Constrains, Stable, Invariant...more of the substitution axiom, but Union
does not work well with them
(* Title: HOL/UNITY/Constrains
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Safety relations: restricted to the set of reachable states.
*)
(**MOVE TO EQUALITIES.ML**)
Goal "(A Un B <= C) = (A <= C & B <= C)";
by (Blast_tac 1);
qed "Un_subset_iff";
Goal "(C <= A Int B) = (C <= A & C <= B)";
by (Blast_tac 1);
qed "Int_subset_iff";
(*** Constrains ***)
(*constrains (Acts prg) B B'
==> constrains (Acts prg) (reachable prg Int B) (reachable prg Int B')*)
bind_thm ("constrains_reachable_Int",
subset_refl RS
rewrite_rule [stable_def] stable_reachable RS
constrains_Int);
Goalw [Constrains_def]
"constrains (Acts prg) A A' ==> Constrains prg A A'";
by (etac constrains_reachable_Int 1);
qed "constrains_imp_Constrains";
val prems = Goal
"(!!act s s'. [| act: Acts prg; (s,s') : act; s: A |] ==> s': A') \
\ ==> Constrains prg A A'";
by (rtac constrains_imp_Constrains 1);
by (blast_tac (claset() addIs (constrainsI::prems)) 1);
qed "ConstrainsI";
Goalw [Constrains_def, constrains_def] "Constrains prg {} B";
by (Blast_tac 1);
qed "Constrains_empty";
Goal "Constrains prg A UNIV";
by (blast_tac (claset() addIs [ConstrainsI]) 1);
qed "Constrains_UNIV";
AddIffs [Constrains_empty, Constrains_UNIV];
Goalw [Constrains_def]
"[| Constrains prg A A'; A'<=B' |] ==> Constrains prg A B'";
by (blast_tac (claset() addIs [constrains_weaken_R]) 1);
qed "Constrains_weaken_R";
Goalw [Constrains_def]
"[| Constrains prg A A'; B<=A |] ==> Constrains prg B A'";
by (blast_tac (claset() addIs [constrains_weaken_L]) 1);
qed "Constrains_weaken_L";
Goalw [Constrains_def]
"[| Constrains prg A A'; B<=A; A'<=B' |] ==> Constrains prg B B'";
by (blast_tac (claset() addIs [constrains_weaken]) 1);
qed "Constrains_weaken";
(** Union **)
Goalw [Constrains_def]
"[| Constrains prg A A'; Constrains prg B B' |] \
\ ==> Constrains prg (A Un B) (A' Un B')";
by (blast_tac (claset() addIs [constrains_Un RS constrains_weaken]) 1);
qed "Constrains_Un";
Goalw [Constrains_def]
"ALL i:I. Constrains prg (A i) (A' i) \
\ ==> Constrains prg (UN i:I. A i) (UN i:I. A' i)";
by (dtac ball_constrains_UN 1);
by (blast_tac (claset() addIs [constrains_weaken]) 1);
qed "ball_Constrains_UN";
(** Intersection **)
Goalw [Constrains_def]
"[| Constrains prg A A'; Constrains prg B B' |] \
\ ==> Constrains prg (A Int B) (A' Int B')";
by (blast_tac (claset() addIs [constrains_Int RS constrains_weaken]) 1);
qed "Constrains_Int";
Goalw [Constrains_def]
"[| ALL i:I. Constrains prg (A i) (A' i) |] \
\ ==> Constrains prg (INT i:I. A i) (INT i:I. A' i)";
by (dtac ball_constrains_INT 1);
by (blast_tac (claset() addIs [constrains_reachable_Int, constrains_weaken]) 1);
qed "ball_Constrains_INT";
Goalw [Constrains_def]
"[| Constrains prg A A'; id: Acts prg |] ==> reachable prg Int A <= A'";
by (dtac constrains_imp_subset 1);
by (assume_tac 1);
by (full_simp_tac (simpset() addsimps [Int_subset_iff, Int_lower1]) 1);
qed "Constrains_imp_subset";
Goalw [Constrains_def]
"[| id: Acts prg; Constrains prg A B; Constrains prg B C |] \
\ ==> Constrains prg A C";
by (blast_tac (claset() addIs [constrains_trans, constrains_weaken]) 1);
qed "Constrains_trans";
(*** Stable ***)
Goal "Stable prg A = stable (Acts prg) (reachable prg Int A)";
by (simp_tac (simpset() addsimps [Stable_def, Constrains_def, stable_def]) 1);
qed "Stable_eq_stable";
Goalw [Stable_def] "Constrains prg A A ==> Stable prg A";
by (assume_tac 1);
qed "StableI";
Goalw [Stable_def] "Stable prg A ==> Constrains prg A A";
by (assume_tac 1);
qed "StableD";
Goalw [Stable_def]
"[| Stable prg A; Stable prg A' |] ==> Stable prg (A Un A')";
by (blast_tac (claset() addIs [Constrains_Un]) 1);
qed "Stable_Un";
Goalw [Stable_def]
"[| Stable prg A; Stable prg A' |] ==> Stable prg (A Int A')";
by (blast_tac (claset() addIs [Constrains_Int]) 1);
qed "Stable_Int";
Goalw [Stable_def]
"[| Stable prg C; Constrains prg A (C Un A') |] \
\ ==> Constrains prg (C Un A) (C Un A')";
by (blast_tac (claset() addIs [Constrains_Un RS Constrains_weaken]) 1);
qed "Stable_Constrains_Un";
Goalw [Stable_def]
"[| Stable prg C; Constrains prg (C Int A) A' |] \
\ ==> Constrains prg (C Int A) (C Int A')";
by (blast_tac (claset() addIs [Constrains_Int RS Constrains_weaken]) 1);
qed "Stable_Constrains_Int";
Goalw [Stable_def]
"(ALL i:I. Stable prg (A i)) ==> Stable prg (INT i:I. A i)";
by (etac ball_Constrains_INT 1);
qed "ball_Stable_INT";
Goal "Stable prg (reachable prg)";
by (simp_tac (simpset() addsimps [Stable_eq_stable, stable_reachable]) 1);
qed "Stable_reachable";
(*** The Elimination Theorem. The "free" m has become universally quantified!
Should the premise be !!m instead of ALL m ? Would make it harder to use
in forward proof. ***)
Goalw [Constrains_def, constrains_def]
"[| ALL m. Constrains prg {s. s x = m} (B m) |] \
\ ==> Constrains prg {s. s x : M} (UN m:M. B m)";
by (Blast_tac 1);
qed "Elimination";
(*As above, but for the trivial case of a one-variable state, in which the
state is identified with its one variable.*)
Goalw [Constrains_def, constrains_def]
"(ALL m. Constrains prg {m} (B m)) ==> Constrains prg M (UN m:M. B m)";
by (Blast_tac 1);
qed "Elimination_sing";
Goalw [Constrains_def, constrains_def]
"[| Constrains prg A (A' Un B); Constrains prg B B'; id: Acts prg |] \
\ ==> Constrains prg A (A' Un B')";
by (Blast_tac 1);
qed "Constrains_cancel";
(*** Specialized laws for handling Invariants ***)
(** Natural deduction rules for "Invariant prg A" **)
Goal "[| Init prg<=A; Stable prg A |] ==> Invariant prg A";
by (asm_simp_tac (simpset() addsimps [Invariant_def]) 1);
qed "InvariantI";
Goal "Invariant prg A ==> Init prg<=A & Stable prg A";
by (asm_full_simp_tac (simpset() addsimps [Invariant_def]) 1);
qed "InvariantD";
bind_thm ("InvariantE", InvariantD RS conjE);
(*The set of all reachable states is an Invariant...*)
Goal "Invariant prg (reachable prg)";
by (simp_tac (simpset() addsimps [Invariant_def]) 1);
by (blast_tac (claset() addIs (Stable_reachable::reachable.intrs)) 1);
qed "Invariant_reachable";
(*...in fact the strongest Invariant!*)
Goal "Invariant prg A ==> reachable prg <= A";
by (full_simp_tac
(simpset() addsimps [Stable_def, Constrains_def, constrains_def,
Invariant_def]) 1);
by (rtac subsetI 1);
by (etac reachable.induct 1);
by (REPEAT (blast_tac (claset() addIs reachable.intrs) 1));
qed "Invariant_includes_reachable";
Goal "Invariant prg INV ==> reachable prg Int INV = reachable prg";
by (dtac Invariant_includes_reachable 1);
by (Blast_tac 1);
qed "reachable_Int_INV";
Goal "[| Invariant prg INV; Constrains prg (INV Int A) A' |] \
\ ==> Constrains prg A A'";
by (asm_full_simp_tac
(simpset() addsimps [Constrains_def, reachable_Int_INV,
Int_assoc RS sym]) 1);
qed "Invariant_ConstrainsI";
bind_thm ("Invariant_StableI", Invariant_ConstrainsI RS StableI);
Goal "[| Invariant prg INV; Constrains prg A A' |] \
\ ==> Constrains prg A (INV Int A')";
by (asm_full_simp_tac
(simpset() addsimps [Constrains_def, reachable_Int_INV,
Int_assoc RS sym]) 1);
qed "Invariant_ConstrainsD";
bind_thm ("Invariant_StableD", StableD RSN (2,Invariant_ConstrainsD));
(** Conjoining Invariants **)
Goal "[| Invariant prg A; Invariant prg B |] ==> Invariant prg (A Int B)";
by (auto_tac (claset(),
simpset() addsimps [Invariant_def, Stable_Int]));
qed "Invariant_Int";
(*Delete the nearest invariance assumption (which will be the second one
used by Invariant_Int) *)
val Invariant_thin =
read_instantiate_sg (sign_of thy)
[("V", "Invariant ?Prg ?A")] thin_rl;
(*Combines two invariance ASSUMPTIONS into one. USEFUL??*)
val Invariant_Int_tac = dtac Invariant_Int THEN'
assume_tac THEN'
etac Invariant_thin;
(*Combines two invariance THEOREMS into one.*)
val Invariant_Int_rule = foldr1 (fn (th1,th2) => [th1,th2] MRS Invariant_Int);