How to understand Coq type constructor var (t: T)

问题

I am reading about mechanization of linear logic in Coq http://www.cs.cmu.edu/~iliano/projects/metaCLF2/inc/dl/papers/lsfa17.pdf and https://github.com/brunofx86/LL and I have trouble to understand the type constructors of the inductive type term from https://github.com/brunofx86/LL/blob/master/FOLL/LL/SyntaxLL.v:

Inductive term  :=
    |var (t: T) (* variables *)
    |cte (e:A) (* constants from the domain DT.A *)
    |fc1 (n:nat) (t: term) (* family of functions of 1 argument *)
    |fc2 (n:nat) (t1 t2: term). (* family of functions of 2 argument *)

I have two question regarding this sample (I am reading https://softwarefoundations.cis.upenn.edu/lf-current/Basics.html along this paper):

what is the (super)type of term? Software Foundations always specifies the (super)type of the new type, like Inductive color : Type;
the main question - how to understand type constructor var (t: T)?. Software Foundation in its first chapter provides only two types of type constructors: constant white : color and function primary : rgb → color. But var (t: T) is very strange notation - it is not valid function type definition as it have no explicit return type and also it has no arrow.

回答1:

Regarding your main question, the syntax var (t : T) when defining a constructor is just some alternative (shorter) syntax for var : forall t : T, term, which can just as well be written var : T -> term (since there is no occurrence of variable t in term).

Actually, you can check this by processing the definition, then the following command:

Print term.

(* and Coq displays the inductive type with the default syntax, that is:

  Inductive term : Type :=
    var : T -> term
  | cte : A -> term
  | fc1 : nat -> term -> term
  | fc2 : nat -> term -> term -> term
*)

Next (as shown by the Coq output above), the type of the datatype term is indeed Type.

I recall that in Coq, all types have also a type, and this latter will always be Prop, Set or Type. A "type of type" is usually called a sort. (The sort Prop deals with logical propositions while the sorts Set and Type deal with so-called "informative" types.)

Finally, it can be noted that Type does not refer to a fixed type, but to a given Type_i where the index i >=0 is automatically determined and checked by Coq's kernel. For more information on this topic, see for example the first section of the Universes chapter of CPDT

来源：https://stackoverflow.com/questions/49718746/how-to-understand-coq-type-constructor-var-t-t

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functional-programming

logic

coq