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