(* Title: HOL/UNITY/Reach.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Reachability in Directed Graphs. From Chandy and Misra, section 6.4.
[ this example took only four days!]
*)
(*TO SIMPDATA.ML?? FOR CLASET?? *)
val major::prems = goal thy
"[| if P then Q else R; \
\ [| P; Q |] ==> S; \
\ [| ~ P; R |] ==> S |] ==> S";
by (cut_facts_tac [major] 1);
by (blast_tac (claset() addSDs [if_bool_eq_disj RS iffD1] addIs prems) 1);
qed "ifE";
AddSEs [ifE];
val cmd_defs = [racts_def, asgt_def, fun_upd_def];
Goalw [racts_def] "id : racts";
by (Simp_tac 1);
qed "id_in_racts";
AddIffs [id_in_racts];
(*All vertex sets are finite*)
AddIffs [[subset_UNIV, finite_graph] MRS finite_subset];
(** Constrains/Ensures tactics: NEED TO BE GENERALIZED OVER ALL PROGRAMS **)
(*proves "constrains" properties when the program is specified*)
val constrains_tac =
SELECT_GOAL
(EVERY [rtac constrainsI 1,
rewtac racts_def,
REPEAT_FIRST (eresolve_tac [insertE, emptyE]),
rewrite_goals_tac [racts_def, asgt_def],
ALLGOALS (SELECT_GOAL Auto_tac)]);
(*proves "ensures" properties when the program is specified*)
fun ensures_tac sact =
SELECT_GOAL
(EVERY [REPEAT (resolve_tac [LeadsTo_Basis, leadsTo_Basis, ensuresI] 1),
res_inst_tac [("act", sact)] transient_mem 2,
Simp_tac 2,
constrains_tac 1,
rewrite_goals_tac [racts_def, asgt_def],
Auto_tac]);
Goalw [stable_def, invariant_def]
"stable racts invariant";
by (constrains_tac 1);
by (blast_tac (claset() addIs [r_into_rtrancl,rtrancl_trans]) 1);
qed "stable_invariant";
Goalw [rinit_def, invariant_def] "rinit <= invariant";
by Auto_tac;
qed "rinit_invariant";
Goal "reachable (rinit,racts) <= invariant";
by (simp_tac (simpset() addsimps
[strongest_invariant, stable_invariant, rinit_invariant]) 1);
qed "reachable_subset_invariant";
val reachable_subset_invariant' =
rewrite_rule [invariant_def] reachable_subset_invariant;
(*** Fixedpoint ***)
(*If it reaches a fixedpoint, it has found a solution*)
Goalw [fixedpoint_def, invariant_def]
"fixedpoint Int invariant = { %v. (init, v) : edges^* }";
by (rtac equalityI 1);
by (blast_tac (claset() addIs [r_into_rtrancl,rtrancl_trans]) 2);
by (auto_tac (claset() addSIs [ext], simpset()));
by (etac rtrancl_induct 1);
by Auto_tac;
qed "fixedpoint_invariant_correct";
Goalw (cmd_defs @ [FP_def, fixedpoint_def, stable_def, constrains_def])
"FP racts <= fixedpoint";
by Auto_tac;
by (dtac bspec 1);
by (Blast_tac 1);
by (asm_full_simp_tac (simpset() addsimps [Image_singleton, image_iff]) 1);
by (dtac fun_cong 1);
by Auto_tac;
val lemma1 = result();
Goalw (cmd_defs @ [FP_def, fixedpoint_def, stable_def, constrains_def])
"fixedpoint <= FP racts";
by (auto_tac (claset() addIs [ext], simpset()));
val lemma2 = result();
Goal "FP racts = fixedpoint";
by (rtac ([lemma1,lemma2] MRS equalityI) 1);
qed "FP_fixedpoint";
(*If we haven't reached a fixedpoint then there is some edge for which u but
not v holds. Progress will be proved via an ENSURES assertion that the
metric will decrease for each suitable edge. A union over all edges proves
a LEADSTO assertion that the metric decreases if we are not at a fixedpoint.
*)
Goal "Compl fixedpoint = (UN (u,v): edges. {s. s u & ~ s v})";
by (simp_tac (simpset() addsimps
([Compl_FP, UN_UN_flatten, FP_fixedpoint RS sym,
racts_def, asgt_def])) 1);
by Safe_tac;
by (rtac fun_upd_idem 1);
by (Blast_tac 1);
by (Full_simp_tac 1);
by (REPEAT (dtac bspec 1 THEN Simp_tac 1));
by (dtac subsetD 1);
by (Simp_tac 1);
by (asm_full_simp_tac (simpset() addsimps [fun_upd_idem_iff]) 1);
qed "Compl_fixedpoint";
Goal "A - fixedpoint = (UN (u,v): edges. A Int {s. s u & ~ s v})";
by (simp_tac (simpset() addsimps [Diff_eq, Compl_fixedpoint]) 1);
by (Blast_tac 1);
qed "Diff_fixedpoint";
(*** Progress ***)
Goalw [metric_def] "~ s x ==> Suc (metric (s(x:=True))) = metric s";
by (subgoal_tac "{v. ~ (s(x:=True)) v} = {v. ~ s v} - {x}" 1);
by Auto_tac;
by (asm_simp_tac (simpset() addsimps [card_Suc_Diff]) 1);
qed "Suc_metric";
Goal "~ s x ==> metric (s(x:=True)) < metric s";
by (etac (Suc_metric RS subst) 1);
by (Blast_tac 1);
qed "metric_less";
AddSIs [metric_less];
Goal "metric (s(y:=s x | s y)) <= metric s";
by (case_tac "s x --> s y" 1);
by (auto_tac (claset() addIs [less_imp_le],
simpset() addsimps [fun_upd_idem]));
qed "metric_le";
Goal "(u,v): edges ==> \
\ ensures racts ((metric-``{m}) Int {s. s u & ~ s v}) \
\ (metric-``(lessThan m))";
by (ensures_tac "asgt u v" 1);
by (cut_facts_tac [metric_le] 1);
by (fast_tac (claset() addSDs [le_imp_less_or_eq]) 1);
qed "edges_ensures";
Goal "leadsTo racts ((metric-``{m}) - fixedpoint) (metric-``(lessThan m))";
by (simp_tac (simpset() addsimps [Diff_fixedpoint]) 1);
by (rtac leadsTo_UN 1);
by (split_all_tac 1);
by (asm_simp_tac (simpset() addsimps [edges_ensures RS leadsTo_Basis]) 1);
qed "leadsTo_Diff_fixedpoint";
Goal "leadsTo racts (metric-``{m}) (metric-``(lessThan m) Un fixedpoint)";
by (rtac (leadsTo_Diff_fixedpoint RS leadsTo_weaken_R RS leadsTo_Diff) 1);
by (ALLGOALS (blast_tac (claset() addIs [subset_imp_leadsTo])));
qed "leadsTo_Un_fixedpoint";
(*Execution in any state leads to a fixedpoint (i.e. can terminate)*)
Goal "leadsTo racts UNIV fixedpoint";
by (rtac lessThan_induct 1);
by Auto_tac;
by (rtac leadsTo_Un_fixedpoint 1);
qed "leadsTo_fixedpoint";
Goal "LeadsTo(rinit,racts) UNIV { %v. (init, v) : edges^* }";
by (stac (fixedpoint_invariant_correct RS sym) 1);
by (rtac (leadsTo_fixedpoint RS leadsTo_imp_LeadsTo RS LeadsTo_weaken_R) 1);
by (cut_facts_tac [reachable_subset_invariant] 1);
by (Blast_tac 1);
qed "LeadsTo_correct";