--- a/src/HOL/UNITY/SubstAx.ML Thu Aug 13 17:44:50 1998 +0200
+++ b/src/HOL/UNITY/SubstAx.ML Thu Aug 13 18:06:40 1998 +0200
@@ -7,32 +7,22 @@
*)
-(*constrains (Acts prg) B B'
- ==> constrains (Acts prg) (reachable prg Int B) (reachable prg Int B')*)
-bind_thm ("constrains_reachable",
- rewrite_rule [stable_def] stable_reachable RS constrains_Int);
-
(*** Specialized laws for handling invariants ***)
-Goal "invariant prg INV ==> reachable prg Int INV = reachable prg";
-bd invariant_includes_reachable 1;
-by (Blast_tac 1);
-qed "reachable_Int_INV";
-
-Goal "[| invariant prg INV; LeadsTo prg (INV Int A) A' |] \
+Goal "[| Invariant prg INV; LeadsTo prg (INV Int A) A' |] \
\ ==> LeadsTo prg A A'";
by (asm_full_simp_tac
(simpset() addsimps [LeadsTo_def, reachable_Int_INV,
Int_assoc RS sym]) 1);
-qed "invariant_LeadsToI";
+qed "Invariant_LeadsToI";
-Goal "[| invariant prg INV; LeadsTo prg A A' |] \
+Goal "[| Invariant prg INV; LeadsTo prg A A' |] \
\ ==> LeadsTo prg A (INV Int A')";
by (asm_full_simp_tac
(simpset() addsimps [LeadsTo_def, reachable_Int_INV,
Int_assoc RS sym]) 1);
-qed "invariant_LeadsToD";
+qed "Invariant_LeadsToD";
(*** Introduction rules: Basis, Trans, Union ***)
@@ -42,9 +32,6 @@
by (blast_tac (claset() addIs [psp_stable2, stable_reachable]) 1);
qed "leadsTo_imp_LeadsTo";
-(* ensures (Acts prg) A B ==> LeadsTo prg A B *)
-bind_thm ("LeadsTo_Basis", leadsTo_Basis RS leadsTo_imp_LeadsTo);
-
Goal "[| LeadsTo prg A B; LeadsTo prg B C |] ==> LeadsTo prg A C";
by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1);
by (blast_tac (claset() addIs [leadsTo_Trans]) 1);
@@ -138,33 +125,37 @@
qed "LeadsTo_weaken";
-(** More rules using the premise "invariant prg" **)
+(** More rules using the premise "Invariant prg" **)
-Goal "[| invariant prg INV; \
-\ constrains (Acts prg) (INV Int (A-A')) (A Un A'); \
+Goalw [LeadsTo_def, Constrains_def]
+ "[| Constrains prg (A-A') (A Un A'); transient (Acts prg) (A-A') |] \
+\ ==> LeadsTo prg A A'";
+by (rtac (ensuresI RS leadsTo_Basis) 1);
+by (blast_tac (claset() addIs [transient_strengthen]) 2);
+by (blast_tac (claset() addIs [constrains_weaken]) 1);
+qed "LeadsTo_Basis";
+
+Goal "[| Invariant prg INV; \
+\ Constrains prg (INV Int (A-A')) (A Un A'); \
\ transient (Acts prg) (INV Int (A-A')) |] \
\ ==> LeadsTo prg A A'";
-br invariant_LeadsToI 1;
-ba 1;
-by (rtac (ensuresI RS LeadsTo_Basis) 1);
-by (ALLGOALS
- (full_simp_tac (simpset() addsimps [Int_Diff, Int_Un_distrib RS sym,
- Diff_Int_distrib RS sym])));
-be invariantE 1;
-by (blast_tac (claset() addSDs [stable_constrains_Int]
- addIs [constrains_weaken]) 1);
-qed "invariant_LeadsTo_Basis";
+by (rtac Invariant_LeadsToI 1);
+by (assume_tac 1);
+by (rtac LeadsTo_Basis 1);
+by (blast_tac (claset() addIs [transient_strengthen]) 2);
+by (blast_tac (claset() addIs [Invariant_ConstrainsD RS Constrains_weaken]) 1);
+qed "Invariant_LeadsTo_Basis";
-Goal "[| invariant prg INV; \
+Goal "[| Invariant prg INV; \
\ LeadsTo prg A A'; id: Acts prg; \
\ INV Int B <= A; INV Int A' <= B' |] \
\ ==> LeadsTo prg B B'";
-br invariant_LeadsToI 1;
-ba 1;
-bd invariant_LeadsToD 1;
-ba 1;
+by (rtac Invariant_LeadsToI 1);
+by (assume_tac 1);
+by (dtac Invariant_LeadsToD 1);
+by (assume_tac 1);
by (blast_tac (claset()addIs [LeadsTo_weaken]) 1);
-qed "invariant_LeadsTo_weaken";
+qed "Invariant_LeadsTo_weaken";
(*Set difference: maybe combine with leadsTo_weaken_L??
@@ -250,43 +241,39 @@
(** PSP: Progress-Safety-Progress **)
(*Special case of PSP: Misra's "stable conjunction". Doesn't need id:Acts. *)
-Goal "[| LeadsTo prg A A'; stable (Acts prg) B |] \
-\ ==> LeadsTo prg (A Int B) (A' Int B)";
-by (asm_full_simp_tac (simpset() addsimps [LeadsTo_def, Int_assoc RS sym,
- psp_stable]) 1);
+Goal
+ "[| LeadsTo prg A A'; Stable prg B |] ==> LeadsTo prg (A Int B) (A' Int B)";
+by (full_simp_tac (simpset() addsimps [LeadsTo_def, Stable_eq_stable]) 1);
+by (dtac psp_stable 1);
+by (assume_tac 1);
+by (asm_full_simp_tac (simpset() addsimps Int_ac) 1);
qed "PSP_stable";
-Goal "[| LeadsTo prg A A'; stable (Acts prg) B |] \
+Goal "[| LeadsTo prg A A'; Stable prg B |] \
\ ==> LeadsTo prg (B Int A) (B Int A')";
by (asm_simp_tac (simpset() addsimps (PSP_stable::Int_ac)) 1);
qed "PSP_stable2";
-
-Goal "[| LeadsTo prg A A'; constrains (Acts prg) B B'; id: Acts prg |] \
+Goalw [LeadsTo_def, Constrains_def]
+ "[| LeadsTo prg A A'; Constrains prg B B'; id: Acts prg |] \
\ ==> LeadsTo prg (A Int B) ((A' Int B) Un (B' - B))";
-by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1);
-by (dtac psp 1);
-by (etac constrains_reachable 1);
-by (etac leadsTo_weaken 2);
-by (ALLGOALS Blast_tac);
+by (blast_tac (claset() addDs [psp] addIs [leadsTo_weaken]) 1);
qed "PSP";
-Goal "[| LeadsTo prg A A'; constrains (Acts prg) B B'; id: Acts prg |] \
+Goal "[| LeadsTo prg A A'; Constrains prg B B'; id: Acts prg |] \
\ ==> LeadsTo prg (B Int A) ((B Int A') Un (B' - B))";
by (asm_simp_tac (simpset() addsimps (PSP::Int_ac)) 1);
qed "PSP2";
-Goalw [unless_def]
- "[| LeadsTo prg A A'; unless (Acts prg) B B'; id: Acts prg |] \
+Goalw [Unless_def]
+ "[| LeadsTo prg A A'; Unless prg B B'; id: Acts prg |] \
\ ==> LeadsTo prg (A Int B) ((A' Int B) Un B')";
by (dtac PSP 1);
by (assume_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [Un_Diff_Diff, Int_Diff_Un]) 2);
-by (asm_full_simp_tac (simpset() addsimps [Diff_Int_distrib]) 2);
-by (etac LeadsTo_Diff 2);
-by (blast_tac (claset() addIs [subset_imp_LeadsTo]) 2);
-by Auto_tac;
-qed "PSP_unless";
+by (blast_tac (claset() addIs [LeadsTo_Diff, LeadsTo_weaken,
+ subset_imp_LeadsTo]) 2);
+by (assume_tac 1);
+qed "PSP_Unless";
(*** Induction rules ***)
@@ -294,7 +281,7 @@
(** Meta or object quantifier ????? **)
Goal "[| wf r; \
\ ALL m. LeadsTo prg (A Int f-``{m}) \
-\ ((A Int f-``(r^-1 ^^ {m})) Un B); \
+\ ((A Int f-``(r^-1 ^^ {m})) Un B); \
\ id: Acts prg |] \
\ ==> LeadsTo prg A B";
by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1);
@@ -353,49 +340,42 @@
(*** Completion: Binary and General Finite versions ***)
-Goal "[| LeadsTo prg A A'; stable (Acts prg) A'; \
-\ LeadsTo prg B B'; stable (Acts prg) B'; id: Acts prg |] \
+Goal "[| LeadsTo prg A A'; Stable prg A'; \
+\ LeadsTo prg B B'; Stable prg B'; id: Acts prg |] \
\ ==> LeadsTo prg (A Int B) (A' Int B')";
-by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1);
-by (blast_tac (claset() addIs [stable_completion RS leadsTo_weaken]
- addSIs [stable_Int, stable_reachable]) 1);
+by (full_simp_tac (simpset() addsimps [LeadsTo_def, Stable_eq_stable]) 1);
+by (blast_tac (claset() addIs [stable_completion, leadsTo_weaken]) 1);
qed "Stable_completion";
-Goal "[| finite I; id: Acts prg |] \
+Goal "[| finite I; id: Acts prg |] \
\ ==> (ALL i:I. LeadsTo prg (A i) (A' i)) --> \
-\ (ALL i:I. stable (Acts prg) (A' i)) --> \
+\ (ALL i:I. Stable prg (A' i)) --> \
\ LeadsTo prg (INT i:I. A i) (INT i:I. A' i)";
by (etac finite_induct 1);
by (Asm_simp_tac 1);
-by (asm_simp_tac
- (simpset() addsimps [Stable_completion, stable_def,
- ball_constrains_INT]) 1);
+by (asm_simp_tac (simpset() addsimps [Stable_completion, ball_Stable_INT]) 1);
qed_spec_mp "Finite_stable_completion";
-Goal "[| LeadsTo prg A (A' Un C); constrains (Acts prg) A' (A' Un C); \
-\ LeadsTo prg B (B' Un C); constrains (Acts prg) B' (B' Un C); \
+Goal "[| LeadsTo prg A (A' Un C); Constrains prg A' (A' Un C); \
+\ LeadsTo prg B (B' Un C); Constrains prg B' (B' Un C); \
\ id: Acts prg |] \
\ ==> LeadsTo prg (A Int B) ((A' Int B') Un C)";
-by (full_simp_tac (simpset() addsimps [LeadsTo_def, Int_Un_distrib]) 1);
-by (dtac completion 1);
-by (assume_tac 2);
-by (ALLGOALS
- (asm_simp_tac
- (simpset() addsimps [constrains_reachable, Int_Un_distrib RS sym])));
-by (blast_tac (claset() addIs [leadsTo_weaken]) 1);
+by (full_simp_tac (simpset() addsimps [LeadsTo_def, Constrains_def,
+ Int_Un_distrib]) 1);
+by (blast_tac (claset() addIs [completion, leadsTo_weaken]) 1);
qed "Completion";
Goal "[| finite I; id: Acts prg |] \
\ ==> (ALL i:I. LeadsTo prg (A i) (A' i Un C)) --> \
-\ (ALL i:I. constrains (Acts prg) (A' i) (A' i Un C)) --> \
+\ (ALL i:I. Constrains prg (A' i) (A' i Un C)) --> \
\ LeadsTo prg (INT i:I. A i) ((INT i:I. A' i) Un C)";
by (etac finite_induct 1);
by (ALLGOALS Asm_simp_tac);
by (Clarify_tac 1);
-by (dtac ball_constrains_INT 1);
+by (dtac ball_Constrains_INT 1);
by (asm_full_simp_tac (simpset() addsimps [Completion]) 1);
qed "Finite_completion";
@@ -408,22 +388,22 @@
(*proves "constrains" properties when the program is specified*)
fun constrains_tac (main_def::cmd_defs) =
SELECT_GOAL
- (EVERY [TRY (rtac stableI 1),
+ (EVERY [REPEAT (resolve_tac [StableI, stableI,
+ constrains_imp_Constrains] 1),
rtac constrainsI 1,
full_simp_tac (simpset() addsimps [main_def]) 1,
- REPEAT_FIRST (eresolve_tac [disjE]),
+ REPEAT_FIRST (etac disjE ),
rewrite_goals_tac cmd_defs,
ALLGOALS (SELECT_GOAL Auto_tac)]);
-(*proves "ensures" properties when the program is specified*)
+(*proves "ensures/leadsTo" properties when the program is specified*)
fun ensures_tac (main_def::cmd_defs) sact =
SELECT_GOAL
- (EVERY [REPEAT (invariant_Int_tac 1),
- etac invariant_LeadsTo_Basis 1
+ (EVERY [REPEAT (Invariant_Int_tac 1),
+ etac Invariant_LeadsTo_Basis 1
ORELSE (*subgoal may involve LeadsTo, leadsTo or ensures*)
- REPEAT (ares_tac [leadsTo_imp_LeadsTo, leadsTo_Basis,
- ensuresI] 1),
+ REPEAT (ares_tac [LeadsTo_Basis, ensuresI] 1),
res_inst_tac [("act", sact)] transient_mem 2,
simp_tac (simpset() addsimps (Domain_partial_func::cmd_defs)) 3,
simp_tac (simpset() addsimps [main_def]) 2,