Theory Def_Init_Small

(* Author: Tobias Nipkow *)

theory Def_Init_Small
imports Star Def_Init_Exp Def_Init

subsection "Initialization-Sensitive Small Step Semantics"

  small_step :: "(com × state)  (com × state)  bool" (infix "" 55)
Assign:  "aval a s = Some i  (x ::= a, s)  (SKIP, s(x := Some i))" |

Seq1:   "(SKIP;;c,s)  (c,s)" |
Seq2:   "(c1,s)  (c1',s')  (c1;;c2,s)  (c1';;c2,s')" |

IfTrue:  "bval b s = Some True  (IF b THEN c1 ELSE c2,s)  (c1,s)" |
IfFalse: "bval b s = Some False  (IF b THEN c1 ELSE c2,s)  (c2,s)" |

While:   "(WHILE b DO c,s)  (IF b THEN c;; WHILE b DO c ELSE SKIP,s)"

lemmas small_step_induct = small_step.induct[split_format(complete)]

abbreviation small_steps :: "com * state  com * state  bool" (infix "→*" 55)
where "x →* y == star small_step x y"

subsection "Soundness wrt Small Steps"

theorem progress:
  "D (dom s) c A'  c  SKIP  cs'. (c,s)  cs'"
proof (induction c arbitrary: s A')
  case Assign thus ?case by auto (metis aval_Some small_step.Assign)
  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  A  A'  M'. D A' c M' & M <= M'"
proof (induction c arbitrary: A A' M)
  case Seq thus ?case by auto (metis D.intros(3))
  case (If b c1 c2)
  then obtain M1 M2 where "vars b  A" "D A c1 M1" "D A c2 M2" "M = M1  M2"
    by auto
  with If.IH A  A' obtain M1' M2'
    where "D A' c1 M1'" "D A' c2 M2'" and "M1  M1'" "M2  M2'" by metis
  hence "D A' (IF b THEN c1 ELSE c2) (M1'  M2')" and "M  M1'  M2'"
    using vars b  A A  A' M = M1  M2 by(fastforce intro: D.intros)+
  thus ?case by metis
  case While thus ?case by auto (metis D.intros(5) subset_trans)
qed (auto intro: D.intros)

theorem D_preservation:
  "(c,s)  (c',s')  D (dom s) c A  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 A': "vars b  dom s" "A = dom s" "D (dom s) c A'" by blast
  then obtain A'' where "D A' c A''" by (metis D_incr D_mono)
  with A' have "D (dom s) (IF b THEN c;; WHILE b DO c ELSE SKIP) (dom s)"
    by (metis D.If[OF vars b  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)
  case Seq2 thus ?case by auto (metis D_mono D.intros(3))
qed (auto intro: D.intros)

theorem D_sound:
  "(c,s) →* (c',s')  D (dom s) c A'
    (cs''. (c',s')  cs'')  c' = SKIP"
apply(induction arbitrary: A' rule:star_induct)
apply (metis progress)
by (metis D_preservation)