# Theory Guard_OtwayRees

```(*  Title:      HOL/Auth/Guard/Guard_OtwayRees.thy
Author:     Frederic Blanqui, University of Cambridge Computer Laboratory
Copyright   2002  University of Cambridge
*)

section‹Otway-Rees Protocol›

theory Guard_OtwayRees imports Guard_Shared begin

subsection‹messages used in the protocol›

abbreviation
nil :: "msg" where
"nil == Number 0"

abbreviation
or1 :: "agent => agent => nat => event" where
"or1 A B NA ==
Says A B ⦃Nonce NA, Agent A, Agent B, Ciph A ⦃Nonce NA, Agent A, Agent B⦄⦄"

abbreviation
or1' :: "agent => agent => agent => nat => msg => event" where
"or1' A' A B NA X == Says A' B ⦃Nonce NA, Agent A, Agent B, X⦄"

abbreviation
or2 :: "agent => agent => nat => nat => msg => event" where
"or2 A B NA NB X ==
Says B Server ⦃Nonce NA, Agent A, Agent B, X,
Ciph B ⦃Nonce NA, Nonce NB, Agent A, Agent B⦄⦄"

abbreviation
or2' :: "agent => agent => agent => nat => nat => event" where
"or2' B' A B NA NB ==
Says B' Server ⦃Nonce NA, Agent A, Agent B,
Ciph A ⦃Nonce NA, Agent A, Agent B⦄,
Ciph B ⦃Nonce NA, Nonce NB, Agent A, Agent B⦄⦄"

abbreviation
or3 :: "agent => agent => nat => nat => key => event" where
"or3 A B NA NB K ==
Says Server B ⦃Nonce NA, Ciph A ⦃Nonce NA, Key K⦄,
Ciph B ⦃Nonce NB, Key K⦄⦄"

abbreviation
or3':: "agent => msg => agent => agent => nat => nat => key => event" where
"or3' S Y A B NA NB K ==
Says S B ⦃Nonce NA, Y, Ciph B ⦃Nonce NB, Key K⦄⦄"

abbreviation
or4 :: "agent => agent => nat => msg => event" where
"or4 A B NA X == Says B A ⦃Nonce NA, X, nil⦄"

abbreviation
or4' :: "agent => agent => nat => key => event" where
"or4' B' A NA K == Says B' A ⦃Nonce NA, Ciph A ⦃Nonce NA, Key K⦄, nil⦄"

subsection‹definition of the protocol›

inductive_set or :: "event list set"
where

Nil: "[] ∈ or"

| Fake: "[| evs ∈ or; X ∈ synth (analz (spies evs)) |] ==> Says Spy B X # evs ∈ or"

| OR1: "[| evs1 ∈ or; Nonce NA ∉ used evs1 |] ==> or1 A B NA # evs1 ∈ or"

| OR2: "[| evs2 ∈ or; or1' A' A B NA X ∈ set evs2; Nonce NB ∉ used evs2 |]
==> or2 A B NA NB X # evs2 ∈ or"

| OR3: "[| evs3 ∈ or; or2' B' A B NA NB ∈ set evs3; Key K ∉ used evs3 |]
==> or3 A B NA NB K # evs3 ∈ or"

| OR4: "[| evs4 ∈ or; or2 A B NA NB X ∈ set evs4; or3' S Y A B NA NB K ∈ set evs4 |]
==> or4 A B NA X # evs4 ∈ or"

subsection‹declarations for tactics›

declare knows_Spy_partsEs [elim]
declare Fake_parts_insert [THEN subsetD, dest]
declare initState.simps [simp del]

subsection‹general properties of or›

lemma or_has_no_Gets: "evs ∈ or ⟹ ∀A X. Gets A X ∉ set evs"
by (erule or.induct, auto)

lemma or_is_Gets_correct [iff]: "Gets_correct or"
by (auto simp: Gets_correct_def dest: or_has_no_Gets)

lemma or_is_one_step [iff]: "one_step or"
by (unfold one_step_def, clarify, ind_cases "ev#evs ∈ or" for ev evs, auto)

lemma or_has_only_Says' [rule_format]: "evs ∈ or ⟹
ev ∈ set evs ⟶ (∃A B X. ev=Says A B X)"
by (erule or.induct, auto)

lemma or_has_only_Says [iff]: "has_only_Says or"
by (auto simp: has_only_Says_def dest: or_has_only_Says')

subsection‹or is regular›

lemma or1'_parts_spies [dest]: "or1' A' A B NA X ∈ set evs
⟹ X ∈ parts (spies evs)"
by blast

lemma or2_parts_spies [dest]: "or2 A B NA NB X ∈ set evs
⟹ X ∈ parts (spies evs)"
by blast

lemma or3_parts_spies [dest]: "Says S B ⦃NA, Y, Ciph B ⦃NB, K⦄⦄ ∈ set evs
⟹ K ∈ parts (spies evs)"
by blast

lemma or_is_regular [iff]: "regular or"
apply (simp only: regular_def, clarify)
apply (erule or.induct, simp_all add: initState.simps knows.simps)
by (auto dest: parts_sub)

subsection‹guardedness of KAB›

lemma Guard_KAB [rule_format]: "[| evs ∈ or; A ∉ bad; B ∉ bad |] ==>
or3 A B NA NB K ∈ set evs ⟶ GuardK K {shrK A,shrK B} (spies evs)"
apply (erule or.induct)
(* Nil *)
apply simp_all
(* Fake *)
apply (clarify, erule in_synth_GuardK, erule GuardK_analz, simp)
(* OR1 *)
apply blast
(* OR2 *)
apply safe
apply (blast dest: Says_imp_spies, blast)
(* OR3 *)
apply blast
apply (drule_tac A=Server in Key_neq, simp+, rule No_Key, simp)
apply (drule_tac A=Server in Key_neq, simp+, rule No_Key, simp)
(* OR4 *)
by (blast dest: Says_imp_spies in_GuardK_kparts)

subsection‹guardedness of NB›

lemma Guard_NB [rule_format]: "[| evs ∈ or; B ∉ bad |] ==>
or2 A B NA NB X ∈ set evs ⟶ Guard NB {shrK B} (spies evs)"
apply (erule or.induct)
(* Nil *)
apply simp_all
(* Fake *)
apply safe
apply (erule in_synth_Guard, erule Guard_analz, simp)
(* OR1 *)
apply (drule_tac n=NB in Nonce_neq, simp+, rule No_Nonce, simp)
apply (drule_tac n=NB in Nonce_neq, simp+, rule No_Nonce, simp)
(* OR2 *)
apply blast
apply (drule_tac n=NA in Nonce_neq, simp+, rule No_Nonce, simp)
apply (blast intro!: No_Nonce dest: used_parts)
apply (drule_tac n=NA in Nonce_neq, simp+, rule No_Nonce, simp)
apply (blast intro!: No_Nonce dest: used_parts)
apply (blast dest: Says_imp_spies)
apply (blast dest: Says_imp_spies)
apply (case_tac "Ba=B", clarsimp)
apply (drule_tac n=NB and A=B in Nonce_neq, simp+)
apply (drule Says_imp_spies)
apply (drule_tac n'=NAa in in_Guard_kparts_neq, simp+, rule No_Nonce, simp)
(* OR3 *)
apply (drule Says_imp_spies)
apply (frule_tac n'=NAa in in_Guard_kparts_neq, simp+, rule No_Nonce, simp)
apply (case_tac "Aa=B", clarsimp)
apply (case_tac "NAa=NB", clarsimp)
apply (drule Says_imp_spies)
apply (drule_tac Y="⦃Nonce NB, Agent Aa, Agent Ba⦄"
and K="shrK Aa" in in_Guard_kparts_Crypt, simp+)
apply (simp add: No_Nonce)
apply (case_tac "Ba=B", clarsimp)
apply (case_tac "NBa=NB", clarify)
apply (drule Says_imp_spies)
apply (drule_tac Y="⦃Nonce NAa, Nonce NB, Agent Aa, Agent Ba⦄"
and K="shrK Ba" in in_Guard_kparts_Crypt, simp+)
apply (simp add: No_Nonce)
(* OR4 *)
by (blast dest: Says_imp_spies)+

end
```