src/HOL/Auth/Event.thy
author paulson
Thu Jul 24 16:36:29 2003 +0200 (2003-07-24)
changeset 14126 28824746d046
parent 13956 8fe7e12290e1
child 14200 d8598e24f8fa
permissions -rw-r--r--
Tidying and replacement of some axioms by specifications
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(*  Title:      HOL/Auth/Event
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1996  University of Cambridge
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Datatype of events; function "spies"; freshness
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"bad" agents have been broken by the Spy; their private keys and internal
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    stores are visible to him
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*)
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header{*Theory of Events for Security Protocols*}
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theory Event = Message:
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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
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  event = Says  agent agent msg
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        | Gets  agent       msg
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        | Notes agent       msg
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consts 
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  bad    :: "agent set"				(*compromised agents*)
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  knows  :: "agent => event list => msg set"
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text{*The constant "spies" is retained for compatibility's sake*}
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syntax
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  spies  :: "event list => msg set"
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translations
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  "spies"   => "knows Spy"
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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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  bad_properties: "Spy \<in> bad & Server \<notin> bad"
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    by (rule exI [of _ "{Spy}"], simp)
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lemmas Spy_in_bad = bad_properties [THEN conjunct1, iff]
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lemmas Server_not_bad = bad_properties [THEN conjunct2, iff]
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primrec
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  knows_Nil:   "knows A [] = initState A"
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  knows_Cons:
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    "knows A (ev # evs) =
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       (if A = Spy then 
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	(case ev of
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	   Says A' B X => insert X (knows Spy evs)
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	 | Gets A' X => knows Spy evs
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	 | Notes A' X  => 
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	     if A' \<in> bad then insert X (knows Spy evs) else knows Spy evs)
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	else
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	(case ev of
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	   Says A' B X => 
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	     if A'=A then insert X (knows A evs) else knows A evs
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	 | Gets A' X    => 
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	     if A'=A then insert X (knows A evs) else knows A evs
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	 | Notes A' X    => 
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	     if A'=A then insert X (knows A evs) else knows A evs))"
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(*
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  Case A=Spy on the Gets event
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  enforces the fact that if a message is received then it must have been sent,
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  therefore the oops case must use Notes
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*)
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consts
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  (*Set of items that might be visible to somebody:
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    complement of the set of fresh items*)
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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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		      | Gets A X   => used evs
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		      | Notes A X  => parts {X} \<union> used evs)"
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    --{*The case for @{term Gets} seems anomalous, but @{term Gets} always
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        follows @{term Says} in real protocols.  Seems difficult to change.
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        See @{text Gets_correct} in theory @{text "Guard/Extensions.thy"}. *}
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lemma Notes_imp_used [rule_format]: "Notes A X \<in> set evs --> 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 --> 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 --> 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{*The point of letting the Spy see "bad" agents' notes is to prevent
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  redundant case-splits on whether A=Spy and whether A: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: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_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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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 --> 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 --> A: bad --> 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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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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text{*Compatibility for the old "spies" function*}
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lemmas spies_partsEs = knows_Spy_partsEs
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lemmas Says_imp_spies = Says_imp_knows_Spy
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lemmas parts_insert_spies = parts_insert_knows_A [of _ Spy]
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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 --> knows A (Gets A X # evs) = insert X (knows A evs)"
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by simp
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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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text{*Agents know what they say*}
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lemma Says_imp_knows [rule_format]: "Says A B X \<in> set evs --> 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 --> 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 --> Gets A X \<in> set evs --> 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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text{*What agents DIFFERENT FROM Spy know 
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  was either said, or noted, or got, or known initially*}
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lemma knows_imp_Says_Gets_Notes_initState [rule_format]:
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     "[| X \<in> knows A evs; A \<noteq> Spy |] ==> EX B.  
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  Says A B X \<in> set evs | Gets A X \<in> set evs | Notes A X \<in> set evs | X \<in> initState A"
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apply (erule rev_mp)
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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{*What the Spy knows -- for the time being --
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  was either said or noted, or known initially*}
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lemma knows_Spy_imp_Says_Notes_initState [rule_format]:
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     "[| X \<in> knows Spy evs |] ==> EX A B.  
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  Says A B X \<in> set evs | Notes A X \<in> set evs | X \<in> initState Spy"
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apply (erule rev_mp)
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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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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) ==> 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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by simp
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lemma used_Gets [simp]: "used (Gets A X # evs) = used evs"
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by simp
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lemma used_nil_subset: "used [] \<subseteq> used evs"
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apply simp
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apply (blast intro: initState_into_used)
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done
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text{*NOTE REMOVAL--laws above are cleaner, as they don't involve "case"*}
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declare knows_Cons [simp del]
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        used_Nil [simp del] used_Cons [simp del]
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text{*For proving theorems of the form @{term "X \<notin> analz (knows Spy evs) --> P"}
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  New events added by induction to "evs" are discarded.  Provided 
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  this information isn't needed, the proof will be much shorter, since
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  it will omit complicated reasoning about @{term analz}.*}
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lemmas analz_mono_contra =
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       knows_Spy_subset_knows_Spy_Says [THEN analz_mono, THEN contra_subsetD]
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       knows_Spy_subset_knows_Spy_Notes [THEN analz_mono, THEN contra_subsetD]
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       knows_Spy_subset_knows_Spy_Gets [THEN analz_mono, THEN contra_subsetD]
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ML
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{*
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val analz_mono_contra_tac = 
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  let val analz_impI = inst "P" "?Y \<notin> analz (knows Spy ?evs)" impI
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  in
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    rtac analz_impI THEN' 
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    REPEAT1 o 
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      (dresolve_tac (thms"analz_mono_contra"))
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    THEN' mp_tac
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  end
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*}
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lemma knows_subset_knows_Cons: "knows A evs \<subseteq> knows A (e # evs)"
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by (induct e, auto simp: knows_Cons)
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lemma initState_subset_knows: "initState A \<subseteq> knows A evs"
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apply (induct_tac evs, simp) 
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apply (blast intro: knows_subset_knows_Cons [THEN subsetD])
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done
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text{*For proving @{text new_keys_not_used}*}
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lemma keysFor_parts_insert:
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     "[| K \<in> keysFor (parts (insert X G));  X \<in> synth (analz H) |] 
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      ==> K \<in> keysFor (parts (G \<union> H)) | Key (invKey K) \<in> parts H"; 
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by (force 
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    dest!: parts_insert_subset_Un [THEN keysFor_mono, THEN [2] rev_subsetD]
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           analz_subset_parts [THEN keysFor_mono, THEN [2] rev_subsetD]
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    intro: analz_subset_parts [THEN subsetD] parts_mono [THEN [2] rev_subsetD])
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method_setup analz_mono_contra = {*
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    Method.no_args
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      (Method.METHOD (fn facts => REPEAT_FIRST analz_mono_contra_tac)) *}
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    "for proving theorems of the form X \<notin> analz (knows Spy evs) --> P"
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subsubsection{*Useful for case analysis on whether a hash is a spoof or not*}
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ML
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{*
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val knows_Cons     = thm "knows_Cons"
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val used_Nil       = thm "used_Nil"
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val used_Cons      = thm "used_Cons"
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val Notes_imp_used = thm "Notes_imp_used";
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val Says_imp_used = thm "Says_imp_used";
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val MPair_used = thm "MPair_used";
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val Says_imp_knows_Spy = thm "Says_imp_knows_Spy";
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val Notes_imp_knows_Spy = thm "Notes_imp_knows_Spy";
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val knows_Spy_partsEs = thms "knows_Spy_partsEs";
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val spies_partsEs = thms "spies_partsEs";
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val Says_imp_spies = thm "Says_imp_spies";
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val parts_insert_spies = thm "parts_insert_spies";
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val Says_imp_knows = thm "Says_imp_knows";
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val Notes_imp_knows = thm "Notes_imp_knows";
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val Gets_imp_knows_agents = thm "Gets_imp_knows_agents";
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val knows_imp_Says_Gets_Notes_initState = thm "knows_imp_Says_Gets_Notes_initState";
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val knows_Spy_imp_Says_Notes_initState = thm "knows_Spy_imp_Says_Notes_initState";
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   327
val usedI = thm "usedI";
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val initState_into_used = thm "initState_into_used";
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   329
val used_Says = thm "used_Says";
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   330
val used_Notes = thm "used_Notes";
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   331
val used_Gets = thm "used_Gets";
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   332
val used_nil_subset = thm "used_nil_subset";
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   333
val analz_mono_contra = thms "analz_mono_contra";
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   334
val knows_subset_knows_Cons = thm "knows_subset_knows_Cons";
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   335
val initState_subset_knows = thm "initState_subset_knows";
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   336
val keysFor_parts_insert = thm "keysFor_parts_insert";
paulson@13926
   337
paulson@13926
   338
paulson@13922
   339
val synth_analz_mono = thm "synth_analz_mono";
paulson@13922
   340
paulson@13935
   341
val knows_Spy_subset_knows_Spy_Says = thm "knows_Spy_subset_knows_Spy_Says";
paulson@13935
   342
val knows_Spy_subset_knows_Spy_Notes = thm "knows_Spy_subset_knows_Spy_Notes";
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   343
val knows_Spy_subset_knows_Spy_Gets = thm "knows_Spy_subset_knows_Spy_Gets";
paulson@13935
   344
paulson@13935
   345
paulson@13922
   346
val synth_analz_mono_contra_tac = 
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  let val syan_impI = inst "P" "?Y \<notin> synth (analz (knows Spy ?evs))" impI
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  in
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    rtac syan_impI THEN' 
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   350
    REPEAT1 o 
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   351
      (dresolve_tac 
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   352
       [knows_Spy_subset_knows_Spy_Says RS synth_analz_mono RS contra_subsetD,
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   353
        knows_Spy_subset_knows_Spy_Notes RS synth_analz_mono RS contra_subsetD,
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   354
	knows_Spy_subset_knows_Spy_Gets RS synth_analz_mono RS contra_subsetD])
paulson@13922
   355
    THEN'
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   356
    mp_tac
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   357
  end;
paulson@13922
   358
*}
paulson@13922
   359
paulson@13922
   360
method_setup synth_analz_mono_contra = {*
paulson@13922
   361
    Method.no_args
paulson@13922
   362
      (Method.METHOD (fn facts => REPEAT_FIRST synth_analz_mono_contra_tac)) *}
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    "for proving theorems of the form X \<notin> synth (analz (knows Spy evs)) --> P"
paulson@3512
   364
paulson@3512
   365
end