Coq中不同类型的重载符号 [英] Overloading notation for different types in Coq

问题描述

我希望能够为不同的归纳定义定义相同的Coq表示法，并根据其自变量类型区分这些表示法。

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.

这里是一个最小的示例：

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).

如您所见，有一个相互归纳的定义，我希望能够对不同类型的术语（ term1 和 term2 ）使用相同的符号。

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).

尝试编译时遇到的错误是术语 tvar1的类型为 term1，而预期类型为 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"..

有没有办法使它工作？

推荐答案

我写了一封Coq邮件列表，并收到了答案使用类型类解决了我的问题：

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.

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