Theory Needham_Schroeder_No_Attacker_Example

theory Needham_Schroeder_No_Attacker_Example
imports Needham_Schroeder_Base
begin

inductive_set ns_public :: "event list set"
  where
         (*Initial trace is empty*)
   Nil:  "[] ∈ ns_public"
         (*Alice initiates a protocol run, sending a nonce to Bob*)
 | NS1:  "[|evs1 ∈ ns_public;  Nonce NA ∉ used evs1|]
          ==> Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>)
                # evs1  ∈  ns_public"
         (*Bob responds to Alice's message with a further nonce*)
 | NS2:  "[|evs2 ∈ ns_public;  Nonce NB ∉ used evs2;
           Says A' B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) ∈ set evs2|]
          ==> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>)
                # evs2  ∈  ns_public"

         (*Alice proves her existence by sending NB back to Bob.*)
 | NS3:  "[|evs3 ∈ ns_public;
           Says A  B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) ∈ set evs3;
           Says B' A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) ∈ set evs3|]
          ==> Says A B (Crypt (pubEK B) (Nonce NB)) # evs3 ∈ ns_public"

code_pred [skip_proof] ns_publicp .
thm ns_publicp.equation

code_pred [generator_cps] ns_publicp .
thm ns_publicp.generator_cps_equation

lemma "ns_publicp evs ==> ¬ (Says Alice Bob (Crypt (pubEK Bob) (Nonce NB))) : set evs"
(*quickcheck[random, iterations = 1000000, size = 20, timeout = 3600, verbose]
quickcheck[exhaustive, size = 8, timeout = 3600, verbose]
quickcheck[narrowing, size = 7, timeout = 3600, verbose]*)
quickcheck[smart_exhaustive, depth = 5, timeout = 30, expect = counterexample]
oops

lemma
  "[|ns_publicp evs|]            
       ==> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) : set evs
       ==> A ≠ Spy ==> B ≠ Spy ==> A ≠ B 
           ==> Nonce NB ∉ analz (spies evs)"
quickcheck[smart_exhaustive, depth = 6, timeout = 30]
quickcheck[narrowing, size = 6, timeout = 30, verbose]
oops

end