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(* Title: HOL/UNITY/Simple/Deadlock.thy
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 \ (A \ B) co A; F \ (B \ A) co B |] ==> F \ stable (A \ B)"
unfolding constrains_def stable_def by blast
(*a simplification step*)
lemma Collect_le_Int_equals:
"(\i \ atMost n. A(Suc i) \ A i) = (\i \ atMost (Suc n). A i)"
by (induct n) (auto simp add: atMost_Suc)
(*Dual of the required property. Converse inclusion fails.*)
lemma UN_Int_Compl_subset:
"(\i \ lessThan n. A i) \ (- A n) \
(\<Union>i \<in> lessThan n. (A i) \<inter> (- A (Suc i)))"
by (induct n) (auto simp: lessThan_Suc)
(*Converse inclusion fails.*)
lemma INT_Un_Compl_subset:
"(\i \ lessThan n. -A i \ A (Suc i)) \
(\<Inter>i \<in> lessThan n. -A i) \<union> A n"
by (induct n) (auto simp: lessThan_Suc)
(*Specialized rewriting*)
lemma INT_le_equals_Int_lemma:
"A 0 \ (-(A n) \ (\i \ lessThan n. -A i \ 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:
"(\i \ atMost n. A i) =
A 0 \<inter> (\<Inter>i \<in> lessThan n. -A i \<union> A(Suc i))"
by (induct n)
(simp_all add: Int_ac Int_Un_distrib Int_Un_distrib2
INT_le_equals_Int_lemma lessThan_Suc atMost_Suc)
lemma INT_le_Suc_equals_Int:
"(\i \ 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 \ (A 0 \ A (Suc n)) co (A 0)"
and allprem:
"!!i. i \ atMost n ==> F \ (A(Suc i) \ A i) co (-A i \ A(Suc i))"
shows "F \ stable (\i \ 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
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