# Theory Abs_Int2

theory Abs_Int2
imports Abs_Int1
```(* Author: Tobias Nipkow *)

theory Abs_Int2
imports Abs_Int1
begin

instantiation prod :: (order,order) order
begin

definition "less_eq_prod p1 p2 = (fst p1 ≤ fst p2 ∧ snd p1 ≤ snd p2)"
definition "less_prod p1 p2 = (p1 ≤ p2 ∧ ¬ p2 ≤ (p1::'a*'b))"

instance
proof (standard, goal_cases)
case 1 show ?case by(rule less_prod_def)
next
case 2 show ?case by(simp add: less_eq_prod_def)
next
case 3 thus ?case unfolding less_eq_prod_def by(metis order_trans)
next
case 4 thus ?case by(simp add: less_eq_prod_def)(metis eq_iff surjective_pairing)
qed

end

subsection "Backward Analysis of Expressions"

subclass (in bounded_lattice) semilattice_sup_top ..

locale Val_lattice_gamma = Gamma_semilattice where γ = γ
for γ :: "'av::bounded_lattice ⇒ val set" +
assumes inter_gamma_subset_gamma_inf:
"γ a1 ∩ γ a2 ⊆ γ(a1 ⊓ a2)"
and gamma_bot[simp]: "γ ⊥ = {}"
begin

lemma in_gamma_inf: "x ∈ γ a1 ⟹ x ∈ γ a2 ⟹ x ∈ γ(a1 ⊓ a2)"
by (metis IntI inter_gamma_subset_gamma_inf set_mp)

lemma gamma_inf: "γ(a1 ⊓ a2) = γ a1 ∩ γ a2"
by(rule equalityI[OF _ inter_gamma_subset_gamma_inf])
(metis inf_le1 inf_le2 le_inf_iff mono_gamma)

end

locale Val_inv = Val_lattice_gamma where γ = γ
for γ :: "'av::bounded_lattice ⇒ val set" +
fixes test_num' :: "val ⇒ 'av ⇒ bool"
and inv_plus' :: "'av ⇒ 'av ⇒ 'av ⇒ 'av * 'av"
and inv_less' :: "bool ⇒ 'av ⇒ 'av ⇒ 'av * 'av"
assumes test_num': "test_num' i a = (i ∈ γ a)"
and inv_plus': "inv_plus' a a1 a2 = (a⇩1',a⇩2') ⟹
i1 ∈ γ a1 ⟹ i2 ∈ γ a2 ⟹ i1+i2 ∈ γ a ⟹ i1 ∈ γ a⇩1' ∧ i2 ∈ γ a⇩2'"
and inv_less': "inv_less' (i1<i2) a1 a2 = (a⇩1',a⇩2') ⟹
i1 ∈ γ a1 ⟹ i2 ∈ γ a2 ⟹ i1 ∈ γ a⇩1' ∧ i2 ∈ γ a⇩2'"

locale Abs_Int_inv = Val_inv where γ = γ
for γ :: "'av::bounded_lattice ⇒ val set"
begin

lemma in_gamma_sup_UpI:
"s ∈ γ⇩o S1 ∨ s ∈ γ⇩o S2 ⟹ s ∈ γ⇩o(S1 ⊔ S2)"
by (metis (hide_lams, no_types) sup_ge1 sup_ge2 mono_gamma_o subsetD)

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

lemma aval''_correct: "s ∈ γ⇩o S ⟹ aval a s ∈ γ(aval'' a S)"
by(cases S)(auto simp add: aval'_correct split: option.splits)

subsubsection "Backward analysis"

fun inv_aval' :: "aexp ⇒ 'av ⇒ 'av st option ⇒ 'av st option" where
"inv_aval' (N n) a S = (if test_num' n a then S else None)" |
"inv_aval' (V x) a S = (case S of None ⇒ None | Some S ⇒
let a' = fun S x ⊓ a in
if a' = ⊥ then None else Some(update S x a'))" |
"inv_aval' (Plus e1 e2) a S =
(let (a1,a2) = inv_plus' a (aval'' e1 S) (aval'' e2 S)
in inv_aval' e1 a1 (inv_aval' 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) sup of two abstract states, one of
which contains @{const bot} values, may produce too large a result, thus
making the analysis less precise.›

fun inv_bval' :: "bexp ⇒ bool ⇒ 'av st option ⇒ 'av st option" where
"inv_bval' (Bc v) res S = (if v=res then S else None)" |
"inv_bval' (Not b) res S = inv_bval' b (¬ res) S" |
"inv_bval' (And b1 b2) res S =
(if res then inv_bval' b1 True (inv_bval' b2 True S)
else inv_bval' b1 False S ⊔ inv_bval' b2 False S)" |
"inv_bval' (Less e1 e2) res S =
(let (a1,a2) = inv_less' res (aval'' e1 S) (aval'' e2 S)
in inv_aval' e1 a1 (inv_aval' e2 a2 S))"

lemma inv_aval'_correct: "s ∈ γ⇩o S ⟹ aval e s ∈ γ a ⟹ s ∈ γ⇩o (inv_aval' 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⇩s S'" using ‹s ∈ γ⇩o S›
by(auto simp: in_gamma_option_iff)
moreover hence "s x ∈ γ (fun S' x)"
moreover have "s x ∈ γ a" using V(2) by simp
ultimately show ?case
(metis mono_gamma emptyE in_gamma_inf gamma_bot subset_empty)
next
case (Plus e1 e2) thus ?case
using inv_plus'[OF _ aval''_correct aval''_correct]
by (auto split: prod.split)
qed

lemma inv_bval'_correct: "s ∈ γ⇩o S ⟹ bv = bval b s ⟹ s ∈ γ⇩o(inv_bval' 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
by simp (metis And(1) And(2) in_gamma_sup_UpI)
next
case (Less e1 e2) thus ?case
apply hypsubst_thin
apply (auto split: prod.split)
apply (metis (lifting) inv_aval'_correct aval''_correct inv_less')
done
qed

definition "step' = Step
(λx e S. case S of None ⇒ None | Some S ⇒ Some(update S x (aval' e S)))
(λb S. inv_bval' b True S)"

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

lemma strip_step'[simp]: "strip(step' S c) = strip c"

lemma top_on_inv_aval': "⟦ top_on_opt S X;  vars e ⊆ -X ⟧ ⟹ top_on_opt (inv_aval' e a S) X"
by(induction e arbitrary: a S) (auto simp: Let_def split: option.splits prod.split)

lemma top_on_inv_bval': "⟦top_on_opt S X; vars b ⊆ -X⟧ ⟹ top_on_opt (inv_bval' b r S) X"
by(induction b arbitrary: r S) (auto simp: top_on_inv_aval' top_on_sup split: prod.split)

lemma top_on_step': "top_on_acom C (- vars C) ⟹ top_on_acom (step' ⊤ C) (- vars C)"
unfolding step'_def
by(rule top_on_Step)
(auto simp add: top_on_top top_on_inv_bval' split: option.split)

subsubsection "Correctness"

lemma step_step': "step (γ⇩o S) (γ⇩c C) ≤ γ⇩c (step' S C)"
unfolding step_def step'_def
by(rule gamma_Step_subcomm)
(auto simp: intro!: aval'_correct inv_bval'_correct in_gamma_update split: option.splits)

lemma AI_correct: "AI c = Some C ⟹ CS c ≤ γ⇩c C"
assume 1: "pfp (step' ⊤) (bot c) = Some C"
have pfp': "step' ⊤ C ≤ C" by(rule pfp_pfp[OF 1])
have 2: "step (γ⇩o ⊤) (γ⇩c C) ≤ γ⇩c C"  ― ‹transfer the pfp'›
proof(rule order_trans)
show "step (γ⇩o ⊤) (γ⇩c C) ≤ γ⇩c (step' ⊤ C)" by(rule step_step')
show "... ≤ γ⇩c C" by (metis mono_gamma_c[OF pfp'])
qed
have 3: "strip (γ⇩c C) = c" by(simp add: strip_pfp[OF _ 1] step'_def)
have "lfp c (step (γ⇩o ⊤)) ≤ γ⇩c C"
by(rule lfp_lowerbound[simplified,where f="step (γ⇩o ⊤)", OF 3 2])
thus "lfp c (step UNIV) ≤ γ⇩c C" by simp
qed

end

subsubsection "Monotonicity"

locale Abs_Int_inv_mono = Abs_Int_inv +
assumes mono_plus': "a1 ≤ b1 ⟹ a2 ≤ b2 ⟹ plus' a1 a2 ≤ plus' b1 b2"
and mono_inv_plus': "a1 ≤ b1 ⟹ a2 ≤ b2 ⟹ r ≤ r' ⟹
inv_plus' r a1 a2 ≤ inv_plus' r' b1 b2"
and mono_inv_less': "a1 ≤ b1 ⟹ a2 ≤ b2 ⟹
inv_less' bv a1 a2 ≤ inv_less' bv b1 b2"
begin

lemma mono_aval':
"S1 ≤ S2 ⟹ aval' e S1 ≤ aval' e S2"
by(induction e) (auto simp: mono_plus' mono_fun)

lemma mono_aval'':
"S1 ≤ S2 ⟹ aval'' e S1 ≤ aval'' e S2"
apply(cases S1)
apply simp
apply(cases S2)
apply simp

lemma mono_inv_aval': "r1 ≤ r2 ⟹ S1 ≤ S2 ⟹ inv_aval' e r1 S1 ≤ inv_aval' e r2 S2"
apply(induction e arbitrary: r1 r2 S1 S2)
apply(auto simp: test_num' Let_def inf_mono split: option.splits prod.splits)
apply (metis mono_gamma subsetD)
apply (metis le_bot inf_mono le_st_iff)
apply (metis inf_mono mono_update le_st_iff)
apply(metis mono_aval'' mono_inv_plus'[simplified less_eq_prod_def] fst_conv snd_conv)
done

lemma mono_inv_bval': "S1 ≤ S2 ⟹ inv_bval' b bv S1 ≤ inv_bval' b bv S2"
apply(induction b arbitrary: bv S1 S2)
apply(simp)
apply(simp)
apply simp
apply(metis order_trans[OF _ sup_ge1] order_trans[OF _ sup_ge2])
apply (simp split: prod.splits)
apply(metis mono_aval'' mono_inv_aval' mono_inv_less'[simplified less_eq_prod_def] fst_conv snd_conv)
done

theorem mono_step': "S1 ≤ S2 ⟹ C1 ≤ C2 ⟹ step' S1 C1 ≤ step' S2 C2"
unfolding step'_def
by(rule mono2_Step) (auto simp: mono_aval' mono_inv_bval' split: option.split)

lemma mono_step'_top: "C1 ≤ C2 ⟹ step' ⊤ C1 ≤ step' ⊤ C2"
by (metis mono_step' order_refl)

end

end
```