(* Title: HOL/UNITY/Simple/Reachability.thy
Author: Tanja Vos, Cambridge University Computer Laboratory
Copyright 2000 University of Cambridge
Reachability in Graphs.
From Chandy and Misra, "Parallel Program Design" (1989), sections 6.2
and 11.3.
*)
theory Reachability imports "../Detects" Reach begin
type_synonym edge = "vertex * vertex"
record state =
reach :: "vertex => bool"
nmsg :: "edge => nat"
consts root :: "vertex"
E :: "edge set"
V :: "vertex set"
inductive_set REACHABLE :: "edge set"
where
base: "v \<in> V ==> ((v,v) \<in> REACHABLE)"
| step: "((u,v) \<in> REACHABLE) & (v,w) \<in> E ==> ((u,w) \<in> REACHABLE)"
definition reachable :: "vertex => state set" where
"reachable p == {s. reach s p}"
definition nmsg_eq :: "nat => edge => state set" where
"nmsg_eq k == %e. {s. nmsg s e = k}"
definition nmsg_gt :: "nat => edge => state set" where
"nmsg_gt k == %e. {s. k < nmsg s e}"
definition nmsg_gte :: "nat => edge => state set" where
"nmsg_gte k == %e. {s. k \<le> nmsg s e}"
definition nmsg_lte :: "nat => edge => state set" where
"nmsg_lte k == %e. {s. nmsg s e \<le> k}"
definition final :: "state set" where
"final == (\<Inter>v\<in>V. reachable v <==> {s. (root, v) \<in> REACHABLE}) \<inter>
(INTER E (nmsg_eq 0))"
axiomatization
where
Graph1: "root \<in> V" and
Graph2: "(v,w) \<in> E ==> (v \<in> V) & (w \<in> V)" and
MA1: "F \<in> Always (reachable root)" and
MA2: "v \<in> V ==> F \<in> Always (- reachable v \<union> {s. ((root,v) \<in> REACHABLE)})" and
MA3: "[|v \<in> V;w \<in> V|] ==> F \<in> Always (-(nmsg_gt 0 (v,w)) \<union> (reachable v))" and
MA4: "(v,w) \<in> E ==>
F \<in> Always (-(reachable v) \<union> (nmsg_gt 0 (v,w)) \<union> (reachable w))" and
MA5: "[|v \<in> V; w \<in> V|]
==> F \<in> Always (nmsg_gte 0 (v,w) \<inter> nmsg_lte (Suc 0) (v,w))" and
MA6: "[|v \<in> V|] ==> F \<in> Stable (reachable v)" and
MA6b: "[|v \<in> V;w \<in> W|] ==> F \<in> Stable (reachable v \<inter> nmsg_lte k (v,w))" and
MA7: "[|v \<in> V;w \<in> V|] ==> F \<in> UNIV LeadsTo nmsg_eq 0 (v,w)"
lemmas E_imp_in_V_L = Graph2 [THEN conjunct1]
lemmas E_imp_in_V_R = Graph2 [THEN conjunct2]
lemma lemma2:
"(v,w) \<in> E ==> F \<in> reachable v LeadsTo nmsg_eq 0 (v,w) \<inter> reachable v"
apply (rule MA7 [THEN PSP_Stable, THEN LeadsTo_weaken_L])
apply (rule_tac [3] MA6)
apply (auto simp add: E_imp_in_V_L E_imp_in_V_R)
done
lemma Induction_base: "(v,w) \<in> E ==> F \<in> reachable v LeadsTo reachable w"
apply (rule MA4 [THEN Always_LeadsTo_weaken])
apply (rule_tac [2] lemma2)
apply (auto simp add: nmsg_eq_def nmsg_gt_def)
done
lemma REACHABLE_LeadsTo_reachable:
"(v,w) \<in> REACHABLE ==> F \<in> reachable v LeadsTo reachable w"
apply (erule REACHABLE.induct)
apply (rule subset_imp_LeadsTo, blast)
apply (blast intro: LeadsTo_Trans Induction_base)
done
lemma Detects_part1: "F \<in> {s. (root,v) \<in> REACHABLE} LeadsTo reachable v"
apply (rule single_LeadsTo_I)
apply (simp split: if_split_asm)
apply (rule MA1 [THEN Always_LeadsToI])
apply (erule REACHABLE_LeadsTo_reachable [THEN LeadsTo_weaken_L], auto)
done
lemma Reachability_Detected:
"v \<in> V ==> F \<in> (reachable v) Detects {s. (root,v) \<in> REACHABLE}"
apply (unfold Detects_def, auto)
prefer 2 apply (blast intro: MA2 [THEN Always_weaken])
apply (rule Detects_part1 [THEN LeadsTo_weaken_L], blast)
done
lemma LeadsTo_Reachability:
"v \<in> V ==> F \<in> UNIV LeadsTo (reachable v <==> {s. (root,v) \<in> REACHABLE})"
by (erule Reachability_Detected [THEN Detects_Imp_LeadstoEQ])
(* ------------------------------------ *)
(* Some lemmas about <==> *)
lemma Eq_lemma1:
"(reachable v <==> {s. (root,v) \<in> REACHABLE}) =
{s. ((s \<in> reachable v) = ((root,v) \<in> REACHABLE))}"
by (unfold Equality_def, blast)
lemma Eq_lemma2:
"(reachable v <==> (if (root,v) \<in> REACHABLE then UNIV else {})) =
{s. ((s \<in> reachable v) = ((root,v) \<in> REACHABLE))}"
by (unfold Equality_def, auto)
(* ------------------------------------ *)
(* Some lemmas about final (I don't need all of them!) *)
lemma final_lemma1:
"(\<Inter>v \<in> V. \<Inter>w \<in> V. {s. ((s \<in> reachable v) = ((root,v) \<in> REACHABLE)) &
s \<in> nmsg_eq 0 (v,w)})
\<subseteq> final"
apply (unfold final_def Equality_def, auto)
apply (frule E_imp_in_V_R)
apply (frule E_imp_in_V_L, blast)
done
lemma final_lemma2:
"E\<noteq>{}
==> (\<Inter>v \<in> V. \<Inter>e \<in> E. {s. ((s \<in> reachable v) = ((root,v) \<in> REACHABLE))}
\<inter> nmsg_eq 0 e) \<subseteq> final"
apply (unfold final_def Equality_def)
apply (auto split!: if_split)
apply (frule E_imp_in_V_L, blast)
done
lemma final_lemma3:
"E\<noteq>{}
==> (\<Inter>v \<in> V. \<Inter>e \<in> E.
(reachable v <==> {s. (root,v) \<in> REACHABLE}) \<inter> nmsg_eq 0 e)
\<subseteq> final"
apply (frule final_lemma2)
apply (simp (no_asm_use) add: Eq_lemma2)
done
lemma final_lemma4:
"E\<noteq>{}
==> (\<Inter>v \<in> V. \<Inter>e \<in> E.
{s. ((s \<in> reachable v) = ((root,v) \<in> REACHABLE))} \<inter> nmsg_eq 0 e)
= final"
apply (rule subset_antisym)
apply (erule final_lemma2)
apply (unfold final_def Equality_def, blast)
done
lemma final_lemma5:
"E\<noteq>{}
==> (\<Inter>v \<in> V. \<Inter>e \<in> E.
((reachable v) <==> {s. (root,v) \<in> REACHABLE}) \<inter> nmsg_eq 0 e)
= final"
apply (frule final_lemma4)
apply (simp (no_asm_use) add: Eq_lemma2)
done
lemma final_lemma6:
"(\<Inter>v \<in> V. \<Inter>w \<in> V.
(reachable v <==> {s. (root,v) \<in> REACHABLE}) \<inter> nmsg_eq 0 (v,w))
\<subseteq> final"
apply (simp (no_asm) add: Eq_lemma2 Int_def)
apply (rule final_lemma1)
done
lemma final_lemma7:
"final =
(\<Inter>v \<in> V. \<Inter>w \<in> V.
((reachable v) <==> {s. (root,v) \<in> REACHABLE}) \<inter>
(-{s. (v,w) \<in> E} \<union> (nmsg_eq 0 (v,w))))"
apply (unfold final_def)
apply (rule subset_antisym, blast)
apply (auto split: if_split_asm)
apply (blast dest: E_imp_in_V_L E_imp_in_V_R)+
done
(* ------------------------------------ *)
(* ------------------------------------ *)
(* Stability theorems *)
lemma not_REACHABLE_imp_Stable_not_reachable:
"[| v \<in> V; (root,v) \<notin> REACHABLE |] ==> F \<in> Stable (- reachable v)"
apply (drule MA2 [THEN AlwaysD], auto)
done
lemma Stable_reachable_EQ_R:
"v \<in> V ==> F \<in> Stable (reachable v <==> {s. (root,v) \<in> REACHABLE})"
apply (simp (no_asm) add: Equality_def Eq_lemma2)
apply (blast intro: MA6 not_REACHABLE_imp_Stable_not_reachable)
done
lemma lemma4:
"((nmsg_gte 0 (v,w) \<inter> nmsg_lte (Suc 0) (v,w)) \<inter>
(- nmsg_gt 0 (v,w) \<union> A))
\<subseteq> A \<union> nmsg_eq 0 (v,w)"
apply (unfold nmsg_gte_def nmsg_lte_def nmsg_gt_def nmsg_eq_def, auto)
done
lemma lemma5:
"reachable v \<inter> nmsg_eq 0 (v,w) =
((nmsg_gte 0 (v,w) \<inter> nmsg_lte (Suc 0) (v,w)) \<inter>
(reachable v \<inter> nmsg_lte 0 (v,w)))"
by (unfold nmsg_gte_def nmsg_lte_def nmsg_gt_def nmsg_eq_def, auto)
lemma lemma6:
"- nmsg_gt 0 (v,w) \<union> reachable v \<subseteq> nmsg_eq 0 (v,w) \<union> reachable v"
apply (unfold nmsg_gte_def nmsg_lte_def nmsg_gt_def nmsg_eq_def, auto)
done
lemma Always_reachable_OR_nmsg_0:
"[|v \<in> V; w \<in> V|] ==> F \<in> Always (reachable v \<union> nmsg_eq 0 (v,w))"
apply (rule Always_Int_I [OF MA5 MA3, THEN Always_weaken])
apply (rule_tac [5] lemma4, auto)
done
lemma Stable_reachable_AND_nmsg_0:
"[|v \<in> V; w \<in> V|] ==> F \<in> Stable (reachable v \<inter> nmsg_eq 0 (v,w))"
apply (subst lemma5)
apply (blast intro: MA5 Always_imp_Stable [THEN Stable_Int] MA6b)
done
lemma Stable_nmsg_0_OR_reachable:
"[|v \<in> V; w \<in> V|] ==> F \<in> Stable (nmsg_eq 0 (v,w) \<union> reachable v)"
by (blast intro!: Always_weaken [THEN Always_imp_Stable] lemma6 MA3)
lemma not_REACHABLE_imp_Stable_not_reachable_AND_nmsg_0:
"[| v \<in> V; w \<in> V; (root,v) \<notin> REACHABLE |]
==> F \<in> Stable (- reachable v \<inter> nmsg_eq 0 (v,w))"
apply (rule Stable_Int [OF MA2 [THEN Always_imp_Stable]
Stable_nmsg_0_OR_reachable,
THEN Stable_eq])
prefer 4 apply blast
apply auto
done
lemma Stable_reachable_EQ_R_AND_nmsg_0:
"[| v \<in> V; w \<in> V |]
==> F \<in> Stable ((reachable v <==> {s. (root,v) \<in> REACHABLE}) \<inter>
nmsg_eq 0 (v,w))"
by (simp add: Equality_def Eq_lemma2 Stable_reachable_AND_nmsg_0
not_REACHABLE_imp_Stable_not_reachable_AND_nmsg_0)
(* ------------------------------------ *)
(* LeadsTo final predicate (Exercise 11.2 page 274) *)
lemma UNIV_lemma: "UNIV \<subseteq> (\<Inter>v \<in> V. UNIV)"
by blast
lemmas UNIV_LeadsTo_completion =
LeadsTo_weaken_L [OF Finite_stable_completion UNIV_lemma]
lemma LeadsTo_final_E_empty: "E={} ==> F \<in> UNIV LeadsTo final"
apply (unfold final_def, simp)
apply (rule UNIV_LeadsTo_completion)
apply safe
apply (erule LeadsTo_Reachability [simplified])
apply (drule Stable_reachable_EQ_R, simp)
done
lemma Leadsto_reachability_AND_nmsg_0:
"[| v \<in> V; w \<in> V |]
==> F \<in> UNIV LeadsTo
((reachable v <==> {s. (root,v) \<in> REACHABLE}) \<inter> nmsg_eq 0 (v,w))"
apply (rule LeadsTo_Reachability [THEN LeadsTo_Trans], blast)
apply (subgoal_tac
"F \<in> (reachable v <==> {s. (root,v) \<in> REACHABLE}) \<inter>
UNIV LeadsTo (reachable v <==> {s. (root,v) \<in> REACHABLE}) \<inter>
nmsg_eq 0 (v,w) ")
apply simp
apply (rule PSP_Stable2)
apply (rule MA7)
apply (rule_tac [3] Stable_reachable_EQ_R, auto)
done
lemma LeadsTo_final_E_NOT_empty: "E\<noteq>{} ==> F \<in> UNIV LeadsTo final"
apply (rule LeadsTo_weaken_L [OF LeadsTo_weaken_R UNIV_lemma])
apply (rule_tac [2] final_lemma6)
apply (rule Finite_stable_completion)
apply blast
apply (rule UNIV_LeadsTo_completion)
apply (blast intro: Stable_INT Stable_reachable_EQ_R_AND_nmsg_0
Leadsto_reachability_AND_nmsg_0)+
done
lemma LeadsTo_final: "F \<in> UNIV LeadsTo final"
apply (cases "E={}")
apply (rule_tac [2] LeadsTo_final_E_NOT_empty)
apply (rule LeadsTo_final_E_empty, auto)
done
(* ------------------------------------ *)
(* Stability of final (Exercise 11.2 page 274) *)
lemma Stable_final_E_empty: "E={} ==> F \<in> Stable final"
apply (unfold final_def, simp)
apply (rule Stable_INT)
apply (drule Stable_reachable_EQ_R, simp)
done
lemma Stable_final_E_NOT_empty: "E\<noteq>{} ==> F \<in> Stable final"
apply (subst final_lemma7)
apply (rule Stable_INT)
apply (rule Stable_INT)
apply (simp (no_asm) add: Eq_lemma2)
apply safe
apply (rule Stable_eq)
apply (subgoal_tac [2]
"({s. (s \<in> reachable v) = ((root,v) \<in> REACHABLE) } \<inter> nmsg_eq 0 (v,w)) =
({s. (s \<in> reachable v) = ( (root,v) \<in> REACHABLE) } \<inter> (- UNIV \<union> nmsg_eq 0 (v,w)))")
prefer 2 apply blast
prefer 2 apply blast
apply (rule Stable_reachable_EQ_R_AND_nmsg_0
[simplified Eq_lemma2 Collect_const])
apply (blast, blast)
apply (rule Stable_eq)
apply (rule Stable_reachable_EQ_R [simplified Eq_lemma2 Collect_const])
apply simp
apply (subgoal_tac
"({s. (s \<in> reachable v) = ((root,v) \<in> REACHABLE) }) =
({s. (s \<in> reachable v) = ( (root,v) \<in> REACHABLE) } \<inter>
(- {} \<union> nmsg_eq 0 (v,w)))")
apply blast+
done
lemma Stable_final: "F \<in> Stable final"
apply (cases "E={}")
prefer 2 apply (blast intro: Stable_final_E_NOT_empty)
apply (blast intro: Stable_final_E_empty)
done
end