TPTP Problem File: SWV735_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWV735_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Needham-Schroeder shared-key protocol line 218
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : ns_218 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 178 ( 66 unt; 50 typ; 0 def)
% Number of atoms : 232 ( 112 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 179 ( 75 ~; 25 |; 16 &)
% ( 20 <=>; 43 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 32 ( 21 >; 11 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 44 ( 44 usr; 21 con; 0-5 aty)
% Number of variables : 355 ( 334 !; 2 ?; 355 :)
% ( 19 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:15:46
%------------------------------------------------------------------------------
%----Should-be-implicit typings (7)
tff(ty_tc_Event_Oevent,type,
event: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_Message_Oagent,type,
agent1: $tType ).
tff(ty_tc_Message_Omsg,type,
msg: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (43)
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Event_Obad,type,
bad: fun(agent1,bool) ).
tff(sy_c_Event_Oevent_OSays,type,
says: ( agent1 * agent1 * msg ) > event ).
tff(sy_c_Event_Oknows,type,
knows: ( agent1 * list(event) ) > fun(msg,bool) ).
tff(sy_c_Event_Oused,type,
used: list(event) > fun(msg,bool) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : ( list(A) > fun(A,bool) ) ).
tff(sy_c_Message_Oagent_OServer,type,
server: agent1 ).
tff(sy_c_Message_Oagent_OSpy,type,
spy: agent1 ).
tff(sy_c_Message_Omsg_OAgent,type,
agent: agent1 > msg ).
tff(sy_c_Message_Omsg_OCrypt,type,
crypt: ( nat * msg ) > msg ).
tff(sy_c_Message_Omsg_OKey,type,
key: nat > msg ).
tff(sy_c_Message_Omsg_OMPair,type,
mPair: ( msg * msg ) > msg ).
tff(sy_c_Message_Omsg_ONonce,type,
nonce: nat > msg ).
tff(sy_c_Message_Oparts,type,
parts: fun(msg,bool) > fun(msg,bool) ).
tff(sy_c_Message_OsymKeys,type,
symKeys: fun(nat,bool) ).
tff(sy_c_NS__Shared__Mirabelle__iywubrjwsc_Ons__shared,type,
nS_Sha254967238shared: fun(list(event),bool) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Public_OshrK,type,
shrK: agent1 > nat ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * fun(A,bool) ) > fun(A,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fdisj,type,
fdisj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_fimplies,type,
fimplies: fun(bool,fun(bool,bool)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : fun(A,fun(fun(A,bool),bool)) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_A,type,
a: agent1 ).
tff(sy_v_A_H,type,
a1: agent1 ).
tff(sy_v_B,type,
b: agent1 ).
tff(sy_v_K,type,
k: nat ).
tff(sy_v_KAB,type,
kab: nat ).
tff(sy_v_KABa,type,
kABa: nat ).
tff(sy_v_Kab,type,
kab1: nat ).
tff(sy_v_NA,type,
na: nat ).
tff(sy_v_X,type,
x: msg ).
tff(sy_v_evs2,type,
evs2: list(event) ).
%----Relevant facts (100)
tff(fact_0_Oops__parts__spies,axiom,
! [Evsa: list(event),Xa: msg,Ka: msg,B2: msg,NA: msg,A1: agent1] :
( pp(aa(fun(event,bool),bool,aa(event,fun(fun(event,bool),bool),member(event),says(server,A1,crypt(shrK(A1),mPair(NA,mPair(B2,mPair(Ka,Xa)))))),set(event,Evsa)))
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Ka),parts(knows(spy,Evsa)))) ) ).
tff(fact_1_NS3__msg__in__parts__spies,axiom,
! [Evsa: list(event),Xa: msg,Ka: msg,B2: msg,N: msg,KA: nat,A1: agent1,S: agent1] :
( pp(aa(fun(event,bool),bool,aa(event,fun(fun(event,bool),bool),member(event),says(S,A1,crypt(KA,mPair(N,mPair(B2,mPair(Ka,Xa)))))),set(event,Evsa)))
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),parts(knows(spy,Evsa)))) ) ).
tff(fact_2_usedI,axiom,
! [Evsa: list(event),C2: msg] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),C2),parts(knows(spy,Evsa))))
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),C2),used(Evsa))) ) ).
tff(fact_3_shrK__in__used,axiom,
! [Evsa: list(event),A1: agent1] : pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),key(shrK(A1))),used(Evsa))) ).
tff(fact_4_shrK__in__knows,axiom,
! [Evsa: list(event),A1: agent1] : pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),key(shrK(A1))),knows(A1,Evsa))) ).
tff(fact_5_parts__insert__Agent,axiom,
! [H: fun(msg,bool),Agt: agent1] : parts(insert(msg,agent(Agt),H)) = insert(msg,agent(Agt),parts(H)) ).
tff(fact_6_parts__insert__Nonce,axiom,
! [H: fun(msg,bool),N: nat] : parts(insert(msg,nonce(N),H)) = insert(msg,nonce(N),parts(H)) ).
tff(fact_7_parts__insert__Crypt,axiom,
! [H: fun(msg,bool),Xa: msg,Ka: nat] : parts(insert(msg,crypt(Ka,Xa),H)) = insert(msg,crypt(Ka,Xa),parts(insert(msg,Xa,H))) ).
tff(fact_8_parts__insert__Key,axiom,
! [H: fun(msg,bool),Ka: nat] : parts(insert(msg,key(Ka),H)) = insert(msg,key(Ka),parts(H)) ).
tff(fact_9_parts__insert__MPair,axiom,
! [H: fun(msg,bool),Y1: msg,Xa: msg] : parts(insert(msg,mPair(Xa,Y1),H)) = insert(msg,mPair(Xa,Y1),parts(insert(msg,Xa,insert(msg,Y1,H)))) ).
tff(fact_10_sym__shrK,axiom,
! [Xa: agent1] : pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),shrK(Xa)),symKeys)) ).
tff(fact_11_A__trusts__NS2,axiom,
! [Evsa: list(event),Xa: msg,Ka: nat,B2: agent1,NA: msg,A1: agent1] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),crypt(shrK(A1),mPair(NA,mPair(agent(B2),mPair(key(Ka),Xa))))),parts(knows(spy,Evsa))))
=> ( ~ pp(aa(fun(agent1,bool),bool,aa(agent1,fun(fun(agent1,bool),bool),member(agent1),A1),bad))
=> ( pp(aa(fun(list(event),bool),bool,aa(list(event),fun(fun(list(event),bool),bool),member(list(event)),Evsa),nS_Sha254967238shared))
=> pp(aa(fun(event,bool),bool,aa(event,fun(fun(event,bool),bool),member(event),says(server,A1,crypt(shrK(A1),mPair(NA,mPair(agent(B2),mPair(key(Ka),Xa)))))),set(event,Evsa))) ) ) ) ).
tff(fact_12_singleton__conv2,axiom,
! [A: $tType,A2: A] : collect(A,aa(A,fun(A,bool),fequal(A),A2)) = insert(A,A2,bot_bot(fun(A,bool))) ).
tff(fact_13_singleton__conv,axiom,
! [A: $tType,A2: A] : collect(A,combc(A,A,bool,fequal(A),A2)) = insert(A,A2,bot_bot(fun(A,bool))) ).
tff(fact_14_emptyE,axiom,
! [A: $tType,A2: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),bot_bot(fun(A,bool)))) ).
tff(fact_15_Collect__empty__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( collect(A,P1) = bot_bot(fun(A,bool)) )
<=> ! [X5: A] : ~ pp(aa(A,bool,P1,X5)) ) ).
tff(fact_16_empty__iff,axiom,
! [A: $tType,C2: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),bot_bot(fun(A,bool)))) ).
tff(fact_17_empty__Collect__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( bot_bot(fun(A,bool)) = collect(A,P1) )
<=> ! [X5: A] : ~ pp(aa(A,bool,P1,X5)) ) ).
tff(fact_18_all__not__in__conv,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ! [X5: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X5),A1))
<=> ( A1 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_19_insertCI,axiom,
! [A: $tType,B1: A,B2: fun(A,bool),A2: A] :
( ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),B2))
=> ( A2 = B1 ) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),insert(A,B1,B2))) ) ).
tff(fact_20_insertE,axiom,
! [A: $tType,A1: fun(A,bool),B1: A,A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),insert(A,B1,A1)))
=> ( ( A2 != B1 )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1)) ) ) ).
tff(fact_21_insert__iff,axiom,
! [A: $tType,A1: fun(A,bool),B1: A,A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),insert(A,B1,A1)))
<=> ( ( A2 = B1 )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1)) ) ) ).
tff(fact_22_insert__absorb2,axiom,
! [A: $tType,A1: fun(A,bool),X3: A] : insert(A,X3,insert(A,X3,A1)) = insert(A,X3,A1) ).
tff(fact_23_parts_OInj,axiom,
! [H: fun(msg,bool),Xa: msg] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),H))
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),parts(H))) ) ).
tff(fact_24_parts__idem,axiom,
! [H: fun(msg,bool)] : parts(parts(H)) = parts(H) ).
tff(fact_25_msg_Osimps_I6_J,axiom,
! [Msg23: msg,Msg13: msg,Msg22: msg,Msg12: msg] :
( ( mPair(Msg12,Msg22) = mPair(Msg13,Msg23) )
<=> ( ( Msg12 = Msg13 )
& ( Msg22 = Msg23 ) ) ) ).
tff(fact_26_msg_Osimps_I4_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( key(Nat2) = key(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_27_msg_Osimps_I7_J,axiom,
! [Msg4: msg,Nat3: nat,Msg3: msg,Nat2: nat] :
( ( crypt(Nat2,Msg3) = crypt(Nat3,Msg4) )
<=> ( ( Nat2 = Nat3 )
& ( Msg3 = Msg4 ) ) ) ).
tff(fact_28_msg_Osimps_I3_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( nonce(Nat2) = nonce(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_29_msg_Osimps_I1_J,axiom,
! [Agent4: agent1,Agent3: agent1] :
( ( agent(Agent3) = agent(Agent4) )
<=> ( Agent3 = Agent4 ) ) ).
tff(fact_30_event_Osimps_I1_J,axiom,
! [Msg4: msg,Agent21: agent1,Agent11: agent1,Msg3: msg,Agent2: agent1,Agent1: agent1] :
( ( says(Agent1,Agent2,Msg3) = says(Agent11,Agent21,Msg4) )
<=> ( ( Agent1 = Agent11 )
& ( Agent2 = Agent21 )
& ( Msg3 = Msg4 ) ) ) ).
tff(fact_31_shrK__injective,axiom,
! [Y2: agent1,X3: agent1] :
( ( shrK(X3) = shrK(Y2) )
<=> ( X3 = Y2 ) ) ).
tff(fact_32_MPair__parts,axiom,
! [H: fun(msg,bool),Y1: msg,Xa: msg] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),mPair(Xa,Y1)),parts(H)))
=> ~ ( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),parts(H)))
=> ~ pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Y1),parts(H))) ) ) ).
tff(fact_33_MPair__used,axiom,
! [H: list(event),Y1: msg,Xa: msg] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),mPair(Xa,Y1)),used(H)))
=> ~ ( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),used(H)))
=> ~ pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Y1),used(H))) ) ) ).
tff(fact_34_parts__emptyE,axiom,
! [Xa: msg] : ~ pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),parts(bot_bot(fun(msg,bool))))) ).
tff(fact_35_parts__empty,axiom,
parts(bot_bot(fun(msg,bool))) = bot_bot(fun(msg,bool)) ).
tff(fact_36_Spy__in__bad,axiom,
pp(aa(fun(agent1,bool),bool,aa(agent1,fun(fun(agent1,bool),bool),member(agent1),spy),bad)) ).
tff(fact_37_Spy__spies__bad__shrK,axiom,
! [Evsa: list(event),A1: agent1] :
( pp(aa(fun(agent1,bool),bool,aa(agent1,fun(fun(agent1,bool),bool),member(agent1),A1),bad))
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),key(shrK(A1))),knows(spy,Evsa))) ) ).
tff(fact_38_Server__not__bad,axiom,
~ pp(aa(fun(agent1,bool),bool,aa(agent1,fun(fun(agent1,bool),bool),member(agent1),server),bad)) ).
tff(fact_39_equals0D,axiom,
! [A: $tType,A2: A,A1: fun(A,bool)] :
( ( A1 = bot_bot(fun(A,bool)) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1)) ) ).
tff(fact_40_ex__in__conv,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ? [X5: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X5),A1))
<=> ( A1 != bot_bot(fun(A,bool)) ) ) ).
tff(fact_41_empty__def,axiom,
! [A: $tType] : bot_bot(fun(A,bool)) = collect(A,combk(bool,A,fFalse)) ).
tff(fact_42_insert__absorb,axiom,
! [A: $tType,A1: fun(A,bool),A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1))
=> ( insert(A,A2,A1) = A1 ) ) ).
tff(fact_43_insertI2,axiom,
! [A: $tType,B1: A,B2: fun(A,bool),A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),B2))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),insert(A,B1,B2))) ) ).
tff(fact_44_insert__eq__iff,axiom,
! [A: $tType,B2: fun(A,bool),B1: A,A1: fun(A,bool),A2: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),B2))
=> ( ( insert(A,A2,A1) = insert(A,B1,B2) )
<=> ( ( ( A2 = B1 )
=> ( A1 = B2 ) )
& ( ( A2 != B1 )
=> ? [C3: fun(A,bool)] :
( ( A1 = insert(A,B1,C3) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),C3))
& ( B2 = insert(A,A2,C3) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),C3)) ) ) ) ) ) ) ).
tff(fact_45_insert__ident,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool),X3: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),B2))
=> ( ( insert(A,X3,A1) = insert(A,X3,B2) )
<=> ( A1 = B2 ) ) ) ) ).
tff(fact_46_insert__code,axiom,
! [A: $tType,X3: A,A1: fun(A,bool),Y2: A] :
( pp(aa(A,bool,insert(A,Y2,A1),X3))
<=> ( ( Y2 = X3 )
| pp(aa(A,bool,A1,X3)) ) ) ).
tff(fact_47_insert__commute,axiom,
! [A: $tType,A1: fun(A,bool),Y2: A,X3: A] : insert(A,X3,insert(A,Y2,A1)) = insert(A,Y2,insert(A,X3,A1)) ).
tff(fact_48_insert__Collect,axiom,
! [A: $tType,P1: fun(A,bool),A2: A] : insert(A,A2,collect(A,P1)) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fimplies,combb(bool,bool,A,fNot,combc(A,A,bool,fequal(A),A2))),P1)) ).
tff(fact_49_insert__compr,axiom,
! [A: $tType,B2: fun(A,bool),A2: A] : insert(A,A2,B2) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,combc(A,A,bool,fequal(A),A2)),combc(A,fun(A,bool),bool,member(A),B2))) ).
tff(fact_50_insertI1,axiom,
! [A: $tType,B2: fun(A,bool),A2: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),insert(A,A2,B2))) ).
tff(fact_51_parts__partsD,axiom,
! [H: fun(msg,bool),Xa: msg] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),parts(parts(H))))
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),parts(H))) ) ).
tff(fact_52_symKeys__neq__imp__neq,axiom,
! [K: nat,Ka: nat] :
( ~ ( pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),Ka),symKeys))
<=> pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),K),symKeys)) )
=> ( Ka != K ) ) ).
tff(fact_53_insert__compr__raw,axiom,
! [A: $tType,X: A,Xa1: fun(A,bool)] : insert(A,X,Xa1) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,combc(A,A,bool,fequal(A),X)),combc(A,fun(A,bool),bool,member(A),Xa1))) ).
tff(fact_54_singleton__inject,axiom,
! [A: $tType,B1: A,A2: A] :
( ( insert(A,A2,bot_bot(fun(A,bool))) = insert(A,B1,bot_bot(fun(A,bool))) )
=> ( A2 = B1 ) ) ).
tff(fact_55_singletonE,axiom,
! [A: $tType,A2: A,B1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),insert(A,A2,bot_bot(fun(A,bool)))))
=> ( B1 = A2 ) ) ).
tff(fact_56_doubleton__eq__iff,axiom,
! [A: $tType,D: A,C2: A,B1: A,A2: A] :
( ( insert(A,A2,insert(A,B1,bot_bot(fun(A,bool)))) = insert(A,C2,insert(A,D,bot_bot(fun(A,bool)))) )
<=> ( ( ( A2 = C2 )
& ( B1 = D ) )
| ( ( A2 = D )
& ( B1 = C2 ) ) ) ) ).
tff(fact_57_singleton__iff,axiom,
! [A: $tType,A2: A,B1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),insert(A,A2,bot_bot(fun(A,bool)))))
<=> ( B1 = A2 ) ) ).
tff(fact_58_insert__not__empty,axiom,
! [A: $tType,A1: fun(A,bool),A2: A] : insert(A,A2,A1) != bot_bot(fun(A,bool)) ).
tff(fact_59_empty__not__insert,axiom,
! [A: $tType,A1: fun(A,bool),A2: A] : bot_bot(fun(A,bool)) != insert(A,A2,A1) ).
tff(fact_60_parts__cut__eq,axiom,
! [H: fun(msg,bool),Xa: msg] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),parts(H)))
=> ( parts(insert(msg,Xa,H)) = parts(H) ) ) ).
tff(fact_61_parts__insertI,axiom,
! [A2: msg,G1: fun(msg,bool),C2: msg] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),C2),parts(G1)))
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),C2),parts(insert(msg,A2,G1)))) ) ).
tff(fact_62_parts_OSnd,axiom,
! [H: fun(msg,bool),Y1: msg,Xa: msg] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),mPair(Xa,Y1)),parts(H)))
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Y1),parts(H))) ) ).
tff(fact_63_parts_OFst,axiom,
! [H: fun(msg,bool),Y1: msg,Xa: msg] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),mPair(Xa,Y1)),parts(H)))
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),parts(H))) ) ).
tff(fact_64_msg_Osimps_I41_J,axiom,
! [Nat1: nat,Msg2: msg,Msg1: msg] : mPair(Msg1,Msg2) != key(Nat1) ).
tff(fact_65_msg_Osimps_I40_J,axiom,
! [Msg2: msg,Msg1: msg,Nat1: nat] : key(Nat1) != mPair(Msg1,Msg2) ).
tff(fact_66_spies__partsEs_I2_J,axiom,
! [H: fun(msg,bool),Xa: msg,Ka: nat] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),crypt(Ka,Xa)),parts(H)))
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),parts(H))) ) ).
tff(fact_67_parts_OBody,axiom,
! [H: fun(msg,bool),Xa: msg,Ka: nat] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),crypt(Ka,Xa)),parts(H)))
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),parts(H))) ) ).
tff(fact_68_msg_Osimps_I48_J,axiom,
! [Msg: msg,Nat: nat,Msg21: msg,Msg11: msg] : mPair(Msg11,Msg21) != crypt(Nat,Msg) ).
tff(fact_69_msg_Osimps_I49_J,axiom,
! [Msg21: msg,Msg11: msg,Msg: msg,Nat: nat] : crypt(Nat,Msg) != mPair(Msg11,Msg21) ).
tff(fact_70_msg_Osimps_I43_J,axiom,
! [Nat1: nat,Msg: msg,Nat: nat] : crypt(Nat,Msg) != key(Nat1) ).
tff(fact_71_msg_Osimps_I42_J,axiom,
! [Msg: msg,Nat: nat,Nat1: nat] : key(Nat1) != crypt(Nat,Msg) ).
tff(fact_72_msg_Osimps_I35_J,axiom,
! [Nat1: nat,Msg2: msg,Msg1: msg] : mPair(Msg1,Msg2) != nonce(Nat1) ).
tff(fact_73_msg_Osimps_I34_J,axiom,
! [Msg2: msg,Msg1: msg,Nat1: nat] : nonce(Nat1) != mPair(Msg1,Msg2) ).
tff(fact_74_msg_Osimps_I30_J,axiom,
! [Nat: nat,Nat1: nat] : nonce(Nat1) != key(Nat) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X4: A] : aa(A,B,F,X4) = aa(A,B,G,X4)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A1))
<=> pp(aa(A,bool,A1,X3)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_78_msg_Osimps_I31_J,axiom,
! [Nat1: nat,Nat: nat] : key(Nat) != nonce(Nat1) ).
tff(fact_79_msg_Osimps_I17_J,axiom,
! [Agent: agent1,Msg2: msg,Msg1: msg] : mPair(Msg1,Msg2) != agent(Agent) ).
tff(fact_80_msg_Osimps_I16_J,axiom,
! [Msg2: msg,Msg1: msg,Agent: agent1] : agent(Agent) != mPair(Msg1,Msg2) ).
tff(fact_81_msg_Osimps_I12_J,axiom,
! [Nat: nat,Agent: agent1] : agent(Agent) != key(Nat) ).
tff(fact_82_msg_Osimps_I13_J,axiom,
! [Agent: agent1,Nat: nat] : key(Nat) != agent(Agent) ).
tff(fact_83_msg_Osimps_I37_J,axiom,
! [Nat1: nat,Msg: msg,Nat: nat] : crypt(Nat,Msg) != nonce(Nat1) ).
tff(fact_84_msg_Osimps_I36_J,axiom,
! [Msg: msg,Nat: nat,Nat1: nat] : nonce(Nat1) != crypt(Nat,Msg) ).
tff(fact_85_msg_Osimps_I19_J,axiom,
! [Agent: agent1,Msg: msg,Nat: nat] : crypt(Nat,Msg) != agent(Agent) ).
tff(fact_86_msg_Osimps_I18_J,axiom,
! [Msg: msg,Nat: nat,Agent: agent1] : agent(Agent) != crypt(Nat,Msg) ).
tff(fact_87_MPair__used__D,axiom,
! [H: list(event),Y1: msg,Xa: msg] :
( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),mPair(Xa,Y1)),used(H)))
=> ( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Xa),used(H)))
& pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),Y1),used(H))) ) ) ).
tff(fact_88_msg_Osimps_I11_J,axiom,
! [Agent: agent1,Nat: nat] : nonce(Nat) != agent(Agent) ).
tff(fact_89_msg_Osimps_I10_J,axiom,
! [Nat: nat,Agent: agent1] : agent(Agent) != nonce(Nat) ).
tff(fact_90_agent_Osimps_I4_J,axiom,
server != spy ).
tff(fact_91_agent_Osimps_I5_J,axiom,
spy != server ).
tff(fact_92_Collect__conv__if,axiom,
! [A: $tType,A2: A,P1: fun(A,bool)] :
( ( pp(aa(A,bool,P1,A2))
=> ( collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,combc(A,A,bool,fequal(A),A2)),P1)) = insert(A,A2,bot_bot(fun(A,bool))) ) )
& ( ~ pp(aa(A,bool,P1,A2))
=> ( collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,combc(A,A,bool,fequal(A),A2)),P1)) = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_93_Collect__conv__if2,axiom,
! [A: $tType,A2: A,P1: fun(A,bool)] :
( ( pp(aa(A,bool,P1,A2))
=> ( collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,aa(A,fun(A,bool),fequal(A),A2)),P1)) = insert(A,A2,bot_bot(fun(A,bool))) ) )
& ( ~ pp(aa(A,bool,P1,A2))
=> ( collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,aa(A,fun(A,bool),fequal(A),A2)),P1)) = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_94_Spy__see__shrK,axiom,
! [A1: agent1,Evsa: list(event)] :
( pp(aa(fun(list(event),bool),bool,aa(list(event),fun(fun(list(event),bool),bool),member(list(event)),Evsa),nS_Sha254967238shared))
=> ( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),key(shrK(A1))),parts(knows(spy,Evsa))))
<=> pp(aa(fun(agent1,bool),bool,aa(agent1,fun(fun(agent1,bool),bool),member(agent1),A1),bad)) ) ) ).
tff(fact_95_pushes_I5_J,axiom,
! [A1: fun(msg,bool),Y1: msg,Xa: msg,Ka: nat] : insert(msg,key(Ka),insert(msg,mPair(Xa,Y1),A1)) = insert(msg,mPair(Xa,Y1),insert(msg,key(Ka),A1)) ).
tff(fact_96_pushes_I12_J,axiom,
! [A1: fun(msg,bool),Y1: msg,X2: msg,Ka: msg,Xa: nat] : insert(msg,crypt(Xa,Ka),insert(msg,mPair(X2,Y1),A1)) = insert(msg,mPair(X2,Y1),insert(msg,crypt(Xa,Ka),A1)) ).
tff(fact_97_pushes_I6_J,axiom,
! [A1: fun(msg,bool),K: msg,Xa: nat,Ka: nat] : insert(msg,key(Ka),insert(msg,crypt(Xa,K),A1)) = insert(msg,crypt(Xa,K),insert(msg,key(Ka),A1)) ).
tff(fact_98_pushes_I2_J,axiom,
! [A1: fun(msg,bool),N: nat,Ka: nat] : insert(msg,key(Ka),insert(msg,nonce(N),A1)) = insert(msg,nonce(N),insert(msg,key(Ka),A1)) ).
tff(fact_99_pushes_I1_J,axiom,
! [A1: fun(msg,bool),C1: agent1,Ka: nat] : insert(msg,key(Ka),insert(msg,agent(C1),A1)) = insert(msg,agent(C1),insert(msg,key(Ka),A1)) ).
%----Helper facts (21)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa(bool,bool,fNot,P))
| ~ pp(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa(bool,bool,fNot,P)) ) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,combb(B,C,A,P,Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q))
| pp(P)
| pp(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X1: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X1),Y))
| ( X1 = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X1: A] :
( ( X1 != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X1),Y)) ) ).
tff(help_fimplies_1_1_U,axiom,
! [Q: bool,P: bool] :
( pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
| ~ pp(P)
| pp(Q) ) ).
%----Conjectures (7)
tff(conj_0,hypothesis,
! [X: agent1] : kab1 != shrK(X) ).
tff(conj_1,hypothesis,
pp(aa(fun(list(event),bool),bool,aa(list(event),fun(fun(list(event),bool),bool),member(list(event)),evs2),nS_Sha254967238shared)) ).
tff(conj_2,hypothesis,
( ( pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),crypt(kab,x)),parts(knows(spy,evs2))))
& pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),key(k)),parts(insert(msg,x,bot_bot(fun(msg,bool)))))) )
=> pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),key(k)),parts(knows(spy,evs2)))) ) ).
tff(conj_3,hypothesis,
~ pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),key(kABa)),used(evs2))) ).
tff(conj_4,hypothesis,
pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),kABa),symKeys)) ).
tff(conj_5,hypothesis,
pp(aa(fun(event,bool),bool,aa(event,fun(fun(event,bool),bool),member(event),says(a1,server,mPair(agent(a),mPair(agent(b),nonce(na))))),set(event,evs2))) ).
tff(conj_6,conjecture,
( ( ( ( kab != shrK(a) )
| ( x != mPair(nonce(na),mPair(agent(b),mPair(key(kABa),crypt(shrK(b),mPair(key(kABa),agent(a)))))) ) )
& ( ( kab != shrK(b) )
| ( x != mPair(key(kABa),agent(a)) ) )
& ~ pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),crypt(kab,x)),parts(knows(spy,evs2)))) )
| ~ pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),key(k)),parts(insert(msg,x,bot_bot(fun(msg,bool))))))
| ( k = kABa )
| pp(aa(fun(msg,bool),bool,aa(msg,fun(fun(msg,bool),bool),member(msg),key(k)),parts(knows(spy,evs2)))) ) ).
%------------------------------------------------------------------------------