theory Send

imports Main

begin

typedecl principal

consts A :: principal

consts S :: principal

consts B :: principal

consts Says :: "principal \<Rightarrow> bool \<Rightarrow> bool" ("_ says _")

consts Sends :: "principal \<Rightarrow> principal \<Rightarrow> bool \<Rightarrow> bool" ("_ sends _ : _")

consts Enc :: "bool \<Rightarrow> bool \<Rightarrow> bool"

consts CAB :: "bool"

consts KAB :: "bool"

consts KAS :: "bool" 

consts KBS :: "bool"

consts M :: "bool"

axiomatization where

  saysI[intro]: "F \<Longrightarrow> P says F" and

  saysE[elim]:  "\<lbrakk>P says (F1 \<longrightarrow> F2); P says F1\<rbrakk> \<Longrightarrow> P says F2" and

  says_encI[intro]: "\<lbrakk>P says F1; P says F2\<rbrakk> \<Longrightarrow> P says (Enc F1 F2)" and

  says_encE[elim]: "\<lbrakk>P says (Enc F1 F2); P says F2\<rbrakk> \<Longrightarrow> P says F1" and

  sendsE[elim]: "\<lbrakk>P sends Q : F; P says F\<rbrakk> \<Longrightarrow> Q says F"

lemma

  assumes start: "A says CAB"

  and     msg1:  "A sends S : CAB"

  and    serv1:  "S says (CAB \<longrightarrow> Enc KAB KAS)"

  and    serv2:  "S says (CAB \<longrightarrow> Enc (Enc KAB KBS) KAS)"

  and    msg2a:  "S sends A : (Enc KAB KAS)"

  and    msg2b:  "S sends A : (Enc (Enc KAB KBS) KAS)"

  and     msg3:  "A sends B : (Enc KAB KBS)"

  and     msg4:  "A sends B : (Enc M KAB)"

  and     keyA:  "A says KAS"

  and     keyB:  "B says KBS"

  and     keyS:  "S says KAS"

  and       MA:  "A says (Enc M KAB)"

  shows "S says M"

using assms

by (metis saysE says_encE sendsE)

by (metis saysE says_encE sendsE)

by metis

by (metis saysE says_encE sendsE)

by (metis saysE says_encE sendsE)

end