(* Title: HOL/UNITY/Comp.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Composition
From Chandy and Sanders, "Reasoning About Program Composition"
*)
(*** component ***)
Goalw [component_def]
"(F component G) = (Init G <= Init F & Acts F <= Acts G)";
by (force_tac (claset() addSIs [exI, program_equalityI],
simpset() addsimps [Acts_Join]) 1);
qed "component_eq_subset";
Goalw [component_def] "SKIP component F";
by (force_tac (claset() addIs [Join_SKIP_left], simpset()) 1);
qed "component_SKIP";
Goalw [component_def] "F component F";
by (blast_tac (claset() addIs [Join_SKIP_right]) 1);
qed "component_refl";
AddIffs [component_SKIP, component_refl];
Goal "F component SKIP ==> F = SKIP";
by (auto_tac (claset() addSIs [program_equalityI],
simpset() addsimps [component_eq_subset]));
qed "SKIP_minimal";
Goalw [component_def] "F component (F Join G)";
by (Blast_tac 1);
qed "component_Join1";
Goalw [component_def] "G component (F Join G)";
by (simp_tac (simpset() addsimps [Join_commute]) 1);
by (Blast_tac 1);
qed "component_Join2";
Goalw [component_def] "i : I ==> (F i) component (JN i:I. (F i))";
by (blast_tac (claset() addIs [JN_absorb]) 1);
qed "component_JN";
Goalw [component_def] "[| F component G; G component H |] ==> F component H";
by (blast_tac (claset() addIs [Join_assoc RS sym]) 1);
qed "component_trans";
Goal "[| F component G; G component F |] ==> F=G";
by (full_simp_tac (simpset() addsimps [component_eq_subset]) 1);
by (blast_tac (claset() addSIs [program_equalityI]) 1);
qed "component_antisym";
Goalw [component_def]
"F component H = (EX G. F Join G = H & Disjoint F G)";
by (blast_tac (claset() addSIs [Diff_Disjoint, Join_Diff2]) 1);
qed "component_eq";
Goal "((F Join G) component H) = (F component H & G component H)";
by (simp_tac (simpset() addsimps [component_eq_subset, Acts_Join]) 1);
by (Blast_tac 1);
qed "Join_component_iff";
(*** existential properties ***)
Goalw [ex_prop_def]
"[| ex_prop X; finite GG |] ==> GG Int X ~= {} --> (JN G:GG. G) : X";
by (etac finite_induct 1);
by (auto_tac (claset(), simpset() addsimps [Int_insert_left]));
qed_spec_mp "ex1";
Goalw [ex_prop_def]
"ALL GG. finite GG & GG Int X ~= {} --> (JN G:GG. G) : X ==> ex_prop X";
by (Clarify_tac 1);
by (dres_inst_tac [("x", "{F,G}")] spec 1);
by Auto_tac;
qed "ex2";
(*Chandy & Sanders take this as a definition*)
Goal "ex_prop X = (ALL GG. finite GG & GG Int X ~= {} --> (JN G:GG. G) : X)";
by (blast_tac (claset() addIs [ex1,ex2]) 1);
qed "ex_prop_finite";
(*Their "equivalent definition" given at the end of section 3*)
Goal "ex_prop X = (ALL G. G:X = (ALL H. G component H --> H: X))";
by Auto_tac;
by (rewrite_goals_tac [ex_prop_def, component_def]);
by (Blast_tac 1);
by Safe_tac;
by (stac Join_commute 2);
by (ALLGOALS Blast_tac);
qed "ex_prop_equiv";
(*** universal properties ***)
Goalw [uv_prop_def]
"[| uv_prop X; finite GG |] ==> GG <= X --> (JN G:GG. G) : X";
by (etac finite_induct 1);
by (auto_tac (claset(), simpset() addsimps [Int_insert_left]));
qed_spec_mp "uv1";
Goalw [uv_prop_def]
"ALL GG. finite GG & GG <= X --> (JN G:GG. G) : X ==> uv_prop X";
by (rtac conjI 1);
by (Clarify_tac 2);
by (dres_inst_tac [("x", "{F,G}")] spec 2);
by (dres_inst_tac [("x", "{}")] spec 1);
by Auto_tac;
qed "uv2";
(*Chandy & Sanders take this as a definition*)
Goal "uv_prop X = (ALL GG. finite GG & GG <= X --> (JN G:GG. G) : X)";
by (blast_tac (claset() addIs [uv1,uv2]) 1);
qed "uv_prop_finite";
(*** guarantees ***)
val prems = Goal
"(!!G. [| F Join G : X; Disjoint F G |] ==> F Join G : Y) \
\ ==> F : X guar Y";
by (simp_tac (simpset() addsimps [guarantees_def, component_eq]) 1);
by (blast_tac (claset() addIs prems) 1);
qed "guaranteesI";
Goalw [guarantees_def, component_def]
"[| F : X guar Y; F Join G : X |] ==> F Join G : Y";
by (Blast_tac 1);
qed "guaranteesD";
(*This equation is more intuitive than the official definition*)
Goal "(F : X guar Y) = \
\ (ALL G. F Join G : X & Disjoint F G --> F Join G : Y)";
by (simp_tac (simpset() addsimps [guarantees_def, component_eq]) 1);
by (Blast_tac 1);
qed "guarantees_eq";
Goalw [guarantees_def] "[| F: X guar X'; Y <= X; X' <= Y' |] ==> F: Y guar Y'";
by (Blast_tac 1);
qed "guarantees_weaken";
Goalw [guarantees_def] "[| F: X guar Y; F component F' |] ==> F': X guar Y";
by (blast_tac (claset() addIs [component_trans]) 1);
qed "guarantees_weaken_prog";
Goalw [guarantees_def] "X <= Y ==> X guar Y = UNIV";
by (Blast_tac 1);
qed "subset_imp_guarantees_UNIV";
(*Equivalent to subset_imp_guarantees_UNIV but more intuitive*)
Goalw [guarantees_def] "X <= Y ==> F : X guar Y";
by (Blast_tac 1);
qed "subset_imp_guarantees";
(*Remark at end of section 4.1*)
Goalw [guarantees_def] "ex_prop Y = (Y = UNIV guar Y)";
by (simp_tac (simpset() addsimps [ex_prop_equiv]) 1);
by (blast_tac (claset() addEs [equalityE]) 1);
qed "ex_prop_equiv2";
(** Distributive laws. Re-orient to perform miniscoping **)
Goalw [guarantees_def]
"(UN X:XX. X) guar Y = (INT X:XX. X guar Y)";
by (Blast_tac 1);
qed "guarantees_UN_left";
Goalw [guarantees_def]
"(X Un Y) guar Z = (X guar Z) Int (Y guar Z)";
by (Blast_tac 1);
qed "guarantees_Un_left";
Goalw [guarantees_def]
"X guar (INT Y:YY. Y) = (INT Y:YY. X guar Y)";
by (Blast_tac 1);
qed "guarantees_INT_right";
Goalw [guarantees_def]
"Z guar (X Int Y) = (Z guar X) Int (Z guar Y)";
by (Blast_tac 1);
qed "guarantees_Int_right";
Goalw [guarantees_def] "(X guar Y) = (UNIV guar (-X Un Y))";
by (Blast_tac 1);
qed "shunting";
Goalw [guarantees_def] "(X guar Y) = -Y guar -X";
by (Blast_tac 1);
qed "contrapositive";
(** The following two can be expressed using intersection and subset, which
is more faithful to the text but looks cryptic.
**)
Goalw [guarantees_def]
"[| F : V guar X; F : (X Int Y) guar Z |]\
\ ==> F : (V Int Y) guar Z";
by (Blast_tac 1);
qed "combining1";
Goalw [guarantees_def]
"[| F : V guar (X Un Y); F : Y guar Z |]\
\ ==> F : V guar (X Un Z)";
by (Blast_tac 1);
qed "combining2";
(** The following two follow Chandy-Sanders, but the use of object-quantifiers
does not suit Isabelle... **)
(*Premise should be (!!i. i: I ==> F: X guar Y i) *)
Goalw [guarantees_def]
"ALL i:I. F : X guar (Y i) ==> F : X guar (INT i:I. Y i)";
by (Blast_tac 1);
qed "all_guarantees";
(*Premises should be [| F: X guar Y i; i: I |] *)
Goalw [guarantees_def]
"EX i:I. F : X guar (Y i) ==> F : X guar (UN i:I. Y i)";
by (Blast_tac 1);
qed "ex_guarantees";
(*** Additional guarantees laws, by lcp ***)
Goalw [guarantees_def]
"[| F: U guar V; G: X guar Y |] ==> F Join G: (U Int X) guar (V Int Y)";
by (simp_tac (simpset() addsimps [Join_component_iff]) 1);
by (Blast_tac 1);
qed "guarantees_Join_Int";
Goalw [guarantees_def]
"[| F: U guar V; G: X guar Y |] ==> F Join G: (U Un X) guar (V Un Y)";
by (simp_tac (simpset() addsimps [Join_component_iff]) 1);
by (Blast_tac 1);
qed "guarantees_Join_Un";
Goal "((JOIN I F) component H) = (ALL i: I. F i component H)";
by (simp_tac (simpset() addsimps [component_eq_subset, Acts_JN]) 1);
by (Blast_tac 1);
qed "JN_component_iff";
Goalw [guarantees_def]
"[| ALL i:I. F i : X i guar Y i |] \
\ ==> (JOIN I F) : (INTER I X) guar (INTER I Y)";
by (simp_tac (simpset() addsimps [JN_component_iff]) 1);
by (Blast_tac 1);
qed "guarantees_JN_INT";
Goalw [guarantees_def]
"[| ALL i:I. F i : X i guar Y i |] \
\ ==> (JOIN I F) : (UNION I X) guar (UNION I Y)";
by (simp_tac (simpset() addsimps [JN_component_iff]) 1);
by (Blast_tac 1);
qed "guarantees_JN_UN";
(*** well-definedness ***)
Goalw [welldef_def] "F Join G: welldef ==> F: welldef";
by Auto_tac;
qed "Join_welldef_D1";
Goalw [welldef_def] "F Join G: welldef ==> G: welldef";
by Auto_tac;
qed "Join_welldef_D2";
(*** refinement ***)
Goalw [refines_def] "F refines F wrt X";
by (Blast_tac 1);
qed "refines_refl";
Goalw [refines_def]
"[| H refines G wrt X; G refines F wrt X |] ==> H refines F wrt X";
by Auto_tac;
qed "refines_trans";
Goalw [strict_ex_prop_def]
"strict_ex_prop X \
\ ==> (ALL H. F Join H : X --> G Join H : X) = (F:X --> G:X)";
by (Blast_tac 1);
qed "strict_ex_refine_lemma";
Goalw [strict_ex_prop_def]
"strict_ex_prop X \
\ ==> (ALL H. F Join H : welldef & F Join H : X --> G Join H : X) = \
\ (F: welldef Int X --> G:X)";
by Safe_tac;
by (eres_inst_tac [("x","SKIP"), ("P", "%H. ?PP H --> ?RR H")] allE 1);
by (auto_tac (claset() addDs [Join_welldef_D1, Join_welldef_D2], simpset()));
qed "strict_ex_refine_lemma_v";
Goal "[| strict_ex_prop X; \
\ ALL H. F Join H : welldef Int X --> G Join H : welldef |] \
\ ==> (G refines F wrt X) = (G iso_refines F wrt X)";
by (res_inst_tac [("x","SKIP")] allE 1
THEN assume_tac 1);
by (asm_full_simp_tac
(simpset() addsimps [refines_def, iso_refines_def,
strict_ex_refine_lemma_v]) 1);
qed "ex_refinement_thm";
Goalw [strict_uv_prop_def]
"strict_uv_prop X \
\ ==> (ALL H. F Join H : X --> G Join H : X) = (F:X --> G:X)";
by (Blast_tac 1);
qed "strict_uv_refine_lemma";
Goalw [strict_uv_prop_def]
"strict_uv_prop X \
\ ==> (ALL H. F Join H : welldef & F Join H : X --> G Join H : X) = \
\ (F: welldef Int X --> G:X)";
by Safe_tac;
by (eres_inst_tac [("x","SKIP"), ("P", "%H. ?PP H --> ?RR H")] allE 1);
by (auto_tac (claset() addDs [Join_welldef_D1, Join_welldef_D2],
simpset()));
qed "strict_uv_refine_lemma_v";
Goal "[| strict_uv_prop X; \
\ ALL H. F Join H : welldef Int X --> G Join H : welldef |] \
\ ==> (G refines F wrt X) = (G iso_refines F wrt X)";
by (res_inst_tac [("x","SKIP")] allE 1
THEN assume_tac 1);
by (asm_full_simp_tac (simpset() addsimps [refines_def, iso_refines_def,
strict_uv_refine_lemma_v]) 1);
qed "uv_refinement_thm";