ladder-calculus/coq/terms.v

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(* 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.