src/HOL/IMP/Def_Init_Sound_Small.thy

tuned;

(* Author: Tobias Nipkow *)

theory Def_Init_Sound_Small
imports Def_Init Def_Init_Small
begin

subsection "Soundness wrt Small Steps"

theorem progress:
  "D (dom s) c A' \<Longrightarrow> c \<noteq> SKIP \<Longrightarrow> EX cs'. (c,s) \<rightarrow> cs'"
proof (induction c arbitrary: s A')
  case Assign thus ?case by auto (metis aval_Some small_step.Assign)
next
  case (If b c1 c2)
  then obtain bv where "bval b s = Some bv" by (auto dest!:bval_Some)
  then show ?case
    by(cases bv)(auto intro: small_step.IfTrue small_step.IfFalse)
qed (fastforce intro: small_step.intros)+

lemma D_mono:  "D A c M \<Longrightarrow> A \<subseteq> A' \<Longrightarrow> EX M'. D A' c M' & M <= M'"
proof (induction c arbitrary: A A' M)
  case Seq thus ?case by auto (metis D.intros(3))
next
  case (If b c1 c2)
  then obtain M1 M2 where "vars b \<subseteq> A" "D A c1 M1" "D A c2 M2" "M = M1 \<inter> M2"
    by auto
  with If.IH `A \<subseteq> A'` obtain M1' M2'
    where "D A' c1 M1'" "D A' c2 M2'" and "M1 \<subseteq> M1'" "M2 \<subseteq> M2'" by metis
  hence "D A' (IF b THEN c1 ELSE c2) (M1' \<inter> M2')" and "M \<subseteq> M1' \<inter> M2'"
    using `vars b \<subseteq> A` `A \<subseteq> A'` `M = M1 \<inter> M2` by(fastforce intro: D.intros)+
  thus ?case by metis
next
  case While thus ?case by auto (metis D.intros(5) subset_trans)
qed (auto intro: D.intros)

theorem D_preservation:
  "(c,s) \<rightarrow> (c',s') \<Longrightarrow> D (dom s) c A \<Longrightarrow> EX A'. D (dom s') c' A' & A <= A'"
proof (induction arbitrary: A rule: small_step_induct)
  case (While b c s)
  then obtain A' where "vars b \<subseteq> dom s" "A = dom s" "D (dom s) c A'" by blast
  moreover
  then obtain A'' where "D A' c A''" by (metis D_incr D_mono)
  ultimately have "D (dom s) (IF b THEN c; WHILE b DO c ELSE SKIP) (dom s)"
    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)
  thus ?case by (metis D_incr `A = dom s`)
next
  case Seq2 thus ?case by auto (metis D_mono D.intros(3))
qed (auto intro: D.intros)

theorem D_sound:
  "(c,s) \<rightarrow>* (c',s') \<Longrightarrow> D (dom s) c A' \<Longrightarrow> c' \<noteq> SKIP
   \<Longrightarrow> \<exists>cs''. (c',s') \<rightarrow> cs''"
apply(induction arbitrary: A' rule:star_induct)
apply (metis progress)
by (metis D_preservation)

end