(* 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_morph : 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_abs_morph : 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.

(*
Coercion type_var : string >-> type_term.
Coercion expr_var : string >-> expr_term.
*)

Declare Scope ladder_type_scope.
Declare Scope ladder_expr_scope.
Declare Custom Entry ladder_type.

Notation "[ e ]" := e (e custom ladder_type at level 80) : ladder_type_scope.

(* TODO: allow any variable names in notation, not just α,β,γ *)
Notation "'∀α.' τ" := (type_univ "α" τ) (in custom ladder_type at level 80) : ladder_type_scope.
Notation "'∀β.' τ" := (type_univ "β" τ) (in custom ladder_type at level 80) : ladder_type_scope.
Notation "'∀γ.' τ" := (type_univ "γ" τ) (in custom ladder_type at level 80) : ladder_type_scope.
Notation "'<' σ τ '>'" := (type_spec σ τ) (in custom ladder_type at level 80, left associativity) : ladder_type_scope.
Notation "'(' τ ')'" := τ (in custom ladder_type at level 70) : ladder_type_scope.
Notation "σ '->' τ" := (type_fun σ τ) (in custom ladder_type at level 75, right associativity) : ladder_type_scope.
Notation "σ '->morph' τ" := (type_morph σ τ) (in custom ladder_type at level 75, right associativity) : ladder_type_scope.
Notation "σ '~' τ" := (type_ladder σ τ) (in custom ladder_type at level 70, right associativity) : ladder_type_scope.
Notation "'α'" := (type_var "α") (in custom ladder_type at level 60, right associativity) : ladder_type_scope.
Notation "'β'" := (type_var "β") (in custom ladder_type at level 60, right associativity) : ladder_type_scope.
Notation "'γ'" := (type_var "γ") (in custom ladder_type at level 60, right associativity) : ladder_type_scope.

Open Scope ladder_type_scope.

Definition t1 : type_term := [ ∀α.∀β.(α~β~γ)->β->(α->α)->β ].

Compute t1.
Close Scope ladder_type_scope.



End Terms.