# Theory Abs_Int1

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

theory Abs_Int1
imports Abs_State
begin

subsection "Computable Abstract Interpretation"

text{* Abstract interpretation over type @{text st} instead of functions. *}

context Gamma_semilattice
begin

fun aval' :: "aexp ⇒ 'av st ⇒ 'av" where
"aval' (N i) S = num' i" |
"aval' (V x) S = fun S x" |
"aval' (Plus a1 a2) S = plus' (aval' a1 S) (aval' a2 S)"

lemma aval'_correct: "s : γ⇩s S ⟹ aval a s : γ(aval' a S)"
by (induction a) (auto simp: gamma_num' gamma_plus' γ_st_def)

lemma gamma_Step_subcomm: fixes C1 C2 :: "'a::semilattice_sup acom"
assumes "!!x e S. f1 x e (γ⇩o S) ⊆ γ⇩o (f2 x e S)"
"!!b S. g1 b (γ⇩o S) ⊆ γ⇩o (g2 b S)"
shows "Step f1 g1 (γ⇩o S) (γ⇩c C) ≤ γ⇩c (Step f2 g2 S C)"
proof(induction C arbitrary: S)
qed (auto simp: assms intro!: mono_gamma_o sup_ge1 sup_ge2)

lemma in_gamma_update: "⟦ s : γ⇩s S; i : γ a ⟧ ⟹ s(x := i) : γ⇩s(update S x a)"

end

locale Abs_Int = Gamma_semilattice where γ=γ
for γ :: "'av::semilattice_sup_top ⇒ val set"
begin

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

text{* 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 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_mono = Abs_Int +
assumes mono_plus': "a1 ≤ b1 ⟹ a2 ≤ b2 ⟹ plus' a1 a2 ≤ plus' b1 b2"
begin

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

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

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

lemma AI_least_pfp: assumes "AI c = Some C" "step' ⊤ C' ≤ C'" "strip C' = c"
shows "C ≤ C'"
by(rule pfp_bot_least[OF _ _ assms(2,3) assms(1)[unfolded AI_def]])

end

subsubsection "Termination"

locale Measure1 =
fixes m :: "'av::order_top ⇒ nat"
fixes h :: "nat"
assumes h: "m x ≤ h"
begin

definition m_s :: "'av st ⇒ vname set ⇒ nat" ("m⇩s") where
"m_s S X = (∑ x ∈ X. m(fun S x))"

lemma m_s_h: "finite X ⟹ m_s S X ≤ h * card X"
by(simp add: m_s_def) (metis mult.commute of_nat_id sum_bounded_above[OF h])

definition m_o :: "'av st option ⇒ vname set ⇒ nat" ("m⇩o") where
"m_o opt X = (case opt of None ⇒ h * card X + 1 | Some S ⇒ m_s S X)"

lemma m_o_h: "finite X ⟹ m_o opt X ≤ (h*card X + 1)"
by(auto simp add: m_o_def m_s_h le_SucI split: option.split dest:m_s_h)

definition m_c :: "'av st option acom ⇒ nat" ("m⇩c") where
"m_c C = sum_list (map (λa. m_o a (vars C)) (annos C))"

text{* Upper complexity bound: *}
lemma m_c_h: "m_c C ≤ size(annos C) * (h * card(vars C) + 1)"
proof-
let ?X = "vars C" let ?n = "card ?X" let ?a = "size(annos C)"
have "m_c C = (∑i<?a. m_o (annos C ! i) ?X)"
also have "… ≤ (∑i<?a. h * ?n + 1)"
apply(rule sum_mono) using m_o_h[OF finite_Cvars] by simp
also have "… = ?a * (h * ?n + 1)" by simp
finally show ?thesis .
qed

end

fun top_on_st :: "'a::order_top st ⇒ vname set ⇒ bool" ("top'_on⇩s") where
"top_on_st S X = (∀x∈X. fun S x = ⊤)"

fun top_on_opt :: "'a::order_top st option ⇒ vname set ⇒ bool" ("top'_on⇩o") where
"top_on_opt (Some S)  X = top_on_st S X" |
"top_on_opt None X = True"

definition top_on_acom :: "'a::order_top st option acom ⇒ vname set ⇒ bool" ("top'_on⇩c") where
"top_on_acom C X = (∀a ∈ set(annos C). top_on_opt a X)"

lemma top_on_top: "top_on_opt (⊤::_ st option) X"
by(auto simp: top_option_def fun_top)

lemma top_on_bot: "top_on_acom (bot c) X"

lemma top_on_post: "top_on_acom C X ⟹ top_on_opt (post C) X"

lemma top_on_acom_simps:
"top_on_acom (SKIP {Q}) X = top_on_opt Q X"
"top_on_acom (x ::= e {Q}) X = top_on_opt Q X"
"top_on_acom (C1;;C2) X = (top_on_acom C1 X ∧ top_on_acom C2 X)"
"top_on_acom (IF b THEN {P1} C1 ELSE {P2} C2 {Q}) X =
(top_on_opt P1 X ∧ top_on_acom C1 X ∧ top_on_opt P2 X ∧ top_on_acom C2 X ∧ top_on_opt Q X)"
"top_on_acom ({I} WHILE b DO {P} C {Q}) X =
(top_on_opt I X ∧ top_on_acom C X ∧ top_on_opt P X ∧ top_on_opt Q X)"

lemma top_on_sup:
"top_on_opt o1 X ⟹ top_on_opt o2 X ⟹ top_on_opt (o1 ⊔ o2 :: _ st option) X"
apply(induction o1 o2 rule: sup_option.induct)
apply(auto)
by transfer simp

lemma top_on_Step: fixes C :: "('a::semilattice_sup_top)st option acom"
assumes "!!x e S. ⟦top_on_opt S X; x ∉ X; vars e ⊆ -X⟧ ⟹ top_on_opt (f x e S) X"
"!!b S. top_on_opt S X ⟹ vars b ⊆ -X ⟹ top_on_opt (g b S) X"
shows "⟦ vars C ⊆ -X; top_on_opt S X; top_on_acom C X ⟧ ⟹ top_on_acom (Step f g S C) X"
proof(induction C arbitrary: S)
qed (auto simp: top_on_acom_simps vars_acom_def top_on_post top_on_sup assms)

locale Measure = Measure1 +
assumes m2: "x < y ⟹ m x > m y"
begin

lemma m1: "x ≤ y ⟹ m x ≥ m y"
by(auto simp: le_less m2)

lemma m_s2_rep: assumes "finite(X)" and "S1 = S2 on -X" and "∀x. S1 x ≤ S2 x" and "S1 ≠ S2"
shows "(∑x∈X. m (S2 x)) < (∑x∈X. m (S1 x))"
proof-
from assms(3) have 1: "∀x∈X. m(S1 x) ≥ m(S2 x)" by (simp add: m1)
from assms(2,3,4) have "EX x:X. S1 x < S2 x"
by(simp add: fun_eq_iff) (metis Compl_iff le_neq_trans)
hence 2: "∃x∈X. m(S1 x) > m(S2 x)" by (metis m2)
from sum_strict_mono_ex1[OF `finite X` 1 2]
show "(∑x∈X. m (S2 x)) < (∑x∈X. m (S1 x))" .
qed

lemma m_s2: "finite(X) ⟹ fun S1 = fun S2 on -X
⟹ S1 < S2 ⟹ m_s S1 X > m_s S2 X"
apply (transfer fixing: m)
done

lemma m_o2: "finite X ⟹ top_on_opt o1 (-X) ⟹ top_on_opt o2 (-X) ⟹
o1 < o2 ⟹ m_o o1 X > m_o o2 X"
proof(induction o1 o2 rule: less_eq_option.induct)
case 1 thus ?case by (auto simp: m_o_def m_s2 less_option_def)
next
case 2 thus ?case by(auto simp: m_o_def less_option_def le_imp_less_Suc m_s_h)
next
case 3 thus ?case by (auto simp: less_option_def)
qed

lemma m_o1: "finite X ⟹ top_on_opt o1 (-X) ⟹ top_on_opt o2 (-X) ⟹
o1 ≤ o2 ⟹ m_o o1 X ≥ m_o o2 X"
by(auto simp: le_less m_o2)

lemma m_c2: "top_on_acom C1 (-vars C1) ⟹ top_on_acom C2 (-vars C2) ⟹
C1 < C2 ⟹ m_c C1 > m_c C2"
proof(auto simp add: le_iff_le_annos size_annos_same[of C1 C2] vars_acom_def less_acom_def)
let ?X = "vars(strip C2)"
assume top: "top_on_acom C1 (- vars(strip C2))"  "top_on_acom C2 (- vars(strip C2))"
and strip_eq: "strip C1 = strip C2"
and 0: "∀i<size(annos C2). annos C1 ! i ≤ annos C2 ! i"
hence 1: "∀i<size(annos C2). m_o (annos C1 ! i) ?X ≥ m_o (annos C2 ! i) ?X"
apply (auto simp: all_set_conv_all_nth vars_acom_def top_on_acom_def)
by (metis finite_cvars m_o1 size_annos_same2)
fix i assume i: "i < size(annos C2)" "¬ annos C2 ! i ≤ annos C1 ! i"
have topo1: "top_on_opt (annos C1 ! i) (- ?X)"
using i(1) top(1) by(simp add: top_on_acom_def size_annos_same[OF strip_eq])
have topo2: "top_on_opt (annos C2 ! i) (- ?X)"
using i(1) top(2) by(simp add: top_on_acom_def size_annos_same[OF strip_eq])
from i have "m_o (annos C1 ! i) ?X > m_o (annos C2 ! i) ?X" (is "?P i")
by (metis 0 less_option_def m_o2[OF finite_cvars topo1] topo2)
hence 2: "∃i < size(annos C2). ?P i" using `i < size(annos C2)` by blast
have "(∑i<size(annos C2). m_o (annos C2 ! i) ?X)
< (∑i<size(annos C2). m_o (annos C1 ! i) ?X)"
apply(rule sum_strict_mono_ex1) using 1 2 by (auto)
thus ?thesis
by(simp add: m_c_def vars_acom_def strip_eq sum_list_sum_nth atLeast0LessThan size_annos_same[OF strip_eq])
qed

end

locale Abs_Int_measure =
Abs_Int_mono where γ=γ + Measure where m=m
for γ :: "'av::semilattice_sup_top ⇒ val set" and m :: "'av ⇒ nat"
begin

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_option_def fun_top split: option.splits)

lemma AI_Some_measure: "∃C. AI c = Some C"
unfolding AI_def
apply(rule pfp_termination[where I = "λC. top_on_acom C (- vars C)" and m="m_c"])
apply(simp_all add: m_c2 mono_step'_top bot_least top_on_bot)
using top_on_step' apply(auto simp add: vars_acom_def)
done

end

end
```