=> ( ( cons_Secrecy_KS @ ( hd_Secrecy_KS @ List ) @ ( tl_Secrecy_KS @ List ) )
= List ) ) ).
% list.collapse
thf(fact_318_list_Ocollapse,axiom,
! [List: list_S529945141ession] :
( ( List != nil_Se901661471ession )
=> ( ( cons_S112077679ession @ ( hd_Sec357328890ession @ List ) @ ( tl_Sec1905266422ession @ List ) )
= List ) ) ).
% list.collapse
thf(fact_319_hd__Cons__tl,axiom,
! [Xs: list_Secrecy_KS] :
( ( Xs != nil_Secrecy_KS )
=> ( ( cons_Secrecy_KS @ ( hd_Secrecy_KS @ Xs ) @ ( tl_Secrecy_KS @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_320_hd__Cons__tl,axiom,
! [Xs: list_S529945141ession] :
( ( Xs != nil_Se901661471ession )
=> ( ( cons_S112077679ession @ ( hd_Sec357328890ession @ Xs ) @ ( tl_Sec1905266422ession @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_321_list_Oexpand,axiom,
! [List: list_S529945141ession,List2: list_S529945141ession] :
( ( ( List = nil_Se901661471ession )
= ( List2 = nil_Se901661471ession ) )
=> ( ( ( List != nil_Se901661471ession )
=> ( ( List2 != nil_Se901661471ession )
=> ( ( ( hd_Sec357328890ession @ List )
= ( hd_Sec357328890ession @ List2 ) )
& ( ( tl_Sec1905266422ession @ List )
= ( tl_Sec1905266422ession @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_322_list_Oexpand,axiom,
! [List: list_Secrecy_KS,List2: list_Secrecy_KS] :
( ( ( List = nil_Secrecy_KS )
= ( List2 = nil_Secrecy_KS ) )
=> ( ( ( List != nil_Secrecy_KS )
=> ( ( List2 != nil_Secrecy_KS )
=> ( ( ( hd_Secrecy_KS @ List )
= ( hd_Secrecy_KS @ List2 ) )
& ( ( tl_Secrecy_KS @ List )
= ( tl_Secrecy_KS @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_323_distinct__adj__singleton,axiom,
! [X3: secrecy_KS] : ( distin1495927511ecy_KS @ ( cons_Secrecy_KS @ X3 @ nil_Secrecy_KS ) ) ).
% distinct_adj_singleton
thf(fact_324_distinct__adj__singleton,axiom,
! [X3: secrecy_Expression] : ( distin1076152135ession @ ( cons_S112077679ession @ X3 @ nil_Se901661471ession ) ) ).
% distinct_adj_singleton
thf(fact_325_tl__Nil,axiom,
! [Xs: list_Secrecy_KS] :
( ( ( tl_Secrecy_KS @ Xs )
= nil_Secrecy_KS )
= ( ( Xs = nil_Secrecy_KS )
| ? [X4: secrecy_KS] :
( Xs
= ( cons_Secrecy_KS @ X4 @ nil_Secrecy_KS ) ) ) ) ).
% tl_Nil
thf(fact_326_tl__Nil,axiom,
! [Xs: list_S529945141ession] :
( ( ( tl_Sec1905266422ession @ Xs )
= nil_Se901661471ession )
= ( ( Xs = nil_Se901661471ession )
| ? [X4: secrecy_Expression] :
( Xs
= ( cons_S112077679ession @ X4 @ nil_Se901661471ession ) ) ) ) ).
% tl_Nil
thf(fact_327_Nil__tl,axiom,
! [Xs: list_Secrecy_KS] :
( ( nil_Secrecy_KS
= ( tl_Secrecy_KS @ Xs ) )
= ( ( Xs = nil_Secrecy_KS )
| ? [X4: secrecy_KS] :
( Xs
= ( cons_Secrecy_KS @ X4 @ nil_Secrecy_KS ) ) ) ) ).
% Nil_tl
thf(fact_328_Nil__tl,axiom,
! [Xs: list_S529945141ession] :
( ( nil_Se901661471ession
= ( tl_Sec1905266422ession @ Xs ) )
= ( ( Xs = nil_Se901661471ession )
| ? [X4: secrecy_Expression] :
( Xs
= ( cons_S112077679ession @ X4 @ nil_Se901661471ession ) ) ) ) ).
% Nil_tl
thf(fact_329_distinct__adj__ConsD,axiom,
! [X3: secrecy_Expression,Xs: list_S529945141ession] :
( ( distin1076152135ession @ ( cons_S112077679ession @ X3 @ Xs ) )
=> ( distin1076152135ession @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_330_distinct__adj__Nil,axiom,
distin1076152135ession @ nil_Se901661471ession ).
% distinct_adj_Nil
thf(fact_331_distinct__adj__Nil,axiom,
distin1495927511ecy_KS @ nil_Secrecy_KS ).
% distinct_adj_Nil
thf(fact_332_list_Osel_I3_J,axiom,
! [X21: secrecy_Expression,X22: list_S529945141ession] :
( ( tl_Sec1905266422ession @ ( cons_S112077679ession @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_333_list_Osel_I2_J,axiom,
( ( tl_Sec1905266422ession @ nil_Se901661471ession )
= nil_Se901661471ession ) ).
% list.sel(2)
thf(fact_334_list_Osel_I2_J,axiom,
( ( tl_Secrecy_KS @ nil_Secrecy_KS )
= nil_Secrecy_KS ) ).
% list.sel(2)
thf(fact_335_distinct__adj__appendD1,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( distin1076152135ession @ ( append2089414218ession @ Xs @ Ys2 ) )
=> ( distin1076152135ession @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_336_distinct__adj__appendD2,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( distin1076152135ession @ ( append2089414218ession @ Xs @ Ys2 ) )
=> ( distin1076152135ession @ Ys2 ) ) ).
% distinct_adj_appendD2
thf(fact_337_last__tl,axiom,
! [Xs: list_S529945141ession] :
( ( ( Xs = nil_Se901661471ession )
| ( ( tl_Sec1905266422ession @ Xs )
!= nil_Se901661471ession ) )
=> ( ( last_S1532390982ession @ ( tl_Sec1905266422ession @ Xs ) )
= ( last_S1532390982ession @ Xs ) ) ) ).
% last_tl
thf(fact_338_last__tl,axiom,
! [Xs: list_Secrecy_KS] :
( ( ( Xs = nil_Secrecy_KS )
| ( ( tl_Secrecy_KS @ Xs )
!= nil_Secrecy_KS ) )
=> ( ( last_Secrecy_KS @ ( tl_Secrecy_KS @ Xs ) )
= ( last_Secrecy_KS @ Xs ) ) ) ).
% last_tl
thf(fact_339_list_Oexhaust__sel,axiom,
! [List: list_Secrecy_KS] :
( ( List != nil_Secrecy_KS )
=> ( List
= ( cons_Secrecy_KS @ ( hd_Secrecy_KS @ List ) @ ( tl_Secrecy_KS @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_340_list_Oexhaust__sel,axiom,
! [List: list_S529945141ession] :
( ( List != nil_Se901661471ession )
=> ( List
= ( cons_S112077679ession @ ( hd_Sec357328890ession @ List ) @ ( tl_Sec1905266422ession @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_341_distinct__adj__Cons,axiom,
! [X3: secrecy_KS,Xs: list_Secrecy_KS] :
( ( distin1495927511ecy_KS @ ( cons_Secrecy_KS @ X3 @ Xs ) )
= ( ( Xs = nil_Secrecy_KS )
| ( ( X3
!= ( hd_Secrecy_KS @ Xs ) )
& ( distin1495927511ecy_KS @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_342_distinct__adj__Cons,axiom,
! [X3: secrecy_Expression,Xs: list_S529945141ession] :
( ( distin1076152135ession @ ( cons_S112077679ession @ X3 @ Xs ) )
= ( ( Xs = nil_Se901661471ession )
| ( ( X3
!= ( hd_Sec357328890ession @ Xs ) )
& ( distin1076152135ession @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_343_successively__append__iff,axiom,
! [P: secrecy_Expression > secrecy_Expression > $o,Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( succes1388047762ession @ P @ ( append2089414218ession @ Xs @ Ys2 ) )
= ( ( succes1388047762ession @ P @ Xs )
& ( succes1388047762ession @ P @ Ys2 )
& ( ( Xs = nil_Se901661471ession )
| ( Ys2 = nil_Se901661471ession )
| ( P @ ( last_S1532390982ession @ Xs ) @ ( hd_Sec357328890ession @ Ys2 ) ) ) ) ) ).
% successively_append_iff
thf(fact_344_successively__append__iff,axiom,
! [P: secrecy_KS > secrecy_KS > $o,Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( succes590251810ecy_KS @ P @ ( append_Secrecy_KS @ Xs @ Ys2 ) )
= ( ( succes590251810ecy_KS @ P @ Xs )
& ( succes590251810ecy_KS @ P @ Ys2 )
& ( ( Xs = nil_Secrecy_KS )
| ( Ys2 = nil_Secrecy_KS )
| ( P @ ( last_Secrecy_KS @ Xs ) @ ( hd_Secrecy_KS @ Ys2 ) ) ) ) ) ).
% successively_append_iff
thf(fact_345_Cons__in__shuffles__iff,axiom,
! [Z: secrecy_KS,Zs2: list_Secrecy_KS,Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( member193705948ecy_KS @ ( cons_Secrecy_KS @ Z @ Zs2 ) @ ( shuffles_Secrecy_KS @ Xs @ Ys2 ) )
= ( ( ( Xs != nil_Secrecy_KS )
& ( ( hd_Secrecy_KS @ Xs )
= Z )
& ( member193705948ecy_KS @ Zs2 @ ( shuffles_Secrecy_KS @ ( tl_Secrecy_KS @ Xs ) @ Ys2 ) ) )
| ( ( Ys2 != nil_Secrecy_KS )
& ( ( hd_Secrecy_KS @ Ys2 )
= Z )
& ( member193705948ecy_KS @ Zs2 @ ( shuffles_Secrecy_KS @ Xs @ ( tl_Secrecy_KS @ Ys2 ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_346_Cons__in__shuffles__iff,axiom,
! [Z: secrecy_Expression,Zs2: list_S529945141ession,Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( member249235532ession @ ( cons_S112077679ession @ Z @ Zs2 ) @ ( shuffl552897610ession @ Xs @ Ys2 ) )
= ( ( ( Xs != nil_Se901661471ession )
& ( ( hd_Sec357328890ession @ Xs )
= Z )
& ( member249235532ession @ Zs2 @ ( shuffl552897610ession @ ( tl_Sec1905266422ession @ Xs ) @ Ys2 ) ) )
| ( ( Ys2 != nil_Se901661471ession )
& ( ( hd_Sec357328890ession @ Ys2 )
= Z )
& ( member249235532ession @ Zs2 @ ( shuffl552897610ession @ Xs @ ( tl_Sec1905266422ession @ Ys2 ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_347_Nil__in__shuffles,axiom,
! [Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( member249235532ession @ nil_Se901661471ession @ ( shuffl552897610ession @ Xs @ Ys2 ) )
= ( ( Xs = nil_Se901661471ession )
& ( Ys2 = nil_Se901661471ession ) ) ) ).
% Nil_in_shuffles
thf(fact_348_Nil__in__shuffles,axiom,
! [Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( member193705948ecy_KS @ nil_Secrecy_KS @ ( shuffles_Secrecy_KS @ Xs @ Ys2 ) )
= ( ( Xs = nil_Secrecy_KS )
& ( Ys2 = nil_Secrecy_KS ) ) ) ).
% Nil_in_shuffles
thf(fact_349_shufflesE,axiom,
! [Zs2: list_Secrecy_KS,Xs: list_Secrecy_KS,Ys2: list_Secrecy_KS] :
( ( member193705948ecy_KS @ Zs2 @ ( shuffles_Secrecy_KS @ Xs @ Ys2 ) )
=> ( ( ( Zs2 = Xs )
=> ( Ys2 != nil_Secrecy_KS ) )
=> ( ( ( Zs2 = Ys2 )
=> ( Xs != nil_Secrecy_KS ) )
=> ( ! [X: secrecy_KS,Xs5: list_Secrecy_KS] :
( ( Xs
= ( cons_Secrecy_KS @ X @ Xs5 ) )
=> ! [Z2: secrecy_KS,Zs3: list_Secrecy_KS] :
( ( Zs2
= ( cons_Secrecy_KS @ Z2 @ Zs3 ) )
=> ( ( X = Z2 )
=> ~ ( member193705948ecy_KS @ Zs3 @ ( shuffles_Secrecy_KS @ Xs5 @ Ys2 ) ) ) ) )
=> ~ ! [Y3: secrecy_KS,Ys5: list_Secrecy_KS] :
( ( Ys2
= ( cons_Secrecy_KS @ Y3 @ Ys5 ) )
=> ! [Z2: secrecy_KS,Zs3: list_Secrecy_KS] :
( ( Zs2
= ( cons_Secrecy_KS @ Z2 @ Zs3 ) )
=> ( ( Y3 = Z2 )
=> ~ ( member193705948ecy_KS @ Zs3 @ ( shuffles_Secrecy_KS @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_350_shufflesE,axiom,
! [Zs2: list_S529945141ession,Xs: list_S529945141ession,Ys2: list_S529945141ession] :
( ( member249235532ession @ Zs2 @ ( shuffl552897610ession @ Xs @ Ys2 ) )
=> ( ( ( Zs2 = Xs )
=> ( Ys2 != nil_Se901661471ession ) )
=> ( ( ( Zs2 = Ys2 )
=> ( Xs != nil_Se901661471ession ) )
=> ( ! [X: secrecy_Expression,Xs5: list_S529945141ession] :
( ( Xs
= ( cons_S112077679ession @ X @ Xs5 ) )
=> ! [Z2: secrecy_Expression,Zs3: list_S529945141ession] :
( ( Zs2
= ( cons_S112077679ession @ Z2 @ Zs3 ) )
=> ( ( X = Z2 )
=> ~ ( member249235532ession @ Zs3 @ ( shuffl552897610ession @ Xs5 @ Ys2 ) ) ) ) )
=> ~ ! [Y3: secrecy_Expression,Ys5: list_S529945141ession] :
( ( Ys2
= ( cons_S112077679ession @ Y3 @ Ys5 ) )
=> ! [Z2: secrecy_Expression,Zs3: list_S529945141ession] :
( ( Zs2
= ( cons_S112077679ession @ Z2 @ Zs3 ) )
=> ( ( Y3 = Z2 )
=> ~ ( member249235532ession @ Zs3 @ ( shuffl552897610ession @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
% Conjectures (1)
thf(conj_0,conjecture,
knowle945212990e_know @ a @ ( secrecy_kKS @ m ) ).
%------------------------------------------------------------------------------