Theory OptionMonad

(*
 * Copyright 2014, NICTA
 *
 * This software may be distributed and modified according to the terms of
 * the BSD 2-Clause license. Note that NO WARRANTY is provided.
 * See "LICENSE_BSD2.txt" for details.
 *
 * @TAG(NICTA_BSD)
 *)
(*
 * Contributions by:
 *   2012 Lars Noschinski <noschinl@in.tum.de>
 *     Option monad while loop formalisation.
 *)

theory OptionMonad
imports Lib
begin

type_synonym ('s,'a) lookup = "'s ⇒ 'a option"

text ‹Similar to map_option but the second function returns option as well›
definition
  opt_map :: "('s,'a) lookup ⇒ ('a ⇒ 'b option) ⇒ ('s,'b) lookup" (infixl "|>" 54)
where
  "f |> g ≡ λs. case f s of None ⇒ None | Some x ⇒ g x"

lemma opt_map_cong [fundef_cong]:
  "⟦ f = f'; ⋀v s. f s = Some v ⟹ g v = g' v⟧ ⟹ f |> g = f' |> g'"
  by (rule ext) (simp add: opt_map_def split: option.splits)

lemma in_opt_map_eq:
  "((f |> g) s = Some v) = (∃v'. f s = Some v' ∧ g v' = Some v)"
  by (simp add: opt_map_def split: option.splits)

lemma opt_mapE:
  "⟦ (f |> g) s = Some v; ⋀v'. ⟦f s = Some v'; g v' = Some v ⟧ ⟹ P ⟧ ⟹ P"
  by (auto simp: in_opt_map_eq)


definition
  obind :: "('s,'a) lookup ⇒ ('a ⇒ ('s,'b) lookup) ⇒ ('s,'b) lookup" (infixl "|>>" 53)
where
  "f |>> g ≡ λs. case f s of None ⇒ None | Some x ⇒ g x s"

definition
  "ofail = K None"

definition
  "oreturn = K o Some"

definition
  "oassert P ≡ if P then oreturn () else ofail"

text ‹
  If the result can be an exception.
  Corresponding bindE would be analogous to lifting in NonDetMonad.
›

definition
  "oreturnOk x = K (Some (Inr x))"

definition
  "othrow e = K (Some (Inl e))"

definition
  "oguard G ≡ (λs. if G s then Some () else None)"

definition
  "ocondition c L R ≡ (λs. if c s then L s else R s)"

definition
  "oskip ≡ oreturn ()"

text ‹Monad laws›
lemma oreturn_bind [simp]: "(oreturn x |>> f) = f x"
  by (auto simp add: oreturn_def obind_def K_def intro!: ext)

lemma obind_return [simp]: "(m |>> oreturn) = m"
  by (auto simp add: oreturn_def obind_def K_def intro!: ext split: option.splits)
 
lemma obind_assoc: 
  "(m |>> f) |>> g  =  m |>> (λx. f x |>> g)"
  by (auto simp add: oreturn_def obind_def K_def intro!: ext split: option.splits)


text ‹Binding fail›

lemma obind_fail [simp]:
  "f |>> (λ_. ofail) = ofail"
  by (auto simp add: ofail_def obind_def K_def intro!: ext split: option.splits)

lemma ofail_bind [simp]:
  "ofail |>> m = ofail"
  by (auto simp add: ofail_def obind_def K_def intro!: ext split: option.splits)



text ‹Function package setup›
lemma opt_bind_cong [fundef_cong]:
  "⟦ f = f'; ⋀v s. f' s = Some v ⟹ g v s = g' v s ⟧ ⟹ f |>> g = f' |>> g'"
  by (rule ext) (simp add: obind_def split: option.splits)

lemma opt_bind_cong_apply [fundef_cong]:
  "⟦ f s = f' s; ⋀v. f' s = Some v ⟹ g v s = g' v s ⟧ ⟹ (f |>> g) s = (f' |>> g') s"
  by (simp add: obind_def split: option.splits)

lemma oassert_bind_cong [fundef_cong]:
  "⟦ P = P'; P' ⟹ m = m' ⟧ ⟹ oassert P |>> m = oassert P' |>> m'"
  by (auto simp: oassert_def)

lemma oassert_bind_cong_apply [fundef_cong]:
  "⟦ P = P'; P' ⟹ m () s = m' () s ⟧ ⟹ (oassert P |>> m) s = (oassert P' |>> m') s"
  by (auto simp: oassert_def)

lemma oreturn_bind_cong [fundef_cong]:
  "⟦ x = x'; m x' = m' x' ⟧ ⟹ oreturn x |>> m = oreturn x' |>> m'"
  by simp

lemma oreturn_bind_cong_apply [fundef_cong]:
  "⟦ x = x'; m x' s = m' x' s ⟧ ⟹ (oreturn x |>> m) s = (oreturn x' |>> m') s"
  by simp

lemma oreturn_bind_cong2 [fundef_cong]:
  "⟦ x = x'; m x' = m' x' ⟧ ⟹ (oreturn $ x) |>> m = (oreturn $ x') |>> m'"
  by simp

lemma oreturn_bind_cong2_apply [fundef_cong]:
  "⟦ x = x'; m x' s = m' x' s ⟧ ⟹ ((oreturn $ x) |>> m) s = ((oreturn $ x') |>> m') s"
  by simp

lemma ocondition_cong [fundef_cong]:
"⟦c = c'; ⋀s. c' s ⟹ l s = l' s; ⋀s. ¬c' s ⟹ r s = r' s⟧
  ⟹ ocondition c l r = ocondition c' l' r'"
  by (auto simp: ocondition_def)


text ‹Decomposition›

lemma ocondition_K_true [simp]:
  "ocondition (λ_. True) T F = T"
  by (simp add: ocondition_def)

lemma ocondition_K_false [simp]:
  "ocondition (λ_. False) T F = F"
  by (simp add: ocondition_def)

lemma ocondition_False:
    "⟦ ⋀s. ¬ P s ⟧ ⟹ ocondition P L R = R"
  by (rule ext, clarsimp simp: ocondition_def)

lemma ocondition_True:
    "⟦ ⋀s. P s ⟧ ⟹ ocondition P L R = L"
  by (rule ext, clarsimp simp: ocondition_def)

lemma in_oreturn [simp]:
  "(oreturn x s = Some v) = (v = x)"
  by (auto simp: oreturn_def K_def)

lemma oreturnE:
  "⟦oreturn x s = Some v; v = x ⟹ P x⟧ ⟹ P v"
  by simp

lemma in_ofail [simp]:
  "ofail s ≠ Some v"
  by (auto simp: ofail_def K_def)

lemma ofailE:
  "ofail s = Some v ⟹ P"
  by simp

lemma in_oassert_eq [simp]:
  "(oassert P s = Some v) = P"
  by (simp add: oassert_def)

lemma oassertE:
  "⟦ oassert P s = Some v; P ⟹ Q ⟧ ⟹ Q"
  by simp

lemma in_obind_eq:
  "((f |>> g) s = Some v) = (∃v'. f s = Some v' ∧ g v' s = Some v)"
  by (simp add: obind_def split: option.splits)

lemma obindE:
  "⟦ (f |>> g) s = Some v; 
     ⋀v'. ⟦f s = Some v'; g v' s = Some v⟧ ⟹ P⟧ ⟹ P"
  by (auto simp: in_obind_eq)

lemma in_othrow_eq [simp]:
  "(othrow e s = Some v) = (v = Inl e)"
  by (auto simp: othrow_def K_def) 

lemma othrowE:
  "⟦othrow e s = Some v; v = Inl e ⟹ P (Inl e)⟧ ⟹ P v"
  by simp

lemma in_oreturnOk_eq [simp]:
  "(oreturnOk x s = Some v) = (v = Inr x)"
  by (auto simp: oreturnOk_def K_def) 

lemma oreturnOkE:
  "⟦oreturnOk x s = Some v; v = Inr x ⟹ P (Inr x)⟧ ⟹ P v"
  by simp

lemmas omonadE [elim!] =
  opt_mapE obindE oreturnE ofailE othrowE oreturnOkE oassertE

section ‹"While" loops over option monad.›

text ‹
  This is an inductive definition of a while loop over the plain option monad
  (without passing through a state)
›

inductive_set
  option_while' :: "('a ⇒ bool) ⇒ ('a ⇒ 'a option) ⇒ 'a option rel"
  for C B
where
    final: "¬ C r ⟹ (Some r, Some r) ∈ option_while' C B"
  | fail: "⟦ C r; B r = None ⟧ ⟹ (Some r, None) ∈ option_while' C B"
  | step: "⟦ C r;  B r = Some r'; (Some r', sr'') ∈ option_while' C B ⟧
           ⟹ (Some r, sr'') ∈ option_while' C B"

definition
  "option_while C B r ≡
    (if (∃s. (Some r, s) ∈ option_while' C B) then
      (THE s. (Some r, s) ∈ option_while' C B) else None)"

lemma option_while'_inj:
  assumes "(s,s') ∈ option_while' C B" "(s, s'') ∈ option_while' C B"
  shows "s' = s''"
  using assms by (induct rule: option_while'.induct) (auto elim: option_while'.cases)

lemma option_while'_inj_step:
  "⟦ C s; B s = Some s'; (Some s, t) ∈ option_while' C B ; (Some s', t') ∈ option_while' C B ⟧ ⟹ t = t'"
  by (metis option_while'.step option_while'_inj)

lemma option_while'_THE:
  assumes "(Some r, sr') ∈ option_while' C B"
  shows "(THE s. (Some r, s) ∈ option_while' C B) = sr'"
  using assms by (blast dest: option_while'_inj)

lemma option_while_simps:
  "¬ C s ⟹ option_while C B s = Some s"
  "C s ⟹ B s = None ⟹ option_while C B s = None"
  "C s ⟹ B s = Some s' ⟹ option_while C B s = option_while C B s'"
  "(Some s, ss') ∈ option_while' C B ⟹ option_while C B s = ss'"
  using option_while'_inj_step[of C s B s']
  by (auto simp: option_while_def option_while'_THE
      intro: option_while'.intros
      dest: option_while'_inj
      elim: option_while'.cases)

lemma option_while_rule:
  assumes "option_while C B s = Some s'"
  assumes "I s"
  assumes istep: "⋀s s'. C s ⟹ I s ⟹ B s = Some s' ⟹ I s'"
  shows "I s' ∧ ¬ C s'" 
proof -
  { fix ss ss' assume "(ss, ss') ∈ option_while' C B" "ss = Some s" "ss' = Some s'"
    then have ?thesis using ‹I s›
      by (induct arbitrary: s) (auto intro: istep) }
  then show ?thesis using assms(1)
    by (auto simp: option_while_def option_while'_THE split: if_split_asm)
qed

lemma option_while'_term:
  assumes "I r"
  assumes "wf M"
  assumes step_less: "⋀r r'. ⟦I r; C r; B r = Some r'⟧ ⟹ (r',r) ∈ M"
  assumes step_I: "⋀r r'. ⟦I r; C r; B r = Some r'⟧ ⟹ I r'"
  obtains sr' where "(Some r, sr') ∈ option_while' C B"
  apply atomize_elim
  using assms(2,1)
proof induct
  case (less r)
  show ?case
  proof (cases "C r" "B r" rule: bool.exhaust[case_product option.exhaust])
    case (True_Some r')
    then have "(r',r) ∈ M" "I r'"
      by (auto intro: less step_less step_I)
    then obtain sr' where "(Some r', sr') ∈ option_while' C B"
      by atomize_elim (rule less)
    then have "(Some r, sr') ∈ option_while' C B"
      using True_Some by (auto intro: option_while'.intros)
    then show ?thesis ..
  qed (auto intro: option_while'.intros)
qed

lemma option_while_rule':
  assumes "option_while C B s = ss'"
  assumes "wf M"
  assumes "I (Some s)"
  assumes less: "⋀s s'. C s ⟹ I (Some s) ⟹ B s = Some s' ⟹ (s', s) ∈ M"
  assumes step: "⋀s s'. C s ⟹ I (Some s) ⟹ B s = Some s' ⟹ I (Some s')"
  assumes final: "⋀s. C s ⟹ I (Some s) ⟹ B s = None ⟹ I None"
  shows "I ss' ∧ (case ss' of Some s' ⇒ ¬ C s' | _ ⇒ True)" 
proof -
  define ss where "ss = Some s"
  obtain ss1' where "(Some s, ss1') ∈ option_while' C B"
    using assms(3,2,4,5) by (rule option_while'_term)
  then have *: "(ss, ss') ∈ option_while' C B" using ‹option_while C B s = ss'›
    by (auto simp: option_while_simps ss_def)
  show ?thesis
  proof (cases ss')
    case (Some s') with * ss_def show ?thesis using ‹I _›
      by (induct arbitrary:s) (auto intro: step)
  next
    case None with * ss_def show ?thesis using ‹I _›
      by (induct arbitrary:s) (auto intro: step final)
  qed
qed

section ‹Lift @{term option_while} to the @{typ "('a,'s) lookup"} monad›

definition
  owhile :: "('a ⇒ 's ⇒ bool) ⇒ ('a ⇒ ('s,'a) lookup) ⇒ 'a ⇒ ('s,'a) lookup"
where
 "owhile c b a ≡ λs. option_while (λa. c a s) (λa. b a s) a"

lemma owhile_unroll:
  "owhile C B r = ocondition (C r) (B r |>> owhile C B) (oreturn r)"
  by (auto simp: ocondition_def obind_def oreturn_def owhile_def
           option_while_simps K_def split: option.split)

text ‹rule for terminating loops›

lemma owhile_rule:
  assumes "I r s"
  assumes "wf M"
  assumes less: "⋀r r'. ⟦I r s; C r s; B r s = Some r'⟧ ⟹ (r',r) ∈ M"
  assumes step: "⋀r r'. ⟦I r s; C r s; B r s = Some r'⟧ ⟹ I r' s"
  assumes fail: "⋀r r'. ⟦I r s; C r s; B r s = None⟧ ⟹ Q None"
  assumes final: "⋀r. ⟦I r s; ¬C r s⟧ ⟹ Q (Some r)"
  shows "Q (owhile C B r s)"
proof -
  let ?rs' = "owhile C B r s"
  have "(case ?rs' of Some r ⇒ I r s | _ ⇒ Q None)
      ∧ (case ?rs' of Some r' ⇒ ¬ C r' s | _ ⇒ True)"
    by (rule option_while_rule'[where B="λr. B r s" and s=r, OF _ ‹wf _›])
       (auto simp: owhile_def intro: assms)
  then show ?thesis by (auto intro: final split: option.split_asm)
qed

end