coq: add subtype relations

coq/_CoqProject

@@ -2,6 +2,7 @@
 terms.v
 equiv.v
 subst.v
+subtype.v
 typing.v
 smallstep.v
 bbencode.v

coq/subtype.v

@@ -0,0 +1,96 @@
+(*
+ * 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.