(* Author: Tobias Nipkow *)

theory Abs_Int_den0

imports Abs_State_den

begin

subsection "Computable Abstract Interpretation"

text{* Abstract interpretation over type @{text astate} instead of

functions. *}

locale Abs_Int = Val_abs +

fixes pfp :: "('a astate \<Rightarrow> 'a astate) \<Rightarrow> 'a astate \<Rightarrow> 'a astate"

assumes pfp: "f(pfp f x0) \<sqsubseteq> pfp f x0"

assumes above: "x0 \<sqsubseteq> pfp f x0"

begin

fun aval' :: "aexp \<Rightarrow> 'a astate \<Rightarrow> 'a" where

"aval' (N n) _ = num' n" |

"aval' (V x) S = lookup S x" |

"aval' (Plus e1 e2) S = plus' (aval' e1 S) (aval' e2 S)"

abbreviation astate_in_rep (infix "<:" 50) where

"s <: S == ALL x. s x <: lookup S x"

lemma astate_in_rep_le: "(s::state) <: S \<Longrightarrow> S \<sqsubseteq> T \<Longrightarrow> s <: T"

by (metis in_mono le_astate_def le_rep lookup_def top)

lemma aval'_sound: "s <: S \<Longrightarrow> aval a s <: aval' a S"

by (induct a) (auto simp: rep_num' rep_plus')

fun AI :: "com \<Rightarrow> 'a astate \<Rightarrow> 'a astate" where

"AI SKIP S = S" |

"AI (x ::= a) S = update S x (aval' a S)" |

"AI (c1;c2) S = AI c2 (AI c1 S)" |

"AI (IF b THEN c1 ELSE c2) S = (AI c1 S) \<squnion> (AI c2 S)" |

"AI (WHILE b DO c) S = pfp (AI c) S"

lemma AI_sound: "(c,s) \<Rightarrow> t \<Longrightarrow> s <: S0 \<Longrightarrow> t <: AI c S0"

proof(induction c arbitrary: s t S0)

  case SKIP thus ?case by fastforce

next

  case Assign thus ?case

    by (auto simp: lookup_update aval'_sound)

next

  case Semi thus ?case by auto

next

  case If thus ?case

    by (metis AI.simps(4) IfE astate_in_rep_le join_ge1 join_ge2)

next

  case (While b c)

  let ?P = "pfp (AI c) S0"

  { fix s t have "(WHILE b DO c,s) \<Rightarrow> t \<Longrightarrow> s <: ?P \<Longrightarrow> t <: ?P"

    proof(induction "WHILE b DO c" s t rule: big_step_induct)

      case WhileFalse thus ?case by simp

    next

      case WhileTrue thus ?case using While.IH pfp astate_in_rep_le by metis

    qed

  }

  with astate_in_rep_le[OF `s <: S0` above]

  show ?case by (metis While(2) AI.simps(5))

qed

end

end