# Theory Pcpodef

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theory Pcpodef
imports Adm
uses (pcpodef_package.ML)
begin

    (*  Title:      HOLCF/Pcpodef.thy
        ID:         $Id: Pcpodef.thy,v 1.3 2005/07/26 16:24:29 huffman Exp $
        Author:     Brian Huffman
    *)

    header {* Subtypes of pcpos *}

    theory Pcpodef
    imports Adm
    uses ("pcpodef_package.ML")
    begin

    subsection {* Proving a subtype is a partial order *}

    text {*
      A subtype of a partial order is itself a partial order,
      if the ordering is defined in the standard way.
    *}

    theorem typedef_po:
      fixes Abs :: "'a::po => 'b::sq_ord"
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
      shows "OFCLASS('b, po_class)"
     apply (intro_classes, unfold less)
       apply (rule refl_less)
      apply (rule type_definition.Rep_inject [OF type, THEN iffD1])
      apply (erule (1) antisym_less)
     apply (erule (1) trans_less)
    done

    subsection {* Proving a subtype is complete *}

    text {*
      A subtype of a cpo is itself a cpo if the ordering is
      defined in the standard way, and the defining subset
      is closed with respect to limits of chains.  A set is
      closed if and only if membership in the set is an
      admissible predicate.
    *}

    lemma monofun_Rep:
      assumes less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
      shows "monofun Rep"
    by (rule monofunI, unfold less)

    lemmas ch2ch_Rep = ch2ch_monofun [OF monofun_Rep]
    lemmas ub2ub_Rep = ub2ub_monofun [OF monofun_Rep]

    lemma Abs_inverse_lub_Rep:
      fixes Abs :: "'a::cpo => 'b::po"
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
        and adm:  "adm (λx. x ∈ A)"
      shows "chain S ==> Rep (Abs (\<Squnion>i. Rep (S i))) = (\<Squnion>i. Rep (S i))"
     apply (rule type_definition.Abs_inverse [OF type])
     apply (erule admD [OF adm ch2ch_Rep [OF less], rule_format])
     apply (rule type_definition.Rep [OF type])
    done

    theorem typedef_lub:
      fixes Abs :: "'a::cpo => 'b::po"
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
        and adm: "adm (λx. x ∈ A)"
      shows "chain S ==> range S <<| Abs (\<Squnion>i. Rep (S i))"
     apply (frule ch2ch_Rep [OF less])
     apply (rule is_lubI)
      apply (rule ub_rangeI)
      apply (simp only: less Abs_inverse_lub_Rep [OF type less adm])
      apply (erule is_ub_thelub)
     apply (simp only: less Abs_inverse_lub_Rep [OF type less adm])
     apply (erule is_lub_thelub)
     apply (erule ub2ub_Rep [OF less])
    done

    lemmas typedef_thelub = typedef_lub [THEN thelubI, standard]

    theorem typedef_cpo:
      fixes Abs :: "'a::cpo => 'b::po"
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
        and adm: "adm (λx. x ∈ A)"
      shows "OFCLASS('b, cpo_class)"
    proof
      fix S::"nat => 'b" assume "chain S"
      hence "range S <<| Abs (\<Squnion>i. Rep (S i))"
        by (rule typedef_lub [OF type less adm])
      thus "∃x. range S <<| x" ..
    qed

    subsubsection {* Continuity of @{term Rep} and @{term Abs} *}

    text {* For any sub-cpo, the @{term Rep} function is continuous. *}

    theorem typedef_cont_Rep:
      fixes Abs :: "'a::cpo => 'b::cpo"
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
        and adm: "adm (λx. x ∈ A)"
      shows "cont Rep"
     apply (rule contI)
     apply (simp only: typedef_thelub [OF type less adm])
     apply (simp only: Abs_inverse_lub_Rep [OF type less adm])
     apply (rule thelubE [OF _ refl])
     apply (erule ch2ch_Rep [OF less])
    done

    text {*
      For a sub-cpo, we can make the @{term Abs} function continuous
      only if we restrict its domain to the defining subset by
      composing it with another continuous function.
    *}

    theorem typedef_is_lubI:
      assumes less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
      shows "range (λi. Rep (S i)) <<| Rep x ==> range S <<| x"
     apply (rule is_lubI)
      apply (rule ub_rangeI)
      apply (subst less)
      apply (erule is_ub_lub)
     apply (subst less)
     apply (erule is_lub_lub)
     apply (erule ub2ub_Rep [OF less])
    done

    theorem typedef_cont_Abs:
      fixes Abs :: "'a::cpo => 'b::cpo"
      fixes f :: "'c::cpo => 'a::cpo"
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
        and adm: "adm (λx. x ∈ A)" (* not used *)
        and f_in_A: "!!x. f x ∈ A"
        and cont_f: "cont f"
      shows "cont (λx. Abs (f x))"
     apply (rule contI)
     apply (rule typedef_is_lubI [OF less])
     apply (simp only: type_definition.Abs_inverse [OF type f_in_A])
     apply (erule cont_f [THEN contE])
    done

    subsection {* Proving a subtype is pointed *}

    text {*
      A subtype of a cpo has a least element if and only if
      the defining subset has a least element.
    *}

    theorem typedef_pcpo_generic:
      fixes Abs :: "'a::cpo => 'b::cpo"
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
        and z_in_A: "z ∈ A"
        and z_least: "!!x. x ∈ A ==> z \<sqsubseteq> x"
      shows "OFCLASS('b, pcpo_class)"
     apply (intro_classes)
     apply (rule_tac x="Abs z" in exI, rule allI)
     apply (unfold less)
     apply (subst type_definition.Abs_inverse [OF type z_in_A])
     apply (rule z_least [OF type_definition.Rep [OF type]])
    done

    text {*
      As a special case, a subtype of a pcpo has a least element
      if the defining subset contains @{term ⊥}.
    *}

    theorem typedef_pcpo:
      fixes Abs :: "'a::pcpo => 'b::cpo"
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
        and UU_in_A: "⊥ ∈ A"
      shows "OFCLASS('b, pcpo_class)"
    by (rule typedef_pcpo_generic [OF type less UU_in_A], rule minimal)

    subsubsection {* Strictness of @{term Rep} and @{term Abs} *}

    text {*
      For a sub-pcpo where @{term ⊥} is a member of the defining
      subset, @{term Rep} and @{term Abs} are both strict.
    *}

    theorem typedef_Abs_strict:
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
        and UU_in_A: "⊥ ∈ A"
      shows "Abs ⊥ = ⊥"
     apply (rule UU_I, unfold less)
     apply (simp add: type_definition.Abs_inverse [OF type UU_in_A])
    done

    theorem typedef_Rep_strict:
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
        and UU_in_A: "⊥ ∈ A"
      shows "Rep ⊥ = ⊥"
     apply (rule typedef_Abs_strict [OF type less UU_in_A, THEN subst])
     apply (rule type_definition.Abs_inverse [OF type UU_in_A])
    done

    theorem typedef_Abs_defined:
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
        and UU_in_A: "⊥ ∈ A"
      shows "[|x ≠ ⊥; x ∈ A|] ==> Abs x ≠ ⊥"
     apply (rule typedef_Abs_strict [OF type less UU_in_A, THEN subst])
     apply (simp add: type_definition.Abs_inject [OF type] UU_in_A)
    done

    theorem typedef_Rep_defined:
      assumes type: "type_definition Rep Abs A"
        and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
        and UU_in_A: "⊥ ∈ A"
      shows "x ≠ ⊥ ==> Rep x ≠ ⊥"
     apply (rule typedef_Rep_strict [OF type less UU_in_A, THEN subst])
     apply (simp add: type_definition.Rep_inject [OF type])
    done

    subsection {* HOLCF type definition package *}

    use "pcpodef_package.ML"

    end

### Proving a subtype is a partial order

theorem typedef_po:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y |]
      ==> OFCLASS('b, po_class)

### Proving a subtype is complete

lemma monofun_Rep:

      op << == %x y. Rep x << Rep y ==> monofun Rep

lemmas ch2ch_Rep:

      [| op << == %x y. f x << f y; chain Y |] ==> chain (%i. f (Y i))

lemmas ch2ch_Rep:

      [| op << == %x y. f x << f y; chain Y |] ==> chain (%i. f (Y i))

lemmas ub2ub_Rep:

      [| op << == %x y. f x << f y; range Y <| u |] ==> range (%i. f (Y i)) <| f u

lemmas ub2ub_Rep:

      [| op << == %x y. f x << f y; range Y <| u |] ==> range (%i. f (Y i)) <| f u

lemma Abs_inverse_lub_Rep:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; adm (%x. x ∈ A);
         chain S |]
      ==> Rep (Abs (LUB i. Rep (S i))) = (LUB i. Rep (S i))

theorem typedef_lub:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; adm (%x. x ∈ A);
         chain S |]
      ==> range S <<| Abs (LUB i. Rep (S i))

lemmas typedef_thelub:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; adm (%x. x ∈ A);
         chain S |]
      ==> lub (range S) = Abs (LUB i. Rep (S i))

lemmas typedef_thelub:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; adm (%x. x ∈ A);
         chain S |]
      ==> lub (range S) = Abs (LUB i. Rep (S i))

theorem typedef_cpo:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; adm (%x. x ∈ A) |]
      ==> OFCLASS('b, cpo_class)

#### Continuity of @{term Rep} and @{term Abs}

theorem typedef_cont_Rep:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; adm (%x. x ∈ A) |]
      ==> cont Rep

theorem typedef_is_lubI:

      [| op << == %x y. Rep x << Rep y; range (%i. Rep (S i)) <<| Rep x |]
      ==> range S <<| x

theorem typedef_cont_Abs:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; adm (%x. x ∈ A);
         !!x. f x ∈ A; cont f |]
      ==> cont (%x. Abs (f x))

### Proving a subtype is pointed

theorem typedef_pcpo_generic:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; z ∈ A;
         !!x. x ∈ A ==> z << x |]
      ==> OFCLASS('b, pcpo_class)

theorem typedef_pcpo:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; UU ∈ A |]
      ==> OFCLASS('b, pcpo_class)

#### Strictness of @{term Rep} and @{term Abs}

theorem typedef_Abs_strict:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; UU ∈ A |]
      ==> Abs UU = UU

theorem typedef_Rep_strict:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; UU ∈ A |]
      ==> Rep UU = UU

theorem typedef_Abs_defined:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; UU ∈ A; x ≠ UU;
         x ∈ A |]
      ==> Abs x ≠ UU

theorem typedef_Rep_defined:

      [| type_definition Rep Abs A; op << == %x y. Rep x << Rep y; UU ∈ A; x ≠ UU |]
      ==> Rep x ≠ UU

### HOLCF type definition package