(* 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 ladder_type : Type :=
  | type_unit : ladder_type
  | type_id : string -> ladder_type
  | type_var : string -> ladder_type
  | type_num : nat -> ladder_type
  | type_abs : string -> ladder_type -> ladder_type
  | type_app : ladder_type -> ladder_type -> ladder_type
  | type_fun : ladder_type -> ladder_type -> ladder_type
  | type_rung : ladder_type -> ladder_type -> ladder_type
.

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

Coercion type_var : string >-> ladder_type.
Coercion expr_var : string >-> expr.


(*
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.