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header{*Theory of Events for Security Protocols that use smartcards*}
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theory EventSC imports "../Message" begin
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consts (*Initial states of agents -- parameter of the construction*)
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initState :: "agent => msg set"
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datatype card = Card agent
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text{*Four new events express the traffic between an agent and his card*}
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datatype
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event = Says agent agent msg
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| Notes agent msg
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| Gets agent msg
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| Inputs agent card msg (*Agent sends to card and\<dots>*)
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| C_Gets card msg (*\<dots> card receives it*)
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| Outpts card agent msg (*Card sends to agent and\<dots>*)
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| A_Gets agent msg (*agent receives it*)
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consts
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bad :: "agent set" (*compromised agents*)
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knows :: "agent => event list => msg set" (*agents' knowledge*)
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stolen :: "card set" (* stolen smart cards *)
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cloned :: "card set" (* cloned smart cards*)
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secureM :: "bool"(*assumption of secure means between agents and their cards*)
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abbreviation
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insecureM :: bool (*certain protocols make no assumption of secure means*)
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"insecureM == \<not>secureM"
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text{*Spy has access to his own key for spoof messages, but Server is secure*}
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specification (bad)
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Spy_in_bad [iff]: "Spy \<in> bad"
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Server_not_bad [iff]: "Server \<notin> bad"
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apply (rule exI [of _ "{Spy}"], simp) done
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specification (stolen)
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(*The server's card is secure by assumption\<dots>*)
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Card_Server_not_stolen [iff]: "Card Server \<notin> stolen"
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Card_Spy_not_stolen [iff]: "Card Spy \<notin> stolen"
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apply blast done
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specification (cloned)
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(*\<dots> the spy's card is secure because she already can use it freely*)
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Card_Server_not_cloned [iff]: "Card Server \<notin> cloned"
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Card_Spy_not_cloned [iff]: "Card Spy \<notin> cloned"
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apply blast done
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primrec (*This definition is extended over the new events, subject to the
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assumption of secure means*)
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knows_Nil: "knows A [] = initState A"
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knows_Cons: "knows A (ev # evs) =
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(case ev of
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Says A' B X =>
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if (A=A' | A=Spy) then insert X (knows A evs) else knows A evs
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| Notes A' X =>
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if (A=A' | (A=Spy & A'\<in>bad)) then insert X (knows A evs)
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else knows A evs
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| Gets A' X =>
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if (A=A' & A \<noteq> Spy) then insert X (knows A evs)
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else knows A evs
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| Inputs A' C X =>
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if secureM then
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if A=A' then insert X (knows A evs) else knows A evs
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else
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if (A=A' | A=Spy) then insert X (knows A evs) else knows A evs
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| C_Gets C X => knows A evs
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| Outpts C A' X =>
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if secureM then
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if A=A' then insert X (knows A evs) else knows A evs
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else
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if A=Spy then insert X (knows A evs) else knows A evs
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| A_Gets A' X =>
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if (A=A' & A \<noteq> Spy) then insert X (knows A evs)
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else knows A evs)"
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consts
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(*The set of items that might be visible to someone is easily extended
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over the new events*)
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used :: "event list => msg set"
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primrec
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used_Nil: "used [] = (UN B. parts (initState B))"
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used_Cons: "used (ev # evs) =
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(case ev of
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Says A B X => parts {X} \<union> (used evs)
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| Notes A X => parts {X} \<union> (used evs)
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| Gets A X => used evs
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| Inputs A C X => parts{X} \<union> (used evs)
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| C_Gets C X => used evs
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| Outpts C A X => parts{X} \<union> (used evs)
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| A_Gets A X => used evs)"
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--{*@{term Gets} always follows @{term Says} in real protocols.
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Likewise, @{term C_Gets} will always have to follow @{term Inputs}
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and @{term A_Gets} will always have to follow @{term Outpts}*}
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lemma Notes_imp_used [rule_format]: "Notes A X \<in> set evs \<longrightarrow> X \<in> used evs"
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apply (induct_tac evs)
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apply (auto split: event.split)
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done
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lemma Says_imp_used [rule_format]: "Says A B X \<in> set evs \<longrightarrow> X \<in> used evs"
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apply (induct_tac evs)
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apply (auto split: event.split)
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done
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lemma MPair_used [rule_format]:
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"MPair X Y \<in> used evs \<longrightarrow> X \<in> used evs & Y \<in> used evs"
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apply (induct_tac evs)
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apply (auto split: event.split)
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done
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subsection{*Function @{term knows}*}
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(*Simplifying
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parts(insert X (knows Spy evs)) = parts{X} \<union> parts(knows Spy evs).
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This version won't loop with the simplifier.*)
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lemmas parts_insert_knows_A = parts_insert [of _ "knows A evs", standard]
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lemma knows_Spy_Says [simp]:
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"knows Spy (Says A B X # evs) = insert X (knows Spy evs)"
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by simp
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text{*Letting the Spy see "bad" agents' notes avoids redundant case-splits
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on whether @{term "A=Spy"} and whether @{term "A\<in>bad"}*}
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lemma knows_Spy_Notes [simp]:
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"knows Spy (Notes A X # evs) =
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(if A\<in>bad then insert X (knows Spy evs) else knows Spy evs)"
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by simp
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lemma knows_Spy_Gets [simp]: "knows Spy (Gets A X # evs) = knows Spy evs"
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by simp
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lemma knows_Spy_Inputs_secureM [simp]:
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"secureM \<Longrightarrow> knows Spy (Inputs A C X # evs) =
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(if A=Spy then insert X (knows Spy evs) else knows Spy evs)"
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by simp
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lemma knows_Spy_Inputs_insecureM [simp]:
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"insecureM \<Longrightarrow> knows Spy (Inputs A C X # evs) = insert X (knows Spy evs)"
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by simp
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lemma knows_Spy_C_Gets [simp]: "knows Spy (C_Gets C X # evs) = knows Spy evs"
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by simp
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lemma knows_Spy_Outpts_secureM [simp]:
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"secureM \<Longrightarrow> knows Spy (Outpts C A X # evs) =
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(if A=Spy then insert X (knows Spy evs) else knows Spy evs)"
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by simp
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lemma knows_Spy_Outpts_insecureM [simp]:
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"insecureM \<Longrightarrow> knows Spy (Outpts C A X # evs) = insert X (knows Spy evs)"
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by simp
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lemma knows_Spy_A_Gets [simp]: "knows Spy (A_Gets A X # evs) = knows Spy evs"
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by simp
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lemma knows_Spy_subset_knows_Spy_Says:
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"knows Spy evs \<subseteq> knows Spy (Says A B X # evs)"
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by (simp add: subset_insertI)
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lemma knows_Spy_subset_knows_Spy_Notes:
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"knows Spy evs \<subseteq> knows Spy (Notes A X # evs)"
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by force
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lemma knows_Spy_subset_knows_Spy_Gets:
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"knows Spy evs \<subseteq> knows Spy (Gets A X # evs)"
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by (simp add: subset_insertI)
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lemma knows_Spy_subset_knows_Spy_Inputs:
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"knows Spy evs \<subseteq> knows Spy (Inputs A C X # evs)"
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by auto
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lemma knows_Spy_equals_knows_Spy_Gets:
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"knows Spy evs = knows Spy (C_Gets C X # evs)"
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by (simp add: subset_insertI)
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lemma knows_Spy_subset_knows_Spy_Outpts: "knows Spy evs \<subseteq> knows Spy (Outpts C A X # evs)"
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by auto
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lemma knows_Spy_subset_knows_Spy_A_Gets: "knows Spy evs \<subseteq> knows Spy (A_Gets A X # evs)"
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by (simp add: subset_insertI)
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text{*Spy sees what is sent on the traffic*}
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lemma Says_imp_knows_Spy [rule_format]:
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"Says A B X \<in> set evs \<longrightarrow> X \<in> knows Spy evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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done
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lemma Notes_imp_knows_Spy [rule_format]:
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"Notes A X \<in> set evs \<longrightarrow> A\<in> bad \<longrightarrow> X \<in> knows Spy evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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done
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(*Nothing can be stated on a Gets event*)
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lemma Inputs_imp_knows_Spy_secureM [rule_format (no_asm)]:
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"Inputs Spy C X \<in> set evs \<longrightarrow> secureM \<longrightarrow> X \<in> knows Spy evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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done
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lemma Inputs_imp_knows_Spy_insecureM [rule_format (no_asm)]:
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"Inputs A C X \<in> set evs \<longrightarrow> insecureM \<longrightarrow> X \<in> knows Spy evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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done
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(*Nothing can be stated on a C_Gets event*)
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lemma Outpts_imp_knows_Spy_secureM [rule_format (no_asm)]:
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"Outpts C Spy X \<in> set evs \<longrightarrow> secureM \<longrightarrow> X \<in> knows Spy evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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done
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lemma Outpts_imp_knows_Spy_insecureM [rule_format (no_asm)]:
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"Outpts C A X \<in> set evs \<longrightarrow> insecureM \<longrightarrow> X \<in> knows Spy evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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done
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(*Nothing can be stated on an A_Gets event*)
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text{*Elimination rules: derive contradictions from old Says events containing
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items known to be fresh*}
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lemmas knows_Spy_partsEs =
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Says_imp_knows_Spy [THEN parts.Inj, THEN revcut_rl, standard]
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parts.Body [THEN revcut_rl, standard]
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subsection{*Knowledge of Agents*}
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lemma knows_Says: "knows A (Says A B X # evs) = insert X (knows A evs)"
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by simp
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lemma knows_Notes: "knows A (Notes A X # evs) = insert X (knows A evs)"
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by simp
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lemma knows_Gets:
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"A \<noteq> Spy \<longrightarrow> knows A (Gets A X # evs) = insert X (knows A evs)"
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by simp
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lemma knows_Inputs: "knows A (Inputs A C X # evs) = insert X (knows A evs)"
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by simp
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lemma knows_C_Gets: "knows A (C_Gets C X # evs) = knows A evs"
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by simp
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lemma knows_Outpts_secureM:
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"secureM \<longrightarrow> knows A (Outpts C A X # evs) = insert X (knows A evs)"
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by simp
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lemma knows_Outpts_secureM:
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"insecureM \<longrightarrow> knows Spy (Outpts C A X # evs) = insert X (knows Spy evs)"
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by simp
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(*somewhat equivalent to knows_Spy_Outpts_insecureM*)
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lemma knows_subset_knows_Says: "knows A evs \<subseteq> knows A (Says A' B X # evs)"
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by (simp add: subset_insertI)
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lemma knows_subset_knows_Notes: "knows A evs \<subseteq> knows A (Notes A' X # evs)"
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by (simp add: subset_insertI)
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lemma knows_subset_knows_Gets: "knows A evs \<subseteq> knows A (Gets A' X # evs)"
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by (simp add: subset_insertI)
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lemma knows_subset_knows_Inputs: "knows A evs \<subseteq> knows A (Inputs A' C X # evs)"
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by (simp add: subset_insertI)
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lemma knows_subset_knows_C_Gets: "knows A evs \<subseteq> knows A (C_Gets C X # evs)"
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by (simp add: subset_insertI)
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lemma knows_subset_knows_Outpts: "knows A evs \<subseteq> knows A (Outpts C A' X # evs)"
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by (simp add: subset_insertI)
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lemma knows_subset_knows_Gets: "knows A evs \<subseteq> knows A (A_Gets A' X # evs)"
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by (simp add: subset_insertI)
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text{*Agents know what they say*}
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lemma Says_imp_knows [rule_format]: "Says A B X \<in> set evs \<longrightarrow> X \<in> knows A evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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apply blast
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done
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text{*Agents know what they note*}
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lemma Notes_imp_knows [rule_format]: "Notes A X \<in> set evs \<longrightarrow> X \<in> knows A evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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apply blast
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done
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text{*Agents know what they receive*}
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lemma Gets_imp_knows_agents [rule_format]:
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"A \<noteq> Spy \<longrightarrow> Gets A X \<in> set evs \<longrightarrow> X \<in> knows A evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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done
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(*Agents know what they input to their smart card*)
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lemma Inputs_imp_knows_agents [rule_format (no_asm)]:
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"Inputs A (Card A) X \<in> set evs \<longrightarrow> X \<in> knows A evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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apply blast
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done
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(*Nothing to prove about C_Gets*)
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(*Agents know what they obtain as output of their smart card,
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if the means is secure...*)
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lemma Outpts_imp_knows_agents_secureM [rule_format (no_asm)]:
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"secureM \<longrightarrow> Outpts (Card A) A X \<in> set evs \<longrightarrow> X \<in> knows A evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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done
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(*otherwise only the spy knows the outputs*)
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lemma Outpts_imp_knows_agents_insecureM [rule_format (no_asm)]:
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"insecureM \<longrightarrow> Outpts (Card A) A X \<in> set evs \<longrightarrow> X \<in> knows Spy evs"
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apply (induct_tac "evs")
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apply (simp_all (no_asm_simp) split add: event.split)
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done
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(*end lemmas about agents' knowledge*)
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lemma parts_knows_Spy_subset_used: "parts (knows Spy evs) \<subseteq> used evs"
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apply (induct_tac "evs", force)
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apply (simp add: parts_insert_knows_A knows_Cons add: event.split, blast)
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done
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lemmas usedI = parts_knows_Spy_subset_used [THEN subsetD, intro]
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lemma initState_into_used: "X \<in> parts (initState B) \<Longrightarrow> X \<in> used evs"
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apply (induct_tac "evs")
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apply (simp_all add: parts_insert_knows_A split add: event.split, blast)
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done
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lemma used_Says [simp]: "used (Says A B X # evs) = parts{X} \<union> used evs"
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by simp
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lemma used_Notes [simp]: "used (Notes A X # evs) = parts{X} \<union> used evs"
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367 |
by simp
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lemma used_Gets [simp]: "used (Gets A X # evs) = used evs"
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370 |
by simp
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lemma used_Inputs [simp]: "used (Inputs A C X # evs) = parts{X} \<union> used evs"
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by simp
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lemma used_C_Gets [simp]: "used (C_Gets C X # evs) = used evs"
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by simp
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377 |
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lemma used_Outpts [simp]: "used (Outpts C A X # evs) = parts{X} \<union> used evs"
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by simp
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380 |
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lemma used_A_Gets [simp]: "used (A_Gets A X # evs) = used evs"
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382 |
by simp
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383 |
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lemma used_nil_subset: "used [] \<subseteq> used evs"
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385 |
apply simp
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386 |
apply (blast intro: initState_into_used)
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387 |
done
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388 |
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389 |
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390 |
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391 |
(*Novel lemmas*)
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lemma Says_parts_used [rule_format (no_asm)]:
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"Says A B X \<in> set evs \<longrightarrow> (parts {X}) \<subseteq> used evs"
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394 |
apply (induct_tac "evs")
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395 |
apply (simp_all (no_asm_simp) split add: event.split)
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396 |
apply blast
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397 |
done
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398 |
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399 |
lemma Notes_parts_used [rule_format (no_asm)]:
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400 |
"Notes A X \<in> set evs \<longrightarrow> (parts {X}) \<subseteq> used evs"
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401 |
apply (induct_tac "evs")
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402 |
apply (simp_all (no_asm_simp) split add: event.split)
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403 |
apply blast
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404 |
done
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405 |
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406 |
lemma Outpts_parts_used [rule_format (no_asm)]:
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407 |
"Outpts C A X \<in> set evs \<longrightarrow> (parts {X}) \<subseteq> used evs"
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408 |
apply (induct_tac "evs")
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409 |
apply (simp_all (no_asm_simp) split add: event.split)
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410 |
apply blast
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411 |
done
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412 |
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413 |
lemma Inputs_parts_used [rule_format (no_asm)]:
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414 |
"Inputs A C X \<in> set evs \<longrightarrow> (parts {X}) \<subseteq> used evs"
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415 |
apply (induct_tac "evs")
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416 |
apply (simp_all (no_asm_simp) split add: event.split)
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|
417 |
apply blast
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418 |
done
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419 |
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420 |
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421 |
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422 |
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423 |
text{*NOTE REMOVAL--laws above are cleaner, as they don't involve "case"*}
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424 |
declare knows_Cons [simp del]
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425 |
used_Nil [simp del] used_Cons [simp del]
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426 |
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427 |
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428 |
lemma knows_subset_knows_Cons: "knows A evs \<subseteq> knows A (e # evs)"
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429 |
by (induct e, auto simp: knows_Cons)
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430 |
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431 |
lemma initState_subset_knows: "initState A \<subseteq> knows A evs"
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432 |
apply (induct_tac evs, simp)
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433 |
apply (blast intro: knows_subset_knows_Cons [THEN subsetD])
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|
434 |
done
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435 |
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436 |
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437 |
text{*For proving @{text new_keys_not_used}*}
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438 |
lemma keysFor_parts_insert:
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439 |
"\<lbrakk> K \<in> keysFor (parts (insert X G)); X \<in> synth (analz H) \<rbrakk>
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440 |
\<Longrightarrow> K \<in> keysFor (parts (G \<union> H)) \<or> Key (invKey K) \<in> parts H";
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|
441 |
by (force
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442 |
dest!: parts_insert_subset_Un [THEN keysFor_mono, THEN [2] rev_subsetD]
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443 |
analz_subset_parts [THEN keysFor_mono, THEN [2] rev_subsetD]
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|
444 |
intro: analz_subset_parts [THEN subsetD] parts_mono [THEN [2] rev_subsetD])
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445 |
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|
446 |
end
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