(*
 * This module defines the subtype relationship
 *
 * We distinguish between *representational* subtypes,
 * where any high-level type is a subtype of its underlying
 * representation type and *convertible* subtypes that
 * are compatible at high level, but have a different representation
 * that requires a conversion.
 *)

From Coq Require Import Strings.String.
Require Import terms.
Require Import equiv.
Include Terms.
Include Equiv.

Module Subtype.

(** Subtyping *)

Reserved Notation "s ':<=' t" (at level 50).
Reserved Notation "s '~<=' t" (at level 50).

(* Representational Subtype *)
Inductive is_repr_subtype : type_term -> type_term -> Prop :=
	| TSubRepr_Refl : forall t t', (t === t') -> (t :<= t')
	| TSubRepr_Trans : forall x y z, (x :<= y) -> (y :<= z) -> (x :<= z)
	| TSubRepr_Ladder : forall x' x y, (x :<= y) -> ((type_ladder x' x) :<= y)
where "s ':<=' t" := (is_repr_subtype s t).

(* Convertible Subtype *)
Inductive is_conv_subtype : type_term -> type_term -> Prop :=
	| TSubConv_Refl : forall t t', (t === t') -> (t ~<= t')
	| TSubConv_Trans : forall x y z, (x ~<= y) -> (y ~<= z) -> (x ~<= z)
	| TSubConv_Ladder : forall x' x y, (x ~<= y) -> ((type_ladder x' x) ~<= y)
	| TSubConv_Morph : forall x y y', (type_ladder x y) ~<= (type_ladder x y')
where "s '~<=' t" := (is_conv_subtype s t).


(* Every Representational Subtype is a Convertible Subtype *)

Lemma syn_sub_is_sem_sub : forall x y, (x :<= y) -> (x ~<= y).
Proof.
	intros.
	induction H.
	apply TSubConv_Refl.
	apply H.
	apply TSubConv_Trans with (x:=x) (y:=y) (z:=z).
	apply IHis_repr_subtype1.
	apply IHis_repr_subtype2.
	apply TSubConv_Ladder.
	apply IHis_repr_subtype.
Qed.




(* EXAMPLES *)

Open Scope ladder_type_scope.
Open Scope ladder_expr_scope.

Example sub0 :
  [   < $"Seq"$ < $"Digit"$ $"10"$ > >
    ~ < $"Seq"$ $"Char"$ > ]
:<=
  [   < $"Seq"$ $"Char"$ > ]
.
Proof.
  apply TSubRepr_Ladder.
  apply TSubRepr_Refl.
	apply TEq_Refl.
Qed.


Example sub1 :
    [ < $"Seq"$ < $"Digit"$ $"10"$ > ~ $"Char"$ > ]
:<= [ < $"Seq"$ $"Char"$ > ]
.
Proof.
  set [ < $"Seq"$ < $"Digit"$ $"10"$ > > ~ < $"Seq"$ $"Char"$ > ].
	set [ < $"Seq"$ < $"Digit"$ $"10"$ > ~ $"Char"$ > ].
	set [ < $"Seq"$ $"Char"$ > ].
	set (t0 === t).
	set (t :<= t0).
	set (t :<= t1).
	apply TSubRepr_Trans with t.
	apply TSubRepr_Refl.
	apply TEq_Distribute.
	apply L_DistributeOverSpec2.
	apply TSubRepr_Ladder.
	apply TSubRepr_Refl.
	apply TEq_Refl.
Qed.

End Subtype.