src/HOL/IMP/Def_Init_Sound_Small.thy
 author kleing Thu May 23 11:39:40 2013 +1000 (2013-05-23) changeset 52120 e6433b34364b parent 52046 bc01725d7918 permissions -rw-r--r--
slightly clearer formulation
1 (* Author: Tobias Nipkow *)
3 theory Def_Init_Sound_Small
4 imports Def_Init Def_Init_Small
5 begin
7 subsection "Soundness wrt Small Steps"
9 theorem progress:
10   "D (dom s) c A' \<Longrightarrow> c \<noteq> SKIP \<Longrightarrow> EX cs'. (c,s) \<rightarrow> cs'"
11 proof (induction c arbitrary: s A')
12   case Assign thus ?case by auto (metis aval_Some small_step.Assign)
13 next
14   case (If b c1 c2)
15   then obtain bv where "bval b s = Some bv" by (auto dest!:bval_Some)
16   then show ?case
17     by(cases bv)(auto intro: small_step.IfTrue small_step.IfFalse)
18 qed (fastforce intro: small_step.intros)+
20 lemma D_mono:  "D A c M \<Longrightarrow> A \<subseteq> A' \<Longrightarrow> EX M'. D A' c M' & M <= M'"
21 proof (induction c arbitrary: A A' M)
22   case Seq thus ?case by auto (metis D.intros(3))
23 next
24   case (If b c1 c2)
25   then obtain M1 M2 where "vars b \<subseteq> A" "D A c1 M1" "D A c2 M2" "M = M1 \<inter> M2"
26     by auto
27   with If.IH `A \<subseteq> A'` obtain M1' M2'
28     where "D A' c1 M1'" "D A' c2 M2'" and "M1 \<subseteq> M1'" "M2 \<subseteq> M2'" by metis
29   hence "D A' (IF b THEN c1 ELSE c2) (M1' \<inter> M2')" and "M \<subseteq> M1' \<inter> M2'"
30     using `vars b \<subseteq> A` `A \<subseteq> A'` `M = M1 \<inter> M2` by(fastforce intro: D.intros)+
31   thus ?case by metis
32 next
33   case While thus ?case by auto (metis D.intros(5) subset_trans)
34 qed (auto intro: D.intros)
36 theorem D_preservation:
37   "(c,s) \<rightarrow> (c',s') \<Longrightarrow> D (dom s) c A \<Longrightarrow> EX A'. D (dom s') c' A' & A <= A'"
38 proof (induction arbitrary: A rule: small_step_induct)
39   case (While b c s)
40   then obtain A' where "vars b \<subseteq> dom s" "A = dom s" "D (dom s) c A'" by blast
41   moreover
42   then obtain A'' where "D A' c A''" by (metis D_incr D_mono)
43   ultimately have "D (dom s) (IF b THEN c;; WHILE b DO c ELSE SKIP) (dom s)"
44     by (metis D.If[OF `vars b \<subseteq> dom s` D.Seq[OF `D (dom s) c A'` D.While[OF _ `D A' c A''`]] D.Skip] D_incr Int_absorb1 subset_trans)
45   thus ?case by (metis D_incr `A = dom s`)
46 next
47   case Seq2 thus ?case by auto (metis D_mono D.intros(3))
48 qed (auto intro: D.intros)
50 theorem D_sound:
51   "(c,s) \<rightarrow>* (c',s') \<Longrightarrow> D (dom s) c A'
52    \<Longrightarrow> (\<exists>cs''. (c',s') \<rightarrow> cs'') \<or> c' = SKIP"
53 apply(induction arbitrary: A' rule:star_induct)
54 apply (metis progress)
55 by (metis D_preservation)
57 end