Theory Opt

(*  Title:      HOL/MicroJava/BV/Opt.thy
    Author:     Tobias Nipkow
    Copyright   2000 TUM

More about options.
*)

section ‹More about Options›

theory Opt imports Err begin

definition le :: "'a ord ⇒ 'a option ord"
where
  "le r o⇩1 o⇩2 =
  (case o⇩2 of None ⇒ o⇩1=None | Some y ⇒ (case o⇩1 of None ⇒ True | Some x ⇒ x ⊑⇩r y))"

definition opt :: "'a set ⇒ 'a option set"
where
  "opt A = insert None {Some y |y. y ∈ A}"

definition sup :: "'a ebinop ⇒ 'a option ebinop"
where
  "sup f o⇩1 o⇩2 =  
  (case o⇩1 of None ⇒ OK o⇩2 
           | Some x ⇒ (case o⇩2 of None ⇒ OK o⇩1
                                 | Some y ⇒ (case f x y of Err ⇒ Err | OK z ⇒ OK (Some z))))"

definition esl :: "'a esl ⇒ 'a option esl"
where
  "esl = (λ(A,r,f). (opt A, le r, sup f))"


lemma unfold_le_opt:
  "o⇩1 ⊑⇘le r⇙ o⇩2 = 
  (case o⇩2 of None ⇒ o⇩1=None | 
              Some y ⇒ (case o⇩1 of None ⇒ True | Some x ⇒ x ⊑⇩r y))"
(*<*)
apply (unfold lesub_def le_def)
apply (rule refl)
done
(*>*)

lemma le_opt_refl: "order r A ⟹  x ∈ opt A ⟹ x ⊑⇘le r⇙ x"
(*<*) by (auto simp add: unfold_le_opt opt_def split: option.split) (*<*)

lemma le_opt_trans [rule_format]:
  "order r A ⟹ x ∈ opt A ⟹ y ∈ opt A ⟹ z ∈ opt A ⟹ x ⊑⇘le r⇙ y ⟶ y ⊑⇘le r⇙ z ⟶ x ⊑⇘le r⇙ z"
(*<*)
apply (simp add: unfold_le_opt opt_def split: option.split)
apply (blast intro: order_trans)
done
(*>*)

lemma le_opt_antisym [rule_format]:
  "order r A ⟹ x ∈ opt A ⟹ y ∈ opt A ⟹ z ∈ opt A ⟹ x ⊑⇘le r⇙ y ⟶ y ⊑⇘le r⇙ x ⟶ x=y"
(*<*)
apply (simp add: unfold_le_opt opt_def split: option.split)
apply (blast intro: order_antisym)
done
(*>*)

lemma order_le_opt [intro!,simp]: "order r A ⟹ order(le r) (opt A)"
(*<*)
apply (subst order_def)
apply (blast intro: le_opt_refl le_opt_trans le_opt_antisym)
done 
(*>*)

lemma None_bot [iff]:  "None ⊑⇘le r⇙ ox"
(*<*)
apply (unfold lesub_def le_def)
apply (simp split: option.split)
done 
(*>*)

lemma Some_le [iff]: "(Some x ⊑⇘le r⇙ z) = (∃y. z = Some y ∧ x ⊑⇩r y)"
(*<*)
apply (unfold lesub_def le_def)
apply (simp split: option.split)
done 
(*>*)

lemma le_None [iff]: "(x ⊑⇘le r⇙ None) = (x = None)"
(*<*)
apply (unfold lesub_def le_def)
apply (simp split: option.split)
done 
(*>*)

lemma OK_None_bot [iff]: "OK None ⊑⇘Err.le (le r)⇙ x"
(*<*) by (simp add: lesub_def Err.le_def le_def split: option.split err.split) (*>*)

lemma sup_None1 [iff]: "x ⊔⇘sup f⇙ None = OK x"
(*<*) by (simp add: plussub_def sup_def split: option.split) (*>*)

lemma sup_None2 [iff]: "None ⊔⇘sup f⇙ x = OK x"
(*<*) by (simp add: plussub_def sup_def split: option.split) (*>*)

lemma None_in_opt [iff]: "None ∈ opt A"
(*<*) by (simp add: opt_def) (*>*)

lemma Some_in_opt [iff]: "(Some x ∈ opt A) = (x ∈ A)"
(*<*) by (unfold opt_def) auto (*>*)

lemma semilat_opt [intro, simp]:
  "err_semilat L ⟹ err_semilat (Opt.esl L)"
(*<*)
proof -
  assume s: "err_semilat L" 
  obtain A r f where [simp]: "L = (A,r,f)" by (cases L)
  let ?A0 = "err A" and ?r0 = "Err.le r" and ?f0 = "lift2 f"
  from s obtain
    ord: "order ?r0 ?A" and
    clo: "closed ?A0 ?f0" and
    ub1: "∀x∈?A0. ∀y∈?A0. x ⊑⇘?r0⇙ x ⊔⇘?f0⇙ y" and
    ub2: "∀x∈?A0. ∀y∈?A0. y ⊑⇘?r0⇙ x ⊔⇘?f0⇙ y" and
    lub: "∀x∈?A0. ∀y∈?A0. ∀z∈?A0. x ⊑⇘?r0⇙ z ∧ y ⊑⇘?r0⇙ z ⟶ x ⊔⇘?f0⇙ y ⊑⇘?r0⇙ z"
    by (unfold semilat_def sl_def) simp

  let ?A = "err (opt A)" and ?r = "Err.le (Opt.le r)" and ?f = "lift2 (Opt.sup f)"

  from ord have "order ?r ?A" by simp
  moreover
  have "closed ?A ?f"
  proof (unfold closed_def, intro strip)
    fix x y assume x: "x ∈ ?A" and y: "y ∈ ?A" 

    { fix a b assume ab: "x = OK a" "y = OK b"
      with x have a: "⋀c. a = Some c ⟹ c ∈ A" by (clarsimp simp add: opt_def)
      from ab y have b: "⋀d. b = Some d ⟹ d ∈ A" by (clarsimp simp add: opt_def)      
      { fix c d assume "a = Some c" "b = Some d"
        with ab x y have "c ∈ A & d ∈ A" by (simp add: err_def opt_def Bex_def)
        with clo have "f c d ∈ err A" 
          by (simp add: closed_def plussub_def err_def' lift2_def)
        moreover fix z assume "f c d = OK z"
        ultimately have "z ∈ A" by simp
      } note f_closed = this    
      have "sup f a b ∈ ?A"
      proof (cases a)
        case None thus ?thesis
          by (simp add: sup_def opt_def) (cases b, simp, simp add: b Bex_def)
      next
        case Some thus ?thesis
          by (auto simp add: sup_def opt_def Bex_def a b f_closed split: err.split option.split)
      qed
    }
    thus "x ⊔⇘?f⇙ y ∈ ?A" by (simp add: plussub_def lift2_def split: err.split)
  qed
  moreover
  { fix a b c assume "a ∈ opt A" and "b ∈ opt A" and "a ⊔⇘sup f⇙ b = OK c" 
    moreover from ord have "order r A" by simp
    moreover
    { fix x y z assume "x ∈ A" and "y ∈ A" 
      hence "OK x ∈ err A ∧ OK y ∈ err A" by simp
      with ub1 ub2
      have "(OK x) ⊑⇘Err.le r⇙ (OK x) ⊔⇘lift2 f⇙ (OK y) ∧
            (OK y) ⊑⇘Err.le r⇙ (OK x) ⊔⇘lift2 f⇙ (OK y)"
        by blast
      moreover assume "x ⊔⇩f y = OK z"
      ultimately have "x ⊑⇩r z ∧ y ⊑⇩r z"
        by (auto simp add: plussub_def lift2_def Err.le_def lesub_def)
    }
    ultimately have "a ⊑⇘le r⇙ c ∧ b ⊑⇘le r⇙ c"
      by (auto simp add: sup_def le_def lesub_def plussub_def 
               dest: order_refl split: option.splits err.splits)
  }     
  hence "(∀x∈?A. ∀y∈?A. x ⊑⇘?r⇙ x ⊔⇘?f⇙ y) ∧ (∀x∈?A. ∀y∈?A. y ⊑⇘?r⇙ x ⊔⇘?f⇙ y)"
    by (auto simp add: lesub_def plussub_def Err.le_def lift2_def split: err.split)
  moreover
  have "∀x∈?A. ∀y∈?A. ∀z∈?A. x ⊑⇘?r⇙ z ∧ y ⊑⇘?r⇙ z ⟶ x ⊔⇘?f⇙ y ⊑⇘?r⇙ z"
  proof (intro strip, elim conjE)
    fix x y z
    assume xyz: "x ∈ ?A"   "y ∈ ?A"   "z ∈ ?A"
    assume xz: "x ⊑⇘?r⇙ z" and yz: "y ⊑⇘?r⇙ z"
    { fix a b c assume ok: "x = OK a"  "y = OK b"  "z = OK c"
      { fix d e g  assume some: "a = Some d"  "b = Some e"  "c = Some g"
        with ok xyz obtain "OK d:err A" "OK e:err A" "OK g:err A"  by simp
        with lub  
        have "⟦ OK d ⊑⇘Err.le r⇙ OK g; OK e ⊑⇘Err.le r⇙ OK g ⟧ ⟹ OK d ⊔⇘lift2 f⇙ OK e ⊑⇘Err.le r⇙ OK g"
          by blast
        hence "⟦ d ⊑⇩r g; e ⊑⇩r g ⟧ ⟹ ∃y. d ⊔⇩f e = OK y ∧ y ⊑⇩r g" by simp
        with ok some xyz xz yz have "x ⊔⇘?f⇙ y ⊑⇘?r⇙ z"
          by (auto simp add: sup_def le_def lesub_def lift2_def plussub_def Err.le_def)
      } note this [intro!]
      from ok xyz xz yz have "x ⊔⇘?f⇙ y ⊑⇘?r⇙ z"
        by - (cases a, simp, cases b, simp, cases c, simp, blast)
    }    
    with xyz xz yz show "x ⊔⇘?f⇙ y ⊑⇘?r⇙ z"
      by - (cases x, simp, cases y, simp, cases z, simp+)
  qed
  ultimately show "err_semilat (Opt.esl L)"
    by (unfold semilat_def esl_def sl_def) simp
qed 
(*>*)

lemma top_le_opt_Some [iff]: "top (le r) (Some T) = top r T"
(*<*)
apply (unfold top_def)
apply (rule iffI)
 apply blast
apply (rule allI)
apply (case_tac "x")
apply simp+
done 
(*>*)

lemma Top_le_conv:  "⟦ order r A; top r T; x ∈ A; T ∈ A ⟧ ⟹ (T ⊑⇩r x) = (x = T)"
(*<*)
apply (unfold top_def)
apply (blast intro: order_antisym)
done 
(*>*)


lemma acc_le_optI [intro!]: "acc r ⟹ acc(le r)"
(*<*)
apply (unfold acc_def lesub_def le_def lesssub_def)
apply (simp add: wf_eq_minimal split: option.split)
apply clarify
apply (case_tac "∃a. Some a ∈ Q")
 apply (erule_tac x = "{a . Some a ∈ Q}" in allE)
 apply blast
apply (case_tac "x")
 apply blast
apply blast
done 
(*>*)

lemma map_option_in_optionI:
  "⟦ ox ∈ opt S; ∀x∈S. ox = Some x ⟶ f x ∈ S ⟧ 
  ⟹ map_option f ox ∈ opt S"
(*<*)
apply (unfold map_option_case)
apply (simp split: option.split)
apply blast
done 
(*>*)

end