Theory Abs_Int0_ITP

theory Abs_Int0_ITP
imports While_Combinator Collecting_ITP
(* Author: Tobias Nipkow *)

theory Abs_Int0_ITP
imports "~~/src/Tools/Permanent_Interpretation"

subsection "Orderings"

class preord =
fixes le :: "'a => 'a => bool" (infix "\<sqsubseteq>" 50)
assumes le_refl[simp]: "x \<sqsubseteq> x"
and le_trans: "x \<sqsubseteq> y ==> y \<sqsubseteq> z ==> x \<sqsubseteq> z"

definition mono where "mono f = (∀x y. x \<sqsubseteq> y --> f x \<sqsubseteq> f y)"

lemma monoD: "mono f ==> x \<sqsubseteq> y ==> f x \<sqsubseteq> f y" by(simp add: mono_def)

lemma mono_comp: "mono f ==> mono g ==> mono (g o f)"
by(simp add: mono_def)

declare le_trans[trans]


text{* Note: no antisymmetry. Allows implementations where some abstract
element is implemented by two different values @{prop "x ≠ 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 => 'a => 'a" (infixl "\<squnion>" 65)
fixes Top :: "'a" ("\<top>")
assumes join_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
and join_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
and join_least: "x \<sqsubseteq> z ==> y \<sqsubseteq> z ==> x \<squnion> y \<sqsubseteq> z"
and top[simp]: "x \<sqsubseteq> \<top>"

lemma join_le_iff[simp]: "x \<squnion> y \<sqsubseteq> z <-> x \<sqsubseteq> z ∧ y \<sqsubseteq> z"
by (metis join_ge1 join_ge2 join_least le_trans)

lemma le_join_disj: "x \<sqsubseteq> y ∨ x \<sqsubseteq> z ==> x \<sqsubseteq> y \<squnion> z"
by (metis join_ge1 join_ge2 le_trans)


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

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

lemma join_apply[simp]: "(f \<squnion> g) x = f x \<squnion> g x"
by (simp add: join_fun_def)

  case goal2 thus ?case by (metis le_fun_def preord_class.le_trans)
qed (simp_all add: le_fun_def Top_fun_def)


instantiation acom :: (preord) preord

fun le_acom :: "('a::preord)acom => 'a acom => bool" where
"le_acom (SKIP {S}) (SKIP {S'}) = (S \<sqsubseteq> S')" |
"le_acom (x ::= e {S}) (x' ::= e' {S'}) = (x=x' ∧ e=e' ∧ S \<sqsubseteq> S')" |
"le_acom (c1;;c2) (c1';;c2') = (le_acom c1 c1' ∧ le_acom c2 c2')" |
"le_acom (IF b THEN c1 ELSE c2 {S}) (IF b' THEN c1' ELSE c2' {S'}) =
  (b=b' ∧ le_acom c1 c1' ∧ le_acom c2 c2' ∧ S \<sqsubseteq> S')" |
"le_acom ({Inv} WHILE b DO c {P}) ({Inv'} WHILE b' DO c' {P'}) =
  (b=b' ∧ le_acom c c' ∧ Inv \<sqsubseteq> Inv' ∧ P \<sqsubseteq> P')" |
"le_acom _ _ = False"

lemma [simp]: "SKIP {S} \<sqsubseteq> c <-> (∃S'. c = SKIP {S'} ∧ S \<sqsubseteq> S')"
by (cases c) auto

lemma [simp]: "x ::= e {S} \<sqsubseteq> c <-> (∃S'. c = x ::= e {S'} ∧ S \<sqsubseteq> S')"
by (cases c) auto

lemma [simp]: "c1;;c2 \<sqsubseteq> c <-> (∃c1' c2'. c = c1';;c2' ∧ c1 \<sqsubseteq> c1' ∧ c2 \<sqsubseteq> c2')"
by (cases c) auto

lemma [simp]: "IF b THEN c1 ELSE c2 {S} \<sqsubseteq> c <->
  (∃c1' c2' S'. c = IF b THEN c1' ELSE c2' {S'} ∧ c1 \<sqsubseteq> c1' ∧ c2 \<sqsubseteq> c2' ∧ S \<sqsubseteq> S')"
by (cases c) auto

lemma [simp]: "{Inv} WHILE b DO c {P} \<sqsubseteq> w <->
  (∃Inv' c' P'. w = {Inv'} WHILE b DO c' {P'} ∧ c \<sqsubseteq> c' ∧ Inv \<sqsubseteq> Inv' ∧ P \<sqsubseteq> P')"
by (cases w) auto

  case goal1 thus ?case by (induct x) auto
  case goal2 thus ?case
  apply(induct x y arbitrary: z rule: le_acom.induct)
  apply (auto intro: le_trans)


subsubsection "Lifting"

instantiation option :: (preord)preord

fun le_option where
"Some x \<sqsubseteq> Some y = (x \<sqsubseteq> y)" |
"None \<sqsubseteq> y = True" |
"Some _ \<sqsubseteq> None = False"

lemma [simp]: "(x \<sqsubseteq> None) = (x = None)"
by (cases x) simp_all

lemma [simp]: "(Some x \<sqsubseteq> u) = (∃y. u = Some y & x \<sqsubseteq> y)"
by (cases u) auto

instance proof
  case goal1 show ?case by(cases x, simp_all)
  case goal2 thus ?case
    by(cases z, simp, cases y, simp, cases x, auto intro: le_trans)


instantiation option :: (SL_top)SL_top

fun join_option where
"Some x \<squnion> Some y = Some(x \<squnion> y)" |
"None \<squnion> y = y" |
"x \<squnion> None = x"

lemma join_None2[simp]: "x \<squnion> None = x"
by (cases x) simp_all

definition "\<top> = Some \<top>"

instance proof
  case goal1 thus ?case by(cases x, simp, cases y, simp_all)
  case goal2 thus ?case by(cases y, simp, cases x, simp_all)
  case goal3 thus ?case by(cases z, simp, cases y, simp, cases x, simp_all)
  case goal4 thus ?case by(cases x, simp_all add: Top_option_def)


definition bot_acom :: "com => ('a::SL_top)option acom" ("⊥c") where
"⊥c = anno None"

lemma strip_bot_acom[simp]: "strip(⊥c c) = c"
by(simp add: bot_acom_def)

lemma bot_acom[rule_format]: "strip c' = c --> ⊥c c \<sqsubseteq> c'"
apply(induct c arbitrary: c')
apply (simp_all add: bot_acom_def)
 apply(induct_tac c')
  apply simp_all
 apply(induct_tac c')
  apply simp_all
 apply(induct_tac c')
  apply simp_all
 apply(induct_tac c')
  apply simp_all
 apply(induct_tac c')
  apply simp_all

subsubsection "Post-fixed point iteration"

  pfp :: "(('a::preord) => 'a) => 'a => 'a option" where
"pfp f = while_option (λx. ¬ f x \<sqsubseteq> x) f"

lemma pfp_pfp: assumes "pfp f x0 = Some x" shows "f x \<sqsubseteq> x"
using while_option_stop[OF assms[simplified pfp_def]] by simp

lemma pfp_least:
assumes mono: "!!x y. x \<sqsubseteq> y ==> f x \<sqsubseteq> f y"
and "f p \<sqsubseteq> p" and "x0 \<sqsubseteq> p" and "pfp f x0 = Some x" shows "x \<sqsubseteq> p"
  { fix x assume "x \<sqsubseteq> p"
    hence  "f x \<sqsubseteq> f p" by(rule mono)
    from this `f p \<sqsubseteq> p` have "f x \<sqsubseteq> p" by(rule le_trans)
  thus "x \<sqsubseteq> p" using assms(2-) while_option_rule[where P = "%x. x \<sqsubseteq> p"]
    unfolding pfp_def by blast

 lpfpc :: "(('a::SL_top)option acom => 'a option acom) => com => 'a option acom option" where
"lpfpc f c = pfp f (⊥c c)"

lemma lpfpc_pfp: "lpfpc f c0 = Some c ==> f c \<sqsubseteq> c"
by(simp add: pfp_pfp lpfpc_def)

lemma strip_pfp:
assumes "!!x. g(f x) = g x" and "pfp f x0 = Some x" shows "g x = g x0"
using assms while_option_rule[where P = "%x. g x = g x0" and c = f]
unfolding pfp_def by metis

lemma strip_lpfpc: assumes "!!c. strip(f c) = strip c" and "lpfpc f c = Some c'"
shows "strip c' = c"
using assms(1) strip_pfp[OF _ assms(2)[simplified lpfpc_def]]
by(metis strip_bot_acom)

lemma lpfpc_least:
assumes mono: "!!x y. x \<sqsubseteq> y ==> f x \<sqsubseteq> f y"
and "strip p = c0" and "f p \<sqsubseteq> p" and lp: "lpfpc f c0 = Some c" shows "c \<sqsubseteq> p"
using pfp_least[OF _ _ bot_acom[OF `strip p = c0`] lp[simplified lpfpc_def]]
  mono `f p \<sqsubseteq> p`
by blast

subsection "Abstract Interpretation"

definition γ_fun :: "('a => 'b set) => ('c => 'a) => ('c => 'b)set" where
"γ_fun γ F = {f. ∀x. f x ∈ γ(F x)}"

fun γ_option :: "('a => 'b set) => 'a option => 'b set" where
"γ_option γ None = {}" |
"γ_option γ (Some a) = γ a"

text{* The interface for abstract values: *}

locale Val_abs =
fixes γ :: "'av::SL_top => val set"
  assumes mono_gamma: "a \<sqsubseteq> b ==> γ a ⊆ γ b"
  and gamma_Top[simp]: "γ \<top> = UNIV"
fixes num' :: "val => 'av"
and plus' :: "'av => 'av => 'av"
  assumes gamma_num': "n : γ(num' n)"
  and gamma_plus':
 "n1 : γ a1 ==> n2 : γ a2 ==> n1+n2 : γ(plus' a1 a2)"

type_synonym 'av st = "(vname => 'av)"

locale Abs_Int_Fun = Val_abs γ for γ :: "'av::SL_top => val set"

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

fun step' :: "'av st option => 'av st option acom => 'av st option acom"
"step' S (SKIP {P}) = (SKIP {S})" |
"step' S (x ::= e {P}) =
  x ::= e {case S of None => None | Some S => Some(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}) =
   IF b THEN step' S c1 ELSE step' S c2 {post c1 \<squnion> post c2}" |
"step' S ({Inv} WHILE b DO c {P}) =
  {S \<squnion> post c} WHILE b DO (step' Inv c) {Inv}"

definition AI :: "com => 'av st option acom option" where
"AI = lpfpc (step' \<top>)"

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

abbreviation γf :: "'av st => state set"
where f == γ_fun γ"

abbreviation γo :: "'av st option => state set"
where o == γ_option γf"

abbreviation γc :: "'av st option acom => state set acom"
where c == map_acom γo"

lemma gamma_f_Top[simp]: f Top = UNIV"
by(simp add: Top_fun_def γ_fun_def)

lemma gamma_o_Top[simp]: o Top = UNIV"
by (simp add: Top_option_def)

(* FIXME (maybe also le -> sqle?) *)

lemma mono_gamma_f: "f \<sqsubseteq> g ==> γf f ⊆ γf g"
by(auto simp: le_fun_def γ_fun_def dest: mono_gamma)

lemma mono_gamma_o:
  "sa \<sqsubseteq> sa' ==> γo sa ⊆ γo sa'"
by(induction sa sa' rule: le_option.induct)(simp_all add: mono_gamma_f)

lemma mono_gamma_c: "ca \<sqsubseteq> ca' ==> γc ca ≤ γc ca'"
by (induction ca ca' rule: le_acom.induct) (simp_all add:mono_gamma_o)

text{* Soundness: *}

lemma aval'_sound: "s : γf S ==> aval a s : γ(aval' a S)"
by (induct a) (auto simp: gamma_num' gamma_plus' γ_fun_def)

lemma in_gamma_update:
  "[| s : γf S; i : γ a |] ==> s(x := i) : γf(S(x := a))"
by(simp add: γ_fun_def)

lemma step_preserves_le:
  "[| S ⊆ γo S'; c ≤ γc c' |] ==> step S c ≤ γc (step' S' c')"
proof(induction c arbitrary: c' S S')
  case SKIP thus ?case by(auto simp:SKIP_le map_acom_SKIP)
  case Assign thus ?case
    by (fastforce simp: Assign_le  map_acom_Assign intro: aval'_sound in_gamma_update
      split: option.splits del:subsetD)
  case Seq thus ?case apply (auto simp: Seq_le map_acom_Seq)
    by (metis le_post post_map_acom)
  case (If b c1 c2 P)
  then obtain c1' c2' P' where
      "c' = IF b THEN c1' ELSE c2' {P'}"
      "P ⊆ γo P'" "c1 ≤ γc c1'" "c2 ≤ γc c2'"
    by (fastforce simp: If_le map_acom_If)
  moreover have "post c1 ⊆ γo(post c1' \<squnion> post c2')"
    by (metis (no_types) `c1 ≤ γc c1'` join_ge1 le_post mono_gamma_o order_trans post_map_acom)
  moreover have "post c2 ⊆ γo(post c1' \<squnion> post c2')"
    by (metis (no_types) `c2 ≤ γc c2'` join_ge2 le_post mono_gamma_o order_trans post_map_acom)
  ultimately show ?case using `S ⊆ γo S'` by (simp add: If.IH subset_iff)
  case (While I b c1 P)
  then obtain c1' I' P' where
    "c' = {I'} WHILE b DO c1' {P'}"
    "I ⊆ γo I'" "P ⊆ γo P'" "c1 ≤ γc c1'"
    by (fastforce simp: map_acom_While While_le)
  moreover have "S ∪ post c1 ⊆ γo (S' \<squnion> post c1')"
    using `S ⊆ γo S'` le_post[OF `c1 ≤ γc c1'`, 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)

lemma AI_sound: "AI c = Some c' ==> CS c ≤ γc c'"
proof(simp add: CS_def AI_def)
  assume 1: "lpfpc (step' \<top>) c = Some c'"
  have 2: "step' \<top> c' \<sqsubseteq> c'" by(rule lpfpc_pfp[OF 1])
  have 3: "strip (γc (step' \<top> c')) = c"
    by(simp add: strip_lpfpc[OF _ 1])
  have "lfp (step UNIV) c ≤ γc (step' \<top> c')"
  proof(rule lfp_lowerbound[simplified,OF 3])
    show "step UNIV (γc (step' \<top> c')) ≤ γc (step' \<top> c')"
    proof(rule step_preserves_le[OF _ _])
      show "UNIV ⊆ γo \<top>" by simp
      show c (step' \<top> c') ≤ γc c'" by(rule mono_gamma_c[OF 2])
  with 2 show "lfp (step UNIV) c ≤ γc c'"
    by (blast intro: mono_gamma_c order_trans)


subsubsection "Monotonicity"

lemma mono_post: "c \<sqsubseteq> c' ==> post c \<sqsubseteq> post c'"
by(induction c c' rule: le_acom.induct) (auto)

locale Abs_Int_Fun_mono = Abs_Int_Fun +
assumes mono_plus': "a1 \<sqsubseteq> b1 ==> a2 \<sqsubseteq> b2 ==> plus' a1 a2 \<sqsubseteq> plus' b1 b2"

lemma mono_aval': "S \<sqsubseteq> S' ==> aval' e S \<sqsubseteq> aval' e S'"
by(induction e)(auto simp: le_fun_def mono_plus')

lemma mono_update: "a \<sqsubseteq> a' ==> S \<sqsubseteq> S' ==> S(x := a) \<sqsubseteq> S'(x := a')"
by(simp add: le_fun_def)

lemma mono_step': "S \<sqsubseteq> S' ==> c \<sqsubseteq> c' ==> step' S c \<sqsubseteq> step' S' c'"
apply(induction c c' arbitrary: S S' rule: le_acom.induct)
apply (auto simp: Let_def mono_update mono_aval' mono_post le_join_disj
            split: option.split)


text{* Problem: not executable because of the comparison of abstract states,
i.e. functions, in the post-fixedpoint computation. *}