Theory Deadlock

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theory Deadlock
imports UNITY

(*  Title:      HOL/UNITY/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)"
by (unfold constrains_def stable_def, blast)


(*a simplification step*)
lemma Collect_le_Int_equals:
"(\<Inter>i ∈ atMost n. A(Suc i) ∩ A i) = (\<Inter>i ∈ 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 ∈ lessThan n. A i) ∩ (- A n) ⊆
(\<Union>i ∈ lessThan n. (A i) ∩ (- 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 ∈ lessThan n. -A i ∪ A (Suc i)) ⊆
(\<Inter>i ∈ lessThan n. -A i) ∪ A n"

apply (induct_tac "n", simp)
apply (simp add: lessThan_Suc, fast)
done


(*Specialized rewriting*)
lemma INT_le_equals_Int_lemma:
"A 0 ∩ (-(A n) ∩ (\<Inter>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:
"(\<Inter>i ∈ atMost n. A i) =
A 0 ∩ (\<Inter>i ∈ lessThan n. -A i ∪ 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 ∈ atMost (Suc n). A i) =
A 0 ∩ (\<Inter>i ∈ atMost n. -A i ∪ 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 (\<Inter>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