(* Define the abstract syntax of the calculus
 * by inductive definition of type-terms
 * and expression-terms.
 *)

From Coq Require Import Strings.String.
Module Terms.

(* types *)
Inductive type_term : Type :=
  | type_unit : type_term
  | type_id : string -> type_term
  | type_var : string -> type_term
  | type_num : nat -> type_term
  | type_fun : type_term -> type_term -> type_term
  | type_univ : string -> type_term -> type_term
  | type_spec : type_term -> type_term -> type_term
  | type_ladder : type_term -> type_term -> type_term
.

(* expressions *)
Inductive expr_term : Type :=
  | expr_var : string -> expr_term
  | expr_ty_abs : string -> expr_term -> expr_term
  | expr_ty_app : expr_term -> type_term -> expr_term
  | expr_tm_abs : string -> type_term -> expr_term -> expr_term
  | expr_tm_app : expr_term -> expr_term -> expr_term
  | expr_let : string -> type_term -> expr_term -> expr_term -> expr_term
  | expr_ascend : type_term -> expr_term -> expr_term
  | expr_descend : type_term -> expr_term -> expr_term
.

Coercion type_var : string >-> type_term.
Coercion expr_var : string >-> expr_term.

(*
Notation "( x )" := x (at level 70).
Notation "x ~ y" := (type_rung x y) (at level 69, left associativity).
Notation "< x y >" := (type_app x y) (at level 68, left associativity).
Notation "'$' x" := (type_id x) (at level 66).
*)

End Terms.