From Coq Require Import Strings.String.
Require Import terms.

Include Terms.

Module Subst.
(* scoped variable substitution in type terms $\label{coq:subst-type}$ *)
Fixpoint type_subst (v:string) (n:type_term) (t0:type_term) :=
  match t0 with
  | type_var name => if (eqb v name) then n else t0
  | type_fun t1 t2 => (type_fun (type_subst v n t1) (type_subst v n t2))
  | type_univ x t => if (eqb v x) then t0 else type_univ x (type_subst v n t)
  | type_spec t1 t2 => (type_spec (type_subst v n t1) (type_subst v n t2))
  | type_ladder t1 t2 => (type_ladder (type_subst v n t1) (type_subst v n t2))
  | t => t
  end.

(* scoped variable substitution, replaces free occurences of v with n in expression e *)
Fixpoint expr_subst (v:string) (n:expr_term) (e0:expr_term) :=
  match e0 with
  | expr_var name => if (eqb v name) then n else e0
  | expr_ty_abs x e => if (eqb v x) then e0 else expr_ty_abs x (expr_subst v n e)
  | expr_ty_app e t => expr_ty_app (expr_subst v n e) t
  | expr_tm_abs x t e => if (eqb v x) then e0 else expr_tm_abs x t (expr_subst v n e)
  | expr_tm_abs_morph x t e => if (eqb v x) then e0 else expr_tm_abs_morph x t (expr_subst v n e)
  | expr_tm_app e a => expr_tm_app (expr_subst v n e) (expr_subst v n a)
  | expr_let x t a e => expr_let x t (expr_subst v n a) (expr_subst v n e)
  | expr_ascend t e => expr_ascend t (expr_subst v n e)
  | expr_descend t e => expr_descend t (expr_subst v n e)
  end.

(* replace only type variables in expression *)
Fixpoint expr_specialize (v:string) (n:type_term) (e0:expr_term) :=
  match e0 with
  | expr_ty_app e t => expr_ty_app (expr_specialize v n e) (type_subst v n t)
  | expr_ascend t e => expr_ascend (type_subst v n t) (expr_specialize v n e)
  | expr_descend t e => expr_descend (type_subst v n t) (expr_specialize v n e)
  | e => e
  end.

End Subst.