100 lines
2.4 KiB
Coq
100 lines
2.4 KiB
Coq
(*
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* This module defines the subtype relationship
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*
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* We distinguish between *representational* subtypes,
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* where any high-level type is a subtype of its underlying
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* representation type and *convertible* subtypes that
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* are compatible at high level, but have a different representation
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* that requires a conversion.
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*)
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From Coq Require Import Strings.String.
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Require Import terms.
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Require Import equiv.
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Include Terms.
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Include Equiv.
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Module Subtype.
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(** Subtyping *)
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Reserved Notation "s ':<=' t" (at level 50).
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Reserved Notation "s '~<=' t" (at level 50).
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(* Representational Subtype *)
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Inductive is_repr_subtype : type_term -> type_term -> Prop :=
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| TSubRepr_Refl : forall t t', (t === t') -> (t :<= t')
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| TSubRepr_Trans : forall x y z, (x :<= y) -> (y :<= z) -> (x :<= z)
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| TSubRepr_Ladder : forall x' x y, (x :<= y) -> ((type_ladder x' x) :<= y)
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where "s ':<=' t" := (is_repr_subtype s t).
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(* Convertible Subtype *)
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Inductive is_conv_subtype : type_term -> type_term -> Prop :=
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| TSubConv_Refl : forall t t', (t === t') -> (t ~<= t')
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| TSubConv_Trans : forall x y z, (x ~<= y) -> (y ~<= z) -> (x ~<= z)
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| TSubConv_Ladder : forall x' x y, (x ~<= y) -> ((type_ladder x' x) ~<= y)
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| TSubConv_Morph : forall x y y', (type_ladder x y) ~<= (type_ladder x y')
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where "s '~<=' t" := (is_conv_subtype s t).
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(* Every Representational Subtype is a Convertible Subtype *)
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Lemma syn_sub_is_sem_sub : forall x y, (x :<= y) -> (x ~<= y).
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Proof.
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intros.
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induction H.
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apply TSubConv_Refl.
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apply H.
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apply TSubConv_Trans with (x:=x) (y:=y) (z:=z).
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apply IHis_repr_subtype1.
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apply IHis_repr_subtype2.
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apply TSubConv_Ladder.
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apply IHis_repr_subtype.
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Qed.
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(* EXAMPLES *)
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Open Scope ladder_type_scope.
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Open Scope ladder_expr_scope.
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Example sub0 :
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[<
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< $"Seq"$ < $"Digit"$ $"10"$ > >
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~ < $"Seq"$ $"Char"$ >
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>]
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:<=
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[<
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< $"Seq"$ $"Char"$ >
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>]
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.
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Proof.
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apply TSubRepr_Ladder.
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apply TSubRepr_Refl.
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apply TEq_Refl.
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Qed.
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Example sub1 :
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[< < $"Seq"$ < $"Digit"$ $"10"$ > ~ $"Char"$ > >]
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:<= [< < $"Seq"$ $"Char"$ > >]
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.
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Proof.
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set [< < $"Seq"$ < $"Digit"$ $"10"$ > > ~ < $"Seq"$ $"Char"$ > >].
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set [< < $"Seq"$ < $"Digit"$ $"10"$ > ~ $"Char"$ > >].
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set [< < $"Seq"$ $"Char"$ > >].
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set (t0 === t).
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set (t :<= t0).
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set (t :<= t1).
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apply TSubRepr_Trans with t.
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apply TSubRepr_Refl.
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apply TEq_Distribute.
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apply L_DistributeOverSpec2.
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apply TSubRepr_Ladder.
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apply TSubRepr_Refl.
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apply TEq_Refl.
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Qed.
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End Subtype.
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