(* Author: Tobias Nipkow *)

 header "Denotational Abstract Interpretation"

 theory Abs_Int_den0_fun

 imports "~~/src/HOL/ex/Interpretation_with_Defs" Big_Step

 begin

 subsection "Orderings"

 class preord =

 fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50)

 assumes le_refl[simp]: "x \<sqsubseteq> x"

 and le_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"

 text{* Note: no antisymmetry. Allows implementations where some abstract

 element is implemented by two different values @{prop "x \<noteq> y"}

 such that @{prop"x \<sqsubseteq> y"} and @{prop"y \<sqsubseteq> x"}. Antisymmetry is not

 needed because we never compare elements for equality but only for @{text"\<sqsubseteq>"}.

 *}

 class SL_top = preord +

 fixes join :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)

 fixes Top :: "'a"

 assumes join_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"

 and join_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"

 and join_least: "x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<squnion> y \<sqsubseteq> z"

 and top[simp]: "x \<sqsubseteq> Top"

 begin

 lemma join_le_iff[simp]: "x \<squnion> y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<and> y \<sqsubseteq> z"

 by (metis join_ge1 join_ge2 join_least le_trans)

 fun iter :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where

 "iter 0 f _ = Top" |

 "iter (Suc n) f x = (if f x \<sqsubseteq> x then x else iter n f (f x))"

 lemma iter_pfp: "f(iter n f x) \<sqsubseteq> iter n f x"

 apply (induction n arbitrary: x)

  apply (simp)

 apply (simp)

 done

 abbreviation iter' :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where

 "iter' n f x0 == iter n (\<lambda>x. x0 \<squnion> f x) x0"

 lemma iter'_pfp_above:

   "f(iter' n f x0) \<sqsubseteq> iter' n f x0"  "x0 \<sqsubseteq> iter' n f x0"

 using iter_pfp[of "\<lambda>x. x0 \<squnion> f x"] by auto

 text{* So much for soundness. But how good an approximation of the post-fixed

 point does @{const iter} yield? *}

 lemma iter_funpow: "iter n f x \<noteq> Top \<Longrightarrow> \<exists>k. iter n f x = (f^^k) x"

 apply(induction n arbitrary: x)

  apply simp

 apply (auto)

  apply(metis funpow.simps(1) id_def)

 by (metis funpow.simps(2) funpow_swap1 o_apply)

 text{* For monotone @{text f}, @{term "iter f n x0"} yields the least

 post-fixed point above @{text x0}, unless it yields @{text Top}. *}

 lemma iter_least_pfp:

 assumes mono: "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y" and "iter n f x0 \<noteq> Top"

 and "f p \<sqsubseteq> p" and "x0 \<sqsubseteq> p" shows "iter n f x0 \<sqsubseteq> p"

 proof-

   obtain k where "iter n f x0 = (f^^k) x0"

     using iter_funpow[OF `iter n f x0 \<noteq> Top`] by blast

   moreover

   { fix n have "(f^^n) x0 \<sqsubseteq> p"

     proof(induction n)

       case 0 show ?case by(simp add: `x0 \<sqsubseteq> p`)

     next

       case (Suc n) thus ?case

         by (simp add: `x0 \<sqsubseteq> p`)(metis Suc assms(3) le_trans mono)

     qed

   } ultimately show ?thesis by simp

 qed

 end

 text{* The interface of abstract values: *}

 locale Rep =

 fixes rep :: "'a::SL_top \<Rightarrow> 'b set"

 assumes le_rep: "a \<sqsubseteq> b \<Longrightarrow> rep a \<subseteq> rep b"

 begin

 abbreviation in_rep (infix "<:" 50) where "x <: a == x : rep a"

 lemma in_rep_join: "x <: a1 \<or> x <: a2 \<Longrightarrow> x <: a1 \<squnion> a2"

 by (metis SL_top_class.join_ge1 SL_top_class.join_ge2 le_rep subsetD)

 end

 locale Val_abs = Rep rep

   for rep :: "'a::SL_top \<Rightarrow> val set" +

 fixes num' :: "val \<Rightarrow> 'a"

 and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"

 assumes rep_num': "rep(num' n) = {n}"

 and rep_plus': "n1 <: a1 \<Longrightarrow> n2 <: a2 \<Longrightarrow> n1+n2 <: plus' a1 a2"

 instantiation "fun" :: (type, SL_top) SL_top

 begin

 definition "f \<sqsubseteq> g = (ALL x. f x \<sqsubseteq> g x)"

 definition "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"

 definition "Top = (\<lambda>x. Top)"

 lemma join_apply[simp]:

   "(f \<squnion> g) x = f x \<squnion> g x"

 by (simp add: join_fun_def)

 instance

 proof

   case goal2 thus ?case by (metis le_fun_def preord_class.le_trans)

 qed (simp_all add: le_fun_def Top_fun_def)

 end

 subsection "Abstract Interpretation Abstractly"

 text{* Abstract interpretation over abstract values. Abstract states are

 simply functions. The post-fixed point finder is parameterized over. *}

 type_synonym 'a st = "name \<Rightarrow> 'a"

 locale Abs_Int_Fun = Val_abs +

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

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

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

 begin

 fun aval' :: "aexp \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a" where

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

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

 "aval' (Plus a1 a2) S = plus' (aval' a1 S) (aval' a2 S)"

 abbreviation fun_in_rep (infix "<:" 50) where

 "f <: F == \<forall>x. f x <: F x"

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

 by (metis le_fun_def le_rep subsetD)

 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> (name \<Rightarrow> 'a) \<Rightarrow> (name \<Rightarrow> 'a)" where

 "AI SKIP S = S" |

 "AI (x ::= a) S = 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: aval'_sound)

 next

   case Semi thus ?case by auto

 next

   case If thus ?case by(auto simp: in_rep_join)

 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 by(metis While.IH pfp fun_in_rep_le)

     qed

   }

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

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

 qed

 end

 text{* Problem: not executable because of the comparison of abstract states,

 i.e. functions, in the post-fixedpoint computation. Need to implement

 abstract states concretely. *}

 end