# Theory Comp

theory Comp
imports Union Increasing
```(*  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‹Composition›

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. <f(x), g(x)>"

definition
localize  :: "[i=>i, i] => i"  where
"localize(f,F) == mk_program(Init(F), Acts(F),
AllowedActs(F) ∩ (⋃G∈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) ⊆ Init(F) & Acts(F) ⊆ Acts(G) & AllowedActs(G) ⊆ 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"
apply (unfold 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"
apply (unfold 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)"
apply (unfold 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)))"
apply (unfold component_def)
apply (blast intro: JOIN_absorb)
done

lemma component_trans: "[| F component G; G component H |] ==> F component H"
apply (unfold 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 ⊔ G) component H) ⟷ (F component H & 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))"
apply (unfold 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)"
apply (unfold 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) ∈ preserves(v) & programify(G) ∈ 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) = <f(x), g(x)>"
apply (unfold 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)"
apply (unfold 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"
apply (unfold 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"
apply (unfold 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))"
apply (unfold 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) |]
==> F ⊔ G ∈ stable({s ∈ 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. <f(s), g(s)>:r});  G ∈ preserves(f);
F ⊔ G ∈ Increasing(A, r, g);
∀x ∈ state. f(x):A & g(x):A |]
==> F ⊔ G ∈ Stable({s ∈ 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))});
∀k ∈ A. F ⊔ G ∈ Stable({s ∈ state. P(k, g(s))});
G ∈ preserves(f); ∀s ∈ state. f(s):A|]
==> F ⊔ G ∈ Stable({s ∈ state. P(f(s), g(s))})"
apply (unfold 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
```