Overloading notation for different types in Coq

问题:

I would like to be able to define the same Coq notations for different inductive definitions, and distinguish the notations based on the types of their arguments.

Here is a minimal example:

Inductive type : Type :=
| TBool : type.

Inductive term1 : Type :=
| tvar1 : term1.

Inductive term2 : Type :=
| tvar2 : term2.

Definition context := nat -> (option type).

Reserved Notation "G '⊢' t '::' T" (at level 40, t at level 59).

Inductive typing1 : context -> term1 -> type -> Prop :=
 | T_Var1 : forall G T,
      G ⊢ tvar1 :: T
where "G '⊢' t1 '::' T" := (typing1 G t1 T)                            
with typing2 : context -> term2 -> type -> Prop :=
 | T_Var2 : forall G T,
      G ⊢ tvar2 :: T
where "G '⊢' t2 '::' T" := (typing2 G t2 T).

As you see, there is a mutually inductive definition, in which I'd like to be able to use the same notation for different types of terms (term1 and term2).

The error I get when trying to compile this is The term "tvar1" has type "term1" while it is expected to have type "term2"..

Is there a way to get this to work?

回答1:

I wrote to the Coq mailing list and received an answer from Gaëtan Gilbert that solved my problem using type classes:

Inductive type : Type :=
| TBool : type.

Inductive term1 : Type :=
| tvar1 : term1.

Inductive term2 : Type :=
| tvar2 : term2.

Definition context := nat -> (option type).

Class VDash (A B C : Type) := vdash : A -> B -> C -> Prop.
Notation "G '⊢' t '::' T" := (vdash G t T) (at level 40, t at level 59).

Inductive typing1 : VDash context term1 type :=
| T_Var1 : forall G T,
    G ⊢ tvar1 :: T

with typing2 : VDash context term2 type :=
| T_Var2 : forall G T,
    G ⊢ tvar2 :: T.