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

Include Terms.
Include Subst.

Module Smallstep.

Reserved Notation " s '-->α' t " (at level 40).
Reserved Notation " s '-->β' t " (at level 40).
Reserved Notation " s '-->δ' t " (at level 40).

Inductive beta_step : expr -> expr -> Prop :=
  | E_AppLeft : forall e1 e1' e2,
    e1 -->β e1' ->
    (expr_tm_app e1 e2) -->β (expr_tm_app e1' e2)

  | E_AppRight : forall e1 e2 e2',
    e2 -->β e2' ->
    (expr_tm_app e1 e2) -->β (expr_tm_app e1 e2')

  | E_AppTmAbs : forall x τ e a,
    (expr_tm_app (expr_tm_abs x τ e) a) -->β (expr_subst x a e)

  | E_AppTyAbs : forall x e a,
    (expr_ty_app (expr_ty_abs x e) a) -->β (expr_specialize x a e)

  | E_AppLet : forall x t e a,
    (expr_let x t a e) -->β (expr_subst x a e)

where "s '-->β' t" := (beta_step s t).

(*
Inductive multi {X : Type} (R : relation X) : relation X :=
  | multi_refl : forall (x : X), multi R x x
  | multi_step : forall (x y z : X),
                    R x y ->
                    multi R y z ->
                    multi R x z.

Notation " s -->β* t " := (multi beta_step s t) (at level 40).
*)

(*
Inductive delta_expand : expr -> expr -> Prop :=
  | E_ImplicitCast
  (expr_tm_app e1 e2)
*)

End Smallstep.