Theory Abs_Int2

(* Author: Tobias Nipkow *)

section "Backward Analysis of Expressions"

theory Abs_Int2
imports Abs_Int1 "HOL-IMP.Vars"
begin

instantiation prod :: (preord,preord) preord
begin

definition "le_prod p1 p2 = (fst p1 ⊑ fst p2 ∧ snd p1 ⊑ snd p2)"

instance
proof (standard, goal_cases)
  case 1 show ?case by(simp add: le_prod_def)
next
  case 2 thus ?case unfolding le_prod_def by(metis le_trans)
qed

end

hide_const bot

class L_top_bot = SL_top +
fixes meet :: "'a ⇒ 'a ⇒ 'a" (infixl ‹⊓› 65)
and bot :: "'a" (‹⊥›)
assumes meet_le1 [simp]: "x ⊓ y ⊑ x"
and meet_le2 [simp]: "x ⊓ y ⊑ y"
and meet_greatest: "x ⊑ y ⟹ x ⊑ z ⟹ x ⊑ y ⊓ z"
assumes bot[simp]: "⊥ ⊑ x"
begin

lemma mono_meet: "x ⊑ x' ⟹ y ⊑ y' ⟹ x ⊓ y ⊑ x' ⊓ y'"
by (metis meet_greatest meet_le1 meet_le2 le_trans)

end

locale Val_abs1_gamma =
  Gamma where γ = γ for γ :: "'av::L_top_bot ⇒ val set" +
assumes inter_gamma_subset_gamma_meet:
  "γ a1 ∩ γ a2 ⊆ γ(a1 ⊓ a2)"
and gamma_Bot[simp]: "γ ⊥ = {}"
begin

lemma in_gamma_meet: "x : γ a1 ⟹ x : γ a2 ⟹ x : γ(a1 ⊓ a2)"
by (metis IntI inter_gamma_subset_gamma_meet subsetD)

lemma gamma_meet[simp]: "γ(a1 ⊓ a2) = γ a1 ∩ γ a2"
by (metis equalityI inter_gamma_subset_gamma_meet le_inf_iff mono_gamma meet_le1 meet_le2)

end


locale Val_abs1 =
  Val_abs1_gamma where γ = γ
   for γ :: "'av::L_top_bot ⇒ val set" +
fixes test_num' :: "val ⇒ 'av ⇒ bool"
and filter_plus' :: "'av ⇒ 'av ⇒ 'av ⇒ 'av * 'av"
and filter_less' :: "bool ⇒ 'av ⇒ 'av ⇒ 'av * 'av"
assumes test_num': "test_num' n a = (n : γ a)"
and filter_plus': "filter_plus' a a1 a2 = (b1,b2) ⟹
  n1 : γ a1 ⟹ n2 : γ a2 ⟹ n1+n2 : γ a ⟹ n1 : γ b1 ∧ n2 : γ b2"
and filter_less': "filter_less' (n1<n2) a1 a2 = (b1,b2) ⟹
  n1 : γ a1 ⟹ n2 : γ a2 ⟹ n1 : γ b1 ∧ n2 : γ b2"


locale Abs_Int1 =
  Val_abs1 where γ = γ for γ :: "'av::L_top_bot ⇒ val set"
begin

lemma in_gamma_join_UpI: "s : γ⇩o S1 ∨ s : γ⇩o S2 ⟹ s : γ⇩o(S1 ⊔ S2)"
by (metis (no_types) join_ge1 join_ge2 mono_gamma_o rev_subsetD)

fun aval'' :: "aexp ⇒ 'av st option ⇒ 'av" where
"aval'' e None = ⊥" |
"aval'' e (Some sa) = aval' e sa"

lemma aval''_sound: "s : γ⇩o S ⟹ aval a s : γ(aval'' a S)"
by(cases S)(simp add: aval'_sound)+

subsection "Backward analysis"

fun afilter :: "aexp ⇒ 'av ⇒ 'av st option ⇒ 'av st option" where
"afilter (N n) a S = (if test_num' n a then S else None)" |
"afilter (V x) a S = (case S of None ⇒ None | Some S ⇒
  let a' = lookup S x ⊓ a in
  if a' ⊑ ⊥ then None else Some(update S x a'))" |
"afilter (Plus e1 e2) a S =
 (let (a1,a2) = filter_plus' a (aval'' e1 S) (aval'' e2 S)
  in afilter e1 a1 (afilter e2 a2 S))"

text‹The test for @{const bot} in the @{const V}-case is important: @{const
bot} indicates that a variable has no possible values, i.e.\ that the current
program point is unreachable. But then the abstract state should collapse to
@{const None}. Put differently, we maintain the invariant that in an abstract
state of the form @{term"Some s"}, all variables are mapped to non-@{const
bot} values. Otherwise the (pointwise) join of two abstract states, one of
which contains @{const bot} values, may produce too large a result, thus
making the analysis less precise.›


fun bfilter :: "bexp ⇒ bool ⇒ 'av st option ⇒ 'av st option" where
"bfilter (Bc v) res S = (if v=res then S else None)" |
"bfilter (Not b) res S = bfilter b (¬ res) S" |
"bfilter (And b1 b2) res S =
  (if res then bfilter b1 True (bfilter b2 True S)
   else bfilter b1 False S ⊔ bfilter b2 False S)" |
"bfilter (Less e1 e2) res S =
  (let (res1,res2) = filter_less' res (aval'' e1 S) (aval'' e2 S)
   in afilter e1 res1 (afilter e2 res2 S))"

lemma afilter_sound: "s : γ⇩o S ⟹ aval e s : γ a ⟹ s : γ⇩o (afilter e a S)"
proof(induction e arbitrary: a S)
  case N thus ?case by simp (metis test_num')
next
  case (V x)
  obtain S' where "S = Some S'" and "s : γ⇩f S'" using ‹s : γ⇩o S›
    by(auto simp: in_gamma_option_iff)
  moreover hence "s x : γ (lookup S' x)" by(simp add: γ_st_def)
  moreover have "s x : γ a" using V by simp
  ultimately show ?case using V(1)
    by(simp add: lookup_update Let_def γ_st_def)
      (metis mono_gamma emptyE in_gamma_meet gamma_Bot subset_empty)
next
  case (Plus e1 e2) thus ?case
    using filter_plus'[OF _ aval''_sound[OF Plus(3)] aval''_sound[OF Plus(3)]]
    by (auto split: prod.split)
qed

lemma bfilter_sound: "s : γ⇩o S ⟹ bv = bval b s ⟹ s : γ⇩o(bfilter b bv S)"
proof(induction b arbitrary: S bv)
  case Bc thus ?case by simp
next
  case (Not b) thus ?case by simp
next
  case (And b1 b2) thus ?case
    apply hypsubst_thin
    apply (fastforce simp: in_gamma_join_UpI)
    done
next
  case (Less e1 e2) thus ?case
    apply hypsubst_thin
    apply (auto split: prod.split)
    apply (metis afilter_sound filter_less' aval''_sound Less(1))
    done
qed


fun step' :: "'av st option ⇒ 'av st option acom ⇒ 'av st option acom"
 where
"step' S (SKIP {P}) = (SKIP {S})" |
"step' S (x ::= e {P}) =
  x ::= e {case S of None ⇒ None | Some S ⇒ Some(update S x (aval' e S))}" |
"step' S (c1;; c2) = step' S c1;; step' (post c1) c2" |
"step' S (IF b THEN c1 ELSE c2 {P}) =
  (let c1' = step' (bfilter b True S) c1; c2' = step' (bfilter b False S) c2
   in IF b THEN c1' ELSE c2' {post c1 ⊔ post c2})" |
"step' S ({Inv} WHILE b DO c {P}) =
   {S ⊔ post c}
   WHILE b DO step' (bfilter b True Inv) c
   {bfilter b False Inv}"

definition AI :: "com ⇒ 'av st option acom option" where
"AI = lpfp⇩c (step' ⊤)"

lemma strip_step'[simp]: "strip(step' S c) = strip c"
by(induct c arbitrary: S) (simp_all add: Let_def)


subsection "Soundness"

lemma in_gamma_update:
  "⟦ s : γ⇩f S; i : γ a ⟧ ⟹ s(x := i) : γ⇩f(update S x a)"
by(simp add: γ_st_def lookup_update)

lemma step_preserves_le:
  "⟦ S ⊆ γ⇩o S'; cs ≤ γ⇩c ca ⟧ ⟹ step S cs ≤ γ⇩c (step' S' ca)"
proof(induction cs arbitrary: ca S S')
  case SKIP thus ?case by(auto simp:SKIP_le map_acom_SKIP)
next
  case Assign thus ?case
    by (fastforce simp: Assign_le  map_acom_Assign intro: aval'_sound in_gamma_update
      split: option.splits del:subsetD)
next
  case Seq thus ?case apply (auto simp: Seq_le map_acom_Seq)
    by (metis le_post post_map_acom)
next
  case (If b cs1 cs2 P)
  then obtain ca1 ca2 Pa where
      "ca= IF b THEN ca1 ELSE ca2 {Pa}"
      "P ⊆ γ⇩o Pa" "cs1 ≤ γ⇩c ca1" "cs2 ≤ γ⇩c ca2"
    by (fastforce simp: If_le map_acom_If)
  moreover have "post cs1 ⊆ γ⇩o(post ca1 ⊔ post ca2)"
    by (metis (no_types) ‹cs1 ≤ γ⇩c ca1› join_ge1 le_post mono_gamma_o order_trans post_map_acom)
  moreover have "post cs2 ⊆ γ⇩o(post ca1 ⊔ post ca2)"
    by (metis (no_types) ‹cs2 ≤ γ⇩c ca2› join_ge2 le_post mono_gamma_o order_trans post_map_acom)
  ultimately show ?case using ‹S ⊆ γ⇩o S'›
    by (simp add: If.IH subset_iff bfilter_sound)
next
  case (While I b cs1 P)
  then obtain ca1 Ia Pa where
    "ca = {Ia} WHILE b DO ca1 {Pa}"
    "I ⊆ γ⇩o Ia" "P ⊆ γ⇩o Pa" "cs1 ≤ γ⇩c ca1"
    by (fastforce simp: map_acom_While While_le)
  moreover have "S ∪ post cs1 ⊆ γ⇩o (S' ⊔ post ca1)"
    using ‹S ⊆ γ⇩o S'› le_post[OF ‹cs1 ≤ γ⇩c ca1›, simplified]
    by (metis (no_types) join_ge1 join_ge2 le_sup_iff mono_gamma_o order_trans)
  ultimately show ?case by (simp add: While.IH subset_iff bfilter_sound)
qed

lemma AI_sound: "AI c = Some c' ⟹ CS c ≤ γ⇩c c'"
proof(simp add: CS_def AI_def)
  assume 1: "lpfp⇩c (step' ⊤) c = Some c'"
  have 2: "step' ⊤ c' ⊑ c'" by(rule lpfpc_pfp[OF 1])
  have 3: "strip (γ⇩c (step' ⊤ c')) = c"
    by(simp add: strip_lpfpc[OF _ 1])
  have "lfp (step UNIV) c ≤ γ⇩c (step' ⊤ c')"
  proof(rule lfp_lowerbound[simplified,OF 3])
    show "step UNIV (γ⇩c (step' ⊤ c')) ≤ γ⇩c (step' ⊤ c')"
    proof(rule step_preserves_le[OF _ _])
      show "UNIV ⊆ γ⇩o ⊤" by simp
      show "γ⇩c (step' ⊤ c') ≤ γ⇩c c'" by(rule mono_gamma_c[OF 2])
    qed
  qed
  from this 2 show "lfp (step UNIV) c ≤ γ⇩c c'"
    by (blast intro: mono_gamma_c order_trans)
qed


subsection "Commands over a set of variables"

text‹Key invariant: the domains of all abstract states are subsets of the
set of variables of the program.›

definition "domo S = (case S of None ⇒ {} | Some S' ⇒ set(dom S'))"

definition Com :: "vname set ⇒ 'a st option acom set" where
"Com X = {c. ∀S ∈ set(annos c). domo S ⊆ X}"

lemma domo_Top[simp]: "domo ⊤ = {}"
by(simp add: domo_def Top_st_def Top_option_def)

lemma bot_acom_Com[simp]: "⊥⇩c c ∈ Com X"
by(simp add: bot_acom_def Com_def domo_def)

lemma post_in_annos: "post c : set(annos c)"
by(induction c) simp_all

lemma domo_join: "domo (S ⊔ T) ⊆ domo S ∪ domo T"
by(auto simp: domo_def join_st_def split: option.split)

lemma domo_afilter: "vars a ⊆ X ⟹ domo S ⊆ X ⟹ domo(afilter a i S) ⊆ X"
apply(induction a arbitrary: i S)
apply(simp add: domo_def)
apply(simp add: domo_def Let_def update_def lookup_def split: option.splits)
apply blast
apply(simp split: prod.split)
done

lemma domo_bfilter: "vars b ⊆ X ⟹ domo S ⊆ X ⟹ domo(bfilter b bv S) ⊆ X"
apply(induction b arbitrary: bv S)
apply(simp add: domo_def)
apply(simp)
apply(simp)
apply rule
apply (metis le_sup_iff subset_trans[OF domo_join])
apply(simp split: prod.split)
by (metis domo_afilter)

lemma step'_Com:
  "domo S ⊆ X ⟹ vars(strip c) ⊆ X ⟹ c : Com X ⟹ step' S c : Com X"
apply(induction c arbitrary: S)
apply(simp add: Com_def)
apply(simp add: Com_def domo_def update_def split: option.splits)
apply(simp (no_asm_use) add: Com_def ball_Un)
apply (metis post_in_annos)
apply(simp (no_asm_use) add: Com_def ball_Un)
apply rule
apply (metis Un_assoc domo_join order_trans post_in_annos subset_Un_eq)
apply (metis domo_bfilter)
apply(simp (no_asm_use) add: Com_def)
apply rule
apply (metis domo_join le_sup_iff post_in_annos subset_trans)
apply rule
apply (metis domo_bfilter)
by (metis domo_bfilter)

end


subsection "Monotonicity"

locale Abs_Int1_mono = Abs_Int1 +
assumes mono_plus': "a1 ⊑ b1 ⟹ a2 ⊑ b2 ⟹ plus' a1 a2 ⊑ plus' b1 b2"
and mono_filter_plus': "a1 ⊑ b1 ⟹ a2 ⊑ b2 ⟹ r ⊑ r' ⟹
  filter_plus' r a1 a2 ⊑ filter_plus' r' b1 b2"
and mono_filter_less': "a1 ⊑ b1 ⟹ a2 ⊑ b2 ⟹
  filter_less' bv a1 a2 ⊑ filter_less' bv b1 b2"
begin

lemma mono_aval': "S ⊑ S' ⟹ aval' e S ⊑ aval' e S'"
by(induction e) (auto simp: le_st_def lookup_def mono_plus')

lemma mono_aval'': "S ⊑ S' ⟹ aval'' e S ⊑ aval'' e S'"
apply(cases S)
 apply simp
apply(cases S')
 apply simp
by (simp add: mono_aval')

lemma mono_afilter: "r ⊑ r' ⟹ S ⊑ S' ⟹ afilter e r S ⊑ afilter e r' S'"
apply(induction e arbitrary: r r' S S')
apply(auto simp: test_num' Let_def split: option.splits prod.splits)
apply (metis mono_gamma subsetD)
apply(rename_tac list a b c d, drule_tac x = "list" in mono_lookup)
apply (metis mono_meet le_trans)
apply (metis mono_meet mono_lookup mono_update)
apply(metis mono_aval'' mono_filter_plus'[simplified le_prod_def] fst_conv snd_conv)
done

lemma mono_bfilter: "S ⊑ S' ⟹ bfilter b r S ⊑ bfilter b r S'"
apply(induction b arbitrary: r S S')
apply(auto simp: le_trans[OF _ join_ge1] le_trans[OF _ join_ge2] split: prod.splits)
apply(metis mono_aval'' mono_afilter mono_filter_less'[simplified le_prod_def] fst_conv snd_conv)
done

lemma mono_step': "S ⊑ S' ⟹ c ⊑ c' ⟹ step' S c ⊑ step' S' c'"
apply(induction c c' arbitrary: S S' rule: le_acom.induct)
apply (auto simp: mono_post mono_bfilter mono_update mono_aval' Let_def le_join_disj
  split: option.split)
done

lemma mono_step'2: "mono (step' S)"
by(simp add: mono_def mono_step'[OF le_refl])

end

end