TPTP Problem File: ITP090^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP090^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer KnowledgeKeysSecrets problem prob_426__3295732_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_426__3295732_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 453 ( 169 unt; 97 typ; 0 def)
% Number of atoms : 1285 ( 448 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3944 ( 293 ~; 51 |; 130 &;2903 @)
% ( 0 <=>; 567 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 143 ( 143 >; 0 *; 0 +; 0 <<)
% Number of symbols : 85 ( 84 usr; 10 con; 0-3 aty)
% Number of variables : 1016 ( 56 ^; 915 !; 45 ?;1016 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:35:43.041
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__List__Olist_It__Secrecy____types__OExpression_J,type,
list_S529945141ession: $tType ).
thf(ty_n_t__Set__Oset_It__Secrecy____types__OSecrets_J,type,
set_Secrecy_Secrets: $tType ).
thf(ty_n_t__Set__Oset_It__Secrecy____types__OspecID_J,type,
set_Secrecy_specID: $tType ).
thf(ty_n_t__Set__Oset_It__Secrecy____types__OchanID_J,type,
set_Secrecy_chanID: $tType ).
thf(ty_n_t__Set__Oset_It__Secrecy____types__OKeys_J,type,
set_Secrecy_Keys: $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__OchanID,type,
secrecy_chanID: $tType ).
thf(ty_n_t__Secrecy____types__OKeys,type,
secrecy_Keys: $tType ).
thf(ty_n_t__Secrecy____types__OKS,type,
secrecy_KS: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (84)
thf(sy_c_CompLocalSecrets_OLocalSecrets,type,
localSecrets: secrecy_specID > set_Secrecy_KS ).
thf(sy_c_KnowledgeKeysSecrets__Mirabelle__vytwmzrumu_OeoutKnowCorrect,type,
knowle233900195orrect: secrecy_specID > secrecy_KS > $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_Lattices_Osup__class_Osup_001t__Set__Oset_It__Secrecy____types__OKS_J,type,
sup_su1301156759ecy_KS: set_Secrecy_KS > set_Secrecy_KS > set_Secrecy_KS ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Secrecy____types__OKeys_J,type,
sup_su552545507y_Keys: set_Secrecy_Keys > set_Secrecy_Keys > set_Secrecy_Keys ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Secrecy____types__OSecrets_J,type,
sup_su1029080330ecrets: set_Secrecy_Secrets > set_Secrecy_Secrets > set_Secrecy_Secrets ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Secrecy____types__OchanID_J,type,
sup_su370684444chanID: set_Secrecy_chanID > set_Secrecy_chanID > set_Secrecy_chanID ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Secrecy____types__OspecID_J,type,
sup_su1712599557specID: set_Secrecy_specID > set_Secrecy_specID > set_Secrecy_specID ).
thf(sy_c_ListExtras_Omem_001t__Secrecy____types__OExpression,type,
mem_Se1967555118ession: secrecy_Expression > list_S529945141ession > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Secrecy____types__OKS_M_Eo_J,type,
bot_bot_Secrecy_KS_o: secrecy_KS > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Secrecy____types__OKeys_M_Eo_J,type,
bot_bo2072920570Keys_o: secrecy_Keys > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Secrecy____types__OSecrets_M_Eo_J,type,
bot_bo1909946299rets_o: secrecy_Secrets > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Secrecy____types__OchanID_M_Eo_J,type,
bot_bo1012329985anID_o: secrecy_chanID > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Secrecy____types__OspecID_M_Eo_J,type,
bot_bo1767780696ecID_o: secrecy_specID > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Secrecy____types__OKS_J,type,
bot_bo1943466239ecy_KS: set_Secrecy_KS ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Secrecy____types__OKeys_J,type,
bot_bo701486667y_Keys: set_Secrecy_Keys ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Secrecy____types__OSecrets_J,type,
bot_bo1124690338ecrets: set_Secrecy_Secrets ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Secrecy____types__OchanID_J,type,
bot_bo443197316chanID: set_Secrecy_chanID ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Secrecy____types__OspecID_J,type,
bot_bo1785112429specID: set_Secrecy_specID ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Secrecy____types__OchanID_J,type,
ord_le1802272080chanID: set_Secrecy_chanID > set_Secrecy_chanID > $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_OcorrectCompositionKS,type,
correctCompositionKS: secrecy_specID > $o ).
thf(sy_c_Secrecy_Oeout,type,
eout: secrecy_specID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_OeoutM,type,
eoutM: secrecy_specID > set_Secrecy_chanID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_OexprChannel,type,
exprChannel: secrecy_chanID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_Oine,type,
ine: secrecy_specID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_OineM,type,
ineM: secrecy_specID > set_Secrecy_chanID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_Oine__exprChannelSet,type,
ine_exprChannelSet: secrecy_specID > set_Secrecy_chanID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_Oine__exprChannelSingle,type,
ine_ex1303305700Single: secrecy_specID > secrecy_chanID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_OnotSpecKeysSecretsExpr,type,
notSpe1586871396tsExpr: secrecy_specID > list_S529945141ession > $o ).
thf(sy_c_Secrecy_Oout__exprChannelSet,type,
out_exprChannelSet: secrecy_specID > set_Secrecy_chanID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_Oout__exprChannelSingle,type,
out_ex985543062Single: secrecy_specID > secrecy_chanID > secrecy_Expression > $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_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_Set_OCollect_001t__Secrecy____types__OchanID,type,
collec625564175chanID: ( secrecy_chanID > $o ) > set_Secrecy_chanID ).
thf(sy_c_Set_OCollect_001t__Secrecy____types__OspecID,type,
collec1545797240specID: ( secrecy_specID > $o ) > set_Secrecy_specID ).
thf(sy_c_Set_Oinsert_001t__Secrecy____types__OKS,type,
insert_Secrecy_KS: secrecy_KS > set_Secrecy_KS > set_Secrecy_KS ).
thf(sy_c_Set_Oinsert_001t__Secrecy____types__OKeys,type,
insert_Secrecy_Keys: secrecy_Keys > set_Secrecy_Keys > set_Secrecy_Keys ).
thf(sy_c_Set_Oinsert_001t__Secrecy____types__OSecrets,type,
insert565337510ecrets: secrecy_Secrets > set_Secrecy_Secrets > set_Secrecy_Secrets ).
thf(sy_c_Set_Oinsert_001t__Secrecy____types__OchanID,type,
insert751661162chanID: secrecy_chanID > set_Secrecy_chanID > set_Secrecy_chanID ).
thf(sy_c_Set_Oinsert_001t__Secrecy____types__OspecID,type,
insert1671894227specID: secrecy_specID > set_Secrecy_specID > set_Secrecy_specID ).
thf(sy_c_Set_Ois__empty_001t__Secrecy____types__OchanID,type,
is_emp928984296chanID: set_Secrecy_chanID > $o ).
thf(sy_c_Set_Ois__empty_001t__Secrecy____types__OspecID,type,
is_emp1849217361specID: set_Secrecy_specID > $o ).
thf(sy_c_Set_Ois__singleton_001t__Secrecy____types__OKS,type,
is_sin1442426433ecy_KS: set_Secrecy_KS > $o ).
thf(sy_c_Set_Ois__singleton_001t__Secrecy____types__OKeys,type,
is_sin1096471309y_Keys: set_Secrecy_Keys > $o ).
thf(sy_c_Set_Ois__singleton_001t__Secrecy____types__OSecrets,type,
is_sin1002224330ecrets: set_Secrecy_Secrets > $o ).
thf(sy_c_Set_Ois__singleton_001t__Secrecy____types__OchanID,type,
is_sin581274182chanID: set_Secrecy_chanID > $o ).
thf(sy_c_Set_Ois__singleton_001t__Secrecy____types__OspecID,type,
is_sin1501507247specID: set_Secrecy_specID > $o ).
thf(sy_c_Set_Othe__elem_001t__Secrecy____types__OchanID,type,
the_el2054381829chanID: set_Secrecy_chanID > secrecy_chanID ).
thf(sy_c_Set_Othe__elem_001t__Secrecy____types__OspecID,type,
the_el827131246specID: set_Secrecy_specID > secrecy_specID ).
thf(sy_c_inout_OcorrectCompositionIn,type,
correctCompositionIn: secrecy_specID > $o ).
thf(sy_c_inout_OcorrectCompositionLoc,type,
correc965049635ionLoc: secrecy_specID > $o ).
thf(sy_c_inout_OcorrectCompositionOut,type,
correc990227761ionOut: secrecy_specID > $o ).
thf(sy_c_inout_OinStream,type,
inStream: secrecy_specID > set_Secrecy_chanID > $o ).
thf(sy_c_inout_Oins,type,
ins: secrecy_specID > set_Secrecy_chanID ).
thf(sy_c_inout_Oloc,type,
loc: secrecy_specID > set_Secrecy_chanID ).
thf(sy_c_inout_OlocStream,type,
locStream: secrecy_specID > set_Secrecy_chanID > $o ).
thf(sy_c_inout_Oout,type,
out: secrecy_specID > set_Secrecy_chanID ).
thf(sy_c_inout_OoutStream,type,
outStream: secrecy_specID > set_Secrecy_chanID > $o ).
thf(sy_c_inout_Osubcomponents,type,
subcomponents: secrecy_specID > set_Secrecy_specID ).
thf(sy_c_member_001t__Secrecy____types__OKS,type,
member_Secrecy_KS: 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_c_member_001t__Secrecy____types__OchanID,type,
member1967978193chanID: secrecy_chanID > set_Secrecy_chanID > $o ).
thf(sy_c_member_001t__Secrecy____types__OspecID,type,
member740727610specID: secrecy_specID > set_Secrecy_specID > $o ).
thf(sy_v_P,type,
p: secrecy_specID ).
thf(sy_v_PQ,type,
pq: secrecy_specID ).
thf(sy_v_Q,type,
q: secrecy_specID ).
thf(sy_v_key____,type,
key: secrecy_Keys ).
thf(sy_v_m,type,
m: secrecy_KS ).
% Relevant facts (355)
thf(fact_0_not__knowQm,axiom,
~ ( knowle945212990e_know @ q @ m ) ).
% not_knowQm
thf(fact_1_a1,axiom,
( m
= ( secrecy_kKS @ key ) ) ).
% a1
thf(fact_2_knowPQ,axiom,
knowle945212990e_know @ pq @ m ).
% knowPQ
thf(fact_3_sg2,axiom,
member_Secrecy_KS @ ( secrecy_kKS @ key ) @ ( localSecrets @ pq ) ).
% sg2
thf(fact_4_KS_Oinject_I1_J,axiom,
! [X1: secrecy_Keys,Y1: secrecy_Keys] :
( ( ( secrecy_kKS @ X1 )
= ( secrecy_kKS @ Y1 ) )
= ( X1 = Y1 ) ) ).
% KS.inject(1)
thf(fact_5_sg3a,axiom,
~ ( ine @ q @ ( secrecy_kE @ key ) ) ).
% sg3a
thf(fact_6_cCompLoc,axiom,
correc965049635ionLoc @ pq ).
% cCompLoc
thf(fact_7_a12,axiom,
~ ( ine @ pq @ ( secrecy_kE @ key ) ) ).
% a12
thf(fact_8_know_Osimps_I1_J,axiom,
! [A: secrecy_specID,M: secrecy_Keys] :
( ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) )
= ( ( ine @ A @ ( secrecy_kE @ M ) )
| ( member_Secrecy_KS @ ( secrecy_kKS @ M ) @ ( localSecrets @ A ) ) ) ) ).
% know.simps(1)
thf(fact_9_subPQ,axiom,
( ( subcomponents @ pq )
= ( insert1671894227specID @ p @ ( insert1671894227specID @ q @ bot_bo1785112429specID ) ) ) ).
% subPQ
thf(fact_10_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_11_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_12_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_13_Expression_Oinject_I1_J,axiom,
! [X1: secrecy_Keys,Y1: secrecy_Keys] :
( ( ( secrecy_kE @ X1 )
= ( secrecy_kE @ Y1 ) )
= ( X1 = Y1 ) ) ).
% Expression.inject(1)
thf(fact_14_cCompI,axiom,
correctCompositionIn @ pq ).
% cCompI
thf(fact_15_not__know__k__not__ine,axiom,
! [A: secrecy_specID,M: secrecy_Keys] :
( ~ ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) )
=> ~ ( ine @ A @ ( secrecy_kE @ M ) ) ) ).
% not_know_k_not_ine
thf(fact_16_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_17_LocalSecretsComposition__ine1__k,axiom,
! [K: secrecy_Keys,PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID] :
( ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ PQ ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc965049635ionLoc @ PQ )
=> ( ~ ( ine @ Q @ ( secrecy_kE @ K ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ Q ) )
=> ( ine @ P @ ( secrecy_kE @ K ) ) ) ) ) ) ) ) ).
% LocalSecretsComposition_ine1_k
thf(fact_18_LocalSecretsComposition__ine2__k,axiom,
! [K: secrecy_Keys,PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID] :
( ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ PQ ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc965049635ionLoc @ PQ )
=> ( ~ ( ine @ P @ ( secrecy_kE @ K ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ Q ) )
=> ( ine @ Q @ ( secrecy_kE @ K ) ) ) ) ) ) ) ) ).
% LocalSecretsComposition_ine2_k
thf(fact_19_LocalSecretsComposition__neg1__k,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Keys: secrecy_Keys] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc965049635ionLoc @ PQ )
=> ( ~ ( ine @ P @ ( secrecy_kE @ Keys ) )
=> ( ~ ( ine @ Q @ ( secrecy_kE @ Keys ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Keys ) @ ( localSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Keys ) @ ( localSecrets @ Q ) )
=> ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Keys ) @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ).
% LocalSecretsComposition_neg1_k
thf(fact_20_singletonI,axiom,
! [A2: secrecy_KS] : ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ A2 @ bot_bo1943466239ecy_KS ) ) ).
% singletonI
thf(fact_21_singletonI,axiom,
! [A2: secrecy_Keys] : ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ A2 @ bot_bo701486667y_Keys ) ) ).
% singletonI
thf(fact_22_singletonI,axiom,
! [A2: secrecy_Secrets] : ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ A2 @ bot_bo1124690338ecrets ) ) ).
% singletonI
thf(fact_23_singletonI,axiom,
! [A2: secrecy_specID] : ( member740727610specID @ A2 @ ( insert1671894227specID @ A2 @ bot_bo1785112429specID ) ) ).
% singletonI
thf(fact_24_singletonI,axiom,
! [A2: secrecy_chanID] : ( member1967978193chanID @ A2 @ ( insert751661162chanID @ A2 @ bot_bo443197316chanID ) ) ).
% singletonI
thf(fact_25_LocalSecretsComposition1,axiom,
! [Ls: secrecy_KS,P: secrecy_specID,PQ: secrecy_specID,Q: secrecy_specID] :
( ( member_Secrecy_KS @ Ls @ ( localSecrets @ P ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( member_Secrecy_KS @ Ls @ ( localSecrets @ PQ ) ) ) ) ).
% LocalSecretsComposition1
thf(fact_26_know__composition__neg__ine__k,axiom,
! [P: secrecy_specID,Key: secrecy_Keys,Q: secrecy_specID,PQ: secrecy_specID] :
( ~ ( knowle945212990e_know @ P @ ( secrecy_kKS @ Key ) )
=> ( ~ ( knowle945212990e_know @ Q @ ( secrecy_kKS @ Key ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ~ ( ine @ PQ @ ( secrecy_kE @ Key ) ) ) ) ) ) ).
% know_composition_neg_ine_k
thf(fact_27_insertCI,axiom,
! [A2: secrecy_KS,B: set_Secrecy_KS,B2: secrecy_KS] :
( ( ~ ( member_Secrecy_KS @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ B2 @ B ) ) ) ).
% insertCI
thf(fact_28_insertCI,axiom,
! [A2: secrecy_chanID,B: set_Secrecy_chanID,B2: secrecy_chanID] :
( ( ~ ( member1967978193chanID @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member1967978193chanID @ A2 @ ( insert751661162chanID @ B2 @ B ) ) ) ).
% insertCI
thf(fact_29_insertCI,axiom,
! [A2: secrecy_specID,B: set_Secrecy_specID,B2: secrecy_specID] :
( ( ~ ( member740727610specID @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member740727610specID @ A2 @ ( insert1671894227specID @ B2 @ B ) ) ) ).
% insertCI
thf(fact_30_insertCI,axiom,
! [A2: secrecy_Keys,B: set_Secrecy_Keys,B2: secrecy_Keys] :
( ( ~ ( member_Secrecy_Keys @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ B2 @ B ) ) ) ).
% insertCI
thf(fact_31_insertCI,axiom,
! [A2: secrecy_Secrets,B: set_Secrecy_Secrets,B2: secrecy_Secrets] :
( ( ~ ( member2138796287ecrets @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ B2 @ B ) ) ) ).
% insertCI
thf(fact_32_insert__iff,axiom,
! [A2: secrecy_KS,B2: secrecy_KS,A: set_Secrecy_KS] :
( ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_Secrecy_KS @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_33_insert__iff,axiom,
! [A2: secrecy_chanID,B2: secrecy_chanID,A: set_Secrecy_chanID] :
( ( member1967978193chanID @ A2 @ ( insert751661162chanID @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member1967978193chanID @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_34_insert__iff,axiom,
! [A2: secrecy_specID,B2: secrecy_specID,A: set_Secrecy_specID] :
( ( member740727610specID @ A2 @ ( insert1671894227specID @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member740727610specID @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_35_insert__iff,axiom,
! [A2: secrecy_Keys,B2: secrecy_Keys,A: set_Secrecy_Keys] :
( ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_Secrecy_Keys @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_36_insert__iff,axiom,
! [A2: secrecy_Secrets,B2: secrecy_Secrets,A: set_Secrecy_Secrets] :
( ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member2138796287ecrets @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_37_insert__absorb2,axiom,
! [X: secrecy_specID,A: set_Secrecy_specID] :
( ( insert1671894227specID @ X @ ( insert1671894227specID @ X @ A ) )
= ( insert1671894227specID @ X @ A ) ) ).
% insert_absorb2
thf(fact_38_insert__absorb2,axiom,
! [X: secrecy_chanID,A: set_Secrecy_chanID] :
( ( insert751661162chanID @ X @ ( insert751661162chanID @ X @ A ) )
= ( insert751661162chanID @ X @ A ) ) ).
% insert_absorb2
thf(fact_39_empty__iff,axiom,
! [C: secrecy_KS] :
~ ( member_Secrecy_KS @ C @ bot_bo1943466239ecy_KS ) ).
% empty_iff
thf(fact_40_empty__iff,axiom,
! [C: secrecy_Keys] :
~ ( member_Secrecy_Keys @ C @ bot_bo701486667y_Keys ) ).
% empty_iff
thf(fact_41_empty__iff,axiom,
! [C: secrecy_Secrets] :
~ ( member2138796287ecrets @ C @ bot_bo1124690338ecrets ) ).
% empty_iff
thf(fact_42_empty__iff,axiom,
! [C: secrecy_specID] :
~ ( member740727610specID @ C @ bot_bo1785112429specID ) ).
% empty_iff
thf(fact_43_empty__iff,axiom,
! [C: secrecy_chanID] :
~ ( member1967978193chanID @ C @ bot_bo443197316chanID ) ).
% empty_iff
thf(fact_44_empty__Collect__eq,axiom,
! [P: secrecy_specID > $o] :
( ( bot_bo1785112429specID
= ( collec1545797240specID @ P ) )
= ( ! [X2: secrecy_specID] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_45_empty__Collect__eq,axiom,
! [P: secrecy_chanID > $o] :
( ( bot_bo443197316chanID
= ( collec625564175chanID @ P ) )
= ( ! [X2: secrecy_chanID] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_46_Collect__empty__eq,axiom,
! [P: secrecy_specID > $o] :
( ( ( collec1545797240specID @ P )
= bot_bo1785112429specID )
= ( ! [X2: secrecy_specID] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_47_Collect__empty__eq,axiom,
! [P: secrecy_chanID > $o] :
( ( ( collec625564175chanID @ P )
= bot_bo443197316chanID )
= ( ! [X2: secrecy_chanID] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_48_all__not__in__conv,axiom,
! [A: set_Secrecy_KS] :
( ( ! [X2: secrecy_KS] :
~ ( member_Secrecy_KS @ X2 @ A ) )
= ( A = bot_bo1943466239ecy_KS ) ) ).
% all_not_in_conv
thf(fact_49_all__not__in__conv,axiom,
! [A: set_Secrecy_Keys] :
( ( ! [X2: secrecy_Keys] :
~ ( member_Secrecy_Keys @ X2 @ A ) )
= ( A = bot_bo701486667y_Keys ) ) ).
% all_not_in_conv
thf(fact_50_all__not__in__conv,axiom,
! [A: set_Secrecy_Secrets] :
( ( ! [X2: secrecy_Secrets] :
~ ( member2138796287ecrets @ X2 @ A ) )
= ( A = bot_bo1124690338ecrets ) ) ).
% all_not_in_conv
thf(fact_51_all__not__in__conv,axiom,
! [A: set_Secrecy_specID] :
( ( ! [X2: secrecy_specID] :
~ ( member740727610specID @ X2 @ A ) )
= ( A = bot_bo1785112429specID ) ) ).
% all_not_in_conv
thf(fact_52_all__not__in__conv,axiom,
! [A: set_Secrecy_chanID] :
( ( ! [X2: secrecy_chanID] :
~ ( member1967978193chanID @ X2 @ A ) )
= ( A = bot_bo443197316chanID ) ) ).
% all_not_in_conv
thf(fact_53_ex__in__conv,axiom,
! [A: set_Secrecy_KS] :
( ( ? [X2: secrecy_KS] : ( member_Secrecy_KS @ X2 @ A ) )
= ( A != bot_bo1943466239ecy_KS ) ) ).
% ex_in_conv
thf(fact_54_ex__in__conv,axiom,
! [A: set_Secrecy_Keys] :
( ( ? [X2: secrecy_Keys] : ( member_Secrecy_Keys @ X2 @ A ) )
= ( A != bot_bo701486667y_Keys ) ) ).
% ex_in_conv
thf(fact_55_ex__in__conv,axiom,
! [A: set_Secrecy_Secrets] :
( ( ? [X2: secrecy_Secrets] : ( member2138796287ecrets @ X2 @ A ) )
= ( A != bot_bo1124690338ecrets ) ) ).
% ex_in_conv
thf(fact_56_ex__in__conv,axiom,
! [A: set_Secrecy_specID] :
( ( ? [X2: secrecy_specID] : ( member740727610specID @ X2 @ A ) )
= ( A != bot_bo1785112429specID ) ) ).
% ex_in_conv
thf(fact_57_ex__in__conv,axiom,
! [A: set_Secrecy_chanID] :
( ( ? [X2: secrecy_chanID] : ( member1967978193chanID @ X2 @ A ) )
= ( A != bot_bo443197316chanID ) ) ).
% ex_in_conv
thf(fact_58_equals0I,axiom,
! [A: set_Secrecy_KS] :
( ! [Y: secrecy_KS] :
~ ( member_Secrecy_KS @ Y @ A )
=> ( A = bot_bo1943466239ecy_KS ) ) ).
% equals0I
thf(fact_59_equals0I,axiom,
! [A: set_Secrecy_Keys] :
( ! [Y: secrecy_Keys] :
~ ( member_Secrecy_Keys @ Y @ A )
=> ( A = bot_bo701486667y_Keys ) ) ).
% equals0I
thf(fact_60_equals0I,axiom,
! [A: set_Secrecy_Secrets] :
( ! [Y: secrecy_Secrets] :
~ ( member2138796287ecrets @ Y @ A )
=> ( A = bot_bo1124690338ecrets ) ) ).
% equals0I
thf(fact_61_equals0I,axiom,
! [A: set_Secrecy_specID] :
( ! [Y: secrecy_specID] :
~ ( member740727610specID @ Y @ A )
=> ( A = bot_bo1785112429specID ) ) ).
% equals0I
thf(fact_62_equals0I,axiom,
! [A: set_Secrecy_chanID] :
( ! [Y: secrecy_chanID] :
~ ( member1967978193chanID @ Y @ A )
=> ( A = bot_bo443197316chanID ) ) ).
% equals0I
thf(fact_63_equals0D,axiom,
! [A: set_Secrecy_KS,A2: secrecy_KS] :
( ( A = bot_bo1943466239ecy_KS )
=> ~ ( member_Secrecy_KS @ A2 @ A ) ) ).
% equals0D
thf(fact_64_equals0D,axiom,
! [A: set_Secrecy_Keys,A2: secrecy_Keys] :
( ( A = bot_bo701486667y_Keys )
=> ~ ( member_Secrecy_Keys @ A2 @ A ) ) ).
% equals0D
thf(fact_65_equals0D,axiom,
! [A: set_Secrecy_Secrets,A2: secrecy_Secrets] :
( ( A = bot_bo1124690338ecrets )
=> ~ ( member2138796287ecrets @ A2 @ A ) ) ).
% equals0D
thf(fact_66_equals0D,axiom,
! [A: set_Secrecy_specID,A2: secrecy_specID] :
( ( A = bot_bo1785112429specID )
=> ~ ( member740727610specID @ A2 @ A ) ) ).
% equals0D
thf(fact_67_equals0D,axiom,
! [A: set_Secrecy_chanID,A2: secrecy_chanID] :
( ( A = bot_bo443197316chanID )
=> ~ ( member1967978193chanID @ A2 @ A ) ) ).
% equals0D
thf(fact_68_emptyE,axiom,
! [A2: secrecy_KS] :
~ ( member_Secrecy_KS @ A2 @ bot_bo1943466239ecy_KS ) ).
% emptyE
thf(fact_69_emptyE,axiom,
! [A2: secrecy_Keys] :
~ ( member_Secrecy_Keys @ A2 @ bot_bo701486667y_Keys ) ).
% emptyE
thf(fact_70_emptyE,axiom,
! [A2: secrecy_Secrets] :
~ ( member2138796287ecrets @ A2 @ bot_bo1124690338ecrets ) ).
% emptyE
thf(fact_71_emptyE,axiom,
! [A2: secrecy_specID] :
~ ( member740727610specID @ A2 @ bot_bo1785112429specID ) ).
% emptyE
thf(fact_72_emptyE,axiom,
! [A2: secrecy_chanID] :
~ ( member1967978193chanID @ A2 @ bot_bo443197316chanID ) ).
% emptyE
thf(fact_73_mk__disjoint__insert,axiom,
! [A2: secrecy_KS,A: set_Secrecy_KS] :
( ( member_Secrecy_KS @ A2 @ A )
=> ? [B3: set_Secrecy_KS] :
( ( A
= ( insert_Secrecy_KS @ A2 @ B3 ) )
& ~ ( member_Secrecy_KS @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_74_mk__disjoint__insert,axiom,
! [A2: secrecy_chanID,A: set_Secrecy_chanID] :
( ( member1967978193chanID @ A2 @ A )
=> ? [B3: set_Secrecy_chanID] :
( ( A
= ( insert751661162chanID @ A2 @ B3 ) )
& ~ ( member1967978193chanID @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_75_mk__disjoint__insert,axiom,
! [A2: secrecy_specID,A: set_Secrecy_specID] :
( ( member740727610specID @ A2 @ A )
=> ? [B3: set_Secrecy_specID] :
( ( A
= ( insert1671894227specID @ A2 @ B3 ) )
& ~ ( member740727610specID @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_76_mk__disjoint__insert,axiom,
! [A2: secrecy_Keys,A: set_Secrecy_Keys] :
( ( member_Secrecy_Keys @ A2 @ A )
=> ? [B3: set_Secrecy_Keys] :
( ( A
= ( insert_Secrecy_Keys @ A2 @ B3 ) )
& ~ ( member_Secrecy_Keys @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_77_mk__disjoint__insert,axiom,
! [A2: secrecy_Secrets,A: set_Secrecy_Secrets] :
( ( member2138796287ecrets @ A2 @ A )
=> ? [B3: set_Secrecy_Secrets] :
( ( A
= ( insert565337510ecrets @ A2 @ B3 ) )
& ~ ( member2138796287ecrets @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_78_insert__commute,axiom,
! [X: secrecy_specID,Y2: secrecy_specID,A: set_Secrecy_specID] :
( ( insert1671894227specID @ X @ ( insert1671894227specID @ Y2 @ A ) )
= ( insert1671894227specID @ Y2 @ ( insert1671894227specID @ X @ A ) ) ) ).
% insert_commute
thf(fact_79_insert__commute,axiom,
! [X: secrecy_chanID,Y2: secrecy_chanID,A: set_Secrecy_chanID] :
( ( insert751661162chanID @ X @ ( insert751661162chanID @ Y2 @ A ) )
= ( insert751661162chanID @ Y2 @ ( insert751661162chanID @ X @ A ) ) ) ).
% insert_commute
thf(fact_80_insert__eq__iff,axiom,
! [A2: secrecy_KS,A: set_Secrecy_KS,B2: secrecy_KS,B: set_Secrecy_KS] :
( ~ ( member_Secrecy_KS @ A2 @ A )
=> ( ~ ( member_Secrecy_KS @ B2 @ B )
=> ( ( ( insert_Secrecy_KS @ A2 @ A )
= ( insert_Secrecy_KS @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C2: set_Secrecy_KS] :
( ( A
= ( insert_Secrecy_KS @ B2 @ C2 ) )
& ~ ( member_Secrecy_KS @ B2 @ C2 )
& ( B
= ( insert_Secrecy_KS @ A2 @ C2 ) )
& ~ ( member_Secrecy_KS @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_81_insert__eq__iff,axiom,
! [A2: secrecy_chanID,A: set_Secrecy_chanID,B2: secrecy_chanID,B: set_Secrecy_chanID] :
( ~ ( member1967978193chanID @ A2 @ A )
=> ( ~ ( member1967978193chanID @ B2 @ B )
=> ( ( ( insert751661162chanID @ A2 @ A )
= ( insert751661162chanID @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C2: set_Secrecy_chanID] :
( ( A
= ( insert751661162chanID @ B2 @ C2 ) )
& ~ ( member1967978193chanID @ B2 @ C2 )
& ( B
= ( insert751661162chanID @ A2 @ C2 ) )
& ~ ( member1967978193chanID @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_82_insert__eq__iff,axiom,
! [A2: secrecy_specID,A: set_Secrecy_specID,B2: secrecy_specID,B: set_Secrecy_specID] :
( ~ ( member740727610specID @ A2 @ A )
=> ( ~ ( member740727610specID @ B2 @ B )
=> ( ( ( insert1671894227specID @ A2 @ A )
= ( insert1671894227specID @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C2: set_Secrecy_specID] :
( ( A
= ( insert1671894227specID @ B2 @ C2 ) )
& ~ ( member740727610specID @ B2 @ C2 )
& ( B
= ( insert1671894227specID @ A2 @ C2 ) )
& ~ ( member740727610specID @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_83_insert__eq__iff,axiom,
! [A2: secrecy_Keys,A: set_Secrecy_Keys,B2: secrecy_Keys,B: set_Secrecy_Keys] :
( ~ ( member_Secrecy_Keys @ A2 @ A )
=> ( ~ ( member_Secrecy_Keys @ B2 @ B )
=> ( ( ( insert_Secrecy_Keys @ A2 @ A )
= ( insert_Secrecy_Keys @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C2: set_Secrecy_Keys] :
( ( A
= ( insert_Secrecy_Keys @ B2 @ C2 ) )
& ~ ( member_Secrecy_Keys @ B2 @ C2 )
& ( B
= ( insert_Secrecy_Keys @ A2 @ C2 ) )
& ~ ( member_Secrecy_Keys @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_84_insert__eq__iff,axiom,
! [A2: secrecy_Secrets,A: set_Secrecy_Secrets,B2: secrecy_Secrets,B: set_Secrecy_Secrets] :
( ~ ( member2138796287ecrets @ A2 @ A )
=> ( ~ ( member2138796287ecrets @ B2 @ B )
=> ( ( ( insert565337510ecrets @ A2 @ A )
= ( insert565337510ecrets @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C2: set_Secrecy_Secrets] :
( ( A
= ( insert565337510ecrets @ B2 @ C2 ) )
& ~ ( member2138796287ecrets @ B2 @ C2 )
& ( B
= ( insert565337510ecrets @ A2 @ C2 ) )
& ~ ( member2138796287ecrets @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_85_insert__absorb,axiom,
! [A2: secrecy_KS,A: set_Secrecy_KS] :
( ( member_Secrecy_KS @ A2 @ A )
=> ( ( insert_Secrecy_KS @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_86_insert__absorb,axiom,
! [A2: secrecy_chanID,A: set_Secrecy_chanID] :
( ( member1967978193chanID @ A2 @ A )
=> ( ( insert751661162chanID @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_87_insert__absorb,axiom,
! [A2: secrecy_specID,A: set_Secrecy_specID] :
( ( member740727610specID @ A2 @ A )
=> ( ( insert1671894227specID @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_88_insert__absorb,axiom,
! [A2: secrecy_Keys,A: set_Secrecy_Keys] :
( ( member_Secrecy_Keys @ A2 @ A )
=> ( ( insert_Secrecy_Keys @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_89_insert__absorb,axiom,
! [A2: secrecy_Secrets,A: set_Secrecy_Secrets] :
( ( member2138796287ecrets @ A2 @ A )
=> ( ( insert565337510ecrets @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_90_insert__ident,axiom,
! [X: secrecy_KS,A: set_Secrecy_KS,B: set_Secrecy_KS] :
( ~ ( member_Secrecy_KS @ X @ A )
=> ( ~ ( member_Secrecy_KS @ X @ B )
=> ( ( ( insert_Secrecy_KS @ X @ A )
= ( insert_Secrecy_KS @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_91_insert__ident,axiom,
! [X: secrecy_chanID,A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ~ ( member1967978193chanID @ X @ A )
=> ( ~ ( member1967978193chanID @ X @ B )
=> ( ( ( insert751661162chanID @ X @ A )
= ( insert751661162chanID @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_92_insert__ident,axiom,
! [X: secrecy_specID,A: set_Secrecy_specID,B: set_Secrecy_specID] :
( ~ ( member740727610specID @ X @ A )
=> ( ~ ( member740727610specID @ X @ B )
=> ( ( ( insert1671894227specID @ X @ A )
= ( insert1671894227specID @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_93_insert__ident,axiom,
! [X: secrecy_Keys,A: set_Secrecy_Keys,B: set_Secrecy_Keys] :
( ~ ( member_Secrecy_Keys @ X @ A )
=> ( ~ ( member_Secrecy_Keys @ X @ B )
=> ( ( ( insert_Secrecy_Keys @ X @ A )
= ( insert_Secrecy_Keys @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_94_insert__ident,axiom,
! [X: secrecy_Secrets,A: set_Secrecy_Secrets,B: set_Secrecy_Secrets] :
( ~ ( member2138796287ecrets @ X @ A )
=> ( ~ ( member2138796287ecrets @ X @ B )
=> ( ( ( insert565337510ecrets @ X @ A )
= ( insert565337510ecrets @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_95_Set_Oset__insert,axiom,
! [X: secrecy_KS,A: set_Secrecy_KS] :
( ( member_Secrecy_KS @ X @ A )
=> ~ ! [B3: set_Secrecy_KS] :
( ( A
= ( insert_Secrecy_KS @ X @ B3 ) )
=> ( member_Secrecy_KS @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_96_Set_Oset__insert,axiom,
! [X: secrecy_chanID,A: set_Secrecy_chanID] :
( ( member1967978193chanID @ X @ A )
=> ~ ! [B3: set_Secrecy_chanID] :
( ( A
= ( insert751661162chanID @ X @ B3 ) )
=> ( member1967978193chanID @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_97_Set_Oset__insert,axiom,
! [X: secrecy_specID,A: set_Secrecy_specID] :
( ( member740727610specID @ X @ A )
=> ~ ! [B3: set_Secrecy_specID] :
( ( A
= ( insert1671894227specID @ X @ B3 ) )
=> ( member740727610specID @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_98_Set_Oset__insert,axiom,
! [X: secrecy_Keys,A: set_Secrecy_Keys] :
( ( member_Secrecy_Keys @ X @ A )
=> ~ ! [B3: set_Secrecy_Keys] :
( ( A
= ( insert_Secrecy_Keys @ X @ B3 ) )
=> ( member_Secrecy_Keys @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_99_Set_Oset__insert,axiom,
! [X: secrecy_Secrets,A: set_Secrecy_Secrets] :
( ( member2138796287ecrets @ X @ A )
=> ~ ! [B3: set_Secrecy_Secrets] :
( ( A
= ( insert565337510ecrets @ X @ B3 ) )
=> ( member2138796287ecrets @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_100_insertI2,axiom,
! [A2: secrecy_KS,B: set_Secrecy_KS,B2: secrecy_KS] :
( ( member_Secrecy_KS @ A2 @ B )
=> ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ B2 @ B ) ) ) ).
% insertI2
thf(fact_101_insertI2,axiom,
! [A2: secrecy_chanID,B: set_Secrecy_chanID,B2: secrecy_chanID] :
( ( member1967978193chanID @ A2 @ B )
=> ( member1967978193chanID @ A2 @ ( insert751661162chanID @ B2 @ B ) ) ) ).
% insertI2
thf(fact_102_insertI2,axiom,
! [A2: secrecy_specID,B: set_Secrecy_specID,B2: secrecy_specID] :
( ( member740727610specID @ A2 @ B )
=> ( member740727610specID @ A2 @ ( insert1671894227specID @ B2 @ B ) ) ) ).
% insertI2
thf(fact_103_insertI2,axiom,
! [A2: secrecy_Keys,B: set_Secrecy_Keys,B2: secrecy_Keys] :
( ( member_Secrecy_Keys @ A2 @ B )
=> ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ B2 @ B ) ) ) ).
% insertI2
thf(fact_104_insertI2,axiom,
! [A2: secrecy_Secrets,B: set_Secrecy_Secrets,B2: secrecy_Secrets] :
( ( member2138796287ecrets @ A2 @ B )
=> ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ B2 @ B ) ) ) ).
% insertI2
thf(fact_105_insertI1,axiom,
! [A2: secrecy_KS,B: set_Secrecy_KS] : ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ A2 @ B ) ) ).
% insertI1
thf(fact_106_insertI1,axiom,
! [A2: secrecy_chanID,B: set_Secrecy_chanID] : ( member1967978193chanID @ A2 @ ( insert751661162chanID @ A2 @ B ) ) ).
% insertI1
thf(fact_107_insertI1,axiom,
! [A2: secrecy_specID,B: set_Secrecy_specID] : ( member740727610specID @ A2 @ ( insert1671894227specID @ A2 @ B ) ) ).
% insertI1
thf(fact_108_insertI1,axiom,
! [A2: secrecy_Keys,B: set_Secrecy_Keys] : ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ A2 @ B ) ) ).
% insertI1
thf(fact_109_insertI1,axiom,
! [A2: secrecy_Secrets,B: set_Secrecy_Secrets] : ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ A2 @ B ) ) ).
% insertI1
thf(fact_110_insertE,axiom,
! [A2: secrecy_KS,B2: secrecy_KS,A: set_Secrecy_KS] :
( ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_Secrecy_KS @ A2 @ A ) ) ) ).
% insertE
thf(fact_111_insertE,axiom,
! [A2: secrecy_chanID,B2: secrecy_chanID,A: set_Secrecy_chanID] :
( ( member1967978193chanID @ A2 @ ( insert751661162chanID @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member1967978193chanID @ A2 @ A ) ) ) ).
% insertE
thf(fact_112_insertE,axiom,
! [A2: secrecy_specID,B2: secrecy_specID,A: set_Secrecy_specID] :
( ( member740727610specID @ A2 @ ( insert1671894227specID @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member740727610specID @ A2 @ A ) ) ) ).
% insertE
thf(fact_113_insertE,axiom,
! [A2: secrecy_Keys,B2: secrecy_Keys,A: set_Secrecy_Keys] :
( ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_Secrecy_Keys @ A2 @ A ) ) ) ).
% insertE
thf(fact_114_insertE,axiom,
! [A2: secrecy_Secrets,B2: secrecy_Secrets,A: set_Secrecy_Secrets] :
( ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member2138796287ecrets @ A2 @ A ) ) ) ).
% insertE
thf(fact_115_mem__Collect__eq,axiom,
! [A2: secrecy_KS,P: secrecy_KS > $o] :
( ( member_Secrecy_KS @ A2 @ ( collect_Secrecy_KS @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_116_mem__Collect__eq,axiom,
! [A2: secrecy_chanID,P: secrecy_chanID > $o] :
( ( member1967978193chanID @ A2 @ ( collec625564175chanID @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_117_mem__Collect__eq,axiom,
! [A2: secrecy_specID,P: secrecy_specID > $o] :
( ( member740727610specID @ A2 @ ( collec1545797240specID @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_118_mem__Collect__eq,axiom,
! [A2: secrecy_Keys,P: secrecy_Keys > $o] :
( ( member_Secrecy_Keys @ A2 @ ( collect_Secrecy_Keys @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_119_mem__Collect__eq,axiom,
! [A2: secrecy_Secrets,P: secrecy_Secrets > $o] :
( ( member2138796287ecrets @ A2 @ ( collec836099393ecrets @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_120_Collect__mem__eq,axiom,
! [A: set_Secrecy_KS] :
( ( collect_Secrecy_KS
@ ^ [X2: secrecy_KS] : ( member_Secrecy_KS @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_121_Collect__mem__eq,axiom,
! [A: set_Secrecy_chanID] :
( ( collec625564175chanID
@ ^ [X2: secrecy_chanID] : ( member1967978193chanID @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_122_Collect__mem__eq,axiom,
! [A: set_Secrecy_specID] :
( ( collec1545797240specID
@ ^ [X2: secrecy_specID] : ( member740727610specID @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_123_Collect__mem__eq,axiom,
! [A: set_Secrecy_Keys] :
( ( collect_Secrecy_Keys
@ ^ [X2: secrecy_Keys] : ( member_Secrecy_Keys @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_124_Collect__mem__eq,axiom,
! [A: set_Secrecy_Secrets] :
( ( collec836099393ecrets
@ ^ [X2: secrecy_Secrets] : ( member2138796287ecrets @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_125_know__composition__neg1,axiom,
! [P: secrecy_specID,M: secrecy_KS,Q: secrecy_specID,PQ: secrecy_specID] :
( ~ ( knowle945212990e_know @ P @ M )
=> ( ~ ( knowle945212990e_know @ Q @ M )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc965049635ionLoc @ PQ )
=> ( ( correctCompositionIn @ PQ )
=> ~ ( knowle945212990e_know @ PQ @ M ) ) ) ) ) ) ).
% know_composition_neg1
thf(fact_126_singleton__inject,axiom,
! [A2: secrecy_specID,B2: secrecy_specID] :
( ( ( insert1671894227specID @ A2 @ bot_bo1785112429specID )
= ( insert1671894227specID @ B2 @ bot_bo1785112429specID ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_127_singleton__inject,axiom,
! [A2: secrecy_chanID,B2: secrecy_chanID] :
( ( ( insert751661162chanID @ A2 @ bot_bo443197316chanID )
= ( insert751661162chanID @ B2 @ bot_bo443197316chanID ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_128_insert__not__empty,axiom,
! [A2: secrecy_specID,A: set_Secrecy_specID] :
( ( insert1671894227specID @ A2 @ A )
!= bot_bo1785112429specID ) ).
% insert_not_empty
thf(fact_129_insert__not__empty,axiom,
! [A2: secrecy_chanID,A: set_Secrecy_chanID] :
( ( insert751661162chanID @ A2 @ A )
!= bot_bo443197316chanID ) ).
% insert_not_empty
thf(fact_130_doubleton__eq__iff,axiom,
! [A2: secrecy_specID,B2: secrecy_specID,C: secrecy_specID,D: secrecy_specID] :
( ( ( insert1671894227specID @ A2 @ ( insert1671894227specID @ B2 @ bot_bo1785112429specID ) )
= ( insert1671894227specID @ C @ ( insert1671894227specID @ D @ bot_bo1785112429specID ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_131_doubleton__eq__iff,axiom,
! [A2: secrecy_chanID,B2: secrecy_chanID,C: secrecy_chanID,D: secrecy_chanID] :
( ( ( insert751661162chanID @ A2 @ ( insert751661162chanID @ B2 @ bot_bo443197316chanID ) )
= ( insert751661162chanID @ C @ ( insert751661162chanID @ D @ bot_bo443197316chanID ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_132_singleton__iff,axiom,
! [B2: secrecy_KS,A2: secrecy_KS] :
( ( member_Secrecy_KS @ B2 @ ( insert_Secrecy_KS @ A2 @ bot_bo1943466239ecy_KS ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_133_singleton__iff,axiom,
! [B2: secrecy_Keys,A2: secrecy_Keys] :
( ( member_Secrecy_Keys @ B2 @ ( insert_Secrecy_Keys @ A2 @ bot_bo701486667y_Keys ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_134_singleton__iff,axiom,
! [B2: secrecy_Secrets,A2: secrecy_Secrets] :
( ( member2138796287ecrets @ B2 @ ( insert565337510ecrets @ A2 @ bot_bo1124690338ecrets ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_135_singleton__iff,axiom,
! [B2: secrecy_specID,A2: secrecy_specID] :
( ( member740727610specID @ B2 @ ( insert1671894227specID @ A2 @ bot_bo1785112429specID ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_136_singleton__iff,axiom,
! [B2: secrecy_chanID,A2: secrecy_chanID] :
( ( member1967978193chanID @ B2 @ ( insert751661162chanID @ A2 @ bot_bo443197316chanID ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_137_singletonD,axiom,
! [B2: secrecy_KS,A2: secrecy_KS] :
( ( member_Secrecy_KS @ B2 @ ( insert_Secrecy_KS @ A2 @ bot_bo1943466239ecy_KS ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_138_singletonD,axiom,
! [B2: secrecy_Keys,A2: secrecy_Keys] :
( ( member_Secrecy_Keys @ B2 @ ( insert_Secrecy_Keys @ A2 @ bot_bo701486667y_Keys ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_139_singletonD,axiom,
! [B2: secrecy_Secrets,A2: secrecy_Secrets] :
( ( member2138796287ecrets @ B2 @ ( insert565337510ecrets @ A2 @ bot_bo1124690338ecrets ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_140_singletonD,axiom,
! [B2: secrecy_specID,A2: secrecy_specID] :
( ( member740727610specID @ B2 @ ( insert1671894227specID @ A2 @ bot_bo1785112429specID ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_141_singletonD,axiom,
! [B2: secrecy_chanID,A2: secrecy_chanID] :
( ( member1967978193chanID @ B2 @ ( insert751661162chanID @ A2 @ bot_bo443197316chanID ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_142_TBtheorem3a,axiom,
! [P: secrecy_specID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ~ ( ine @ P @ E2 )
=> ( ~ ( ine @ Q @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ~ ( ine @ PQ @ E2 ) ) ) ) ) ).
% TBtheorem3a
thf(fact_143_TBtheorem1a,axiom,
! [PQ: secrecy_specID,E2: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
( ( ine @ PQ @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ine @ P @ E2 )
| ( ine @ Q @ E2 ) ) ) ) ) ).
% TBtheorem1a
thf(fact_144_the__elem__eq,axiom,
! [X: secrecy_specID] :
( ( the_el827131246specID @ ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
= X ) ).
% the_elem_eq
thf(fact_145_the__elem__eq,axiom,
! [X: secrecy_chanID] :
( ( the_el2054381829chanID @ ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
= X ) ).
% the_elem_eq
thf(fact_146_is__singletonI,axiom,
! [X: secrecy_specID] : ( is_sin1501507247specID @ ( insert1671894227specID @ X @ bot_bo1785112429specID ) ) ).
% is_singletonI
thf(fact_147_is__singletonI,axiom,
! [X: secrecy_chanID] : ( is_sin581274182chanID @ ( insert751661162chanID @ X @ bot_bo443197316chanID ) ) ).
% is_singletonI
thf(fact_148_KS2Expression_Osimps_I1_J,axiom,
! [M: secrecy_Keys] :
( ( secrec1016742679ession @ ( secrecy_kKS @ M ) )
= ( secrecy_kE @ M ) ) ).
% KS2Expression.simps(1)
thf(fact_149_Set_Ois__empty__def,axiom,
( is_emp1849217361specID
= ( ^ [A3: set_Secrecy_specID] : A3 = bot_bo1785112429specID ) ) ).
% Set.is_empty_def
thf(fact_150_Set_Ois__empty__def,axiom,
( is_emp928984296chanID
= ( ^ [A3: set_Secrecy_chanID] : A3 = bot_bo443197316chanID ) ) ).
% Set.is_empty_def
thf(fact_151_is__singleton__def,axiom,
( is_sin1501507247specID
= ( ^ [A3: set_Secrecy_specID] :
? [X2: secrecy_specID] :
( A3
= ( insert1671894227specID @ X2 @ bot_bo1785112429specID ) ) ) ) ).
% is_singleton_def
thf(fact_152_is__singleton__def,axiom,
( is_sin581274182chanID
= ( ^ [A3: set_Secrecy_chanID] :
? [X2: secrecy_chanID] :
( A3
= ( insert751661162chanID @ X2 @ bot_bo443197316chanID ) ) ) ) ).
% is_singleton_def
thf(fact_153_is__singletonE,axiom,
! [A: set_Secrecy_specID] :
( ( is_sin1501507247specID @ A )
=> ~ ! [X3: secrecy_specID] :
( A
!= ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) ) ) ).
% is_singletonE
thf(fact_154_is__singletonE,axiom,
! [A: set_Secrecy_chanID] :
( ( is_sin581274182chanID @ A )
=> ~ ! [X3: secrecy_chanID] :
( A
!= ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) ) ) ).
% is_singletonE
thf(fact_155_bot__set__def,axiom,
( bot_bo1785112429specID
= ( collec1545797240specID @ bot_bo1767780696ecID_o ) ) ).
% bot_set_def
thf(fact_156_bot__set__def,axiom,
( bot_bo443197316chanID
= ( collec625564175chanID @ bot_bo1012329985anID_o ) ) ).
% bot_set_def
thf(fact_157_is__singleton__the__elem,axiom,
( is_sin1501507247specID
= ( ^ [A3: set_Secrecy_specID] :
( A3
= ( insert1671894227specID @ ( the_el827131246specID @ A3 ) @ bot_bo1785112429specID ) ) ) ) ).
% is_singleton_the_elem
thf(fact_158_is__singleton__the__elem,axiom,
( is_sin581274182chanID
= ( ^ [A3: set_Secrecy_chanID] :
( A3
= ( insert751661162chanID @ ( the_el2054381829chanID @ A3 ) @ bot_bo443197316chanID ) ) ) ) ).
% is_singleton_the_elem
thf(fact_159_is__singletonI_H,axiom,
! [A: set_Secrecy_KS] :
( ( A != bot_bo1943466239ecy_KS )
=> ( ! [X3: secrecy_KS,Y: secrecy_KS] :
( ( member_Secrecy_KS @ X3 @ A )
=> ( ( member_Secrecy_KS @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_sin1442426433ecy_KS @ A ) ) ) ).
% is_singletonI'
thf(fact_160_is__singletonI_H,axiom,
! [A: set_Secrecy_Keys] :
( ( A != bot_bo701486667y_Keys )
=> ( ! [X3: secrecy_Keys,Y: secrecy_Keys] :
( ( member_Secrecy_Keys @ X3 @ A )
=> ( ( member_Secrecy_Keys @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_sin1096471309y_Keys @ A ) ) ) ).
% is_singletonI'
thf(fact_161_is__singletonI_H,axiom,
! [A: set_Secrecy_Secrets] :
( ( A != bot_bo1124690338ecrets )
=> ( ! [X3: secrecy_Secrets,Y: secrecy_Secrets] :
( ( member2138796287ecrets @ X3 @ A )
=> ( ( member2138796287ecrets @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_sin1002224330ecrets @ A ) ) ) ).
% is_singletonI'
thf(fact_162_is__singletonI_H,axiom,
! [A: set_Secrecy_specID] :
( ( A != bot_bo1785112429specID )
=> ( ! [X3: secrecy_specID,Y: secrecy_specID] :
( ( member740727610specID @ X3 @ A )
=> ( ( member740727610specID @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_sin1501507247specID @ A ) ) ) ).
% is_singletonI'
thf(fact_163_is__singletonI_H,axiom,
! [A: set_Secrecy_chanID] :
( ( A != bot_bo443197316chanID )
=> ( ! [X3: secrecy_chanID,Y: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ A )
=> ( ( member1967978193chanID @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_sin581274182chanID @ A ) ) ) ).
% is_singletonI'
thf(fact_164_TBtheorem4a__empty,axiom,
! [P: secrecy_specID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( ine @ P @ E2 )
| ( ine @ Q @ E2 ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ( loc @ PQ )
= bot_bo443197316chanID )
=> ( ine @ PQ @ E2 ) ) ) ) ) ).
% TBtheorem4a_empty
thf(fact_165_TBtheorem1b,axiom,
! [PQ: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
( ( ineM @ PQ @ M2 @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ineM @ P @ M2 @ E2 )
| ( ineM @ Q @ M2 @ E2 ) ) ) ) ) ).
% TBtheorem1b
thf(fact_166_TBtheorem3b,axiom,
! [P: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ~ ( ineM @ P @ M2 @ E2 )
=> ( ~ ( ineM @ Q @ M2 @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ~ ( ineM @ PQ @ M2 @ E2 ) ) ) ) ) ).
% TBtheorem3b
thf(fact_167_DecrEnc,axiom,
! [K12: secrecy_Keys,K22: secrecy_Keys,E2: list_S529945141ession] :
( ( encrDecrKeys @ K12 @ K22 )
=> ( ( decr @ K22 @ ( enc @ K12 @ E2 ) )
= E2 ) ) ).
% DecrEnc
thf(fact_168_ExtSign,axiom,
! [K12: secrecy_Keys,K22: secrecy_Keys,E2: list_S529945141ession] :
( ( encrDecrKeys @ K12 @ K22 )
=> ( ( ext @ K12 @ ( sign @ K22 @ E2 ) )
= E2 ) ) ).
% ExtSign
thf(fact_169_correctCompositionIn__L2,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
=> ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
| ( member1967978193chanID @ Ch @ ( ins @ Q ) ) ) ) ) ) ).
% correctCompositionIn_L2
thf(fact_170_correctCompositionIn__prop1,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,X: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member1967978193chanID @ X @ ( ins @ PQ ) )
=> ( ( member1967978193chanID @ X @ ( ins @ P ) )
| ( member1967978193chanID @ X @ ( ins @ Q ) ) ) ) ) ) ).
% correctCompositionIn_prop1
thf(fact_171_not__ine__ineM,axiom,
! [P: secrecy_specID,E2: secrecy_Expression,M2: set_Secrecy_chanID] :
( ~ ( ine @ P @ E2 )
=> ~ ( ineM @ P @ M2 @ E2 ) ) ).
% not_ine_ineM
thf(fact_172_ineM__ine,axiom,
! [P: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression] :
( ( ineM @ P @ M2 @ E2 )
=> ( ine @ P @ E2 ) ) ).
% ineM_ine
thf(fact_173_correctCompositionIn__L1,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ~ ( member1967978193chanID @ Ch @ ( loc @ PQ ) )
=> ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
=> ( member1967978193chanID @ Ch @ ( ins @ PQ ) ) ) ) ) ) ).
% correctCompositionIn_L1
thf(fact_174_subcomponents__loc,axiom,
! [X: secrecy_specID] :
( ( correc965049635ionLoc @ X )
=> ( ( ( subcomponents @ X )
= bot_bo1785112429specID )
=> ( ( loc @ X )
= bot_bo443197316chanID ) ) ) ).
% subcomponents_loc
thf(fact_175_TBtheorem4b__notP1,axiom,
! [P: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ineM @ P @ M2 @ E2 )
=> ( ~ ( ineM @ Q @ M2 @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ine_ex1303305700Single @ P @ Ch2 @ E2 )
& ( member1967978193chanID @ Ch2 @ M2 )
& ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ~ ( ineM @ PQ @ M2 @ E2 ) ) ) ) ) ) ).
% TBtheorem4b_notP1
thf(fact_176_TBtheorem4a__notP1,axiom,
! [P: secrecy_specID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ine @ P @ E2 )
=> ( ~ ( ine @ Q @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ine_ex1303305700Single @ P @ Ch2 @ E2 )
& ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ~ ( ine @ PQ @ E2 ) ) ) ) ) ) ).
% TBtheorem4a_notP1
thf(fact_177_TBtheorem4b__notPQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set_Secrecy_chanID,E2: secrecy_Expression,ChSetQ: set_Secrecy_chanID,M2: set_Secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ine_exprChannelSet @ P @ ChSetP @ E2 )
=> ( ( ine_exprChannelSet @ Q @ ChSetQ @ E2 )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSetP )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSetQ )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ~ ( ineM @ PQ @ M2 @ E2 ) ) ) ) ) ) ) ).
% TBtheorem4b_notPQ
thf(fact_178_TBtheorem4b__notP2,axiom,
! [Q: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set_Secrecy_chanID] :
( ~ ( ineM @ Q @ M2 @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ine_exprChannelSet @ P @ ChSet @ E2 )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSet )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ~ ( ineM @ PQ @ M2 @ E2 ) ) ) ) ) ) ).
% TBtheorem4b_notP2
thf(fact_179_TBtheorem4a__notPQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set_Secrecy_chanID,E2: secrecy_Expression,ChSetQ: set_Secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ine_exprChannelSet @ P @ ChSetP @ E2 )
=> ( ( ine_exprChannelSet @ Q @ ChSetQ @ E2 )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSetP )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSetQ )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ~ ( ine @ PQ @ E2 ) ) ) ) ) ) ) ).
% TBtheorem4a_notPQ
thf(fact_180_ine__exprChannelSingle__Set,axiom,
! [P: secrecy_specID,Ch: secrecy_chanID,E2: secrecy_Expression] :
( ( ine_ex1303305700Single @ P @ Ch @ E2 )
=> ( ine_exprChannelSet @ P @ ( insert751661162chanID @ Ch @ bot_bo443197316chanID ) @ E2 ) ) ).
% ine_exprChannelSingle_Set
thf(fact_181_ine__exprChannelSet__Single,axiom,
! [P: secrecy_specID,Ch: secrecy_chanID,E2: secrecy_Expression] :
( ( ine_exprChannelSet @ P @ ( insert751661162chanID @ Ch @ bot_bo443197316chanID ) @ E2 )
=> ( ine_ex1303305700Single @ P @ Ch @ E2 ) ) ).
% ine_exprChannelSet_Single
thf(fact_182_ine__nonempty__exprChannelSet,axiom,
! [P: secrecy_specID,ChSet: set_Secrecy_chanID,E2: secrecy_Expression] :
( ( ine_exprChannelSet @ P @ ChSet @ E2 )
=> ( ( ChSet != bot_bo443197316chanID )
=> ( ine @ P @ E2 ) ) ) ).
% ine_nonempty_exprChannelSet
thf(fact_183_ine__empty__exprChannelSet,axiom,
! [P: secrecy_specID,ChSet: set_Secrecy_chanID,E2: secrecy_Expression] :
( ( ine_exprChannelSet @ P @ ChSet @ E2 )
=> ( ( ChSet = bot_bo443197316chanID )
=> ~ ( ine @ P @ E2 ) ) ) ).
% ine_empty_exprChannelSet
thf(fact_184_TBtheorem4a__notP2,axiom,
! [Q: secrecy_specID,E2: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set_Secrecy_chanID] :
( ~ ( ine @ Q @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ine_exprChannelSet @ P @ ChSet @ E2 )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSet )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ~ ( ine @ PQ @ E2 ) ) ) ) ) ) ).
% TBtheorem4a_notP2
thf(fact_185_locStream__def,axiom,
( locStream
= ( ^ [X2: secrecy_specID] :
( ^ [Y3: set_Secrecy_chanID,Z: set_Secrecy_chanID] : Y3 = Z
@ ( loc @ X2 ) ) ) ) ).
% locStream_def
thf(fact_186_inStream__def,axiom,
( inStream
= ( ^ [X2: secrecy_specID] :
( ^ [Y3: set_Secrecy_chanID,Z: set_Secrecy_chanID] : Y3 = Z
@ ( ins @ X2 ) ) ) ) ).
% inStream_def
thf(fact_187_TBtheorem4b__P,axiom,
! [P: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
( ( ineM @ P @ M2 @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( member1967978193chanID @ Ch2 @ ( ins @ Q ) )
& ( exprChannel @ Ch2 @ E2 )
& ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) )
& ( member1967978193chanID @ Ch2 @ M2 ) )
=> ( ineM @ PQ @ M2 @ E2 ) ) ) ) ) ).
% TBtheorem4b_P
thf(fact_188_TBtheorem4b__PQ,axiom,
! [P: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( ineM @ P @ M2 @ E2 )
| ( ineM @ Q @ M2 @ E2 ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ( member1967978193chanID @ Ch2 @ ( ins @ P ) )
| ( member1967978193chanID @ Ch2 @ ( ins @ Q ) ) )
& ( member1967978193chanID @ Ch2 @ M2 )
& ( exprChannel @ Ch2 @ E2 )
& ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( ineM @ PQ @ M2 @ E2 ) ) ) ) ) ).
% TBtheorem4b_PQ
thf(fact_189_TBtheorem4a__P,axiom,
! [P: secrecy_specID,E2: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
( ( ine @ P @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( member1967978193chanID @ Ch2 @ ( ins @ P ) )
& ( exprChannel @ Ch2 @ E2 )
& ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( ine @ PQ @ E2 ) ) ) ) ) ).
% TBtheorem4a_P
thf(fact_190_TBtheorem4a__PQ,axiom,
! [P: secrecy_specID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( ine @ P @ E2 )
| ( ine @ Q @ E2 ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ( member1967978193chanID @ Ch2 @ ( ins @ P ) )
| ( member1967978193chanID @ Ch2 @ ( ins @ Q ) ) )
& ( exprChannel @ Ch2 @ E2 )
& ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( ine @ PQ @ E2 ) ) ) ) ) ).
% TBtheorem4a_PQ
thf(fact_191_ine__ins__neg1,axiom,
! [P: secrecy_specID,M: secrecy_Expression,X: secrecy_chanID] :
( ~ ( ine @ P @ M )
=> ( ( exprChannel @ X @ M )
=> ~ ( member1967978193chanID @ X @ ( ins @ P ) ) ) ) ).
% ine_ins_neg1
thf(fact_192_ine__def,axiom,
( ine
= ( ^ [SP: secrecy_specID,E3: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member1967978193chanID @ Ch3 @ ( ins @ SP ) )
& ( exprChannel @ Ch3 @ E3 ) ) ) ) ).
% ine_def
thf(fact_193_ineM__L1,axiom,
! [Ch: secrecy_chanID,M2: set_Secrecy_chanID,P: secrecy_specID,E2: secrecy_Expression] :
( ( member1967978193chanID @ Ch @ M2 )
=> ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
=> ( ( exprChannel @ Ch @ E2 )
=> ( ineM @ P @ M2 @ E2 ) ) ) ) ).
% ineM_L1
thf(fact_194_ineM__def,axiom,
( ineM
= ( ^ [SP: secrecy_specID,M3: set_Secrecy_chanID,E3: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member1967978193chanID @ Ch3 @ ( ins @ SP ) )
& ( member1967978193chanID @ Ch3 @ M3 )
& ( exprChannel @ Ch3 @ E3 ) ) ) ) ).
% ineM_def
thf(fact_195_ine__exprChannelSet__def,axiom,
( ine_exprChannelSet
= ( ^ [SP: secrecy_specID,ChSet2: set_Secrecy_chanID,E3: secrecy_Expression] :
( ! [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ ChSet2 )
=> ( ( member1967978193chanID @ X2 @ ( ins @ SP ) )
& ( exprChannel @ X2 @ E3 ) ) )
& ! [X2: secrecy_chanID] :
( ( ~ ( member1967978193chanID @ X2 @ ChSet2 )
& ( member1967978193chanID @ X2 @ ( ins @ SP ) ) )
=> ~ ( exprChannel @ X2 @ E3 ) ) ) ) ) ).
% ine_exprChannelSet_def
thf(fact_196_ine__exprChannelSingle__def,axiom,
( ine_ex1303305700Single
= ( ^ [SP: secrecy_specID,Ch3: secrecy_chanID,E3: secrecy_Expression] :
( ( member1967978193chanID @ Ch3 @ ( ins @ SP ) )
& ( exprChannel @ Ch3 @ E3 )
& ! [X2: secrecy_chanID,T: nat] :
( ( ( member1967978193chanID @ X2 @ ( ins @ SP ) )
& ( X2 != Ch3 ) )
=> ~ ( exprChannel @ X2 @ E3 ) ) ) ) ) ).
% ine_exprChannelSingle_def
thf(fact_197_LocalSecretsComposition__exprChannel__k,axiom,
! [X: secrecy_chanID,Keys: secrecy_Keys,P: secrecy_specID,Q: secrecy_specID] :
( ( exprChannel @ X @ ( secrecy_kE @ Keys ) )
=> ( ~ ( ine @ P @ ( secrecy_kE @ Keys ) )
=> ( ~ ( ine @ Q @ ( secrecy_kE @ Keys ) )
=> ( ~ ( member1967978193chanID @ X @ ( ins @ P ) )
& ~ ( member1967978193chanID @ X @ ( ins @ Q ) ) ) ) ) ) ).
% LocalSecretsComposition_exprChannel_k
thf(fact_198_correctCompositionInLoc__exprChannel,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ch: secrecy_chanID,M: secrecy_Expression] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
=> ( ( exprChannel @ Ch @ M )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ( ins @ PQ ) )
=> ~ ( exprChannel @ X3 @ M ) )
=> ( member1967978193chanID @ Ch @ ( loc @ PQ ) ) ) ) ) ) ) ).
% correctCompositionInLoc_exprChannel
thf(fact_199_TBlemma3b,axiom,
! [P: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID,Ch: secrecy_chanID] :
( ~ ( ineM @ P @ M2 @ E2 )
=> ( ~ ( ineM @ Q @ M2 @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member1967978193chanID @ Ch @ M2 )
=> ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
=> ~ ( exprChannel @ Ch @ E2 ) ) ) ) ) ) ) ).
% TBlemma3b
thf(fact_200_bot__empty__eq,axiom,
( bot_bot_Secrecy_KS_o
= ( ^ [X2: secrecy_KS] : ( member_Secrecy_KS @ X2 @ bot_bo1943466239ecy_KS ) ) ) ).
% bot_empty_eq
thf(fact_201_bot__empty__eq,axiom,
( bot_bo2072920570Keys_o
= ( ^ [X2: secrecy_Keys] : ( member_Secrecy_Keys @ X2 @ bot_bo701486667y_Keys ) ) ) ).
% bot_empty_eq
thf(fact_202_bot__empty__eq,axiom,
( bot_bo1909946299rets_o
= ( ^ [X2: secrecy_Secrets] : ( member2138796287ecrets @ X2 @ bot_bo1124690338ecrets ) ) ) ).
% bot_empty_eq
thf(fact_203_bot__empty__eq,axiom,
( bot_bo1767780696ecID_o
= ( ^ [X2: secrecy_specID] : ( member740727610specID @ X2 @ bot_bo1785112429specID ) ) ) ).
% bot_empty_eq
thf(fact_204_bot__empty__eq,axiom,
( bot_bo1012329985anID_o
= ( ^ [X2: secrecy_chanID] : ( member1967978193chanID @ X2 @ bot_bo443197316chanID ) ) ) ).
% bot_empty_eq
thf(fact_205_Collect__empty__eq__bot,axiom,
! [P: secrecy_specID > $o] :
( ( ( collec1545797240specID @ P )
= bot_bo1785112429specID )
= ( P = bot_bo1767780696ecID_o ) ) ).
% Collect_empty_eq_bot
thf(fact_206_Collect__empty__eq__bot,axiom,
! [P: secrecy_chanID > $o] :
( ( ( collec625564175chanID @ P )
= bot_bo443197316chanID )
= ( P = bot_bo1012329985anID_o ) ) ).
% Collect_empty_eq_bot
thf(fact_207_LocalSecretsComposition__neg__loc__k,axiom,
! [Key: secrecy_Keys,P: secrecy_specID,Ch: secrecy_chanID] :
( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( localSecrets @ P ) )
=> ( ( exprChannel @ Ch @ ( secrecy_kE @ Key ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( specKeysSecrets @ P ) )
=> ~ ( member1967978193chanID @ Ch @ ( loc @ P ) ) ) ) ) ).
% LocalSecretsComposition_neg_loc_k
thf(fact_208_correctCompositionKS__exprChannel__k__Qex,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Key: secrecy_Keys,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( localSecrets @ PQ ) )
=> ( ( member1967978193chanID @ Ch @ ( ins @ Q ) )
=> ( ( exprChannel @ Ch @ ( secrecy_kE @ Key ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( specKeysSecrets @ PQ ) )
=> ( ( correctCompositionIn @ PQ )
=> ? [Ch4: secrecy_chanID] :
( ( member1967978193chanID @ Ch4 @ ( ins @ PQ ) )
& ( exprChannel @ Ch4 @ ( secrecy_kE @ Key ) ) ) ) ) ) ) ) ) ) ).
% correctCompositionKS_exprChannel_k_Qex
thf(fact_209_correctCompositionKS__exprChannel__k__Pex,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Key: secrecy_Keys,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( localSecrets @ PQ ) )
=> ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
=> ( ( exprChannel @ Ch @ ( secrecy_kE @ Key ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( specKeysSecrets @ PQ ) )
=> ( ( correctCompositionIn @ PQ )
=> ? [Ch4: secrecy_chanID] :
( ( member1967978193chanID @ Ch4 @ ( ins @ PQ ) )
& ( exprChannel @ Ch4 @ ( secrecy_kE @ Key ) ) ) ) ) ) ) ) ) ) ).
% correctCompositionKS_exprChannel_k_Pex
thf(fact_210_correctCompositionKS__PQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ PQ ) )
=> ( ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ P ) )
| ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ Q ) ) ) ) ) ) ).
% correctCompositionKS_PQ
thf(fact_211_correctCompositionKS__neg1,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ Q ) )
=> ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ PQ ) ) ) ) ) ) ).
% correctCompositionKS_neg1
thf(fact_212_correctCompositionKS__negP,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ PQ ) )
=> ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ P ) ) ) ) ) ).
% correctCompositionKS_negP
thf(fact_213_correctCompositionKS__negQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ PQ ) )
=> ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ Q ) ) ) ) ) ).
% correctCompositionKS_negQ
thf(fact_214_know__composition,axiom,
! [M: secrecy_KS,P: secrecy_specID,Q: secrecy_specID,PQ: secrecy_specID] :
( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ Q ) )
=> ( ( ( knowle945212990e_know @ P @ M )
| ( knowle945212990e_know @ Q @ M ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( correctCompositionKS @ PQ )
=> ( knowle945212990e_know @ PQ @ M ) ) ) ) ) ) ) ).
% know_composition
thf(fact_215_know__composition1,axiom,
! [M: secrecy_KS,P: secrecy_specID,Q: secrecy_specID,PQ: secrecy_specID] :
( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ Q ) )
=> ( ( knowle945212990e_know @ P @ M )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( correctCompositionKS @ PQ )
=> ( knowle945212990e_know @ PQ @ M ) ) ) ) ) ) ) ).
% know_composition1
thf(fact_216_know__composition2,axiom,
! [M: secrecy_KS,P: secrecy_specID,Q: secrecy_specID,PQ: secrecy_specID] :
( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ Q ) )
=> ( ( knowle945212990e_know @ Q @ M )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( correctCompositionKS @ PQ )
=> ( knowle945212990e_know @ PQ @ M ) ) ) ) ) ) ) ).
% know_composition2
thf(fact_217_correctCompositionKS__exprChannel__k__P,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Key: secrecy_Keys,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( localSecrets @ PQ ) )
=> ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
=> ( ( exprChannel @ Ch @ ( secrecy_kE @ Key ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( specKeysSecrets @ PQ ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
& ( exprChannel @ Ch @ ( secrecy_kE @ Key ) ) ) ) ) ) ) ) ) ) ).
% correctCompositionKS_exprChannel_k_P
thf(fact_218_correctCompositionKS__exprChannel__k__Q,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Key: secrecy_Keys,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( localSecrets @ PQ ) )
=> ( ( member1967978193chanID @ Ch @ ( ins @ Q ) )
=> ( ( exprChannel @ Ch @ ( secrecy_kE @ Key ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( specKeysSecrets @ PQ ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
& ( exprChannel @ Ch @ ( secrecy_kE @ Key ) ) ) ) ) ) ) ) ) ) ).
% correctCompositionKS_exprChannel_k_Q
thf(fact_219_LocalSecretsComposition__neg__k,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,M: secrecy_Keys] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc965049635ionLoc @ PQ )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ M ) @ ( specKeysSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ M ) @ ( specKeysSecrets @ Q ) )
=> ( ~ ( ine @ P @ ( secrecy_kE @ M ) )
=> ( ~ ( ine @ Q @ ( secrecy_kE @ M ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ M ) @ ( sup_su1301156759ecy_KS @ ( localSecrets @ P ) @ ( localSecrets @ Q ) ) )
=> ~ ( member_Secrecy_KS @ ( secrecy_kKS @ M ) @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ) ) ).
% LocalSecretsComposition_neg_k
thf(fact_220_correctCompositionKS__exprChannel__s__Qex,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Secret: secrecy_Secrets,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( localSecrets @ PQ ) )
=> ( ( member1967978193chanID @ Ch @ ( ins @ Q ) )
=> ( ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( specKeysSecrets @ PQ ) )
=> ( ( correctCompositionIn @ PQ )
=> ? [Ch4: secrecy_chanID] :
( ( member1967978193chanID @ Ch4 @ ( ins @ PQ ) )
& ( exprChannel @ Ch4 @ ( secrecy_sE @ Secret ) ) ) ) ) ) ) ) ) ) ).
% correctCompositionKS_exprChannel_s_Qex
thf(fact_221_correctCompositionKS__exprChannel__s__Pex,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Secret: secrecy_Secrets,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( localSecrets @ PQ ) )
=> ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
=> ( ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( specKeysSecrets @ PQ ) )
=> ( ( correctCompositionIn @ PQ )
=> ? [Ch4: secrecy_chanID] :
( ( member1967978193chanID @ Ch4 @ ( ins @ PQ ) )
& ( exprChannel @ Ch4 @ ( secrecy_sE @ Secret ) ) ) ) ) ) ) ) ) ) ).
% correctCompositionKS_exprChannel_s_Pex
thf(fact_222_correctCompositionKS__exprChannel__s__Q,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Secret: secrecy_Secrets,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( localSecrets @ PQ ) )
=> ( ( member1967978193chanID @ Ch @ ( ins @ Q ) )
=> ( ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( specKeysSecrets @ PQ ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
& ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) ) ) ) ) ) ) ) ) ) ).
% correctCompositionKS_exprChannel_s_Q
thf(fact_223_correctCompositionKS__exprChannel__s__P,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Secret: secrecy_Secrets,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( localSecrets @ PQ ) )
=> ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
=> ( ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( specKeysSecrets @ PQ ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
& ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) ) ) ) ) ) ) ) ) ) ).
% correctCompositionKS_exprChannel_s_P
thf(fact_224_UnCI,axiom,
! [C: secrecy_specID,B: set_Secrecy_specID,A: set_Secrecy_specID] :
( ( ~ ( member740727610specID @ C @ B )
=> ( member740727610specID @ C @ A ) )
=> ( member740727610specID @ C @ ( sup_su1712599557specID @ A @ B ) ) ) ).
% UnCI
thf(fact_225_UnCI,axiom,
! [C: secrecy_Keys,B: set_Secrecy_Keys,A: set_Secrecy_Keys] :
( ( ~ ( member_Secrecy_Keys @ C @ B )
=> ( member_Secrecy_Keys @ C @ A ) )
=> ( member_Secrecy_Keys @ C @ ( sup_su552545507y_Keys @ A @ B ) ) ) ).
% UnCI
thf(fact_226_UnCI,axiom,
! [C: secrecy_Secrets,B: set_Secrecy_Secrets,A: set_Secrecy_Secrets] :
( ( ~ ( member2138796287ecrets @ C @ B )
=> ( member2138796287ecrets @ C @ A ) )
=> ( member2138796287ecrets @ C @ ( sup_su1029080330ecrets @ A @ B ) ) ) ).
% UnCI
thf(fact_227_UnCI,axiom,
! [C: secrecy_KS,B: set_Secrecy_KS,A: set_Secrecy_KS] :
( ( ~ ( member_Secrecy_KS @ C @ B )
=> ( member_Secrecy_KS @ C @ A ) )
=> ( member_Secrecy_KS @ C @ ( sup_su1301156759ecy_KS @ A @ B ) ) ) ).
% UnCI
thf(fact_228_UnCI,axiom,
! [C: secrecy_chanID,B: set_Secrecy_chanID,A: set_Secrecy_chanID] :
( ( ~ ( member1967978193chanID @ C @ B )
=> ( member1967978193chanID @ C @ A ) )
=> ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A @ B ) ) ) ).
% UnCI
thf(fact_229_Un__iff,axiom,
! [C: secrecy_specID,A: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( member740727610specID @ C @ ( sup_su1712599557specID @ A @ B ) )
= ( ( member740727610specID @ C @ A )
| ( member740727610specID @ C @ B ) ) ) ).
% Un_iff
thf(fact_230_Un__iff,axiom,
! [C: secrecy_Keys,A: set_Secrecy_Keys,B: set_Secrecy_Keys] :
( ( member_Secrecy_Keys @ C @ ( sup_su552545507y_Keys @ A @ B ) )
= ( ( member_Secrecy_Keys @ C @ A )
| ( member_Secrecy_Keys @ C @ B ) ) ) ).
% Un_iff
thf(fact_231_Un__iff,axiom,
! [C: secrecy_Secrets,A: set_Secrecy_Secrets,B: set_Secrecy_Secrets] :
( ( member2138796287ecrets @ C @ ( sup_su1029080330ecrets @ A @ B ) )
= ( ( member2138796287ecrets @ C @ A )
| ( member2138796287ecrets @ C @ B ) ) ) ).
% Un_iff
thf(fact_232_Un__iff,axiom,
! [C: secrecy_KS,A: set_Secrecy_KS,B: set_Secrecy_KS] :
( ( member_Secrecy_KS @ C @ ( sup_su1301156759ecy_KS @ A @ B ) )
= ( ( member_Secrecy_KS @ C @ A )
| ( member_Secrecy_KS @ C @ B ) ) ) ).
% Un_iff
thf(fact_233_Un__iff,axiom,
! [C: secrecy_chanID,A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A @ B ) )
= ( ( member1967978193chanID @ C @ A )
| ( member1967978193chanID @ C @ B ) ) ) ).
% Un_iff
thf(fact_234_KS_Oinject_I2_J,axiom,
! [X22: secrecy_Secrets,Y22: secrecy_Secrets] :
( ( ( secrecy_sKS @ X22 )
= ( secrecy_sKS @ Y22 ) )
= ( X22 = Y22 ) ) ).
% KS.inject(2)
thf(fact_235_Expression_Oinject_I2_J,axiom,
! [X22: secrecy_Secrets,Y22: secrecy_Secrets] :
( ( ( secrecy_sE @ X22 )
= ( secrecy_sE @ Y22 ) )
= ( X22 = Y22 ) ) ).
% Expression.inject(2)
thf(fact_236_Un__empty,axiom,
! [A: set_Secrecy_KS,B: set_Secrecy_KS] :
( ( ( sup_su1301156759ecy_KS @ A @ B )
= bot_bo1943466239ecy_KS )
= ( ( A = bot_bo1943466239ecy_KS )
& ( B = bot_bo1943466239ecy_KS ) ) ) ).
% Un_empty
thf(fact_237_Un__empty,axiom,
! [A: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( ( sup_su1712599557specID @ A @ B )
= bot_bo1785112429specID )
= ( ( A = bot_bo1785112429specID )
& ( B = bot_bo1785112429specID ) ) ) ).
% Un_empty
thf(fact_238_Un__empty,axiom,
! [A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( ( sup_su370684444chanID @ A @ B )
= bot_bo443197316chanID )
= ( ( A = bot_bo443197316chanID )
& ( B = bot_bo443197316chanID ) ) ) ).
% Un_empty
thf(fact_239_Un__insert__right,axiom,
! [A: set_Secrecy_specID,A2: secrecy_specID,B: set_Secrecy_specID] :
( ( sup_su1712599557specID @ A @ ( insert1671894227specID @ A2 @ B ) )
= ( insert1671894227specID @ A2 @ ( sup_su1712599557specID @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_240_Un__insert__right,axiom,
! [A: set_Secrecy_KS,A2: secrecy_KS,B: set_Secrecy_KS] :
( ( sup_su1301156759ecy_KS @ A @ ( insert_Secrecy_KS @ A2 @ B ) )
= ( insert_Secrecy_KS @ A2 @ ( sup_su1301156759ecy_KS @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_241_Un__insert__right,axiom,
! [A: set_Secrecy_chanID,A2: secrecy_chanID,B: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A @ ( insert751661162chanID @ A2 @ B ) )
= ( insert751661162chanID @ A2 @ ( sup_su370684444chanID @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_242_Un__insert__left,axiom,
! [A2: secrecy_specID,B: set_Secrecy_specID,C3: set_Secrecy_specID] :
( ( sup_su1712599557specID @ ( insert1671894227specID @ A2 @ B ) @ C3 )
= ( insert1671894227specID @ A2 @ ( sup_su1712599557specID @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_243_Un__insert__left,axiom,
! [A2: secrecy_KS,B: set_Secrecy_KS,C3: set_Secrecy_KS] :
( ( sup_su1301156759ecy_KS @ ( insert_Secrecy_KS @ A2 @ B ) @ C3 )
= ( insert_Secrecy_KS @ A2 @ ( sup_su1301156759ecy_KS @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_244_Un__insert__left,axiom,
! [A2: secrecy_chanID,B: set_Secrecy_chanID,C3: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ ( insert751661162chanID @ A2 @ B ) @ C3 )
= ( insert751661162chanID @ A2 @ ( sup_su370684444chanID @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_245_Un__empty__left,axiom,
! [B: set_Secrecy_KS] :
( ( sup_su1301156759ecy_KS @ bot_bo1943466239ecy_KS @ B )
= B ) ).
% Un_empty_left
thf(fact_246_Un__empty__left,axiom,
! [B: set_Secrecy_specID] :
( ( sup_su1712599557specID @ bot_bo1785112429specID @ B )
= B ) ).
% Un_empty_left
thf(fact_247_Un__empty__left,axiom,
! [B: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ bot_bo443197316chanID @ B )
= B ) ).
% Un_empty_left
thf(fact_248_Un__empty__right,axiom,
! [A: set_Secrecy_KS] :
( ( sup_su1301156759ecy_KS @ A @ bot_bo1943466239ecy_KS )
= A ) ).
% Un_empty_right
thf(fact_249_Un__empty__right,axiom,
! [A: set_Secrecy_specID] :
( ( sup_su1712599557specID @ A @ bot_bo1785112429specID )
= A ) ).
% Un_empty_right
thf(fact_250_Un__empty__right,axiom,
! [A: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A @ bot_bo443197316chanID )
= A ) ).
% Un_empty_right
thf(fact_251_not__know__s__not__ine,axiom,
! [A: secrecy_specID,M: secrecy_Secrets] :
( ~ ( knowle945212990e_know @ A @ ( secrecy_sKS @ M ) )
=> ~ ( ine @ A @ ( secrecy_sE @ M ) ) ) ).
% not_know_s_not_ine
thf(fact_252_UnE,axiom,
! [C: secrecy_specID,A: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( member740727610specID @ C @ ( sup_su1712599557specID @ A @ B ) )
=> ( ~ ( member740727610specID @ C @ A )
=> ( member740727610specID @ C @ B ) ) ) ).
% UnE
thf(fact_253_UnE,axiom,
! [C: secrecy_Keys,A: set_Secrecy_Keys,B: set_Secrecy_Keys] :
( ( member_Secrecy_Keys @ C @ ( sup_su552545507y_Keys @ A @ B ) )
=> ( ~ ( member_Secrecy_Keys @ C @ A )
=> ( member_Secrecy_Keys @ C @ B ) ) ) ).
% UnE
thf(fact_254_UnE,axiom,
! [C: secrecy_Secrets,A: set_Secrecy_Secrets,B: set_Secrecy_Secrets] :
( ( member2138796287ecrets @ C @ ( sup_su1029080330ecrets @ A @ B ) )
=> ( ~ ( member2138796287ecrets @ C @ A )
=> ( member2138796287ecrets @ C @ B ) ) ) ).
% UnE
thf(fact_255_UnE,axiom,
! [C: secrecy_KS,A: set_Secrecy_KS,B: set_Secrecy_KS] :
( ( member_Secrecy_KS @ C @ ( sup_su1301156759ecy_KS @ A @ B ) )
=> ( ~ ( member_Secrecy_KS @ C @ A )
=> ( member_Secrecy_KS @ C @ B ) ) ) ).
% UnE
thf(fact_256_UnE,axiom,
! [C: secrecy_chanID,A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A @ B ) )
=> ( ~ ( member1967978193chanID @ C @ A )
=> ( member1967978193chanID @ C @ B ) ) ) ).
% UnE
thf(fact_257_UnI1,axiom,
! [C: secrecy_specID,A: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( member740727610specID @ C @ A )
=> ( member740727610specID @ C @ ( sup_su1712599557specID @ A @ B ) ) ) ).
% UnI1
thf(fact_258_UnI1,axiom,
! [C: secrecy_Keys,A: set_Secrecy_Keys,B: set_Secrecy_Keys] :
( ( member_Secrecy_Keys @ C @ A )
=> ( member_Secrecy_Keys @ C @ ( sup_su552545507y_Keys @ A @ B ) ) ) ).
% UnI1
thf(fact_259_UnI1,axiom,
! [C: secrecy_Secrets,A: set_Secrecy_Secrets,B: set_Secrecy_Secrets] :
( ( member2138796287ecrets @ C @ A )
=> ( member2138796287ecrets @ C @ ( sup_su1029080330ecrets @ A @ B ) ) ) ).
% UnI1
thf(fact_260_UnI1,axiom,
! [C: secrecy_KS,A: set_Secrecy_KS,B: set_Secrecy_KS] :
( ( member_Secrecy_KS @ C @ A )
=> ( member_Secrecy_KS @ C @ ( sup_su1301156759ecy_KS @ A @ B ) ) ) ).
% UnI1
thf(fact_261_UnI1,axiom,
! [C: secrecy_chanID,A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( member1967978193chanID @ C @ A )
=> ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A @ B ) ) ) ).
% UnI1
thf(fact_262_UnI2,axiom,
! [C: secrecy_specID,B: set_Secrecy_specID,A: set_Secrecy_specID] :
( ( member740727610specID @ C @ B )
=> ( member740727610specID @ C @ ( sup_su1712599557specID @ A @ B ) ) ) ).
% UnI2
thf(fact_263_UnI2,axiom,
! [C: secrecy_Keys,B: set_Secrecy_Keys,A: set_Secrecy_Keys] :
( ( member_Secrecy_Keys @ C @ B )
=> ( member_Secrecy_Keys @ C @ ( sup_su552545507y_Keys @ A @ B ) ) ) ).
% UnI2
thf(fact_264_UnI2,axiom,
! [C: secrecy_Secrets,B: set_Secrecy_Secrets,A: set_Secrecy_Secrets] :
( ( member2138796287ecrets @ C @ B )
=> ( member2138796287ecrets @ C @ ( sup_su1029080330ecrets @ A @ B ) ) ) ).
% UnI2
thf(fact_265_UnI2,axiom,
! [C: secrecy_KS,B: set_Secrecy_KS,A: set_Secrecy_KS] :
( ( member_Secrecy_KS @ C @ B )
=> ( member_Secrecy_KS @ C @ ( sup_su1301156759ecy_KS @ A @ B ) ) ) ).
% UnI2
thf(fact_266_UnI2,axiom,
! [C: secrecy_chanID,B: set_Secrecy_chanID,A: set_Secrecy_chanID] :
( ( member1967978193chanID @ C @ B )
=> ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A @ B ) ) ) ).
% UnI2
thf(fact_267_bex__Un,axiom,
! [A: set_Secrecy_KS,B: set_Secrecy_KS,P: secrecy_KS > $o] :
( ( ? [X2: secrecy_KS] :
( ( member_Secrecy_KS @ X2 @ ( sup_su1301156759ecy_KS @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: secrecy_KS] :
( ( member_Secrecy_KS @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: secrecy_KS] :
( ( member_Secrecy_KS @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_268_bex__Un,axiom,
! [A: set_Secrecy_chanID,B: set_Secrecy_chanID,P: secrecy_chanID > $o] :
( ( ? [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ ( sup_su370684444chanID @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_269_ball__Un,axiom,
! [A: set_Secrecy_KS,B: set_Secrecy_KS,P: secrecy_KS > $o] :
( ( ! [X2: secrecy_KS] :
( ( member_Secrecy_KS @ X2 @ ( sup_su1301156759ecy_KS @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: secrecy_KS] :
( ( member_Secrecy_KS @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: secrecy_KS] :
( ( member_Secrecy_KS @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_270_ball__Un,axiom,
! [A: set_Secrecy_chanID,B: set_Secrecy_chanID,P: secrecy_chanID > $o] :
( ( ! [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ ( sup_su370684444chanID @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_271_Un__assoc,axiom,
! [A: set_Secrecy_KS,B: set_Secrecy_KS,C3: set_Secrecy_KS] :
( ( sup_su1301156759ecy_KS @ ( sup_su1301156759ecy_KS @ A @ B ) @ C3 )
= ( sup_su1301156759ecy_KS @ A @ ( sup_su1301156759ecy_KS @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_272_Un__assoc,axiom,
! [A: set_Secrecy_chanID,B: set_Secrecy_chanID,C3: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ ( sup_su370684444chanID @ A @ B ) @ C3 )
= ( sup_su370684444chanID @ A @ ( sup_su370684444chanID @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_273_Un__absorb,axiom,
! [A: set_Secrecy_KS] :
( ( sup_su1301156759ecy_KS @ A @ A )
= A ) ).
% Un_absorb
thf(fact_274_Un__absorb,axiom,
! [A: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A @ A )
= A ) ).
% Un_absorb
thf(fact_275_Un__commute,axiom,
( sup_su1301156759ecy_KS
= ( ^ [A3: set_Secrecy_KS,B4: set_Secrecy_KS] : ( sup_su1301156759ecy_KS @ B4 @ A3 ) ) ) ).
% Un_commute
thf(fact_276_Un__commute,axiom,
( sup_su370684444chanID
= ( ^ [A3: set_Secrecy_chanID,B4: set_Secrecy_chanID] : ( sup_su370684444chanID @ B4 @ A3 ) ) ) ).
% Un_commute
thf(fact_277_Un__left__absorb,axiom,
! [A: set_Secrecy_KS,B: set_Secrecy_KS] :
( ( sup_su1301156759ecy_KS @ A @ ( sup_su1301156759ecy_KS @ A @ B ) )
= ( sup_su1301156759ecy_KS @ A @ B ) ) ).
% Un_left_absorb
thf(fact_278_Un__left__absorb,axiom,
! [A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A @ ( sup_su370684444chanID @ A @ B ) )
= ( sup_su370684444chanID @ A @ B ) ) ).
% Un_left_absorb
thf(fact_279_Un__left__commute,axiom,
! [A: set_Secrecy_KS,B: set_Secrecy_KS,C3: set_Secrecy_KS] :
( ( sup_su1301156759ecy_KS @ A @ ( sup_su1301156759ecy_KS @ B @ C3 ) )
= ( sup_su1301156759ecy_KS @ B @ ( sup_su1301156759ecy_KS @ A @ C3 ) ) ) ).
% Un_left_commute
thf(fact_280_Un__left__commute,axiom,
! [A: set_Secrecy_chanID,B: set_Secrecy_chanID,C3: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A @ ( sup_su370684444chanID @ B @ C3 ) )
= ( sup_su370684444chanID @ B @ ( sup_su370684444chanID @ A @ C3 ) ) ) ).
% Un_left_commute
thf(fact_281_KS2Expression_Osimps_I2_J,axiom,
! [M: secrecy_Secrets] :
( ( secrec1016742679ession @ ( secrecy_sKS @ M ) )
= ( secrecy_sE @ M ) ) ).
% KS2Expression.simps(2)
thf(fact_282_KS_Odistinct_I1_J,axiom,
! [X1: secrecy_Keys,X22: secrecy_Secrets] :
( ( secrecy_kKS @ X1 )
!= ( secrecy_sKS @ X22 ) ) ).
% KS.distinct(1)
thf(fact_283_KS_Oinduct,axiom,
! [P: secrecy_KS > $o,KS: secrecy_KS] :
( ! [X3: secrecy_Keys] : ( P @ ( secrecy_kKS @ X3 ) )
=> ( ! [X3: secrecy_Secrets] : ( P @ ( secrecy_sKS @ X3 ) )
=> ( P @ KS ) ) ) ).
% KS.induct
thf(fact_284_KS_Oexhaust,axiom,
! [Y2: secrecy_KS] :
( ! [X12: secrecy_Keys] :
( Y2
!= ( secrecy_kKS @ X12 ) )
=> ~ ! [X23: secrecy_Secrets] :
( Y2
!= ( secrecy_sKS @ X23 ) ) ) ).
% KS.exhaust
thf(fact_285_Expression_Odistinct_I1_J,axiom,
! [X1: secrecy_Keys,X22: secrecy_Secrets] :
( ( secrecy_kE @ X1 )
!= ( secrecy_sE @ X22 ) ) ).
% Expression.distinct(1)
thf(fact_286_know_Osimps_I2_J,axiom,
! [A: secrecy_specID,M: secrecy_Secrets] :
( ( knowle945212990e_know @ A @ ( secrecy_sKS @ M ) )
= ( ( ine @ A @ ( secrecy_sE @ M ) )
| ( member_Secrecy_KS @ ( secrecy_sKS @ M ) @ ( localSecrets @ A ) ) ) ) ).
% know.simps(2)
thf(fact_287_singleton__Un__iff,axiom,
! [X: secrecy_KS,A: set_Secrecy_KS,B: set_Secrecy_KS] :
( ( ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS )
= ( sup_su1301156759ecy_KS @ A @ B ) )
= ( ( ( A = bot_bo1943466239ecy_KS )
& ( B
= ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) ) )
| ( ( A
= ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) )
& ( B = bot_bo1943466239ecy_KS ) )
| ( ( A
= ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) )
& ( B
= ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_288_singleton__Un__iff,axiom,
! [X: secrecy_specID,A: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( ( insert1671894227specID @ X @ bot_bo1785112429specID )
= ( sup_su1712599557specID @ A @ B ) )
= ( ( ( A = bot_bo1785112429specID )
& ( B
= ( insert1671894227specID @ X @ bot_bo1785112429specID ) ) )
| ( ( A
= ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
& ( B = bot_bo1785112429specID ) )
| ( ( A
= ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
& ( B
= ( insert1671894227specID @ X @ bot_bo1785112429specID ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_289_singleton__Un__iff,axiom,
! [X: secrecy_chanID,A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( ( insert751661162chanID @ X @ bot_bo443197316chanID )
= ( sup_su370684444chanID @ A @ B ) )
= ( ( ( A = bot_bo443197316chanID )
& ( B
= ( insert751661162chanID @ X @ bot_bo443197316chanID ) ) )
| ( ( A
= ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
& ( B = bot_bo443197316chanID ) )
| ( ( A
= ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
& ( B
= ( insert751661162chanID @ X @ bot_bo443197316chanID ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_290_Un__singleton__iff,axiom,
! [A: set_Secrecy_KS,B: set_Secrecy_KS,X: secrecy_KS] :
( ( ( sup_su1301156759ecy_KS @ A @ B )
= ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) )
= ( ( ( A = bot_bo1943466239ecy_KS )
& ( B
= ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) ) )
| ( ( A
= ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) )
& ( B = bot_bo1943466239ecy_KS ) )
| ( ( A
= ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) )
& ( B
= ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_291_Un__singleton__iff,axiom,
! [A: set_Secrecy_specID,B: set_Secrecy_specID,X: secrecy_specID] :
( ( ( sup_su1712599557specID @ A @ B )
= ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
= ( ( ( A = bot_bo1785112429specID )
& ( B
= ( insert1671894227specID @ X @ bot_bo1785112429specID ) ) )
| ( ( A
= ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
& ( B = bot_bo1785112429specID ) )
| ( ( A
= ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
& ( B
= ( insert1671894227specID @ X @ bot_bo1785112429specID ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_292_Un__singleton__iff,axiom,
! [A: set_Secrecy_chanID,B: set_Secrecy_chanID,X: secrecy_chanID] :
( ( ( sup_su370684444chanID @ A @ B )
= ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
= ( ( ( A = bot_bo443197316chanID )
& ( B
= ( insert751661162chanID @ X @ bot_bo443197316chanID ) ) )
| ( ( A
= ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
& ( B = bot_bo443197316chanID ) )
| ( ( A
= ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
& ( B
= ( insert751661162chanID @ X @ bot_bo443197316chanID ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_293_insert__is__Un,axiom,
( insert_Secrecy_KS
= ( ^ [A4: secrecy_KS] : ( sup_su1301156759ecy_KS @ ( insert_Secrecy_KS @ A4 @ bot_bo1943466239ecy_KS ) ) ) ) ).
% insert_is_Un
thf(fact_294_insert__is__Un,axiom,
( insert1671894227specID
= ( ^ [A4: secrecy_specID] : ( sup_su1712599557specID @ ( insert1671894227specID @ A4 @ bot_bo1785112429specID ) ) ) ) ).
% insert_is_Un
thf(fact_295_insert__is__Un,axiom,
( insert751661162chanID
= ( ^ [A4: secrecy_chanID] : ( sup_su370684444chanID @ ( insert751661162chanID @ A4 @ bot_bo443197316chanID ) ) ) ) ).
% insert_is_Un
thf(fact_296_LocalSecretsComposition__neg__loc__s,axiom,
! [Secret: secrecy_Secrets,P: secrecy_specID,Ch: secrecy_chanID] :
( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( localSecrets @ P ) )
=> ( ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( specKeysSecrets @ P ) )
=> ~ ( member1967978193chanID @ Ch @ ( loc @ P ) ) ) ) ) ).
% LocalSecretsComposition_neg_loc_s
thf(fact_297_LocalSecretsComposition__neg__s,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,M: secrecy_Secrets] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc965049635ionLoc @ PQ )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ M ) @ ( specKeysSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ M ) @ ( specKeysSecrets @ Q ) )
=> ( ~ ( ine @ P @ ( secrecy_sE @ M ) )
=> ( ~ ( ine @ Q @ ( secrecy_sE @ M ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ M ) @ ( sup_su1301156759ecy_KS @ ( localSecrets @ P ) @ ( localSecrets @ Q ) ) )
=> ~ ( member_Secrecy_KS @ ( secrecy_sKS @ M ) @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ) ) ).
% LocalSecretsComposition_neg_s
thf(fact_298_know__composition__neg__ine__s,axiom,
! [P: secrecy_specID,Secret: secrecy_Secrets,Q: secrecy_specID,PQ: secrecy_specID] :
( ~ ( knowle945212990e_know @ P @ ( secrecy_sKS @ Secret ) )
=> ( ~ ( knowle945212990e_know @ Q @ ( secrecy_sKS @ Secret ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ~ ( ine @ PQ @ ( secrecy_sE @ Secret ) ) ) ) ) ) ).
% know_composition_neg_ine_s
thf(fact_299_LocalSecretsComposition__neg1__s,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,S: secrecy_Secrets] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc965049635ionLoc @ PQ )
=> ( ~ ( ine @ P @ ( secrecy_sE @ S ) )
=> ( ~ ( ine @ Q @ ( secrecy_sE @ S ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ Q ) )
=> ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ).
% LocalSecretsComposition_neg1_s
thf(fact_300_LocalSecretsComposition__ine2__s,axiom,
! [S: secrecy_Secrets,PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID] :
( ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ PQ ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc965049635ionLoc @ PQ )
=> ( ~ ( ine @ P @ ( secrecy_sE @ S ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ Q ) )
=> ( ine @ Q @ ( secrecy_sE @ S ) ) ) ) ) ) ) ) ).
% LocalSecretsComposition_ine2_s
thf(fact_301_LocalSecretsComposition__ine1__s,axiom,
! [S: secrecy_Secrets,PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID] :
( ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ PQ ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc965049635ionLoc @ PQ )
=> ( ~ ( ine @ Q @ ( secrecy_sE @ S ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ Q ) )
=> ( ine @ P @ ( secrecy_sE @ S ) ) ) ) ) ) ) ) ).
% LocalSecretsComposition_ine1_s
thf(fact_302_LocalSecretsComposition__exprChannel__s,axiom,
! [X: secrecy_chanID,Secrets: secrecy_Secrets,P: secrecy_specID,Q: secrecy_specID] :
( ( exprChannel @ X @ ( secrecy_sE @ Secrets ) )
=> ( ~ ( ine @ P @ ( secrecy_sE @ Secrets ) )
=> ( ~ ( ine @ Q @ ( secrecy_sE @ Secrets ) )
=> ( ~ ( member1967978193chanID @ X @ ( ins @ P ) )
& ~ ( member1967978193chanID @ X @ ( ins @ Q ) ) ) ) ) ) ).
% LocalSecretsComposition_exprChannel_s
thf(fact_303_LocalSecretsComposition__neg,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc965049635ionLoc @ PQ )
=> ( ( correctCompositionKS @ PQ )
=> ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ Q ) )
=> ( ! [M4: secrecy_Keys] :
( ( Ks
= ( secrecy_kKS @ M4 ) )
=> ( ~ ( ine @ P @ ( secrecy_kE @ M4 ) )
& ~ ( ine @ Q @ ( secrecy_kE @ M4 ) ) ) )
=> ( ! [M4: secrecy_Secrets] :
( ( Ks
= ( secrecy_sKS @ M4 ) )
=> ( ~ ( ine @ P @ ( secrecy_sE @ M4 ) )
& ~ ( ine @ Q @ ( secrecy_sE @ M4 ) ) ) )
=> ( ~ ( member_Secrecy_KS @ Ks @ ( sup_su1301156759ecy_KS @ ( localSecrets @ P ) @ ( localSecrets @ Q ) ) )
=> ~ ( member_Secrecy_KS @ Ks @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ) ) ).
% LocalSecretsComposition_neg
thf(fact_304_LocalSecretsComposition__neg1,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc965049635ionLoc @ PQ )
=> ( ! [M4: secrecy_Keys] :
( ( Ks
= ( secrecy_kKS @ M4 ) )
=> ( ~ ( ine @ P @ ( secrecy_kE @ M4 ) )
& ~ ( ine @ Q @ ( secrecy_kE @ M4 ) ) ) )
=> ( ! [M4: secrecy_Secrets] :
( ( Ks
= ( secrecy_sKS @ M4 ) )
=> ( ~ ( ine @ P @ ( secrecy_sE @ M4 ) )
& ~ ( ine @ Q @ ( secrecy_sE @ M4 ) ) ) )
=> ( ~ ( member_Secrecy_KS @ Ks @ ( localSecrets @ P ) )
=> ( ~ ( member_Secrecy_KS @ Ks @ ( localSecrets @ Q ) )
=> ~ ( member_Secrecy_KS @ Ks @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ).
% LocalSecretsComposition_neg1
thf(fact_305_sup__bot_Oright__neutral,axiom,
! [A2: set_Secrecy_KS] :
( ( sup_su1301156759ecy_KS @ A2 @ bot_bo1943466239ecy_KS )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_306_sup__bot_Oright__neutral,axiom,
! [A2: set_Secrecy_specID] :
( ( sup_su1712599557specID @ A2 @ bot_bo1785112429specID )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_307_sup__bot_Oright__neutral,axiom,
! [A2: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A2 @ bot_bo443197316chanID )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_308_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Secrecy_KS,B2: set_Secrecy_KS] :
( ( bot_bo1943466239ecy_KS
= ( sup_su1301156759ecy_KS @ A2 @ B2 ) )
= ( ( A2 = bot_bo1943466239ecy_KS )
& ( B2 = bot_bo1943466239ecy_KS ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_309_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Secrecy_specID,B2: set_Secrecy_specID] :
( ( bot_bo1785112429specID
= ( sup_su1712599557specID @ A2 @ B2 ) )
= ( ( A2 = bot_bo1785112429specID )
& ( B2 = bot_bo1785112429specID ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_310_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Secrecy_chanID,B2: set_Secrecy_chanID] :
( ( bot_bo443197316chanID
= ( sup_su370684444chanID @ A2 @ B2 ) )
= ( ( A2 = bot_bo443197316chanID )
& ( B2 = bot_bo443197316chanID ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_311_sup__bot_Oleft__neutral,axiom,
! [A2: set_Secrecy_KS] :
( ( sup_su1301156759ecy_KS @ bot_bo1943466239ecy_KS @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_312_sup__bot_Oleft__neutral,axiom,
! [A2: set_Secrecy_specID] :
( ( sup_su1712599557specID @ bot_bo1785112429specID @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_313_sup__bot_Oleft__neutral,axiom,
! [A2: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ bot_bo443197316chanID @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_314_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Secrecy_specID,B2: set_Secrecy_specID] :
( ( ( sup_su1712599557specID @ A2 @ B2 )
= bot_bo1785112429specID )
= ( ( A2 = bot_bo1785112429specID )
& ( B2 = bot_bo1785112429specID ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_315_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Secrecy_chanID,B2: set_Secrecy_chanID] :
( ( ( sup_su370684444chanID @ A2 @ B2 )
= bot_bo443197316chanID )
= ( ( A2 = bot_bo443197316chanID )
& ( B2 = bot_bo443197316chanID ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_316_ineM__Un1,axiom,
! [P: secrecy_specID,A: set_Secrecy_chanID,E2: secrecy_Expression,B: set_Secrecy_chanID] :
( ( ineM @ P @ A @ E2 )
=> ( ineM @ P @ ( sup_su370684444chanID @ A @ B ) @ E2 ) ) ).
% ineM_Un1
thf(fact_317_notSpecKeysSecretsExpr__def,axiom,
( notSpe1586871396tsExpr
= ( ^ [P2: secrecy_specID,E4: list_S529945141ession] :
( ! [X2: secrecy_Keys] :
( ( mem_Se1967555118ession @ ( secrecy_kE @ X2 ) @ E4 )
=> ~ ( member_Secrecy_KS @ ( secrecy_kKS @ X2 ) @ ( specKeysSecrets @ P2 ) ) )
& ! [Y4: secrecy_Secrets] :
( ( mem_Se1967555118ession @ ( secrecy_sE @ Y4 ) @ E4 )
=> ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Y4 ) @ ( specKeysSecrets @ P2 ) ) ) ) ) ) ).
% notSpecKeysSecretsExpr_def
thf(fact_318_TBtheorem5a__empty,axiom,
! [P: secrecy_specID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( eout @ P @ E2 )
| ( eout @ Q @ E2 ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( ( loc @ PQ )
= bot_bo443197316chanID )
=> ( eout @ PQ @ E2 ) ) ) ) ) ).
% TBtheorem5a_empty
thf(fact_319_correctCompositionKS__subcomp1,axiom,
! [C3: secrecy_specID,X: secrecy_specID,Xa: secrecy_Keys] :
( ( correctCompositionKS @ C3 )
=> ( ( member740727610specID @ X @ ( subcomponents @ C3 ) )
=> ( ( member_Secrecy_Keys @ Xa @ ( specKeys @ C3 ) )
=> ? [X3: secrecy_specID] :
( ( member740727610specID @ X3 @ ( subcomponents @ C3 ) )
& ( member_Secrecy_Keys @ Xa @ ( specKeys @ X3 ) ) ) ) ) ) ).
% correctCompositionKS_subcomp1
thf(fact_320_TBtheorem2a,axiom,
! [PQ: secrecy_specID,E2: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
( ( eout @ PQ @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( eout @ P @ E2 )
| ( eout @ Q @ E2 ) ) ) ) ) ).
% TBtheorem2a
thf(fact_321_correctCompositionKS__subcomp3,axiom,
! [C3: secrecy_specID,X: secrecy_specID,Xa: secrecy_Keys] :
( ( correctCompositionKS @ C3 )
=> ( ( member740727610specID @ X @ ( subcomponents @ C3 ) )
=> ( ( member_Secrecy_Keys @ Xa @ ( specKeys @ X ) )
=> ( member_Secrecy_Keys @ Xa @ ( specKeys @ C3 ) ) ) ) ) ).
% correctCompositionKS_subcomp3
thf(fact_322_TBtheorem5a__notP1,axiom,
! [P: secrecy_specID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( eout @ P @ E2 )
=> ( ~ ( eout @ Q @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( out_ex985543062Single @ P @ Ch2 @ E2 )
& ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ~ ( eout @ PQ @ E2 ) ) ) ) ) ) ).
% TBtheorem5a_notP1
thf(fact_323_TBtheorem5a__notPQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set_Secrecy_chanID,E2: secrecy_Expression,ChSetQ: set_Secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( out_exprChannelSet @ P @ ChSetP @ E2 )
=> ( ( out_exprChannelSet @ Q @ ChSetQ @ E2 )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSetP )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSetQ )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ~ ( eout @ PQ @ E2 ) ) ) ) ) ) ) ).
% TBtheorem5a_notPQ
thf(fact_324_TBtheorem5a__notP2,axiom,
! [Q: secrecy_specID,E2: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set_Secrecy_chanID] :
( ~ ( eout @ Q @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( out_exprChannelSet @ P @ ChSet @ E2 )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSet )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ~ ( eout @ PQ @ E2 ) ) ) ) ) ) ).
% TBtheorem5a_notP2
thf(fact_325_out__exprChannelSingle__Set,axiom,
! [P: secrecy_specID,Ch: secrecy_chanID,E2: secrecy_Expression] :
( ( out_ex985543062Single @ P @ Ch @ E2 )
=> ( out_exprChannelSet @ P @ ( insert751661162chanID @ Ch @ bot_bo443197316chanID ) @ E2 ) ) ).
% out_exprChannelSingle_Set
thf(fact_326_out__exprChannelSet__Single,axiom,
! [P: secrecy_specID,Ch: secrecy_chanID,E2: secrecy_Expression] :
( ( out_exprChannelSet @ P @ ( insert751661162chanID @ Ch @ bot_bo443197316chanID ) @ E2 )
=> ( out_ex985543062Single @ P @ Ch @ E2 ) ) ).
% out_exprChannelSet_Single
thf(fact_327_TBtheorem5b__notPQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set_Secrecy_chanID,E2: secrecy_Expression,ChSetQ: set_Secrecy_chanID,M2: set_Secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( out_exprChannelSet @ P @ ChSetP @ E2 )
=> ( ( out_exprChannelSet @ Q @ ChSetQ @ E2 )
=> ( ( M2
= ( sup_su370684444chanID @ ChSetP @ ChSetQ ) )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSetP )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSetQ )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ~ ( eoutM @ PQ @ M2 @ E2 ) ) ) ) ) ) ) ) ).
% TBtheorem5b_notPQ
thf(fact_328_eout__know__k,axiom,
! [C3: secrecy_specID,M: secrecy_Keys] :
( ( knowle233900195orrect @ C3 @ ( secrecy_kKS @ M ) )
= ( ( eout @ C3 @ ( secrecy_kE @ M ) )
= ( ( member_Secrecy_Keys @ M @ ( specKeys @ C3 ) )
| ( knowle945212990e_know @ C3 @ ( secrecy_kKS @ M ) ) ) ) ) ).
% eout_know_k
thf(fact_329_not__know__k__not__eout,axiom,
! [M: secrecy_Keys,A: secrecy_specID] :
( ~ ( member_Secrecy_Keys @ M @ ( specKeys @ A ) )
=> ( ~ ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) )
=> ( ( knowle233900195orrect @ A @ ( secrecy_kKS @ M ) )
=> ~ ( eout @ A @ ( secrecy_kE @ M ) ) ) ) ) ).
% not_know_k_not_eout
thf(fact_330_not__eout__eoutM,axiom,
! [P: secrecy_specID,E2: secrecy_Expression,M2: set_Secrecy_chanID] :
( ~ ( eout @ P @ E2 )
=> ~ ( eoutM @ P @ M2 @ E2 ) ) ).
% not_eout_eoutM
thf(fact_331_eoutM__eout,axiom,
! [P: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression] :
( ( eoutM @ P @ M2 @ E2 )
=> ( eout @ P @ E2 ) ) ).
% eoutM_eout
thf(fact_332_TBtheorem2b,axiom,
! [PQ: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
( ( eoutM @ PQ @ M2 @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( eoutM @ P @ M2 @ E2 )
| ( eoutM @ Q @ M2 @ E2 ) ) ) ) ) ).
% TBtheorem2b
thf(fact_333_TBtheorem5b__notP2,axiom,
! [Q: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set_Secrecy_chanID] :
( ~ ( eoutM @ Q @ M2 @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( out_exprChannelSet @ P @ ChSet @ E2 )
=> ( ! [X3: secrecy_chanID] :
( ( member1967978193chanID @ X3 @ ChSet )
=> ( member1967978193chanID @ X3 @ ( loc @ PQ ) ) )
=> ~ ( eoutM @ PQ @ M2 @ E2 ) ) ) ) ) ) ).
% TBtheorem5b_notP2
thf(fact_334_TBtheorem5b__notP1,axiom,
! [P: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( eoutM @ P @ M2 @ E2 )
=> ( ~ ( eoutM @ Q @ M2 @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( out_ex985543062Single @ P @ Ch2 @ E2 )
& ( member1967978193chanID @ Ch2 @ M2 )
& ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ~ ( eoutM @ PQ @ M2 @ E2 ) ) ) ) ) ) ).
% TBtheorem5b_notP1
thf(fact_335_eout__know__nonKS__k,axiom,
! [M: secrecy_Keys,A: secrecy_specID] :
( ~ ( member_Secrecy_Keys @ M @ ( specKeys @ A ) )
=> ( ( eout @ A @ ( secrecy_kE @ M ) )
=> ( ( knowle233900195orrect @ A @ ( secrecy_kKS @ M ) )
=> ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) ) ) ) ) ).
% eout_know_nonKS_k
thf(fact_336_eoutKnowCorrect__L1k,axiom,
! [C3: secrecy_specID,M: secrecy_Keys] :
( ( knowle233900195orrect @ C3 @ ( secrecy_kKS @ M ) )
=> ( ( eout @ C3 @ ( secrecy_kE @ M ) )
=> ( ( member_Secrecy_Keys @ M @ ( specKeys @ C3 ) )
| ( knowle945212990e_know @ C3 @ ( secrecy_kKS @ M ) ) ) ) ) ).
% eoutKnowCorrect_L1k
thf(fact_337_TBtheore54b__P,axiom,
! [P: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
( ( eoutM @ P @ M2 @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( member1967978193chanID @ Ch2 @ ( out @ Q ) )
& ( exprChannel @ Ch2 @ E2 )
& ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) )
& ( member1967978193chanID @ Ch2 @ M2 ) )
=> ( eoutM @ PQ @ M2 @ E2 ) ) ) ) ) ).
% TBtheore54b_P
thf(fact_338_TBtheorem5b__PQ,axiom,
! [P: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( eoutM @ P @ M2 @ E2 )
| ( eoutM @ Q @ M2 @ E2 ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ( member1967978193chanID @ Ch2 @ ( out @ P ) )
| ( member1967978193chanID @ Ch2 @ ( out @ Q ) ) )
& ( member1967978193chanID @ Ch2 @ M2 )
& ( exprChannel @ Ch2 @ E2 )
& ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( eoutM @ PQ @ M2 @ E2 ) ) ) ) ) ).
% TBtheorem5b_PQ
thf(fact_339_TBtheorem5a__PQ,axiom,
! [P: secrecy_specID,E2: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( eout @ P @ E2 )
| ( eout @ Q @ E2 ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ( member1967978193chanID @ Ch2 @ ( out @ P ) )
| ( member1967978193chanID @ Ch2 @ ( out @ Q ) ) )
& ( exprChannel @ Ch2 @ E2 )
& ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( eout @ PQ @ E2 ) ) ) ) ) ).
% TBtheorem5a_PQ
thf(fact_340_eout__def,axiom,
( eout
= ( ^ [SP: secrecy_specID,E3: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member1967978193chanID @ Ch3 @ ( out @ SP ) )
& ( exprChannel @ Ch3 @ E3 ) ) ) ) ).
% eout_def
thf(fact_341_eoutM__def,axiom,
( eoutM
= ( ^ [SP: secrecy_specID,M3: set_Secrecy_chanID,E3: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member1967978193chanID @ Ch3 @ ( out @ SP ) )
& ( member1967978193chanID @ Ch3 @ M3 )
& ( exprChannel @ Ch3 @ E3 ) ) ) ) ).
% eoutM_def
thf(fact_342_out__exprChannelSet__def,axiom,
( out_exprChannelSet
= ( ^ [SP: secrecy_specID,ChSet2: set_Secrecy_chanID,E3: secrecy_Expression] :
( ! [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ ChSet2 )
=> ( ( member1967978193chanID @ X2 @ ( out @ SP ) )
& ( exprChannel @ X2 @ E3 ) ) )
& ! [X2: secrecy_chanID] :
( ( ~ ( member1967978193chanID @ X2 @ ChSet2 )
& ( member1967978193chanID @ X2 @ ( out @ SP ) ) )
=> ~ ( exprChannel @ X2 @ E3 ) ) ) ) ) ).
% out_exprChannelSet_def
thf(fact_343_out__exprChannelSingle__def,axiom,
( out_ex985543062Single
= ( ^ [SP: secrecy_specID,Ch3: secrecy_chanID,E3: secrecy_Expression] :
( ( member1967978193chanID @ Ch3 @ ( out @ SP ) )
& ( exprChannel @ Ch3 @ E3 )
& ! [X2: secrecy_chanID,T: nat] :
( ( ( member1967978193chanID @ X2 @ ( out @ SP ) )
& ( X2 != Ch3 ) )
=> ~ ( exprChannel @ X2 @ E3 ) ) ) ) ) ).
% out_exprChannelSingle_def
thf(fact_344_correctCompositionOut__prop1,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,X: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( member1967978193chanID @ X @ ( out @ PQ ) )
=> ( ( member1967978193chanID @ X @ ( out @ P ) )
| ( member1967978193chanID @ X @ ( out @ Q ) ) ) ) ) ) ).
% correctCompositionOut_prop1
thf(fact_345_TBtheorem45a__P,axiom,
! [P: secrecy_specID,E2: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
( ( eout @ P @ E2 )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( member1967978193chanID @ Ch2 @ ( out @ P ) )
& ( exprChannel @ Ch2 @ E2 )
& ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( eout @ PQ @ E2 ) ) ) ) ) ).
% TBtheorem45a_P
thf(fact_346_outStream__def,axiom,
( outStream
= ( ^ [X2: secrecy_specID] :
( ^ [Y3: set_Secrecy_chanID,Z: set_Secrecy_chanID] : Y3 = Z
@ ( out @ X2 ) ) ) ) ).
% outStream_def
thf(fact_347_eout__know__nonKS__s,axiom,
! [M: secrecy_Secrets,A: secrecy_specID] :
( ~ ( member2138796287ecrets @ M @ ( specSecrets @ A ) )
=> ( ( eout @ A @ ( secrecy_sE @ M ) )
=> ( ( knowle233900195orrect @ A @ ( secrecy_sKS @ M ) )
=> ( knowle945212990e_know @ A @ ( secrecy_sKS @ M ) ) ) ) ) ).
% eout_know_nonKS_s
thf(fact_348_correctCompositionKS__subcomp4,axiom,
! [C3: secrecy_specID,X: secrecy_specID,Xa: secrecy_Secrets] :
( ( correctCompositionKS @ C3 )
=> ( ( member740727610specID @ X @ ( subcomponents @ C3 ) )
=> ( ( member2138796287ecrets @ Xa @ ( specSecrets @ X ) )
=> ( member2138796287ecrets @ Xa @ ( specSecrets @ C3 ) ) ) ) ) ).
% correctCompositionKS_subcomp4
thf(fact_349_correctCompositionKS__subcomp2,axiom,
! [C3: secrecy_specID,X: secrecy_specID,Xa: secrecy_Secrets] :
( ( correctCompositionKS @ C3 )
=> ( ( member740727610specID @ X @ ( subcomponents @ C3 ) )
=> ( ( member2138796287ecrets @ Xa @ ( specSecrets @ C3 ) )
=> ? [X3: secrecy_specID] :
( ( member740727610specID @ X3 @ ( subcomponents @ C3 ) )
& ( member2138796287ecrets @ Xa @ ( specSecrets @ X3 ) ) ) ) ) ) ).
% correctCompositionKS_subcomp2
thf(fact_350_eout__know__s,axiom,
! [C3: secrecy_specID,M: secrecy_Secrets] :
( ( knowle233900195orrect @ C3 @ ( secrecy_sKS @ M ) )
= ( ( eout @ C3 @ ( secrecy_sE @ M ) )
= ( ( member2138796287ecrets @ M @ ( specSecrets @ C3 ) )
| ( knowle945212990e_know @ C3 @ ( secrecy_sKS @ M ) ) ) ) ) ).
% eout_know_s
thf(fact_351_not__know__s__not__eout,axiom,
! [M: secrecy_Secrets,A: secrecy_specID] :
( ~ ( member2138796287ecrets @ M @ ( specSecrets @ A ) )
=> ( ~ ( knowle945212990e_know @ A @ ( secrecy_sKS @ M ) )
=> ( ( knowle233900195orrect @ A @ ( secrecy_sKS @ M ) )
=> ~ ( eout @ A @ ( secrecy_sE @ M ) ) ) ) ) ).
% not_know_s_not_eout
thf(fact_352_eoutKnowCorrect__L1s,axiom,
! [C3: secrecy_specID,M: secrecy_Secrets] :
( ( knowle233900195orrect @ C3 @ ( secrecy_sKS @ M ) )
=> ( ( eout @ C3 @ ( secrecy_sE @ M ) )
=> ( ( member2138796287ecrets @ M @ ( specSecrets @ C3 ) )
| ( knowle945212990e_know @ C3 @ ( secrecy_sKS @ M ) ) ) ) ) ).
% eoutKnowCorrect_L1s
thf(fact_353_adv__not__know2,axiom,
! [P: secrecy_specID,A: secrecy_specID,M: secrecy_Secrets] :
( ( ord_le1802272080chanID @ ( out @ P ) @ ( ins @ A ) )
=> ( ~ ( knowle945212990e_know @ A @ ( secrecy_sKS @ M ) )
=> ~ ( eout @ P @ ( secrecy_sE @ M ) ) ) ) ).
% adv_not_know2
thf(fact_354_adv__not__know1,axiom,
! [P: secrecy_specID,A: secrecy_specID,M: secrecy_Keys] :
( ( ord_le1802272080chanID @ ( out @ P ) @ ( ins @ A ) )
=> ( ~ ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) )
=> ~ ( eout @ P @ ( secrecy_kE @ M ) ) ) ) ).
% adv_not_know1
% Conjectures (1)
thf(conj_0,conjecture,
~ ( member_Secrecy_KS @ ( secrecy_kKS @ key ) @ ( localSecrets @ q ) ) ).
%------------------------------------------------------------------------------