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(* Title: HOL/UNITY/Union.ML
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1998 University of Cambridge
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Unions of programs
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From Misra's Chapter 5: Asynchronous Compositions of Programs
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*)
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Addcongs [UN_cong, INT_cong];
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(** SKIP **)
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Goal "Init SKIP = UNIV";
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by (simp_tac (simpset() addsimps [SKIP_def]) 1);
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qed "Init_SKIP";
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Goal "Acts SKIP = {Id}";
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by (simp_tac (simpset() addsimps [SKIP_def]) 1);
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qed "Acts_SKIP";
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Addsimps [Init_SKIP, Acts_SKIP];
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Goal "reachable SKIP = UNIV";
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by (force_tac (claset() addEs [reachable.induct]
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addIs reachable.intrs, simpset()) 1);
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qed "reachable_SKIP";
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Addsimps [reachable_SKIP];
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(** Join **)
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Goal "Init (F Join G) = Init F Int Init G";
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by (simp_tac (simpset() addsimps [Join_def]) 1);
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qed "Init_Join";
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Goal "Acts (F Join G) = Acts F Un Acts G";
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by (auto_tac (claset(), simpset() addsimps [Join_def]));
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qed "Acts_Join";
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Addsimps [Init_Join];
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(** JN **)
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Goalw [JOIN_def, SKIP_def] "(JN i:{}. F i) = SKIP";
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by Auto_tac;
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qed "JN_empty";
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Addsimps [JN_empty];
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Goal "(JN x:insert a A. B x) = (B a) Join (JN x:A. B x)";
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by (rtac program_equalityI 1);
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by (ALLGOALS (simp_tac (simpset() addsimps [JOIN_def, Join_def])));
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qed "JN_insert";
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Addsimps[JN_empty, JN_insert];
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Goal "Init (JN i:I. F i) = (INT i:I. Init (F i))";
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by (simp_tac (simpset() addsimps [JOIN_def]) 1);
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qed "Init_JN";
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Goal "Acts (JN i:I. F i) = insert Id (UN i:I. Acts (F i))";
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by (auto_tac (claset(), simpset() addsimps [JOIN_def]));
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qed "Acts_JN";
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Addsimps [Init_JN];
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val prems = Goalw [JOIN_def]
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"[| A=B; !!x. x:B ==> F(x) = G(x) |] ==> \
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\ (JN x:A. F(x)) = (JN x:B. G(x))";
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by (asm_simp_tac (simpset() addsimps prems) 1);
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qed "JN_cong";
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Addcongs [JN_cong];
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(** Algebraic laws **)
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Goal "F Join G = G Join F";
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by (simp_tac (simpset() addsimps [Join_def, Un_commute, Int_commute]) 1);
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qed "Join_commute";
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Goal "(F Join G) Join H = F Join (G Join H)";
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by (simp_tac (simpset() addsimps Un_ac@[Join_def, Int_assoc]) 1);
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qed "Join_assoc";
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Goalw [Join_def, SKIP_def] "SKIP Join F = F";
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by (rtac program_equalityI 1);
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by (ALLGOALS (simp_tac (simpset() addsimps [insert_absorb])));
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qed "Join_SKIP_left";
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Goalw [Join_def, SKIP_def] "F Join SKIP = F";
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by (rtac program_equalityI 1);
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by (ALLGOALS (simp_tac (simpset() addsimps [insert_absorb])));
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qed "Join_SKIP_right";
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Addsimps [Join_SKIP_left, Join_SKIP_right];
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Goalw [Join_def] "F Join F = F";
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by (rtac program_equalityI 1);
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by Auto_tac;
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qed "Join_absorb";
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Addsimps [Join_absorb];
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(*** JN laws ***)
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Goal "i : I ==> (JN i:I. A i Join B) = ((JN i:I. A i) Join B)";
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by (auto_tac (claset() addSIs [program_equalityI],
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simpset() addsimps [Acts_JN, Acts_Join]));
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qed "JN_Join_miniscope";
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Goal "k:I ==> A k Join (JN i:I. A i) = (JN i:I. A i)";
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by (auto_tac (claset() addSIs [program_equalityI],
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simpset() addsimps [Acts_JN, Acts_Join]));
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qed "JN_absorb";
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Goal "(JN i: I Un J. A i) = ((JN i: I. A i) Join (JN i:J. A i))";
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by (auto_tac (claset() addSIs [program_equalityI],
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simpset() addsimps [Acts_JN, Acts_Join]));
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qed "JN_Join";
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Goal "i: I ==> (JN i:I. c) = c";
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by (auto_tac (claset() addSIs [program_equalityI],
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simpset() addsimps [Acts_JN]));
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qed "JN_constant";
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Goal "(JN i:I. A i Join B i) = (JN i:I. A i) Join (JN i:I. B i)";
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by (auto_tac (claset() addSIs [program_equalityI],
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simpset() addsimps [Acts_JN, Acts_Join]));
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qed "JN_Join_distrib";
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(*** Safety: co, stable, FP ***)
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Goalw [constrains_def, JOIN_def]
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"i : I ==> \
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\ (JN i:I. F i) : A co B = (ALL i:I. F i : A co B)";
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by (Simp_tac 1);
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by (Blast_tac 1);
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qed "constrains_JN";
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Goal "F Join G : A co B = \
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\ (F : A co B & G : A co B)";
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by (auto_tac
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(claset(),
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simpset() addsimps [constrains_def, Join_def]));
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qed "constrains_Join";
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Goal "[| i : I; ALL i:I. F i : A co B |] \
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\ ==> (JN i:I. F i) : A co B";
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by (full_simp_tac (simpset() addsimps [constrains_def, Acts_JN]) 1);
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by (Blast_tac 1);
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qed "constrains_imp_JN_constrains";
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(**FAILS, I think; see 5.4.1, Substitution Axiom.
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The problem is to relate reachable (F Join G) with
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reachable F and reachable G
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Goalw [Constrains_def]
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"(JN i:I. F i) : A Co B = (ALL i:I. F i : A Co B)";
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by (simp_tac (simpset() addsimps [constrains_JN]) 1);
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by (Blast_tac 1);
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qed "Constrains_JN";
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Goal "F Join G : A Co B = (F Co A B & G Co A B)";
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by (auto_tac
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(claset(),
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simpset() addsimps [Constrains_def, constrains_Join]));
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qed "Constrains_Join";
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**)
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Goal "[| F : A co A'; G : B co B' |] \
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\ ==> F Join G : (A Int B) co (A' Un B')";
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paulson@6295
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by (simp_tac (simpset() addsimps [constrains_Join]) 1);
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paulson@5313
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by (blast_tac (claset() addIs [constrains_weaken]) 1);
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qed "constrains_Join_weaken";
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Goal "i : I ==> \
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\ (JN i:I. F i) : stable A = (ALL i:I. F i : stable A)";
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by (asm_simp_tac (simpset() addsimps [stable_def, constrains_JN]) 1);
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qed "stable_JN";
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Goal "[| ALL i:I. F i : invariant A; i : I |] \
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\ ==> (JN i:I. F i) : invariant A";
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paulson@5970
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by (asm_full_simp_tac (simpset() addsimps [invariant_def, stable_JN]) 1);
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paulson@5970
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by (Blast_tac 1);
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bind_thm ("invariant_JN_I", ballI RS result());
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paulson@5970
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paulson@6295
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Goal "F Join G : stable A = \
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\ (F : stable A & G : stable A)";
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by (simp_tac (simpset() addsimps [stable_def, constrains_Join]) 1);
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qed "stable_Join";
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Goal "[| F : invariant A; G : invariant A |] \
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\ ==> F Join G : invariant A";
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paulson@6295
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by (full_simp_tac (simpset() addsimps [invariant_def, stable_Join]) 1);
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paulson@5970
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by (Blast_tac 1);
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paulson@5970
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qed "invariant_JoinI";
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paulson@5970
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paulson@6295
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Goal "i : I ==> FP (JN i:I. F i) = (INT i:I. FP (F i))";
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paulson@5584
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by (asm_simp_tac (simpset() addsimps [FP_def, stable_JN, INTER_def]) 1);
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paulson@5313
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qed "FP_JN";
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paulson@5313
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paulson@5313
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paulson@5313
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(*** Progress: transient, ensures ***)
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paulson@6295
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Goal "i : I ==> \
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paulson@6295
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\ (JN i:I. F i) : transient A = (EX i:I. F i : transient A)";
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paulson@6295
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by (auto_tac (claset(),
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paulson@6295
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simpset() addsimps [transient_def, JOIN_def]));
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paulson@5313
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qed "transient_JN";
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paulson@5313
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paulson@6295
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Goal "F Join G : transient A = \
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paulson@6295
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\ (F : transient A | G : transient A)";
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paulson@6295
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by (auto_tac (claset(),
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paulson@6295
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simpset() addsimps [bex_Un, transient_def,
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paulson@6295
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Join_def]));
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paulson@5313
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qed "transient_Join";
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paulson@5313
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paulson@6295
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Goal "i : I ==> \
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paulson@6536
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\ (JN i:I. F i) : A ensures B = \
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paulson@6536
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\ ((ALL i:I. F i : (A-B) co (A Un B)) & \
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paulson@6536
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\ (EX i:I. F i : A ensures B))";
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paulson@5313
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by (auto_tac (claset(),
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paulson@5584
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simpset() addsimps [ensures_def, constrains_JN, transient_JN]));
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paulson@5313
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qed "ensures_JN";
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paulson@5313
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paulson@5313
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Goalw [ensures_def]
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paulson@6536
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"F Join G : A ensures B = \
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paulson@6536
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\ (F : (A-B) co (A Un B) & \
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paulson@6536
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\ G : (A-B) co (A Un B) & \
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paulson@6536
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\ (F : A ensures B | G : A ensures B))";
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paulson@5313
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by (auto_tac (claset(),
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paulson@5313
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simpset() addsimps [constrains_Join, transient_Join]));
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paulson@5313
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qed "ensures_Join";
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paulson@5313
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paulson@6295
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Goalw [stable_def, constrains_def, Join_def]
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paulson@6536
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"[| F : stable A; G : A co A' |] \
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paulson@6536
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\ ==> F Join G : A co A'";
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paulson@6295
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by (asm_full_simp_tac (simpset() addsimps [ball_Un]) 1);
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paulson@5313
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by (Blast_tac 1);
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paulson@5313
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qed "stable_constrains_Join";
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paulson@5313
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paulson@5648
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(*Premise for G cannot use Invariant because Stable F A is weaker than
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paulson@5648
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G : stable A *)
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paulson@6295
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Goal "[| F : stable A; G : invariant A |] ==> F Join G : Invariant A";
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paulson@6295
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by (full_simp_tac (simpset() addsimps [Invariant_def, invariant_def,
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paulson@6295
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Stable_eq_stable, stable_Join]) 1);
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paulson@5313
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by (force_tac(claset() addIs [stable_reachable, stable_Int],
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paulson@5313
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simpset() addsimps [Acts_Join]) 1);
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paulson@5313
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qed "stable_Join_Invariant";
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paulson@5313
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paulson@6536
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Goal "[| F : stable A; G : A ensures B |] ==> F Join G : A ensures B";
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paulson@5313
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by (asm_simp_tac (simpset() addsimps [ensures_Join]) 1);
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paulson@5313
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by (asm_full_simp_tac (simpset() addsimps [stable_def, ensures_def]) 1);
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paulson@5313
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by (etac constrains_weaken 1);
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paulson@5313
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by Auto_tac;
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paulson@5313
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266 |
qed "ensures_stable_Join1";
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paulson@5313
|
267 |
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paulson@5313
|
268 |
(*As above, but exchanging the roles of F and G*)
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paulson@6536
|
269 |
Goal "[| F : A ensures B; G : stable A |] ==> F Join G : A ensures B";
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paulson@5313
|
270 |
by (stac Join_commute 1);
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paulson@5313
|
271 |
by (blast_tac (claset() addIs [ensures_stable_Join1]) 1);
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paulson@5313
|
272 |
qed "ensures_stable_Join2";
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paulson@5313
|
273 |
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paulson@5648
|
274 |
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paulson@5804
|
275 |
(** Diff and localTo **)
|
paulson@5648
|
276 |
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paulson@6295
|
277 |
Goalw [Join_def, Diff_def] "F Join (Diff G (Acts F)) = F Join G";
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paulson@5804
|
278 |
by (rtac program_equalityI 1);
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paulson@6295
|
279 |
by Auto_tac;
|
paulson@5804
|
280 |
qed "Join_Diff2";
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paulson@5804
|
281 |
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paulson@6295
|
282 |
Goalw [Diff_def, Disjoint_def] "Disjoint F (Diff G (Acts F))";
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paulson@5804
|
283 |
by Auto_tac;
|
paulson@5804
|
284 |
qed "Diff_Disjoint";
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paulson@5804
|
285 |
|
paulson@5804
|
286 |
Goal "[| F Join G : v localTo F; Disjoint F G |] \
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paulson@5804
|
287 |
\ ==> G : (INT z. stable {s. v s = z})";
|
paulson@5648
|
288 |
by (asm_full_simp_tac
|
paulson@6295
|
289 |
(simpset() addsimps [localTo_def, Diff_def, Disjoint_def,
|
paulson@5804
|
290 |
Acts_Join, stable_def, constrains_def]) 1);
|
paulson@5804
|
291 |
by (Blast_tac 1);
|
paulson@5804
|
292 |
qed "localTo_imp_stable";
|
paulson@5804
|
293 |
|
paulson@5804
|
294 |
Goal "[| F Join G : v localTo F; (s,s') : act; \
|
paulson@6295
|
295 |
\ act : Acts G; Disjoint F G |] ==> v s' = v s";
|
paulson@5804
|
296 |
by (asm_full_simp_tac
|
paulson@6295
|
297 |
(simpset() addsimps [localTo_def, Diff_def, Disjoint_def,
|
paulson@5804
|
298 |
Acts_Join, stable_def, constrains_def]) 1);
|
paulson@5648
|
299 |
by (Blast_tac 1);
|
paulson@5648
|
300 |
qed "localTo_imp_equals";
|
paulson@5804
|
301 |
|
paulson@5804
|
302 |
Goalw [localTo_def, stable_def, constrains_def]
|
paulson@5804
|
303 |
"v localTo F <= (f o v) localTo F";
|
paulson@5804
|
304 |
by (Clarify_tac 1);
|
paulson@5898
|
305 |
by (force_tac (claset() addSEs [allE, ballE], simpset()) 1);
|
paulson@5804
|
306 |
qed "localTo_imp_o_localTo";
|
paulson@5804
|
307 |
|
paulson@5804
|
308 |
|
paulson@5804
|
309 |
(*** Higher-level rules involving localTo and Join ***)
|
paulson@5804
|
310 |
|
paulson@6536
|
311 |
Goal "[| F : {s. P (v s) (w s)} co {s. P' (v s) (w s)}; \
|
paulson@5804
|
312 |
\ F Join G : v localTo F; \
|
paulson@5804
|
313 |
\ F Join G : w localTo F; \
|
paulson@5804
|
314 |
\ Disjoint F G |] \
|
paulson@6536
|
315 |
\ ==> F Join G : {s. P (v s) (w s)} co {s. P' (v s) (w s)}";
|
paulson@6295
|
316 |
by (auto_tac (claset(), simpset() addsimps [constrains_def, Acts_Join]));
|
paulson@5804
|
317 |
by (REPEAT_FIRST (dtac localTo_imp_equals THEN' REPEAT1 o atac));
|
paulson@6295
|
318 |
by Auto_tac;
|
paulson@5804
|
319 |
qed "constrains_localTo_constrains2";
|
paulson@5804
|
320 |
|
paulson@5804
|
321 |
Goalw [stable_def]
|
paulson@5804
|
322 |
"[| F : stable {s. P (v s) (w s)}; \
|
paulson@5804
|
323 |
\ F Join G : v localTo F; \
|
paulson@5804
|
324 |
\ F Join G : w localTo F; \
|
paulson@5804
|
325 |
\ Disjoint F G |] \
|
paulson@5804
|
326 |
\ ==> F Join G : stable {s. P (v s) (w s)}";
|
paulson@5804
|
327 |
by (blast_tac (claset() addIs [constrains_localTo_constrains2]) 1);
|
paulson@5804
|
328 |
qed "stable_localTo_stable2";
|
paulson@5804
|
329 |
|
paulson@5804
|
330 |
|
paulson@5804
|
331 |
Goal "(UN k. {s. f s = k}) = UNIV";
|
paulson@5804
|
332 |
by (Blast_tac 1);
|
paulson@5804
|
333 |
qed "UN_eq_UNIV";
|
paulson@5804
|
334 |
|
paulson@5804
|
335 |
Goal "[| F : stable {s. v s <= w s}; \
|
paulson@5804
|
336 |
\ F Join G : v localTo F; \
|
paulson@5804
|
337 |
\ F Join G : Increasing w; \
|
paulson@5804
|
338 |
\ Disjoint F G |] \
|
paulson@5804
|
339 |
\ ==> F Join G : Stable {s. v s <= w s}";
|
paulson@5804
|
340 |
by (safe_tac (claset() addSDs [localTo_imp_stable]));
|
paulson@5804
|
341 |
by (rewrite_goals_tac [stable_def, Stable_def, Increasing_def]);
|
paulson@6536
|
342 |
by (subgoal_tac "ALL k: UNIV. ?H : ({s. v s = k} Int ?AA) Co ?AA" 1);
|
paulson@5804
|
343 |
by (dtac ball_Constrains_UN 1);
|
paulson@5804
|
344 |
by (full_simp_tac (simpset() addsimps [UN_eq_UNIV]) 1);
|
paulson@5804
|
345 |
by (rtac ballI 1);
|
paulson@6536
|
346 |
by (subgoal_tac "F Join G : ({s. v s = k} Int {s. v s <= w s}) co \
|
paulson@5804
|
347 |
\ ({s. v s = k} Un {s. v s <= w s})" 1);
|
paulson@6295
|
348 |
by (asm_simp_tac (simpset() addsimps [constrains_Join]) 2);
|
paulson@5804
|
349 |
by (blast_tac (claset() addIs [constrains_weaken]) 2);
|
paulson@5804
|
350 |
by (dtac (constrains_imp_Constrains RS Constrains_Int) 1 THEN etac INT_D 1);
|
paulson@5804
|
351 |
by (etac Constrains_weaken 2);
|
paulson@5804
|
352 |
by Auto_tac;
|
paulson@5804
|
353 |
qed "Increasing_localTo_Stable";
|