author | wenzelm |
Sat, 17 Oct 2009 14:43:18 +0200 | |
changeset 32960 | 69916a850301 |
parent 26824 | 32e612e77edb |
child 37936 | 1e4c5015a72e |
permissions | -rw-r--r-- |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
26824
diff
changeset
|
1 |
(* Title: HOL/UNITY/Deadlock.thy |
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
2 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
3 |
Copyright 1998 University of Cambridge |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
4 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
5 |
Deadlock examples from section 5.6 of |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
6 |
Misra, "A Logic for Concurrent Programming", 1994 |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
7 |
*) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
8 |
|
16417 | 9 |
theory Deadlock imports UNITY begin |
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
10 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
11 |
(*Trivial, two-process case*) |
13806 | 12 |
lemma "[| F \<in> (A \<inter> B) co A; F \<in> (B \<inter> A) co B |] ==> F \<in> stable (A \<inter> B)" |
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
13 |
by (unfold constrains_def stable_def, blast) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
14 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
15 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
16 |
(*a simplification step*) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
17 |
lemma Collect_le_Int_equals: |
13806 | 18 |
"(\<Inter>i \<in> atMost n. A(Suc i) \<inter> A i) = (\<Inter>i \<in> atMost (Suc n). A i)" |
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
19 |
apply (induct_tac "n") |
13806 | 20 |
apply (auto simp add: atMost_Suc) |
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
21 |
done |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
22 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
23 |
(*Dual of the required property. Converse inclusion fails.*) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
24 |
lemma UN_Int_Compl_subset: |
13806 | 25 |
"(\<Union>i \<in> lessThan n. A i) \<inter> (- A n) \<subseteq> |
26 |
(\<Union>i \<in> lessThan n. (A i) \<inter> (- A (Suc i)))" |
|
27 |
apply (induct_tac "n", simp) |
|
28 |
apply (simp add: lessThan_Suc, blast) |
|
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
29 |
done |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
30 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
31 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
32 |
(*Converse inclusion fails.*) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
33 |
lemma INT_Un_Compl_subset: |
13806 | 34 |
"(\<Inter>i \<in> lessThan n. -A i \<union> A (Suc i)) \<subseteq> |
35 |
(\<Inter>i \<in> lessThan n. -A i) \<union> A n" |
|
36 |
apply (induct_tac "n", simp) |
|
26824
32e612e77edb
Replaced blast by fast in proof of INT_Un_Compl_subset, since blast looped
berghofe
parents:
16417
diff
changeset
|
37 |
apply (simp add: lessThan_Suc, fast) |
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
38 |
done |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
39 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
40 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
41 |
(*Specialized rewriting*) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
42 |
lemma INT_le_equals_Int_lemma: |
13806 | 43 |
"A 0 \<inter> (-(A n) \<inter> (\<Inter>i \<in> lessThan n. -A i \<union> A (Suc i))) = {}" |
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
44 |
by (blast intro: gr0I dest: INT_Un_Compl_subset [THEN subsetD]) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
45 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
46 |
(*Reverse direction makes it harder to invoke the ind hyp*) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
47 |
lemma INT_le_equals_Int: |
13806 | 48 |
"(\<Inter>i \<in> atMost n. A i) = |
49 |
A 0 \<inter> (\<Inter>i \<in> lessThan n. -A i \<union> A(Suc i))" |
|
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
50 |
apply (induct_tac "n", simp) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
51 |
apply (simp add: Int_ac Int_Un_distrib Int_Un_distrib2 |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
52 |
INT_le_equals_Int_lemma lessThan_Suc atMost_Suc) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
53 |
done |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
54 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
55 |
lemma INT_le_Suc_equals_Int: |
13806 | 56 |
"(\<Inter>i \<in> atMost (Suc n). A i) = |
57 |
A 0 \<inter> (\<Inter>i \<in> atMost n. -A i \<union> A(Suc i))" |
|
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
58 |
by (simp add: lessThan_Suc_atMost INT_le_equals_Int) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
59 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
60 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
61 |
(*The final deadlock example*) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
62 |
lemma |
13806 | 63 |
assumes zeroprem: "F \<in> (A 0 \<inter> A (Suc n)) co (A 0)" |
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
64 |
and allprem: |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
26824
diff
changeset
|
65 |
"!!i. i \<in> atMost n ==> F \<in> (A(Suc i) \<inter> A i) co (-A i \<union> A(Suc i))" |
13806 | 66 |
shows "F \<in> stable (\<Inter>i \<in> atMost (Suc n). A i)" |
13785
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
67 |
apply (unfold stable_def) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
68 |
apply (rule constrains_Int [THEN constrains_weaken]) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
69 |
apply (rule zeroprem) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
70 |
apply (rule constrains_INT) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
71 |
apply (erule allprem) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
72 |
apply (simp add: Collect_le_Int_equals Int_assoc INT_absorb) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
73 |
apply (simp add: INT_le_Suc_equals_Int) |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
74 |
done |
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
75 |
|
e2fcd88be55d
Partial conversion of UNITY to Isar new-style theories
paulson
parents:
11195
diff
changeset
|
76 |
end |