(* Title: HOL/UNITY/SubstAx
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Weak Fairness versions of transient, ensures, LeadsTo.
From Misra, "A Logic for Concurrent Programming", 1994
*)
(*constrains Acts B B' ==> constrains Acts (reachable(Init,Acts) Int B)
(reachable(Init,Acts) Int B') *)
bind_thm ("constrains_reachable",
rewrite_rule [stable_def] stable_reachable RS constrains_Int);
(*** Introduction rules: Basis, Trans, Union ***)
Goal "leadsTo (Acts prg) A B ==> LeadsTo prg A B";
by (simp_tac (simpset() addsimps [LeadsTo_def]) 1);
by (blast_tac (claset() addIs [PSP_stable2, stable_reachable]) 1);
qed "leadsTo_imp_LeadsTo";
Goal "ensures (Acts prg) (reachable prg Int A) (reachable prg Int A') \
\ ==> LeadsTo prg A A'";
by (full_simp_tac (simpset() addsimps [ensures_def, LeadsTo_def]) 1);
by (rtac (stable_reachable RS stable_ensures_Int RS leadsTo_Basis) 1);
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [Diff_Int_distrib])));
by (blast_tac (claset() addIs [constrains_weaken]) 1);
qed "LeadsTo_Basis";
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);
qed "LeadsTo_Trans";
val [prem] = goalw thy [LeadsTo_def]
"(!!A. A : S ==> LeadsTo prg A B) ==> LeadsTo prg (Union S) B";
by (Simp_tac 1);
by (stac Int_Union 1);
by (blast_tac (claset() addIs [leadsTo_UN,
simplify (simpset()) prem]) 1);
qed "LeadsTo_Union";
(*** Derived rules ***)
Goal "id: (Acts prg) ==> LeadsTo prg A UNIV";
by (asm_simp_tac (simpset() addsimps [LeadsTo_def,
Int_lower1 RS subset_imp_leadsTo]) 1);
qed "LeadsTo_UNIV";
Addsimps [LeadsTo_UNIV];
(*Useful with cancellation, disjunction*)
Goal "LeadsTo prg A (A' Un A') ==> LeadsTo prg A A'";
by (asm_full_simp_tac (simpset() addsimps Un_ac) 1);
qed "LeadsTo_Un_duplicate";
Goal "LeadsTo prg A (A' Un C Un C) ==> LeadsTo prg A (A' Un C)";
by (asm_full_simp_tac (simpset() addsimps Un_ac) 1);
qed "LeadsTo_Un_duplicate2";
val prems = goal thy
"(!!i. i : I ==> LeadsTo prg (A i) B) \
\ ==> LeadsTo prg (UN i:I. A i) B";
by (simp_tac (simpset() addsimps [Union_image_eq RS sym]) 1);
by (blast_tac (claset() addIs (LeadsTo_Union::prems)) 1);
qed "LeadsTo_UN";
(*Binary union introduction rule*)
Goal "[| LeadsTo prg A C; LeadsTo prg B C |] ==> LeadsTo prg (A Un B) C";
by (stac Un_eq_Union 1);
by (blast_tac (claset() addIs [LeadsTo_Union]) 1);
qed "LeadsTo_Un";
Goal "[| reachable prg Int A <= B; id: (Acts prg) |] \
\ ==> LeadsTo prg A B";
by (simp_tac (simpset() addsimps [LeadsTo_def]) 1);
by (blast_tac (claset() addIs [subset_imp_leadsTo]) 1);
qed "Int_subset_imp_LeadsTo";
Goal "[| A <= B; id: (Acts prg) |] ==> LeadsTo prg A B";
by (simp_tac (simpset() addsimps [LeadsTo_def]) 1);
by (blast_tac (claset() addIs [subset_imp_leadsTo]) 1);
qed "subset_imp_LeadsTo";
bind_thm ("empty_LeadsTo", empty_subsetI RS subset_imp_LeadsTo);
Addsimps [empty_LeadsTo];
Goal "[| reachable prg Int A = {}; id: (Acts prg) |] \
\ ==> LeadsTo prg A B";
by (asm_simp_tac (simpset() addsimps [Int_subset_imp_LeadsTo]) 1);
qed "Int_empty_LeadsTo";
Goal "[| LeadsTo prg A A'; \
\ reachable prg Int A' <= B' |] \
\ ==> LeadsTo prg A B'";
by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1);
by (blast_tac (claset() addIs [leadsTo_weaken_R]) 1);
qed_spec_mp "LeadsTo_weaken_R";
Goal "[| LeadsTo prg A A'; \
\ reachable prg Int B <= A; id: (Acts prg) |] \
\ ==> LeadsTo prg B A'";
by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1);
by (blast_tac (claset() addIs [leadsTo_weaken_L]) 1);
qed_spec_mp "LeadsTo_weaken_L";
(*Distributes over binary unions*)
Goal "id: (Acts prg) ==> \
\ LeadsTo prg (A Un B) C = \
\ (LeadsTo prg A C & LeadsTo prg B C)";
by (blast_tac (claset() addIs [LeadsTo_Un, LeadsTo_weaken_L]) 1);
qed "LeadsTo_Un_distrib";
Goal "id: (Acts prg) ==> \
\ LeadsTo prg (UN i:I. A i) B = \
\ (ALL i : I. LeadsTo prg (A i) B)";
by (blast_tac (claset() addIs [LeadsTo_UN, LeadsTo_weaken_L]) 1);
qed "LeadsTo_UN_distrib";
Goal "id: (Acts prg) ==> \
\ LeadsTo prg (Union S) B = \
\ (ALL A : S. LeadsTo prg A B)";
by (blast_tac (claset() addIs [LeadsTo_Union, LeadsTo_weaken_L]) 1);
qed "LeadsTo_Union_distrib";
Goal "[| LeadsTo prg A A'; id: (Acts prg); \
\ reachable prg Int B <= A; \
\ reachable prg Int A' <= B' |] \
\ ==> LeadsTo prg B B'";
(*PROOF FAILED: why?*)
by (blast_tac (claset() addIs [LeadsTo_Trans, LeadsTo_weaken_R,
LeadsTo_weaken_L]) 1);
qed "LeadsTo_weaken";
(*Set difference: maybe combine with leadsTo_weaken_L??
This is the most useful form of the "disjunction" rule*)
Goal "[| LeadsTo prg (A-B) C; LeadsTo prg B C; id: (Acts prg) |] \
\ ==> LeadsTo prg A C";
by (blast_tac (claset() addIs [LeadsTo_Un, LeadsTo_weaken]) 1);
qed "LeadsTo_Diff";
(** Meta or object quantifier ???????????????????
see ball_constrains_UN in UNITY.ML***)
val prems = goal thy
"(!! i. i:I ==> LeadsTo prg (A i) (A' i)) \
\ ==> LeadsTo prg (UN i:I. A i) (UN i:I. A' i)";
by (simp_tac (simpset() addsimps [Union_image_eq RS sym]) 1);
by (blast_tac (claset() addIs [LeadsTo_Union, LeadsTo_weaken_R]
addIs prems) 1);
qed "LeadsTo_UN_UN";
(*Version with no index set*)
val prems = goal thy
"(!! i. LeadsTo prg (A i) (A' i)) \
\ ==> LeadsTo prg (UN i. A i) (UN i. A' i)";
by (blast_tac (claset() addIs [LeadsTo_UN_UN]
addIs prems) 1);
qed "LeadsTo_UN_UN_noindex";
(*Version with no index set*)
Goal "ALL i. LeadsTo prg (A i) (A' i) \
\ ==> LeadsTo prg (UN i. A i) (UN i. A' i)";
by (blast_tac (claset() addIs [LeadsTo_UN_UN]) 1);
qed "all_LeadsTo_UN_UN";
(*Binary union version*)
Goal "[| LeadsTo prg A A'; LeadsTo prg B B' |] \
\ ==> LeadsTo prg (A Un B) (A' Un B')";
by (blast_tac (claset() addIs [LeadsTo_Un,
LeadsTo_weaken_R]) 1);
qed "LeadsTo_Un_Un";
(** The cancellation law **)
Goal "[| LeadsTo prg A (A' Un B); LeadsTo prg B B'; \
\ id: (Acts prg) |] \
\ ==> LeadsTo prg A (A' Un B')";
by (blast_tac (claset() addIs [LeadsTo_Un_Un,
subset_imp_LeadsTo, LeadsTo_Trans]) 1);
qed "LeadsTo_cancel2";
Goal "[| LeadsTo prg A (A' Un B); LeadsTo prg (B-A') B'; id: (Acts prg) |] \
\ ==> LeadsTo prg A (A' Un B')";
by (rtac LeadsTo_cancel2 1);
by (assume_tac 2);
by (ALLGOALS Asm_simp_tac);
qed "LeadsTo_cancel_Diff2";
Goal "[| LeadsTo prg A (B Un A'); LeadsTo prg B B'; id: (Acts prg) |] \
\ ==> LeadsTo prg A (B' Un A')";
by (asm_full_simp_tac (simpset() addsimps [Un_commute]) 1);
by (blast_tac (claset() addSIs [LeadsTo_cancel2]) 1);
qed "LeadsTo_cancel1";
Goal "[| LeadsTo prg A (B Un A'); LeadsTo prg (B-A') B'; id: (Acts prg) |] \
\ ==> LeadsTo prg A (B' Un A')";
by (rtac LeadsTo_cancel1 1);
by (assume_tac 2);
by (ALLGOALS Asm_simp_tac);
qed "LeadsTo_cancel_Diff1";
(** The impossibility law **)
Goal "LeadsTo prg A {} ==> reachable prg Int A = {}";
by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1);
by (etac leadsTo_empty 1);
qed "LeadsTo_empty";
(** 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);
qed "R_PSP_stable";
Goal "[| LeadsTo prg A A'; stable (Acts prg) B |] \
\ ==> LeadsTo prg (B Int A) (B Int A')";
by (asm_simp_tac (simpset() addsimps (R_PSP_stable::Int_ac)) 1);
qed "R_PSP_stable2";
Goal "[| LeadsTo prg A A'; constrains (Acts 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);
qed "R_PSP";
Goal "[| LeadsTo prg A A'; constrains (Acts prg) B B'; id: (Acts prg) |] \
\ ==> LeadsTo prg (B Int A) ((B Int A') Un (B' - B))";
by (asm_simp_tac (simpset() addsimps (R_PSP::Int_ac)) 1);
qed "R_PSP2";
Goalw [unless_def]
"[| LeadsTo prg A A'; unless (Acts prg) B B'; id: (Acts prg) |] \
\ ==> LeadsTo prg (A Int B) ((A' Int B) Un B')";
by (dtac R_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 "R_PSP_unless";
(*** Induction rules ***)
(** Meta or object quantifier ????? **)
Goal "[| wf r; \
\ ALL m. LeadsTo prg (A Int f-``{m}) \
\ ((A Int f-``(r^-1 ^^ {m})) Un B); \
\ id: (Acts prg) |] \
\ ==> LeadsTo prg A B";
by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1);
by (etac leadsTo_wf_induct 1);
by (assume_tac 2);
by (blast_tac (claset() addIs [leadsTo_weaken]) 1);
qed "LeadsTo_wf_induct";
Goal "[| wf r; \
\ ALL m:I. LeadsTo prg (A Int f-``{m}) \
\ ((A Int f-``(r^-1 ^^ {m})) Un B); \
\ id: (Acts prg) |] \
\ ==> LeadsTo prg A ((A - (f-``I)) Un B)";
by (etac LeadsTo_wf_induct 1);
by Safe_tac;
by (case_tac "m:I" 1);
by (blast_tac (claset() addIs [LeadsTo_weaken]) 1);
by (blast_tac (claset() addIs [subset_imp_LeadsTo]) 1);
qed "R_bounded_induct";
Goal "[| ALL m. LeadsTo prg (A Int f-``{m}) \
\ ((A Int f-``(lessThan m)) Un B); \
\ id: (Acts prg) |] \
\ ==> LeadsTo prg A B";
by (rtac (wf_less_than RS LeadsTo_wf_induct) 1);
by (assume_tac 2);
by (Asm_simp_tac 1);
qed "R_lessThan_induct";
Goal "[| ALL m:(greaterThan l). LeadsTo prg (A Int f-``{m}) \
\ ((A Int f-``(lessThan m)) Un B); \
\ id: (Acts prg) |] \
\ ==> LeadsTo prg A ((A Int (f-``(atMost l))) Un B)";
by (simp_tac (HOL_ss addsimps [Diff_eq RS sym, vimage_Compl, Compl_greaterThan RS sym]) 1);
by (rtac (wf_less_than RS R_bounded_induct) 1);
by (assume_tac 2);
by (Asm_simp_tac 1);
qed "R_lessThan_bounded_induct";
Goal "[| ALL m:(lessThan l). LeadsTo prg (A Int f-``{m}) \
\ ((A Int f-``(greaterThan m)) Un B); \
\ id: (Acts prg) |] \
\ ==> LeadsTo prg A ((A Int (f-``(atLeast l))) Un B)";
by (res_inst_tac [("f","f"),("f1", "%k. l - k")]
(wf_less_than RS wf_inv_image RS LeadsTo_wf_induct) 1);
by (assume_tac 2);
by (simp_tac (simpset() addsimps [inv_image_def, Image_singleton]) 1);
by (Clarify_tac 1);
by (case_tac "m<l" 1);
by (blast_tac (claset() addIs [not_leE, subset_imp_LeadsTo]) 2);
by (blast_tac (claset() addIs [LeadsTo_weaken_R, diff_less_mono2]) 1);
qed "R_greaterThan_bounded_induct";
(*** 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) |] \
\ ==> 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);
qed "R_stable_completion";
Goal "[| finite I; id: (Acts prg) |] \
\ ==> (ALL i:I. LeadsTo prg (A i) (A' i)) --> \
\ (ALL i:I. stable (Acts 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 [R_stable_completion, stable_def,
ball_constrains_INT]) 1);
qed_spec_mp "R_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); \
\ 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);
qed "R_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)) --> \
\ 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 (asm_full_simp_tac (simpset() addsimps [R_completion]) 1);
qed "R_finite_completion";
(*** Specialized laws for handling invariants ***)
Goalw [transient_def]
"[| reachable prg <= INV; transient (Acts prg) (INV Int A) |] \
\ ==> transient (Acts prg) (reachable prg Int A)";
by (Clarify_tac 1);
by (rtac bexI 1);
by (assume_tac 2);
by (Blast_tac 1);
qed "reachable_transient";
(*Uses the premise "invariant prg". Raw theorem-proving on this
inclusion is slow: the term contains a copy of the program.*)
Goal "[| invariant prg INV; \
\ constrains (Acts prg) (INV Int (A-A')) (A Un A'); \
\ transient (Acts prg) (INV Int (A-A')) |] \
\ ==> ensures (Acts prg) (reachable prg Int A) (reachable prg Int A')";
bd invariant_includes_reachable 1;
by (rtac ensuresI 1);
by (ALLGOALS
(full_simp_tac (simpset() addsimps [Int_Un_distrib RS sym,
Diff_Int_distrib RS sym])));
by (blast_tac (claset() addSIs [reachable_transient]) 2);
br (stable_reachable RS stable_constrains_Int) 1;
by (blast_tac (claset() addIs [constrains_weaken]) 1);
qed "invariant_ensuresI";
bind_thm ("invariant_LeadsTo_Basis", invariant_ensuresI RS LeadsTo_Basis);
Goal "[| invariant prg INV; \
\ LeadsTo prg A A'; id: (Acts prg); \
\ INV Int B <= A; INV Int A' <= B' |] \
\ ==> LeadsTo prg B B'";
by (blast_tac (claset() addDs [invariant_includes_reachable]
addIs [LeadsTo_weaken]) 1);
qed "invariant_LeadsTo_weaken";
(** Constrains/Ensures tactics
main_def defines the main program as a set;
cmd_defs defines the separate commands
**)
(*proves "constrains" properties when the program is specified*)
fun constrains_tac (main_def::cmd_defs) =
SELECT_GOAL
(EVERY [TRY (rtac stableI 1),
rtac constrainsI 1,
full_simp_tac (simpset() addsimps [main_def]) 1,
REPEAT_FIRST (eresolve_tac [disjE]),
rewrite_goals_tac cmd_defs,
ALLGOALS (SELECT_GOAL Auto_tac)]);
(*proves "ensures" 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
ORELSE (*subgoal may involve LeadsTo, leadsTo or ensures*)
REPEAT (ares_tac [leadsTo_imp_LeadsTo, 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,
constrains_tac (main_def::cmd_defs) 1,
rewrite_goals_tac cmd_defs,
ALLGOALS Clarify_tac,
Auto_tac]);