(* 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
*)
open SubstAx;
(*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 ***)
goalw thy [LeadsTo_def]
"!!Acts. leadsTo Acts A B ==> LeadsTo Init Acts A B";
by (blast_tac (claset() addIs [PSP_stable2, stable_reachable]) 1);
qed "leadsTo_imp_LeadsTo";
goalw thy [LeadsTo_def]
"!!Acts. [| constrains Acts (reachable Init Acts Int (A - A')) \
\ (A Un A'); \
\ transient Acts (reachable Init Acts Int (A-A')) |] \
\ ==> LeadsTo Init Acts A A'";
by (rtac (stable_reachable RS stable_ensures_Int RS leadsTo_Basis) 1);
by (assume_tac 2);
by (asm_simp_tac
(simpset() addsimps [Int_Un_distrib RS sym, Diff_Int_distrib RS sym,
stable_constrains_Int]) 1);
qed "LeadsTo_Basis";
goalw thy [LeadsTo_def]
"!!Acts. [| LeadsTo Init Acts A B; LeadsTo Init Acts B C |] \
\ ==> LeadsTo Init Acts A C";
by (blast_tac (claset() addIs [leadsTo_Trans]) 1);
qed "LeadsTo_Trans";
val prems = goalw thy [LeadsTo_def]
"(!!A. A : S ==> LeadsTo Init Acts A B) ==> LeadsTo Init Acts (Union S) B";
by (stac Int_Union 1);
by (blast_tac (claset() addIs (leadsTo_UN::prems)) 1);
qed "LeadsTo_Union";
(*** Derived rules ***)
goal thy "!!Acts. id: Acts ==> LeadsTo Init Acts 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 thy "!!Acts. LeadsTo Init Acts A (A' Un A') ==> LeadsTo Init Acts A A'";
by (asm_full_simp_tac (simpset() addsimps Un_ac) 1);
qed "LeadsTo_Un_duplicate";
goal thy "!!Acts. LeadsTo Init Acts A (A' Un C Un C) ==> LeadsTo Init Acts 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 Init Acts (A i) B) \
\ ==> LeadsTo Init Acts (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 thy
"!!C. [| LeadsTo Init Acts A C; LeadsTo Init Acts B C |] ==> LeadsTo Init Acts (A Un B) C";
by (stac Un_eq_Union 1);
by (blast_tac (claset() addIs [LeadsTo_Union]) 1);
qed "LeadsTo_Un";
goalw thy [LeadsTo_def]
"!!A B. [| reachable Init Acts Int A <= B; id: Acts |] \
\ ==> LeadsTo Init Acts A B";
by (blast_tac (claset() addIs [subset_imp_leadsTo]) 1);
qed "Int_subset_imp_LeadsTo";
goalw thy [LeadsTo_def]
"!!A B. [| A <= B; id: Acts |] \
\ ==> LeadsTo Init Acts A B";
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 thy
"!!A B. [| reachable Init Acts Int A = {}; id: Acts |] \
\ ==> LeadsTo Init Acts A B";
by (asm_simp_tac (simpset() addsimps [Int_subset_imp_LeadsTo]) 1);
qed "Int_empty_LeadsTo";
goalw thy [LeadsTo_def]
"!!Acts. [| LeadsTo Init Acts A A'; \
\ reachable Init Acts Int A' <= B' |] \
\ ==> LeadsTo Init Acts A B'";
by (blast_tac (claset() addIs [leadsTo_weaken_R]) 1);
qed_spec_mp "LeadsTo_weaken_R";
goalw thy [LeadsTo_def]
"!!Acts. [| LeadsTo Init Acts A A'; \
\ reachable Init Acts Int B <= A; id: Acts |] \
\ ==> LeadsTo Init Acts B A'";
by (blast_tac (claset() addIs [leadsTo_weaken_L]) 1);
qed_spec_mp "LeadsTo_weaken_L";
(*Distributes over binary unions*)
goal thy
"!!C. id: Acts ==> \
\ LeadsTo Init Acts (A Un B) C = \
\ (LeadsTo Init Acts A C & LeadsTo Init Acts B C)";
by (blast_tac (claset() addIs [LeadsTo_Un, LeadsTo_weaken_L]) 1);
qed "LeadsTo_Un_distrib";
goal thy
"!!C. id: Acts ==> \
\ LeadsTo Init Acts (UN i:I. A i) B = \
\ (ALL i : I. LeadsTo Init Acts (A i) B)";
by (blast_tac (claset() addIs [LeadsTo_UN, LeadsTo_weaken_L]) 1);
qed "LeadsTo_UN_distrib";
goal thy
"!!C. id: Acts ==> \
\ LeadsTo Init Acts (Union S) B = \
\ (ALL A : S. LeadsTo Init Acts A B)";
by (blast_tac (claset() addIs [LeadsTo_Union, LeadsTo_weaken_L]) 1);
qed "LeadsTo_Union_distrib";
goal thy
"!!Acts. [| LeadsTo Init Acts A A'; id: Acts; \
\ reachable Init Acts Int B <= A; \
\ reachable Init Acts Int A' <= B' |] \
\ ==> LeadsTo Init Acts 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??*)
goal thy
"!!C. [| LeadsTo Init Acts (A-B) C; LeadsTo Init Acts B C; id: Acts |] \
\ ==> LeadsTo Init Acts 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 Init Acts (A i) (A' i)) \
\ ==> LeadsTo Init Acts (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 Init Acts (A i) (A' i)) \
\ ==> LeadsTo Init Acts (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 thy
"!!Acts. ALL i. LeadsTo Init Acts (A i) (A' i) \
\ ==> LeadsTo Init Acts (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 thy "!!Acts. [| LeadsTo Init Acts A A'; LeadsTo Init Acts B B' |] \
\ ==> LeadsTo Init Acts (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 thy
"!!Acts. [| LeadsTo Init Acts A (A' Un B); LeadsTo Init Acts B B'; \
\ id: Acts |] \
\ ==> LeadsTo Init Acts A (A' Un B')";
by (blast_tac (claset() addIs [LeadsTo_Un_Un,
subset_imp_LeadsTo, LeadsTo_Trans]) 1);
qed "LeadsTo_cancel2";
goal thy
"!!Acts. [| LeadsTo Init Acts A (A' Un B); LeadsTo Init Acts (B-A') B'; id: Acts |] \
\ ==> LeadsTo Init Acts A (A' Un B')";
by (rtac LeadsTo_cancel2 1);
by (assume_tac 2);
by (ALLGOALS Asm_simp_tac);
qed "LeadsTo_cancel_Diff2";
goal thy
"!!Acts. [| LeadsTo Init Acts A (B Un A'); LeadsTo Init Acts B B'; id: Acts |] \
\ ==> LeadsTo Init Acts 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 thy
"!!Acts. [| LeadsTo Init Acts A (B Un A'); LeadsTo Init Acts (B-A') B'; id: Acts |] \
\ ==> LeadsTo Init Acts 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 **)
goalw thy [LeadsTo_def]
"!!Acts. LeadsTo Init Acts A {} ==> reachable Init Acts Int A = {}";
by (Full_simp_tac 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. *)
goalw thy [LeadsTo_def]
"!!Acts. [| LeadsTo Init Acts A A'; stable Acts B |] \
\ ==> LeadsTo Init Acts (A Int B) (A' Int B)";
by (asm_simp_tac (simpset() addsimps [Int_assoc RS sym, PSP_stable]) 1);
qed "R_PSP_stable";
goal thy
"!!Acts. [| LeadsTo Init Acts A A'; stable Acts B |] \
\ ==> LeadsTo Init Acts (B Int A) (B Int A')";
by (asm_simp_tac (simpset() addsimps (R_PSP_stable::Int_ac)) 1);
qed "R_PSP_stable2";
goalw thy [LeadsTo_def]
"!!Acts. [| LeadsTo Init Acts A A'; constrains Acts B B'; id: Acts |] \
\ ==> LeadsTo Init Acts (A Int B) ((A' Int B) Un (B' - B))";
by (dtac PSP 1);
by (etac constrains_reachable 1);
by (etac leadsTo_weaken 2);
by (ALLGOALS Blast_tac);
qed "R_PSP";
goal thy
"!!Acts. [| LeadsTo Init Acts A A'; constrains Acts B B'; id: Acts |] \
\ ==> LeadsTo Init Acts (B Int A) ((B Int A') Un (B' - B))";
by (asm_simp_tac (simpset() addsimps (R_PSP::Int_ac)) 1);
qed "R_PSP2";
goalw thy [unless_def]
"!!Acts. [| LeadsTo Init Acts A A'; unless Acts B B'; id: Acts |] \
\ ==> LeadsTo Init Acts (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 ????? **)
goalw thy [LeadsTo_def]
"!!Acts. [| wf r; \
\ ALL m. LeadsTo Init Acts (A Int f-``{m}) \
\ ((A Int f-``(r^-1 ^^ {m})) Un B); \
\ id: Acts |] \
\ ==> LeadsTo Init Acts A B";
by (etac leadsTo_wf_induct 1);
by (assume_tac 2);
by (blast_tac (claset() addIs [leadsTo_weaken]) 1);
qed "LeadsTo_wf_induct";
goal thy
"!!Acts. [| wf r; \
\ ALL m:I. LeadsTo Init Acts (A Int f-``{m}) \
\ ((A Int f-``(r^-1 ^^ {m})) Un B); \
\ id: Acts |] \
\ ==> LeadsTo Init Acts 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 thy
"!!Acts. [| ALL m. LeadsTo Init Acts (A Int f-``{m}) \
\ ((A Int f-``(lessThan m)) Un B); \
\ id: Acts |] \
\ ==> LeadsTo Init Acts 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 thy
"!!Acts. [| ALL m:(greaterThan l). LeadsTo Init Acts (A Int f-``{m}) \
\ ((A Int f-``(lessThan m)) Un B); \
\ id: Acts |] \
\ ==> LeadsTo Init Acts 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 thy
"!!Acts. [| ALL m:(lessThan l). LeadsTo Init Acts (A Int f-``{m}) \
\ ((A Int f-``(greaterThan m)) Un B); \
\ id: Acts |] \
\ ==> LeadsTo Init Acts 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 ***)
goalw thy [LeadsTo_def]
"!!Acts. [| LeadsTo Init Acts A A'; stable Acts A'; \
\ LeadsTo Init Acts B B'; stable Acts B'; id: Acts |] \
\ ==> LeadsTo Init Acts (A Int B) (A' Int B')";
by (blast_tac (claset() addIs [stable_completion RS leadsTo_weaken]
addSIs [stable_Int, stable_reachable]) 1);
qed "R_stable_completion";
goal thy
"!!Acts. [| finite I; id: Acts |] \
\ ==> (ALL i:I. LeadsTo Init Acts (A i) (A' i)) --> \
\ (ALL i:I. stable Acts (A' i)) --> \
\ LeadsTo Init Acts (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";
goalw thy [LeadsTo_def]
"!!Acts. [| LeadsTo Init Acts A (A' Un C); constrains Acts A' (A' Un C); \
\ LeadsTo Init Acts B (B' Un C); constrains Acts B' (B' Un C); \
\ id: Acts |] \
\ ==> LeadsTo Init Acts (A Int B) ((A' Int B') Un C)";
by (full_simp_tac (simpset() addsimps [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 thy
"!!Acts. [| finite I; id: Acts |] \
\ ==> (ALL i:I. LeadsTo Init Acts (A i) (A' i Un C)) --> \
\ (ALL i:I. constrains Acts (A' i) (A' i Un C)) --> \
\ LeadsTo Init Acts (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";