Module Qcert.TypeSystem.TBrandModel

Require Import String.
Require Import List.
Require Import ListSet.
Require Import Sumbool.
Require Import Arith.
Require Import Bool.
Require Import Morphisms.
Require Import Basics.
Require Import EquivDec.
Require Import Eqdep_dec.
Require Import Utils.
Require Import ForeignType.
Require Import RType.
Require Export BrandRelation.
Require Export TBrandContext.
Require Import RSubtype.
Require Import RConsistentSubtype.

Section TBrandModel.

  Definition sub_brand_context {br:brand_relation} {ftype:foreign_type} (m1 m2:brand_context) :=
    subtype (brand_context_Rec m1) (brand_context_Rec m2).

  Section brand_model_types.
    Context {ftype:foreign_type} {br:brand_relation} {m:brand_context}.

    Definition brand_model_domain_t
      := forall a, In a (domain brand_relation_brands) ->
                   In a (domain brand_context_types).

    Definition brand_model_subtype_weak_t :=
      forall a b τa,
        In (a,b) brand_relation_brands ->
        In (a,τa) brand_context_types ->
        exists (τb:rtype),
          In (b,τb) brand_context_types /\
          subtype τa τb.

    Definition brand_model_subtype_t :=
      forall a b τa,
        In (a,b) brand_relation_brands ->
        In (a,τa) brand_context_types ->
        {τb:rtype |
         In (b,τb) brand_context_types /\ subtype τa τb}.

    Section brand_model_types_dec.

      Lemma brand_model_domain_dec :
        {brand_model_domain_t }
        + {~ brand_model_domain_t}.

Proof.

apply (incl_list_dec string_dec).
Defined.

      Lemma brand_model_subtype_weak_dec :
        {brand_model_subtype_weak_t}
        + {~ brand_model_subtype_weak_t}.

Proof.

        unfold brand_model_subtype_weak_t.
        case_eq (forallb
                   (fun ab =>
                      match lookup string_dec brand_context_types (fst ab) with
                      | Some ta => match lookup string_dec brand_context_types (snd ab) with
                                   | Some tb => if subtype_dec ta tb then true else false
                                   | None => false
                                   end
                      | None => true
                      end)
                   brand_relation_brands).
        - left. intros.
          rewrite forallb_forall in H.
          specialize (H _ H0).
          match_case_in H; intros.
          + rewrite H2 in *.
            apply lookup_in in H2; simpl in *.
            generalize (nodup_in_eq brand_context_nodup H1 H2); intros; subst.
            match_case_in H; intros.
            * rewrite H3 in *. match_destr_in H.
              apply lookup_in in H3. simpl in *.
              exists r0; intuition.
            * rewrite H3 in H; discriminate.
          + rewrite H2 in H; simpl in *.
             apply lookup_none_nin in H2. apply in_dom in H1. intuition.
        - right.
          rewrite forallb_not_existb in H; rewrite negb_false_iff in H.
          rewrite existsb_exists in H.
          destruct H as [[? ?] [inn ne]].
          rewrite negb_true_iff in ne.
          match_case_in ne; intros.
          + rewrite H in ne; simpl in *.
            intro fb. specialize (fb _ _ _ inn (lookup_in _ _ H)).
            destruct fb as [? [inn2 sub]].
            match_case_in ne; intros.
            * rewrite H0 in ne.
              apply lookup_in in H0.
              generalize (nodup_in_eq brand_context_nodup H0 inn2); intros; subst.
              match_destr_in ne.
              intuition.
            * apply lookup_none_nin in H0. apply in_dom in inn2.
              intuition.
          + rewrite H in *. discriminate.
      Defined.

    End brand_model_types_dec.

  End brand_model_types.

  Context {ftype:foreign_type}.

  Class brand_model :=
    mkBrand_model {
        brand_model_relation :> brand_relation;
        brand_model_context :> brand_context;
        brand_model_domain_b :
         holds (brand_model_domain_dec);

        brand_model_subtype_weak_b :
           holds (brand_model_subtype_weak_dec)
      }.

Section brand_model_prop.
  Context {m:brand_model}.
Lemma brand_model_domain :
  brand_model_domain_t.

Proof.

generalize (brand_model_domain_b).
unfold holds, is_true; match_destr.
Qed.

Lemma brand_model_subtype_weak :
brand_model_subtype_weak_t.

Proof.

generalize (brand_model_subtype_weak_b).
unfold holds, is_true; match_destr.
Qed.

Lemma brand_model_subtype :
brand_model_subtype_t.

Proof.

  generalize (brand_model_subtype_weak_b).
  unfold holds, is_true; match_destr.
  unfold brand_model_subtype_t, brand_model_subtype_weak_t in *;
    intros.
  case_eq (lookup string_dec brand_context_types b0); intros.
  - exists r.
    destruct (b _ _ _ H0 H1) as [? [inn sub]].
    apply lookup_in in H2.
    generalize (nodup_in_eq brand_context_nodup H2 inn). intros; subst.
    intuition.
  - apply lookup_none_nin in H2. elim H2.
    destruct (b _ _ _ H0 H1) as [? [??]].
    apply in_dom in H3. trivial.
Qed.

  End brand_model_prop.

  Lemma brand_model_ext a b pf1 pf2 pf1' pf2' :
    mkBrand_model a b pf1 pf2 = mkBrand_model a b pf1' pf2'.

Proof.

f_equal; apply UIP_dec; apply bool_dec.
Defined.

  Definition make_brand_model (b:brand_relation) (m:brand_context)
             (pf:
             (is_true (brand_model_domain_dec)
            && is_true (brand_model_subtype_weak_dec)
             ) = true) : brand_model.

Proof.

    unfold is_true in *.
    match_case_in pf; intros ? eqq.
    - case_eq (brand_model_subtype_weak_dec); intros ? eqq2.
      + apply (@mkBrand_model b m); unfold holds, is_true;
        try rewrite eqq; try rewrite eqq2; trivial.
      + rewrite eqq, eqq2 in pf. simpl in pf. discriminate.
    - rewrite eqq in pf. simpl in pf. discriminate.
  Defined.

Definition make_brand_model_fails (b:brand_relation) (m:brand_context) : option brand_model.

Proof.

    generalize (make_brand_model b m); intros;
    destruct (is_true brand_model_domain_dec);
    destruct (is_true brand_model_subtype_weak_dec).
    - specialize (H eq_refl).
      exact (Some H).
    - exact None.
    - exact None.
    - exact None.
  Defined.

Program Definition empty_brand_model := make_brand_model (mkBrand_relation nil _ _) (mkBrand_context nil _).

Next Obligation.

unfold holds. auto.
Qed.

Next Obligation.

unfold holds. auto.
Defined.

End TBrandModel.