(* Title: ZF/UNITY/Constrains.thy
Author: Sidi O Ehmety, Computer Laboratory
Copyright 2001 University of Cambridge
*)
section\<open>Weak Safety Properties\<close>
theory Constrains
imports UNITY
begin
consts traces :: "[i, i] => i"
(* Initial states and program => (final state, reversed trace to it)...
the domain may also be state*list(state) *)
inductive
domains
"traces(init, acts)" <=
"(init \<union> (\<Union>act\<in>acts. field(act)))*list(\<Union>act\<in>acts. field(act))"
intros
(*Initial trace is empty*)
Init: "s: init ==> <s,[]> \<in> traces(init,acts)"
Acts: "[| act:acts; <s,evs> \<in> traces(init,acts); <s,s'>: act |]
==> <s', Cons(s,evs)> \<in> traces(init, acts)"
type_intros list.intros UnI1 UnI2 UN_I fieldI2 fieldI1
consts reachable :: "i=>i"
inductive
domains
"reachable(F)" \<subseteq> "Init(F) \<union> (\<Union>act\<in>Acts(F). field(act))"
intros
Init: "s:Init(F) ==> s:reachable(F)"
Acts: "[| act: Acts(F); s:reachable(F); <s,s'>: act |]
==> s':reachable(F)"
type_intros UnI1 UnI2 fieldI2 UN_I
definition
Constrains :: "[i,i] => i" (infixl "Co" 60) where
"A Co B == {F:program. F:(reachable(F) \<inter> A) co B}"
definition
op_Unless :: "[i, i] => i" (infixl "Unless" 60) where
"A Unless B == (A-B) Co (A \<union> B)"
definition
Stable :: "i => i" where
"Stable(A) == A Co A"
definition
(*Always is the weak form of "invariant"*)
Always :: "i => i" where
"Always(A) == initially(A) \<inter> Stable(A)"
(*** traces and reachable ***)
lemma reachable_type: "reachable(F) \<subseteq> state"
apply (cut_tac F = F in Init_type)
apply (cut_tac F = F in Acts_type)
apply (cut_tac F = F in reachable.dom_subset, blast)
done
lemma st_set_reachable: "st_set(reachable(F))"
apply (unfold st_set_def)
apply (rule reachable_type)
done
declare st_set_reachable [iff]
lemma reachable_Int_state: "reachable(F) \<inter> state = reachable(F)"
by (cut_tac reachable_type, auto)
declare reachable_Int_state [iff]
lemma state_Int_reachable: "state \<inter> reachable(F) = reachable(F)"
by (cut_tac reachable_type, auto)
declare state_Int_reachable [iff]
lemma reachable_equiv_traces:
"F \<in> program ==> reachable(F)={s \<in> state. \<exists>evs. <s,evs>:traces(Init(F), Acts(F))}"
apply (rule equalityI, safe)
apply (blast dest: reachable_type [THEN subsetD])
apply (erule_tac [2] traces.induct)
apply (erule reachable.induct)
apply (blast intro: reachable.intros traces.intros)+
done
lemma Init_into_reachable: "Init(F) \<subseteq> reachable(F)"
by (blast intro: reachable.intros)
lemma stable_reachable: "[| F \<in> program; G \<in> program;
Acts(G) \<subseteq> Acts(F) |] ==> G \<in> stable(reachable(F))"
apply (blast intro: stableI constrainsI st_setI
reachable_type [THEN subsetD] reachable.intros)
done
declare stable_reachable [intro!]
declare stable_reachable [simp]
(*The set of all reachable states is an invariant...*)
lemma invariant_reachable:
"F \<in> program ==> F \<in> invariant(reachable(F))"
apply (unfold invariant_def initially_def)
apply (blast intro: reachable_type [THEN subsetD] reachable.intros)
done
(*...in fact the strongest invariant!*)
lemma invariant_includes_reachable: "F \<in> invariant(A) ==> reachable(F) \<subseteq> A"
apply (cut_tac F = F in Acts_type)
apply (cut_tac F = F in Init_type)
apply (cut_tac F = F in reachable_type)
apply (simp (no_asm_use) add: stable_def constrains_def invariant_def initially_def)
apply (rule subsetI)
apply (erule reachable.induct)
apply (blast intro: reachable.intros)+
done
(*** Co ***)
lemma constrains_reachable_Int: "F \<in> B co B'==>F:(reachable(F) \<inter> B) co (reachable(F) \<inter> B')"
apply (frule constrains_type [THEN subsetD])
apply (frule stable_reachable [OF _ _ subset_refl])
apply (simp_all add: stable_def constrains_Int)
done
(*Resembles the previous definition of Constrains*)
lemma Constrains_eq_constrains:
"A Co B = {F \<in> program. F:(reachable(F) \<inter> A) co (reachable(F) \<inter> B)}"
apply (unfold Constrains_def)
apply (blast dest: constrains_reachable_Int constrains_type [THEN subsetD]
intro: constrains_weaken)
done
lemmas Constrains_def2 = Constrains_eq_constrains [THEN eq_reflection]
lemma constrains_imp_Constrains: "F \<in> A co A' ==> F \<in> A Co A'"
apply (unfold Constrains_def)
apply (blast intro: constrains_weaken_L dest: constrainsD2)
done
lemma ConstrainsI:
"[|!!act s s'. [| act \<in> Acts(F); <s,s'>:act; s \<in> A |] ==> s':A';
F \<in> program|]
==> F \<in> A Co A'"
apply (auto simp add: Constrains_def constrains_def st_set_def)
apply (blast dest: reachable_type [THEN subsetD])
done
lemma Constrains_type:
"A Co B \<subseteq> program"
apply (unfold Constrains_def, blast)
done
lemma Constrains_empty: "F \<in> 0 Co B \<longleftrightarrow> F \<in> program"
by (auto dest: Constrains_type [THEN subsetD]
intro: constrains_imp_Constrains)
declare Constrains_empty [iff]
lemma Constrains_state: "F \<in> A Co state \<longleftrightarrow> F \<in> program"
apply (unfold Constrains_def)
apply (auto dest: Constrains_type [THEN subsetD] intro: constrains_imp_Constrains)
done
declare Constrains_state [iff]
lemma Constrains_weaken_R:
"[| F \<in> A Co A'; A'<=B' |] ==> F \<in> A Co B'"
apply (unfold Constrains_def2)
apply (blast intro: constrains_weaken_R)
done
lemma Constrains_weaken_L:
"[| F \<in> A Co A'; B<=A |] ==> F \<in> B Co A'"
apply (unfold Constrains_def2)
apply (blast intro: constrains_weaken_L st_set_subset)
done
lemma Constrains_weaken:
"[| F \<in> A Co A'; B<=A; A'<=B' |] ==> F \<in> B Co B'"
apply (unfold Constrains_def2)
apply (blast intro: constrains_weaken st_set_subset)
done
(** Union **)
lemma Constrains_Un:
"[| F \<in> A Co A'; F \<in> B Co B' |] ==> F \<in> (A \<union> B) Co (A' \<union> B')"
apply (unfold Constrains_def2, auto)
apply (simp add: Int_Un_distrib)
apply (blast intro: constrains_Un)
done
lemma Constrains_UN:
"[|(!!i. i \<in> I==>F \<in> A(i) Co A'(i)); F \<in> program|]
==> F:(\<Union>i \<in> I. A(i)) Co (\<Union>i \<in> I. A'(i))"
by (auto intro: constrains_UN simp del: UN_simps
simp add: Constrains_def2 Int_UN_distrib)
(** Intersection **)
lemma Constrains_Int:
"[| F \<in> A Co A'; F \<in> B Co B'|]==> F:(A \<inter> B) Co (A' \<inter> B')"
apply (unfold Constrains_def)
apply (subgoal_tac "reachable (F) \<inter> (A \<inter> B) = (reachable (F) \<inter> A) \<inter> (reachable (F) \<inter> B) ")
apply (auto intro: constrains_Int)
done
lemma Constrains_INT:
"[| (!!i. i \<in> I ==>F \<in> A(i) Co A'(i)); F \<in> program |]
==> F:(\<Inter>i \<in> I. A(i)) Co (\<Inter>i \<in> I. A'(i))"
apply (simp (no_asm_simp) del: INT_simps add: Constrains_def INT_extend_simps)
apply (rule constrains_INT)
apply (auto simp add: Constrains_def)
done
lemma Constrains_imp_subset: "F \<in> A Co A' ==> reachable(F) \<inter> A \<subseteq> A'"
apply (unfold Constrains_def)
apply (blast dest: constrains_imp_subset)
done
lemma Constrains_trans:
"[| F \<in> A Co B; F \<in> B Co C |] ==> F \<in> A Co C"
apply (unfold Constrains_def2)
apply (blast intro: constrains_trans constrains_weaken)
done
lemma Constrains_cancel:
"[| F \<in> A Co (A' \<union> B); F \<in> B Co B' |] ==> F \<in> A Co (A' \<union> B')"
apply (unfold Constrains_def2)
apply (simp (no_asm_use) add: Int_Un_distrib)
apply (blast intro: constrains_cancel)
done
(*** Stable ***)
(* Useful because there's no Stable_weaken. [Tanja Vos] *)
lemma stable_imp_Stable:
"F \<in> stable(A) ==> F \<in> Stable(A)"
apply (unfold stable_def Stable_def)
apply (erule constrains_imp_Constrains)
done
lemma Stable_eq: "[| F \<in> Stable(A); A = B |] ==> F \<in> Stable(B)"
by blast
lemma Stable_eq_stable:
"F \<in> Stable(A) \<longleftrightarrow> (F \<in> stable(reachable(F) \<inter> A))"
apply (auto dest: constrainsD2 simp add: Stable_def stable_def Constrains_def2)
done
lemma StableI: "F \<in> A Co A ==> F \<in> Stable(A)"
by (unfold Stable_def, assumption)
lemma StableD: "F \<in> Stable(A) ==> F \<in> A Co A"
by (unfold Stable_def, assumption)
lemma Stable_Un:
"[| F \<in> Stable(A); F \<in> Stable(A') |] ==> F \<in> Stable(A \<union> A')"
apply (unfold Stable_def)
apply (blast intro: Constrains_Un)
done
lemma Stable_Int:
"[| F \<in> Stable(A); F \<in> Stable(A') |] ==> F \<in> Stable (A \<inter> A')"
apply (unfold Stable_def)
apply (blast intro: Constrains_Int)
done
lemma Stable_Constrains_Un:
"[| F \<in> Stable(C); F \<in> A Co (C \<union> A') |]
==> F \<in> (C \<union> A) Co (C \<union> A')"
apply (unfold Stable_def)
apply (blast intro: Constrains_Un [THEN Constrains_weaken_R])
done
lemma Stable_Constrains_Int:
"[| F \<in> Stable(C); F \<in> (C \<inter> A) Co A' |]
==> F \<in> (C \<inter> A) Co (C \<inter> A')"
apply (unfold Stable_def)
apply (blast intro: Constrains_Int [THEN Constrains_weaken])
done
lemma Stable_UN:
"[| (!!i. i \<in> I ==> F \<in> Stable(A(i))); F \<in> program |]
==> F \<in> Stable (\<Union>i \<in> I. A(i))"
apply (simp add: Stable_def)
apply (blast intro: Constrains_UN)
done
lemma Stable_INT:
"[|(!!i. i \<in> I ==> F \<in> Stable(A(i))); F \<in> program |]
==> F \<in> Stable (\<Inter>i \<in> I. A(i))"
apply (simp add: Stable_def)
apply (blast intro: Constrains_INT)
done
lemma Stable_reachable: "F \<in> program ==>F \<in> Stable (reachable(F))"
apply (simp (no_asm_simp) add: Stable_eq_stable Int_absorb)
done
lemma Stable_type: "Stable(A) \<subseteq> program"
apply (unfold Stable_def)
apply (rule Constrains_type)
done
(*** The Elimination Theorem. The "free" m has become universally quantified!
Should the premise be !!m instead of \<forall>m ? Would make it harder to use
in forward proof. ***)
lemma Elimination:
"[| \<forall>m \<in> M. F \<in> ({s \<in> A. x(s) = m}) Co (B(m)); F \<in> program |]
==> F \<in> ({s \<in> A. x(s):M}) Co (\<Union>m \<in> M. B(m))"
apply (unfold Constrains_def, auto)
apply (rule_tac A1 = "reachable (F) \<inter> A"
in UNITY.elimination [THEN constrains_weaken_L])
apply (auto intro: constrains_weaken_L)
done
(* As above, but for the special case of A=state *)
lemma Elimination2:
"[| \<forall>m \<in> M. F \<in> {s \<in> state. x(s) = m} Co B(m); F \<in> program |]
==> F \<in> {s \<in> state. x(s):M} Co (\<Union>m \<in> M. B(m))"
apply (blast intro: Elimination)
done
(** Unless **)
lemma Unless_type: "A Unless B <=program"
apply (unfold op_Unless_def)
apply (rule Constrains_type)
done
(*** Specialized laws for handling Always ***)
(** Natural deduction rules for "Always A" **)
lemma AlwaysI:
"[| Init(F)<=A; F \<in> Stable(A) |] ==> F \<in> Always(A)"
apply (unfold Always_def initially_def)
apply (frule Stable_type [THEN subsetD], auto)
done
lemma AlwaysD: "F \<in> Always(A) ==> Init(F)<=A & F \<in> Stable(A)"
by (simp add: Always_def initially_def)
lemmas AlwaysE = AlwaysD [THEN conjE]
lemmas Always_imp_Stable = AlwaysD [THEN conjunct2]
(*The set of all reachable states is Always*)
lemma Always_includes_reachable: "F \<in> Always(A) ==> reachable(F) \<subseteq> A"
apply (simp (no_asm_use) add: Stable_def Constrains_def constrains_def Always_def initially_def)
apply (rule subsetI)
apply (erule reachable.induct)
apply (blast intro: reachable.intros)+
done
lemma invariant_imp_Always:
"F \<in> invariant(A) ==> F \<in> Always(A)"
apply (unfold Always_def invariant_def Stable_def stable_def)
apply (blast intro: constrains_imp_Constrains)
done
lemmas Always_reachable = invariant_reachable [THEN invariant_imp_Always]
lemma Always_eq_invariant_reachable: "Always(A) = {F \<in> program. F \<in> invariant(reachable(F) \<inter> A)}"
apply (simp (no_asm) add: Always_def invariant_def Stable_def Constrains_def2 stable_def initially_def)
apply (rule equalityI, auto)
apply (blast intro: reachable.intros reachable_type)
done
(*the RHS is the traditional definition of the "always" operator*)
lemma Always_eq_includes_reachable: "Always(A) = {F \<in> program. reachable(F) \<subseteq> A}"
apply (rule equalityI, safe)
apply (auto dest: invariant_includes_reachable
simp add: subset_Int_iff invariant_reachable Always_eq_invariant_reachable)
done
lemma Always_type: "Always(A) \<subseteq> program"
by (unfold Always_def initially_def, auto)
lemma Always_state_eq: "Always(state) = program"
apply (rule equalityI)
apply (auto dest: Always_type [THEN subsetD] reachable_type [THEN subsetD]
simp add: Always_eq_includes_reachable)
done
declare Always_state_eq [simp]
lemma state_AlwaysI: "F \<in> program ==> F \<in> Always(state)"
by (auto dest: reachable_type [THEN subsetD]
simp add: Always_eq_includes_reachable)
lemma Always_eq_UN_invariant: "st_set(A) ==> Always(A) = (\<Union>I \<in> Pow(A). invariant(I))"
apply (simp (no_asm) add: Always_eq_includes_reachable)
apply (rule equalityI, auto)
apply (blast intro: invariantI rev_subsetD [OF _ Init_into_reachable]
rev_subsetD [OF _ invariant_includes_reachable]
dest: invariant_type [THEN subsetD])+
done
lemma Always_weaken: "[| F \<in> Always(A); A \<subseteq> B |] ==> F \<in> Always(B)"
by (auto simp add: Always_eq_includes_reachable)
(*** "Co" rules involving Always ***)
lemmas Int_absorb2 = subset_Int_iff [unfolded iff_def, THEN conjunct1, THEN mp]
lemma Always_Constrains_pre: "F \<in> Always(I) ==> (F:(I \<inter> A) Co A') \<longleftrightarrow> (F \<in> A Co A')"
apply (simp (no_asm_simp) add: Always_includes_reachable [THEN Int_absorb2] Constrains_def Int_assoc [symmetric])
done
lemma Always_Constrains_post: "F \<in> Always(I) ==> (F \<in> A Co (I \<inter> A')) \<longleftrightarrow>(F \<in> A Co A')"
apply (simp (no_asm_simp) add: Always_includes_reachable [THEN Int_absorb2] Constrains_eq_constrains Int_assoc [symmetric])
done
lemma Always_ConstrainsI: "[| F \<in> Always(I); F \<in> (I \<inter> A) Co A' |] ==> F \<in> A Co A'"
by (blast intro: Always_Constrains_pre [THEN iffD1])
(* [| F \<in> Always(I); F \<in> A Co A' |] ==> F \<in> A Co (I \<inter> A') *)
lemmas Always_ConstrainsD = Always_Constrains_post [THEN iffD2]
(*The analogous proof of Always_LeadsTo_weaken doesn't terminate*)
lemma Always_Constrains_weaken:
"[|F \<in> Always(C); F \<in> A Co A'; C \<inter> B<=A; C \<inter> A'<=B'|]==>F \<in> B Co B'"
apply (rule Always_ConstrainsI)
apply (drule_tac [2] Always_ConstrainsD, simp_all)
apply (blast intro: Constrains_weaken)
done
(** Conjoining Always properties **)
lemma Always_Int_distrib: "Always(A \<inter> B) = Always(A) \<inter> Always(B)"
by (auto simp add: Always_eq_includes_reachable)
(* the premise i \<in> I is need since \<Inter>is formally not defined for I=0 *)
lemma Always_INT_distrib: "i \<in> I==>Always(\<Inter>i \<in> I. A(i)) = (\<Inter>i \<in> I. Always(A(i)))"
apply (rule equalityI)
apply (auto simp add: Inter_iff Always_eq_includes_reachable)
done
lemma Always_Int_I: "[| F \<in> Always(A); F \<in> Always(B) |] ==> F \<in> Always(A \<inter> B)"
apply (simp (no_asm_simp) add: Always_Int_distrib)
done
(*Allows a kind of "implication introduction"*)
lemma Always_Diff_Un_eq: "[| F \<in> Always(A) |] ==> (F \<in> Always(C-A \<union> B)) \<longleftrightarrow> (F \<in> Always(B))"
by (auto simp add: Always_eq_includes_reachable)
(*Delete the nearest invariance assumption (which will be the second one
used by Always_Int_I) *)
lemmas Always_thin = thin_rl [of "F \<in> Always(A)"] for F A
(*To allow expansion of the program's definition when appropriate*)
named_theorems program "program definitions"
ML
\<open>
(*Combines two invariance ASSUMPTIONS into one. USEFUL??*)
fun Always_Int_tac ctxt =
dresolve_tac ctxt @{thms Always_Int_I} THEN'
assume_tac ctxt THEN'
eresolve_tac ctxt @{thms Always_thin};
(*Combines a list of invariance THEOREMS into one.*)
val Always_Int_rule = foldr1 (fn (th1,th2) => [th1,th2] MRS @{thm Always_Int_I});
(*proves "co" properties when the program is specified*)
fun constrains_tac ctxt =
SELECT_GOAL
(EVERY [REPEAT (Always_Int_tac ctxt 1),
REPEAT (eresolve_tac ctxt @{thms Always_ConstrainsI} 1
ORELSE
resolve_tac ctxt [@{thm StableI}, @{thm stableI},
@{thm constrains_imp_Constrains}] 1),
resolve_tac ctxt @{thms constrainsI} 1,
(* Three subgoals *)
rewrite_goal_tac ctxt [@{thm st_set_def}] 3,
REPEAT (force_tac ctxt 2),
full_simp_tac (ctxt addsimps (Named_Theorems.get ctxt @{named_theorems program})) 1,
ALLGOALS (clarify_tac ctxt),
REPEAT (FIRSTGOAL (eresolve_tac ctxt @{thms disjE})),
ALLGOALS (clarify_tac ctxt),
REPEAT (FIRSTGOAL (eresolve_tac ctxt @{thms disjE})),
ALLGOALS (clarify_tac ctxt),
ALLGOALS (asm_full_simp_tac ctxt),
ALLGOALS (clarify_tac ctxt)]);
(*For proving invariants*)
fun always_tac ctxt i =
resolve_tac ctxt @{thms AlwaysI} i THEN
force_tac ctxt i
THEN constrains_tac ctxt i;
\<close>
method_setup safety = \<open>
Scan.succeed (SIMPLE_METHOD' o constrains_tac)\<close>
"for proving safety properties"
method_setup always = \<open>
Scan.succeed (SIMPLE_METHOD' o always_tac)\<close>
"for proving invariants"
end