(* Title: HOL/UNITY/Reachability
ID: $Id$
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 = Detects + Reach:
types edge = "(vertex*vertex)"
record state =
reach :: "vertex => bool"
nmsg :: "edge => nat"
consts REACHABLE :: "edge set"
root :: "vertex"
E :: "edge set"
V :: "vertex set"
inductive "REACHABLE"
intros
base: "v : V ==> ((v,v) : REACHABLE)"
step: "((u,v) : REACHABLE) & (v,w) : E ==> ((u,w) : REACHABLE)"
constdefs
reachable :: "vertex => state set"
"reachable p == {s. reach s p}"
nmsg_eq :: "nat => edge => state set"
"nmsg_eq k == %e. {s. nmsg s e = k}"
nmsg_gt :: "nat => edge => state set"
"nmsg_gt k == %e. {s. k < nmsg s e}"
nmsg_gte :: "nat => edge => state set"
"nmsg_gte k == %e. {s. k <= nmsg s e}"
nmsg_lte :: "nat => edge => state set"
"nmsg_lte k == %e. {s. nmsg s e <= k}"
final :: "state set"
"final == (INTER V (%v. reachable v <==> {s. (root, v) : REACHABLE})) Int (INTER E (nmsg_eq 0))"
axioms
Graph1: "root : V"
Graph2: "(v,w) : E ==> (v : V) & (w : V)"
MA1: "F : Always (reachable root)"
MA2: "v: V ==> F : Always (- reachable v Un {s. ((root,v) : REACHABLE)})"
MA3: "[|v:V;w:V|] ==> F : Always (-(nmsg_gt 0 (v,w)) Un (reachable v))"
MA4: "(v,w) : E ==>
F : Always (-(reachable v) Un (nmsg_gt 0 (v,w)) Un (reachable w))"
MA5: "[|v:V; w:V|]
==> F : Always (nmsg_gte 0 (v,w) Int nmsg_lte (Suc 0) (v,w))"
MA6: "[|v:V|] ==> F : Stable (reachable v)"
MA6b: "[|v:V;w:W|] ==> F : Stable (reachable v Int nmsg_lte k (v,w))"
MA7: "[|v:V;w:V|] ==> F : UNIV LeadsTo nmsg_eq 0 (v,w)"
lemmas E_imp_in_V_L = Graph2 [THEN conjunct1, standard]
lemmas E_imp_in_V_R = Graph2 [THEN conjunct2, standard]
lemma lemma2:
"(v,w) : E ==> F : reachable v LeadsTo nmsg_eq 0 (v,w) Int 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) : E ==> F : 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) : REACHABLE ==> F : 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 : {s. (root,v) : REACHABLE} LeadsTo reachable v"
apply (rule single_LeadsTo_I)
apply (simp split add: split_if_asm)
apply (rule MA1 [THEN Always_LeadsToI])
apply (erule REACHABLE_LeadsTo_reachable [THEN LeadsTo_weaken_L], auto)
done
lemma Reachability_Detected:
"v : V ==> F : (reachable v) Detects {s. (root,v) : 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 : V ==> F : UNIV LeadsTo (reachable v <==> {s. (root,v) : REACHABLE})"
by (erule Reachability_Detected [THEN Detects_Imp_LeadstoEQ])
(* ------------------------------------ *)
(* Some lemmas about <==> *)
lemma Eq_lemma1:
"(reachable v <==> {s. (root,v) : REACHABLE}) =
{s. ((s : reachable v) = ((root,v) : REACHABLE))}"
apply (unfold Equality_def, blast)
done
lemma Eq_lemma2:
"(reachable v <==> (if (root,v) : REACHABLE then UNIV else {})) =
{s. ((s : reachable v) = ((root,v) : REACHABLE))}"
apply (unfold Equality_def, auto)
done
(* ------------------------------------ *)
(* Some lemmas about final (I don't need all of them!) *)
lemma final_lemma1:
"(INT v: V. INT w:V. {s. ((s : reachable v) = ((root,v) : REACHABLE)) &
s : nmsg_eq 0 (v,w)})
<= 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~={}
==> (INT v: V. INT e: E. {s. ((s : reachable v) = ((root,v) : REACHABLE))}
Int nmsg_eq 0 e) <= final"
apply (unfold final_def Equality_def)
apply (auto split add: split_if_asm)
apply (frule E_imp_in_V_L, blast)
done
lemma final_lemma3:
"E~={}
==> (INT v: V. INT e: E.
(reachable v <==> {s. (root,v) : REACHABLE}) Int nmsg_eq 0 e)
<= final"
apply (frule final_lemma2)
apply (simp (no_asm_use) add: Eq_lemma2)
done
lemma final_lemma4:
"E~={}
==> (INT v: V. INT e: E.
{s. ((s : reachable v) = ((root,v) : REACHABLE))} Int 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~={}
==> (INT v: V. INT e: E.
((reachable v) <==> {s. (root,v) : REACHABLE}) Int nmsg_eq 0 e)
= final"
apply (frule final_lemma4)
apply (simp (no_asm_use) add: Eq_lemma2)
done
lemma final_lemma6:
"(INT v: V. INT w: V.
(reachable v <==> {s. (root,v) : REACHABLE}) Int nmsg_eq 0 (v,w))
<= final"
apply (simp (no_asm) add: Eq_lemma2 Int_def)
apply (rule final_lemma1)
done
lemma final_lemma7:
"final =
(INT v: V. INT w: V.
((reachable v) <==> {s. (root,v) : REACHABLE}) Int
(-{s. (v,w) : E} Un (nmsg_eq 0 (v,w))))"
apply (unfold final_def)
apply (rule subset_antisym, blast)
apply (auto split add: split_if_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 : V; (root,v) ~: REACHABLE |] ==> F : Stable (- reachable v)"
apply (drule MA2 [THEN AlwaysD], auto)
done
lemma Stable_reachable_EQ_R:
"v : V ==> F : Stable (reachable v <==> {s. (root,v) : 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) Int nmsg_lte (Suc 0) (v,w)) Int
(- nmsg_gt 0 (v,w) Un A))
<= A Un 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 Int nmsg_eq 0 (v,w) =
((nmsg_gte 0 (v,w) Int nmsg_lte (Suc 0) (v,w)) Int
(reachable v Int nmsg_lte 0 (v,w)))"
apply (unfold nmsg_gte_def nmsg_lte_def nmsg_gt_def nmsg_eq_def, auto)
done
lemma lemma6:
"- nmsg_gt 0 (v,w) Un reachable v <= nmsg_eq 0 (v,w) Un 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 : V; w : V|] ==> F : Always (reachable v Un 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 : V; w : V|] ==> F : Stable (reachable v Int 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 : V; w : V|] ==> F : Stable (nmsg_eq 0 (v,w) Un reachable v)"
by (blast intro!: Always_weaken [THEN Always_imp_Stable] lemma6 MA3)
lemma not_REACHABLE_imp_Stable_not_reachable_AND_nmsg_0:
"[| v : V; w:V; (root,v) ~: REACHABLE |]
==> F : Stable (- reachable v Int 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 : V; w:V |]
==> F : Stable ((reachable v <==> {s. (root,v) : REACHABLE}) Int
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 <= (INT v: V. UNIV)"
by blast
lemmas UNIV_LeadsTo_completion =
LeadsTo_weaken_L [OF Finite_stable_completion UNIV_lemma]
lemma LeadsTo_final_E_empty: "E={} ==> F : 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 : V; w:V |]
==> F : UNIV LeadsTo
((reachable v <==> {s. (root,v): REACHABLE}) Int nmsg_eq 0 (v,w))"
apply (rule LeadsTo_Reachability [THEN LeadsTo_Trans], blast)
apply (subgoal_tac
"F : (reachable v <==> {s. (root,v) : REACHABLE}) Int
UNIV LeadsTo (reachable v <==> {s. (root,v) : REACHABLE}) Int
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~={} ==> F : 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 : UNIV LeadsTo final"
apply (case_tac "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 : 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~={} ==> F : 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 : reachable v) = ((root,v) : REACHABLE) } Int nmsg_eq 0 (v,w)) = ({s. (s : reachable v) = ( (root,v) : REACHABLE) } Int (- UNIV Un 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 : reachable v) = ((root,v) : REACHABLE) }) =
({s. (s : reachable v) = ( (root,v) : REACHABLE) } Int
(- {} Un nmsg_eq 0 (v,w)))")
apply blast+
done
lemma Stable_final: "F : Stable final"
apply (case_tac "E={}")
prefer 2 apply (blast intro: Stable_final_E_NOT_empty)
apply (blast intro: Stable_final_E_empty)
done
end