(* Title: HOL/Auth/Event ID: $Id: Event.thy,v 1.36 2007/08/01 18:25:16 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1996 University of Cambridge Datatype of events; function "spies"; freshness "bad" agents have been broken by the Spy; their private keys and internal stores are visible to him *) header{*Theory of Events for Security Protocols*} theory Event imports Message begin consts (*Initial states of agents -- parameter of the construction*) initState :: "agent => msg set" datatype event = Says agent agent msg | Gets agent msg | Notes agent msg consts bad :: "agent set" (*compromised agents*) knows :: "agent => event list => msg set" text{*The constant "spies" is retained for compatibility's sake*} abbreviation (input) spies :: "event list => msg set" where "spies == knows Spy" text{*Spy has access to his own key for spoof messages, but Server is secure*} specification (bad) Spy_in_bad [iff]: "Spy ∈ bad" Server_not_bad [iff]: "Server ∉ bad" by (rule exI [of _ "{Spy}"], simp) primrec knows_Nil: "knows A [] = initState A" knows_Cons: "knows A (ev # evs) = (if A = Spy then (case ev of Says A' B X => insert X (knows Spy evs) | Gets A' X => knows Spy evs | Notes A' X => if A' ∈ bad then insert X (knows Spy evs) else knows Spy evs) else (case ev of Says A' B X => if A'=A then insert X (knows A evs) else knows A evs | Gets A' X => if A'=A then insert X (knows A evs) else knows A evs | Notes A' X => if A'=A then insert X (knows A evs) else knows A evs))" (* Case A=Spy on the Gets event enforces the fact that if a message is received then it must have been sent, therefore the oops case must use Notes *) consts (*Set of items that might be visible to somebody: complement of the set of fresh items*) used :: "event list => msg set" primrec used_Nil: "used [] = (UN B. parts (initState B))" used_Cons: "used (ev # evs) = (case ev of Says A B X => parts {X} ∪ used evs | Gets A X => used evs | Notes A X => parts {X} ∪ used evs)" --{*The case for @{term Gets} seems anomalous, but @{term Gets} always follows @{term Says} in real protocols. Seems difficult to change. See @{text Gets_correct} in theory @{text "Guard/Extensions.thy"}. *} lemma Notes_imp_used [rule_format]: "Notes A X ∈ set evs --> X ∈ used evs" apply (induct_tac evs) apply (auto split: event.split) done lemma Says_imp_used [rule_format]: "Says A B X ∈ set evs --> X ∈ used evs" apply (induct_tac evs) apply (auto split: event.split) done subsection{*Function @{term knows}*} (*Simplifying parts(insert X (knows Spy evs)) = parts{X} ∪ parts(knows Spy evs). This version won't loop with the simplifier.*) lemmas parts_insert_knows_A = parts_insert [of _ "knows A evs", standard] lemma knows_Spy_Says [simp]: "knows Spy (Says A B X # evs) = insert X (knows Spy evs)" by simp text{*Letting the Spy see "bad" agents' notes avoids redundant case-splits on whether @{term "A=Spy"} and whether @{term "A∈bad"}*} lemma knows_Spy_Notes [simp]: "knows Spy (Notes A X # evs) = (if A:bad then insert X (knows Spy evs) else knows Spy evs)" by simp lemma knows_Spy_Gets [simp]: "knows Spy (Gets A X # evs) = knows Spy evs" by simp lemma knows_Spy_subset_knows_Spy_Says: "knows Spy evs ⊆ knows Spy (Says A B X # evs)" by (simp add: subset_insertI) lemma knows_Spy_subset_knows_Spy_Notes: "knows Spy evs ⊆ knows Spy (Notes A X # evs)" by force lemma knows_Spy_subset_knows_Spy_Gets: "knows Spy evs ⊆ knows Spy (Gets A X # evs)" by (simp add: subset_insertI) text{*Spy sees what is sent on the traffic*} lemma Says_imp_knows_Spy [rule_format]: "Says A B X ∈ set evs --> X ∈ knows Spy evs" apply (induct_tac "evs") apply (simp_all (no_asm_simp) split add: event.split) done lemma Notes_imp_knows_Spy [rule_format]: "Notes A X ∈ set evs --> A: bad --> X ∈ knows Spy evs" apply (induct_tac "evs") apply (simp_all (no_asm_simp) split add: event.split) done text{*Elimination rules: derive contradictions from old Says events containing items known to be fresh*} lemmas knows_Spy_partsEs = Says_imp_knows_Spy [THEN parts.Inj, THEN revcut_rl, standard] parts.Body [THEN revcut_rl, standard] lemmas Says_imp_analz_Spy = Says_imp_knows_Spy [THEN analz.Inj] text{*Compatibility for the old "spies" function*} lemmas spies_partsEs = knows_Spy_partsEs lemmas Says_imp_spies = Says_imp_knows_Spy lemmas parts_insert_spies = parts_insert_knows_A [of _ Spy] subsection{*Knowledge of Agents*} lemma knows_Says: "knows A (Says A B X # evs) = insert X (knows A evs)" by simp lemma knows_Notes: "knows A (Notes A X # evs) = insert X (knows A evs)" by simp lemma knows_Gets: "A ≠ Spy --> knows A (Gets A X # evs) = insert X (knows A evs)" by simp lemma knows_subset_knows_Says: "knows A evs ⊆ knows A (Says A' B X # evs)" by (simp add: subset_insertI) lemma knows_subset_knows_Notes: "knows A evs ⊆ knows A (Notes A' X # evs)" by (simp add: subset_insertI) lemma knows_subset_knows_Gets: "knows A evs ⊆ knows A (Gets A' X # evs)" by (simp add: subset_insertI) text{*Agents know what they say*} lemma Says_imp_knows [rule_format]: "Says A B X ∈ set evs --> X ∈ knows A evs" apply (induct_tac "evs") apply (simp_all (no_asm_simp) split add: event.split) apply blast done text{*Agents know what they note*} lemma Notes_imp_knows [rule_format]: "Notes A X ∈ set evs --> X ∈ knows A evs" apply (induct_tac "evs") apply (simp_all (no_asm_simp) split add: event.split) apply blast done text{*Agents know what they receive*} lemma Gets_imp_knows_agents [rule_format]: "A ≠ Spy --> Gets A X ∈ set evs --> X ∈ knows A evs" apply (induct_tac "evs") apply (simp_all (no_asm_simp) split add: event.split) done text{*What agents DIFFERENT FROM Spy know was either said, or noted, or got, or known initially*} lemma knows_imp_Says_Gets_Notes_initState [rule_format]: "[| X ∈ knows A evs; A ≠ Spy |] ==> EX B. Says A B X ∈ set evs | Gets A X ∈ set evs | Notes A X ∈ set evs | X ∈ initState A" apply (erule rev_mp) apply (induct_tac "evs") apply (simp_all (no_asm_simp) split add: event.split) apply blast done text{*What the Spy knows -- for the time being -- was either said or noted, or known initially*} lemma knows_Spy_imp_Says_Notes_initState [rule_format]: "[| X ∈ knows Spy evs |] ==> EX A B. Says A B X ∈ set evs | Notes A X ∈ set evs | X ∈ initState Spy" apply (erule rev_mp) apply (induct_tac "evs") apply (simp_all (no_asm_simp) split add: event.split) apply blast done lemma parts_knows_Spy_subset_used: "parts (knows Spy evs) ⊆ used evs" apply (induct_tac "evs", force) apply (simp add: parts_insert_knows_A knows_Cons add: event.split, blast) done lemmas usedI = parts_knows_Spy_subset_used [THEN subsetD, intro] lemma initState_into_used: "X ∈ parts (initState B) ==> X ∈ used evs" apply (induct_tac "evs") apply (simp_all add: parts_insert_knows_A split add: event.split, blast) done lemma used_Says [simp]: "used (Says A B X # evs) = parts{X} ∪ used evs" by simp lemma used_Notes [simp]: "used (Notes A X # evs) = parts{X} ∪ used evs" by simp lemma used_Gets [simp]: "used (Gets A X # evs) = used evs" by simp lemma used_nil_subset: "used [] ⊆ used evs" apply simp apply (blast intro: initState_into_used) done text{*NOTE REMOVAL--laws above are cleaner, as they don't involve "case"*} declare knows_Cons [simp del] used_Nil [simp del] used_Cons [simp del] text{*For proving theorems of the form @{term "X ∉ analz (knows Spy evs) --> P"} New events added by induction to "evs" are discarded. Provided this information isn't needed, the proof will be much shorter, since it will omit complicated reasoning about @{term analz}.*} lemmas analz_mono_contra = knows_Spy_subset_knows_Spy_Says [THEN analz_mono, THEN contra_subsetD] knows_Spy_subset_knows_Spy_Notes [THEN analz_mono, THEN contra_subsetD] knows_Spy_subset_knows_Spy_Gets [THEN analz_mono, THEN contra_subsetD] lemma knows_subset_knows_Cons: "knows A evs ⊆ knows A (e # evs)" by (induct e, auto simp: knows_Cons) lemma initState_subset_knows: "initState A ⊆ knows A evs" apply (induct_tac evs, simp) apply (blast intro: knows_subset_knows_Cons [THEN subsetD]) done text{*For proving @{text new_keys_not_used}*} lemma keysFor_parts_insert: "[| K ∈ keysFor (parts (insert X G)); X ∈ synth (analz H) |] ==> K ∈ keysFor (parts (G ∪ H)) | Key (invKey K) ∈ parts H"; by (force dest!: parts_insert_subset_Un [THEN keysFor_mono, THEN [2] rev_subsetD] analz_subset_parts [THEN keysFor_mono, THEN [2] rev_subsetD] intro: analz_subset_parts [THEN subsetD] parts_mono [THEN [2] rev_subsetD]) ML {* val analz_mono_contra_tac = let val analz_impI = inst "P" "?Y ∉ analz (knows Spy ?evs)" impI in rtac analz_impI THEN' REPEAT1 o (dresolve_tac @{thms analz_mono_contra}) THEN' mp_tac end *} method_setup analz_mono_contra = {* Method.no_args (Method.SIMPLE_METHOD (REPEAT_FIRST analz_mono_contra_tac)) *} "for proving theorems of the form X ∉ analz (knows Spy evs) --> P" subsubsection{*Useful for case analysis on whether a hash is a spoof or not*} ML {* val synth_analz_mono_contra_tac = let val syan_impI = inst "P" "?Y ∉ synth (analz (knows Spy ?evs))" impI in rtac syan_impI THEN' REPEAT1 o (dresolve_tac [@{thm knows_Spy_subset_knows_Spy_Says} RS @{thm synth_analz_mono} RS @{thm contra_subsetD}, @{thm knows_Spy_subset_knows_Spy_Notes} RS @{thm synth_analz_mono} RS @{thm contra_subsetD}, @{thm knows_Spy_subset_knows_Spy_Gets} RS @{thm synth_analz_mono} RS @{thm contra_subsetD}]) THEN' mp_tac end; *} method_setup synth_analz_mono_contra = {* Method.no_args (Method.SIMPLE_METHOD (REPEAT_FIRST synth_analz_mono_contra_tac)) *} "for proving theorems of the form X ∉ synth (analz (knows Spy evs)) --> P" end
lemma Notes_imp_used:
Notes A X ∈ set evs ==> X ∈ used evs
lemma Says_imp_used:
Says A B X ∈ set evs ==> X ∈ used evs
lemma parts_insert_knows_A:
parts (insert X (knows A evs)) = parts {X} ∪ parts (knows A evs)
lemma knows_Spy_Says:
knows Spy (Says A B X # evs) = insert X (knows Spy evs)
lemma knows_Spy_Notes:
knows Spy (Notes A X # evs) =
(if A ∈ bad then insert X (knows Spy evs) else knows Spy evs)
lemma knows_Spy_Gets:
knows Spy (Gets A X # evs) = knows Spy evs
lemma knows_Spy_subset_knows_Spy_Says:
knows Spy evs ⊆ knows Spy (Says A B X # evs)
lemma knows_Spy_subset_knows_Spy_Notes:
knows Spy evs ⊆ knows Spy (Notes A X # evs)
lemma knows_Spy_subset_knows_Spy_Gets:
knows Spy evs ⊆ knows Spy (Gets A X # evs)
lemma Says_imp_knows_Spy:
Says A B X ∈ set evs ==> X ∈ knows Spy evs
lemma Notes_imp_knows_Spy:
[| Notes A X ∈ set evs; A ∈ bad |] ==> X ∈ knows Spy evs
lemma knows_Spy_partsEs:
[| Says A B X ∈ set evs; X ∈ parts (knows Spy evs) ==> PROP W |] ==> PROP W
[| Crypt K X ∈ parts H; X ∈ parts H ==> PROP W |] ==> PROP W
lemma Says_imp_analz_Spy:
Says A1 B1 X ∈ set evs1 ==> X ∈ analz (knows Spy evs1)
lemma spies_partsEs:
[| Says A B X ∈ set evs; X ∈ parts (knows Spy evs) ==> PROP W |] ==> PROP W
[| Crypt K X ∈ parts H; X ∈ parts H ==> PROP W |] ==> PROP W
lemma Says_imp_spies:
Says A B X ∈ set evs ==> X ∈ knows Spy evs
lemma parts_insert_spies:
parts (insert X (knows Spy evs)) = parts {X} ∪ parts (knows Spy evs)
lemma knows_Says:
knows A (Says A B X # evs) = insert X (knows A evs)
lemma knows_Notes:
knows A (Notes A X # evs) = insert X (knows A evs)
lemma knows_Gets:
A ≠ Spy --> knows A (Gets A X # evs) = insert X (knows A evs)
lemma knows_subset_knows_Says:
knows A evs ⊆ knows A (Says A' B X # evs)
lemma knows_subset_knows_Notes:
knows A evs ⊆ knows A (Notes A' X # evs)
lemma knows_subset_knows_Gets:
knows A evs ⊆ knows A (Gets A' X # evs)
lemma Says_imp_knows:
Says A B X ∈ set evs ==> X ∈ knows A evs
lemma Notes_imp_knows:
Notes A X ∈ set evs ==> X ∈ knows A evs
lemma Gets_imp_knows_agents:
[| A ≠ Spy; Gets A X ∈ set evs |] ==> X ∈ knows A evs
lemma knows_imp_Says_Gets_Notes_initState:
[| X ∈ knows A evs; A ≠ Spy |]
==> ∃B. Says A B X ∈ set evs ∨
Gets A X ∈ set evs ∨ Notes A X ∈ set evs ∨ X ∈ initState A
lemma knows_Spy_imp_Says_Notes_initState:
X ∈ knows Spy evs
==> ∃A B. Says A B X ∈ set evs ∨ Notes A X ∈ set evs ∨ X ∈ initState Spy
lemma parts_knows_Spy_subset_used:
parts (knows Spy evs) ⊆ used evs
lemma usedI:
c ∈ parts (knows Spy evs1) ==> c ∈ used evs1
lemma initState_into_used:
X ∈ parts (initState B) ==> X ∈ used evs
lemma used_Says:
used (Says A B X # evs) = parts {X} ∪ used evs
lemma used_Notes:
used (Notes A X # evs) = parts {X} ∪ used evs
lemma used_Gets:
used (Gets A X # evs) = used evs
lemma used_nil_subset:
used [] ⊆ used evs
lemma analz_mono_contra:
c ∉ analz (knows Spy (Says A2 B2 X2 # evs2)) ==> c ∉ analz (knows Spy evs2)
c ∉ analz (knows Spy (Notes A2 X2 # evs2)) ==> c ∉ analz (knows Spy evs2)
c ∉ analz (knows Spy (Gets A2 X2 # evs2)) ==> c ∉ analz (knows Spy evs2)
lemma knows_subset_knows_Cons:
knows A evs ⊆ knows A (e # evs)
lemma initState_subset_knows:
initState A ⊆ knows A evs
lemma keysFor_parts_insert:
[| K ∈ keysFor (parts (insert X G)); X ∈ synth (analz H) |]
==> K ∈ keysFor (parts (G ∪ H)) ∨ Key (invKey K) ∈ parts H