(* Title: HOL/UNITY/Deadlock
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Deadlock examples from section 5.6 of
Misra, "A Logic for Concurrent Programming", 1994
*)
theory Deadlock imports UNITY begin
(*Trivial, two-process case*)
lemma "[| F \<in> (A \<inter> B) co A; F \<in> (B \<inter> A) co B |] ==> F \<in> stable (A \<inter> B)"
by (unfold constrains_def stable_def, blast)
(*a simplification step*)
lemma Collect_le_Int_equals:
"(\<Inter>i \<in> atMost n. A(Suc i) \<inter> A i) = (\<Inter>i \<in> atMost (Suc n). A i)"
apply (induct_tac "n")
apply (auto simp add: atMost_Suc)
done
(*Dual of the required property. Converse inclusion fails.*)
lemma UN_Int_Compl_subset:
"(\<Union>i \<in> lessThan n. A i) \<inter> (- A n) \<subseteq>
(\<Union>i \<in> lessThan n. (A i) \<inter> (- A (Suc i)))"
apply (induct_tac "n", simp)
apply (simp add: lessThan_Suc, blast)
done
(*Converse inclusion fails.*)
lemma INT_Un_Compl_subset:
"(\<Inter>i \<in> lessThan n. -A i \<union> A (Suc i)) \<subseteq>
(\<Inter>i \<in> lessThan n. -A i) \<union> A n"
apply (induct_tac "n", simp)
apply (simp add: lessThan_Suc, blast)
done
(*Specialized rewriting*)
lemma INT_le_equals_Int_lemma:
"A 0 \<inter> (-(A n) \<inter> (\<Inter>i \<in> lessThan n. -A i \<union> A (Suc i))) = {}"
by (blast intro: gr0I dest: INT_Un_Compl_subset [THEN subsetD])
(*Reverse direction makes it harder to invoke the ind hyp*)
lemma INT_le_equals_Int:
"(\<Inter>i \<in> atMost n. A i) =
A 0 \<inter> (\<Inter>i \<in> lessThan n. -A i \<union> A(Suc i))"
apply (induct_tac "n", simp)
apply (simp add: Int_ac Int_Un_distrib Int_Un_distrib2
INT_le_equals_Int_lemma lessThan_Suc atMost_Suc)
done
lemma INT_le_Suc_equals_Int:
"(\<Inter>i \<in> atMost (Suc n). A i) =
A 0 \<inter> (\<Inter>i \<in> atMost n. -A i \<union> A(Suc i))"
by (simp add: lessThan_Suc_atMost INT_le_equals_Int)
(*The final deadlock example*)
lemma
assumes zeroprem: "F \<in> (A 0 \<inter> A (Suc n)) co (A 0)"
and allprem:
"!!i. i \<in> atMost n ==> F \<in> (A(Suc i) \<inter> A i) co (-A i \<union> A(Suc i))"
shows "F \<in> stable (\<Inter>i \<in> atMost (Suc n). A i)"
apply (unfold stable_def)
apply (rule constrains_Int [THEN constrains_weaken])
apply (rule zeroprem)
apply (rule constrains_INT)
apply (erule allprem)
apply (simp add: Collect_le_Int_equals Int_assoc INT_absorb)
apply (simp add: INT_le_Suc_equals_Int)
done
end