Impressum Comp.thy Sprache: Isabelle

(*  Title:      ZF/UNITY/Comp.thy
    Author:     Sidi O Ehmety, Computer Laboratory
    Copyright   1998  University of Cambridge

From Chandy and Sanders, "Reasoning About Program Composition",
Technical Report 2000-003, University of Florida, 2000.

Revised by Sidi Ehmety on January  2001

Added: a strong form of the order relation over component and localize

Theory ported from HOL.

*)

section\<open>Composition\<close>

theory Comp imports Union Increasing begin

definition
  component :: "[i,i]\o" (infixl \component\ 65) where
  "F component H \ (\G. F \ G = H)"

definition
  strict_component :: "[i,i]\o" (infixl \strict'_component\ 65) where
  "F strict_component H \ F component H \ F\H"

definition
  (* A stronger form of the component relation *)
  component_of :: "[i,i]\o" (infixl \component'_of\ 65) where
  "F component_of H \ (\G. F ok G \ F \ G = H)"

definition
  strict_component_of :: "[i,i]\o" (infixl \strict'_component'_of\ 65) where
  "F strict_component_of H \ F component_of H \ F\H"

definition
  (* Preserves a state function f, in particular a variable *)
  preserves :: "(i\i)\i" where
  "preserves(f) \ {F:program. \z. F: stable({s:state. f(s) = z})}"

definition
  fun_pair :: "[i\i, i \i] \(i\i)" where
  "fun_pair(f, g) \ \x. "

definition
  localize  :: "[i\i, i] \ i" where
  "localize(f,F) \ mk_program(Init(F), Acts(F),
                       AllowedActs(F) \<inter> (\<Union>G\<in>preserves(f). Acts(G)))"


(*** component and strict_component relations ***)

lemma componentI:
     "H component F | H component G \ H component (F \ G)"
apply (unfold component_def, auto)
apply (rule_tac x = "Ga \ G" in exI)
apply (rule_tac [2] x = "G \ F" in exI)
apply (auto simp add: Join_ac)
done

lemma component_eq_subset:
     "G \ program \ (F component G) \
   (Init(G) \<subseteq> Init(F) \<and> Acts(F) \<subseteq> Acts(G) \<and> AllowedActs(G) \<subseteq> AllowedActs(F))"
apply (unfold component_def, auto)
apply (rule exI)
apply (rule program_equalityI, auto)
done

lemma component_SKIP [simp]: "F \ program \ SKIP component F"
  unfolding component_def
apply (rule_tac x = F in exI)
apply (force intro: Join_SKIP_left)
done

lemma component_refl [simp]: "F \ program \ F component F"
  unfolding component_def
apply (rule_tac x = F in exI)
apply (force intro: Join_SKIP_right)
done

lemma SKIP_minimal: "F component SKIP \ programify(F) = SKIP"
apply (rule program_equalityI)
apply (simp_all add: component_eq_subset, blast)
done

lemma component_Join1: "F component (F \ G)"
by (unfold component_def, blast)

lemma component_Join2: "G component (F \ G)"
  unfolding component_def
apply (simp (no_asm) add: Join_commute)
apply blast
done

lemma Join_absorb1: "F component G \ F \ G = G"
by (auto simp add: component_def Join_left_absorb)

lemma Join_absorb2: "G component F \ F \ G = F"
by (auto simp add: Join_ac component_def)

lemma JOIN_component_iff:
     "H \ program\(JOIN(I,F) component H) \ (\i \ I. F(i) component H)"
apply (case_tac "I=0", force)
apply (simp (no_asm_simp) add: component_eq_subset)
apply auto
apply blast
apply (rename_tac "y")
apply (drule_tac c = y and A = "AllowedActs (H)" in subsetD)
apply (blast elim!: not_emptyE)+
done

lemma component_JOIN: "i \ I \ F(i) component (\i \ I. (F(i)))"
  unfolding component_def
apply (blast intro: JOIN_absorb)
done

lemma component_trans: "\F component G; G component H\ \ F component H"
  unfolding component_def
apply (blast intro: Join_assoc [symmetric])
done

lemma component_antisym:
     "\F \ program; G \ program\ \(F component G \ G component F) \ F = G"
apply (simp (no_asm_simp) add: component_eq_subset)
apply clarify
apply (rule program_equalityI, auto)
done

lemma Join_component_iff:
     "H \ program \
      ((F \<squnion> G) component H) \<longleftrightarrow> (F component H \<and> G component H)"
apply (simp (no_asm_simp) add: component_eq_subset)
apply blast
done

lemma component_constrains:
     "\F component G; G \ A co B; F \ program\ \ F \ A co B"
apply (frule constrainsD2)
apply (auto simp add: constrains_def component_eq_subset)
done

(*** preserves ***)

lemma preserves_is_safety_prop [simp]: "safety_prop(preserves(f))"
  unfolding preserves_def safety_prop_def
apply (auto dest: ActsD simp add: stable_def constrains_def)
apply (drule_tac c = act in subsetD, auto)
done

lemma preservesI [rule_format]:
     "\z. F \ stable({s \ state. f(s) = z}) \ F \ preserves(f)"
apply (auto simp add: preserves_def)
apply (blast dest: stableD2)
done

lemma preserves_imp_eq:
     "\F \ preserves(f); act \ Acts(F); \s,t\ \ act\ \ f(s) = f(t)"
  unfolding preserves_def stable_def constrains_def
apply (subgoal_tac "s \ state \ t \ state")
prefer 2 apply (blast dest!: Acts_type [THEN subsetD], force)
done

lemma Join_preserves [iff]:
"(F \ G \ preserves(v)) \
      (programify(F) \<in> preserves(v) \<and> programify(G) \<in> preserves(v))"
by (auto simp add: preserves_def INT_iff)

lemma JOIN_preserves [iff]:
     "(JOIN(I,F): preserves(v)) \ (\i \ I. programify(F(i)):preserves(v))"
by (auto simp add: JOIN_stable preserves_def INT_iff)

lemma SKIP_preserves [iff]: "SKIP \ preserves(v)"
by (auto simp add: preserves_def INT_iff)

lemma fun_pair_apply [simp]: "fun_pair(f,g,x) = "
  unfolding fun_pair_def
apply (simp (no_asm))
done

lemma preserves_fun_pair:
     "preserves(fun_pair(v,w)) = preserves(v) \ preserves(w)"
apply (rule equalityI)
apply (auto simp add: preserves_def stable_def constrains_def, blast+)
done

lemma preserves_fun_pair_iff [iff]:
     "F \ preserves(fun_pair(v, w)) \ F \ preserves(v) \ preserves(w)"
by (simp add: preserves_fun_pair)

lemma fun_pair_comp_distrib:
     "(fun_pair(f, g) comp h)(x) = fun_pair(f comp h, g comp h, x)"
by (simp add: fun_pair_def metacomp_def)

lemma comp_apply [simp]: "(f comp g)(x) = f(g(x))"
by (simp add: metacomp_def)

lemma preserves_type: "preserves(v)<=program"
by (unfold preserves_def, auto)

lemma preserves_into_program [TC]: "F \ preserves(f) \ F \ program"
by (blast intro: preserves_type [THEN subsetD])

lemma subset_preserves_comp: "preserves(f) \ preserves(g comp f)"
apply (auto simp add: preserves_def stable_def constrains_def)
apply (drule_tac x = "f (xb)" in spec)
apply (drule_tac x = act in bspec, auto)
done

lemma imp_preserves_comp: "F \ preserves(f) \ F \ preserves(g comp f)"
by (blast intro: subset_preserves_comp [THEN subsetD])

lemma preserves_subset_stable: "preserves(f) \ stable({s \ state. P(f(s))})"
apply (auto simp add: preserves_def stable_def constrains_def)
apply (rename_tac s' s)
apply (subgoal_tac "f (s) = f (s') ")
apply (force+)
done

lemma preserves_imp_stable:
     "F \ preserves(f) \ F \ stable({s \ state. P(f(s))})"
by (blast intro: preserves_subset_stable [THEN subsetD])

lemma preserves_imp_increasing:
"\F \ preserves(f); \x \ state. f(x):A\ \ F \ Increasing.increasing(A, r, f)"
  unfolding Increasing.increasing_def
apply (auto intro: preserves_into_program)
apply (rule_tac P = "\x. \k, x\:r" in preserves_imp_stable, auto)
done

lemma preserves_id_subset_stable:
"st_set(A) \ preserves(\x. x) \ stable(A)"
apply (unfold preserves_def stable_def constrains_def, auto)
apply (drule_tac x = xb in spec)
apply (drule_tac x = act in bspec)
apply (auto dest: ActsD)
done

lemma preserves_id_imp_stable:
     "\F \ preserves(\x. x); st_set(A)\ \ F \ stable(A)"
by (blast intro: preserves_id_subset_stable [THEN subsetD])

(** Added by Sidi **)
(** component_of **)

(*  component_of is stronger than component *)
lemma component_of_imp_component:
"F component_of H \ F component H"
apply (unfold component_def component_of_def, blast)
done

(* component_of satisfies many of component's properties *)
lemma component_of_refl [simp]: "F \ program \ F component_of F"
  unfolding component_of_def
apply (rule_tac x = SKIP in exI, auto)
done

lemma component_of_SKIP [simp]: "F \ program \SKIP component_of F"
apply (unfold component_of_def, auto)
apply (rule_tac x = F in exI, auto)
done

lemma component_of_trans:
     "\F component_of G; G component_of H\ \ F component_of H"
  unfolding component_of_def
apply (blast intro: Join_assoc [symmetric])
done

(** localize **)
lemma localize_Init_eq [simp]: "Init(localize(v,F)) = Init(F)"
by (unfold localize_def, simp)

lemma localize_Acts_eq [simp]: "Acts(localize(v,F)) = Acts(F)"
by (unfold localize_def, simp)

lemma localize_AllowedActs_eq [simp]:
"AllowedActs(localize(v,F)) = AllowedActs(F) \ (\G \ preserves(v). Acts(G))"
  unfolding localize_def
apply (rule equalityI)
apply (auto dest: Acts_type [THEN subsetD])
done

(** Theorems used in ClientImpl **)

lemma stable_localTo_stable2:
"\F \ stable({s \ state. P(f(s), g(s))}); G \ preserves(f); G \ preserves(g)\
      \<Longrightarrow> F \<squnion> G \<in> stable({s \<in> state. P(f(s), g(s))})"
apply (auto dest: ActsD preserves_into_program simp add: stable_def constrains_def)
apply (case_tac "act \ Acts (F) ")
apply auto
apply (drule preserves_imp_eq)
apply (drule_tac [3] preserves_imp_eq, auto)
done

lemma Increasing_preserves_Stable:
     "\F \ stable({s \ state. :r}); G \ preserves(f);
         F \<squnion> G \<in> Increasing(A, r, g);
         \<forall>x \<in> state. f(x):A \<and> g(x):A\<rbrakk>
      \<Longrightarrow> F \<squnion> G \<in> Stable({s \<in> state. <f(s), g(s)>:r})"
apply (auto simp add: stable_def Stable_def Increasing_def Constrains_def all_conj_distrib)
apply (simp_all add: constrains_type [THEN subsetD] preserves_type [THEN subsetD])
apply (blast intro: constrains_weaken)
(*The G case remains*)
apply (auto dest: ActsD simp add: preserves_def stable_def constrains_def ball_conj_distrib all_conj_distrib)
(*We have a G-action, so delete assumptions about F-actions*)
apply (erule_tac V = "\act \ Acts (F). P (act)" for P in thin_rl)
apply (erule_tac V = "\k\A. \act \ Acts (F) . P (k,act)" for P in thin_rl)
apply (subgoal_tac "f (x) = f (xa) ")
apply (auto dest!: bspec)
done

(** Lemma used in AllocImpl **)

lemma Constrains_UN_left:
     "\\x \ I. F \ A(x) Co B; F \ program\ \ F:(\x \ I. A(x)) Co B"
by (unfold Constrains_def constrains_def, auto)

lemma stable_Join_Stable:
"\F \ stable({s \ state. P(f(s), g(s))});
     \<forall>k \<in> A. F \<squnion> G \<in> Stable({s \<in> state. P(k, g(s))});
     G \<in> preserves(f); \<forall>s \<in> state. f(s):A\<rbrakk>
  \<Longrightarrow> F \<squnion> G \<in> Stable({s \<in> state. P(f(s), g(s))})"
  unfolding stable_def Stable_def preserves_def
apply (rule_tac A = "(\k \ A. {s \ state. f(s)=k} \ {s \ state. P (f (s), g (s))})" in Constrains_weaken_L)
prefer 2 apply blast
apply (rule Constrains_UN_left, auto)
apply (rule_tac A = "{s \ state. f(s)=k} \ {s \ state. P (f (s), g (s))} \ {s \ state. P (k, g (s))}" and A' = "({s \ state. f(s)=k} \ {s \ state. P (f (s), g (s))}) \ {s \ state. P (k, g (s))}" in Constrains_weaken)
prefer 2 apply blast
prefer 2 apply blast
apply (rule Constrains_Int)
apply (rule constrains_imp_Constrains)
apply (auto simp add: constrains_type [THEN subsetD])
apply (blast intro: constrains_weaken)
apply (drule_tac x = k in spec)
apply (blast intro: constrains_weaken)
done

end

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