Theory Event

theory Event
imports Message
(*  Title:      HOL/Auth/Event.thy
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 *}

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 :: "agent => event list => msg set"
where
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
*)


text{*The constant "spies" is retained for compatibility's sake*}

abbreviation (input)
spies :: "event list => msg set" where
"spies == knows Spy"


(*Set of items that might be visible to somebody:
complement of the set of fresh items*)


primrec used :: "event list => msg set"
where
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"] for A evs

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 Says_imp_parts_knows_Spy =
Says_imp_knows_Spy [THEN parts.Inj, elim_format]

lemmas knows_Spy_partsEs =
Says_imp_parts_knows_Spy parts.Body [elim_format]

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_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 (cases 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])


lemmas analz_impI = impI [where P = "Y ∉ analz (knows Spy evs)"] for Y evs

ML
{*
val analz_mono_contra_tac =
rtac @{thm analz_impI} THEN'
REPEAT1 o (dresolve_tac @{thms analz_mono_contra})
THEN' mp_tac
*}


method_setup analz_mono_contra = {*
Scan.succeed (K (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*}

lemmas syan_impI = impI [where P = "Y ∉ synth (analz (knows Spy evs))"] for Y evs

ML
{*
val synth_analz_mono_contra_tac =
rtac @{thm 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
*}


method_setup synth_analz_mono_contra = {*
Scan.succeed (K (SIMPLE_METHOD (REPEAT_FIRST synth_analz_mono_contra_tac))) *}

"for proving theorems of the form X ∉ synth (analz (knows Spy evs)) --> P"

end