# Theory Constrains

theory Constrains
imports UNITY
```(*  Title:      ZF/UNITY/Constrains.thy
Author:     Sidi O Ehmety, Computer Laboratory
*)

section‹Weak Safety Properties›

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 ∪ (⋃act∈acts. field(act)))*list(⋃act∈acts. field(act))"
intros
(*Initial trace is empty*)
Init: "s: init ==> <s,[]> ∈ traces(init,acts)"

Acts: "[| act:acts;  <s,evs> ∈ traces(init,acts);  <s,s'>: act |]
==> <s', Cons(s,evs)> ∈ traces(init, acts)"

type_intros list.intros UnI1 UnI2 UN_I fieldI2 fieldI1

consts reachable :: "i=>i"
inductive
domains
"reachable(F)" ⊆ "Init(F) ∪ (⋃act∈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) ∩ A) co B}"

definition
op_Unless  :: "[i, i] => i"  (infixl "Unless" 60)  where
"A Unless B == (A-B) Co (A ∪ 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) ∩ Stable(A)"

(*** traces and reachable ***)

lemma reachable_type: "reachable(F) ⊆ 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) ∩ state = reachable(F)"
by (cut_tac reachable_type, auto)
declare reachable_Int_state [iff]

lemma state_Int_reachable: "state ∩ reachable(F) = reachable(F)"
by (cut_tac reachable_type, auto)
declare state_Int_reachable [iff]

lemma reachable_equiv_traces:
"F ∈ program ==> reachable(F)={s ∈ state. ∃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) ⊆ reachable(F)"
by (blast intro: reachable.intros)

lemma stable_reachable: "[| F ∈ program; G ∈ program;
Acts(G) ⊆ Acts(F)  |] ==> G ∈ 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 ∈ program ==> F ∈ 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 ∈ invariant(A) ==> reachable(F) ⊆ 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 ∈ B co B'==>F:(reachable(F) ∩ B) co (reachable(F) ∩ B')"
apply (frule constrains_type [THEN subsetD])
apply (frule stable_reachable [OF _ _ subset_refl])
done

(*Resembles the previous definition of Constrains*)
lemma Constrains_eq_constrains:
"A Co B = {F ∈ program. F:(reachable(F) ∩ A) co (reachable(F)  ∩  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 ∈ A co A' ==> F ∈ A Co A'"
apply (unfold Constrains_def)
apply (blast intro: constrains_weaken_L dest: constrainsD2)
done

lemma ConstrainsI:
"[|!!act s s'. [| act ∈ Acts(F); <s,s'>:act; s ∈ A |] ==> s':A';
F ∈ program|]
==> F ∈ 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 ⊆ program"
apply (unfold Constrains_def, blast)
done

lemma Constrains_empty: "F ∈ 0 Co B ⟷ F ∈ program"
by (auto dest: Constrains_type [THEN subsetD]
intro: constrains_imp_Constrains)
declare Constrains_empty [iff]

lemma Constrains_state: "F ∈ A Co state ⟷ F ∈ 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 ∈ A Co A'; A'<=B' |] ==> F ∈ A Co B'"
apply (unfold Constrains_def2)
apply (blast intro: constrains_weaken_R)
done

lemma Constrains_weaken_L:
"[| F ∈ A Co A'; B<=A |] ==> F ∈ B Co A'"
apply (unfold Constrains_def2)
apply (blast intro: constrains_weaken_L st_set_subset)
done

lemma Constrains_weaken:
"[| F ∈ A Co A'; B<=A; A'<=B' |] ==> F ∈ B Co B'"
apply (unfold Constrains_def2)
apply (blast intro: constrains_weaken st_set_subset)
done

(** Union **)
lemma Constrains_Un:
"[| F ∈ A Co A'; F ∈ B Co B' |] ==> F ∈ (A ∪ B) Co (A' ∪ B')"
apply (unfold Constrains_def2, auto)
apply (blast intro: constrains_Un)
done

lemma Constrains_UN:
"[|(!!i. i ∈ I==>F ∈ A(i) Co A'(i)); F ∈ program|]
==> F:(⋃i ∈ I. A(i)) Co (⋃i ∈ I. A'(i))"
by (auto intro: constrains_UN simp del: UN_simps

(** Intersection **)

lemma Constrains_Int:
"[| F ∈ A Co A'; F ∈ B Co B'|]==> F:(A ∩ B) Co (A' ∩ B')"
apply (unfold Constrains_def)
apply (subgoal_tac "reachable (F) ∩ (A ∩ B) = (reachable (F) ∩ A) ∩ (reachable (F) ∩ B) ")
apply (auto intro: constrains_Int)
done

lemma Constrains_INT:
"[| (!!i. i ∈ I ==>F ∈ A(i) Co A'(i)); F ∈ program  |]
==> F:(⋂i ∈ I. A(i)) Co (⋂i ∈ I. A'(i))"
apply (simp (no_asm_simp) del: INT_simps add: Constrains_def INT_extend_simps)
apply (rule constrains_INT)
done

lemma Constrains_imp_subset: "F ∈ A Co A' ==> reachable(F) ∩ A ⊆ A'"
apply (unfold Constrains_def)
apply (blast dest: constrains_imp_subset)
done

lemma Constrains_trans:
"[| F ∈ A Co B; F ∈ B Co C |] ==> F ∈ A Co C"
apply (unfold Constrains_def2)
apply (blast intro: constrains_trans constrains_weaken)
done

lemma Constrains_cancel:
"[| F ∈ A Co (A' ∪ B); F ∈ B Co B' |] ==> F ∈ A Co (A' ∪ B')"
apply (unfold Constrains_def2)
apply (blast intro: constrains_cancel)
done

(*** Stable ***)
(* Useful because there's no Stable_weaken.  [Tanja Vos] *)

lemma stable_imp_Stable:
"F ∈ stable(A) ==> F ∈ Stable(A)"

apply (unfold stable_def Stable_def)
apply (erule constrains_imp_Constrains)
done

lemma Stable_eq: "[| F ∈ Stable(A); A = B |] ==> F ∈ Stable(B)"
by blast

lemma Stable_eq_stable:
"F ∈ Stable(A) ⟷  (F ∈ stable(reachable(F) ∩ A))"
apply (auto dest: constrainsD2 simp add: Stable_def stable_def Constrains_def2)
done

lemma StableI: "F ∈ A Co A ==> F ∈ Stable(A)"
by (unfold Stable_def, assumption)

lemma StableD: "F ∈ Stable(A) ==> F ∈ A Co A"
by (unfold Stable_def, assumption)

lemma Stable_Un:
"[| F ∈ Stable(A); F ∈ Stable(A') |] ==> F ∈ Stable(A ∪ A')"
apply (unfold Stable_def)
apply (blast intro: Constrains_Un)
done

lemma Stable_Int:
"[| F ∈ Stable(A); F ∈ Stable(A') |] ==> F ∈ Stable (A ∩ A')"
apply (unfold Stable_def)
apply (blast intro: Constrains_Int)
done

lemma Stable_Constrains_Un:
"[| F ∈ Stable(C); F ∈ A Co (C ∪ A') |]
==> F ∈ (C ∪ A) Co (C ∪ A')"
apply (unfold Stable_def)
apply (blast intro: Constrains_Un [THEN Constrains_weaken_R])
done

lemma Stable_Constrains_Int:
"[| F ∈ Stable(C); F ∈ (C ∩ A) Co A' |]
==> F ∈ (C ∩ A) Co (C ∩ A')"
apply (unfold Stable_def)
apply (blast intro: Constrains_Int [THEN Constrains_weaken])
done

lemma Stable_UN:
"[| (!!i. i ∈ I ==> F ∈ Stable(A(i))); F ∈ program |]
==> F ∈ Stable (⋃i ∈ I. A(i))"
apply (blast intro: Constrains_UN)
done

lemma Stable_INT:
"[|(!!i. i ∈ I ==> F ∈ Stable(A(i))); F ∈ program |]
==> F ∈ Stable (⋂i ∈ I. A(i))"
apply (blast intro: Constrains_INT)
done

lemma Stable_reachable: "F ∈ program ==>F ∈ Stable (reachable(F))"
apply (simp (no_asm_simp) add: Stable_eq_stable Int_absorb)
done

lemma Stable_type: "Stable(A) ⊆ 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 ∀m ?  Would make it harder to use
in forward proof. ***)

lemma Elimination:
"[| ∀m ∈ M. F ∈ ({s ∈ A. x(s) = m}) Co (B(m)); F ∈ program |]
==> F ∈ ({s ∈ A. x(s):M}) Co (⋃m ∈ M. B(m))"
apply (unfold Constrains_def, auto)
apply (rule_tac A1 = "reachable (F) ∩ 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:
"[| ∀m ∈ M. F ∈ {s ∈ state. x(s) = m} Co B(m); F ∈ program |]
==> F ∈ {s ∈ state. x(s):M} Co (⋃m ∈ 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 ∈ Stable(A) |] ==> F ∈ Always(A)"

apply (unfold Always_def initially_def)
apply (frule Stable_type [THEN subsetD], auto)
done

lemma AlwaysD: "F ∈ Always(A) ==> Init(F)<=A & F ∈ Stable(A)"

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 ∈ Always(A) ==> reachable(F) ⊆ 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 ∈ invariant(A) ==> F ∈ 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 ∈ program. F ∈ invariant(reachable(F) ∩ 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 ∈ program. reachable(F) ⊆ A}"
apply (rule equalityI, safe)
apply (auto dest: invariant_includes_reachable
done

lemma Always_type: "Always(A) ⊆ 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]
done
declare Always_state_eq [simp]

lemma state_AlwaysI: "F ∈ program ==> F ∈ Always(state)"
by (auto dest: reachable_type [THEN subsetD]

lemma Always_eq_UN_invariant: "st_set(A) ==> Always(A) = (⋃I ∈ Pow(A). invariant(I))"
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 ∈ Always(A); A ⊆ B |] ==> F ∈ Always(B)"

(*** "Co" rules involving Always ***)
lemmas Int_absorb2 = subset_Int_iff [unfolded iff_def, THEN conjunct1, THEN mp]

lemma Always_Constrains_pre: "F ∈ Always(I) ==> (F:(I ∩ A) Co A') ⟷ (F ∈ 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 ∈ Always(I) ==> (F ∈ A Co (I ∩ A')) ⟷(F ∈ 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 ∈ Always(I);  F ∈ (I ∩ A) Co A' |] ==> F ∈ A Co A'"
by (blast intro: Always_Constrains_pre [THEN iffD1])

(* [| F ∈ Always(I);  F ∈ A Co A' |] ==> F ∈ A Co (I ∩ A') *)
lemmas Always_ConstrainsD = Always_Constrains_post [THEN iffD2]

(*The analogous proof of Always_LeadsTo_weaken doesn't terminate*)
lemma Always_Constrains_weaken:
"[|F ∈ Always(C); F ∈ A Co A'; C ∩ B<=A; C ∩ A'<=B'|]==>F ∈ 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 ∩ B) = Always(A) ∩ Always(B)"

(* the premise i ∈ I is need since ⋂is formally not defined for I=0 *)
lemma Always_INT_distrib: "i ∈ I==>Always(⋂i ∈ I. A(i)) = (⋂i ∈ I. Always(A(i)))"
apply (rule equalityI)
apply (auto simp add: Inter_iff Always_eq_includes_reachable)
done

lemma Always_Int_I: "[| F ∈ Always(A);  F ∈ Always(B) |] ==> F ∈ Always(A ∩ B)"
done

(*Allows a kind of "implication introduction"*)
lemma Always_Diff_Un_eq: "[| F ∈ Always(A) |] ==> (F ∈ Always(C-A ∪ B)) ⟷ (F ∈ Always(B))"

(*Delete the nearest invariance assumption (which will be the second one
used by Always_Int_I) *)
lemmas Always_thin = thin_rl [of "F ∈ Always(A)"]

(*To allow expansion of the program's definition when appropriate*)
named_theorems program "program definitions"

ML
‹
(*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;
›

method_setup safety = ‹
Scan.succeed (SIMPLE_METHOD' o constrains_tac)›
"for proving safety properties"

method_setup always = ‹
Scan.succeed (SIMPLE_METHOD' o always_tac)›
"for proving invariants"

end
```