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?
amaurremi
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.
来源：https://stackoverflow.com/questions/44034734/overloading-notation-for-different-types-in-coq