From Coq Require Import Lists.List.
Import ListNotations.
Require Import Atom.
Require Import debruijn.
Require Import subtype.
Require Import morph.
Require Import typing.

Open Scope ladder_expr_scope.

Inductive is_value : expr_DeBruijn -> Prop :=
  | V_TAbs : forall e,
    expr_lc [{ Λ e }] ->
    is_value [{ Λ e }]

  | V_Abs : forall σ e,
    expr_lc [{ λ σ ↦ e }] ->
    is_value [{ λ σ ↦ e }]

  | V_Morph : forall σ e,
    is_value [{ λ σ ↦morph e }]

  | V_Ascend : forall τ e,
    is_value e ->
    is_value [{ e as τ }]

  | V_Descend : forall τ e,
    is_value e ->
    is_value [{ e des τ }]
.


Reserved Notation " s '-->eval' t " (at level 40).

Inductive eval : expr_DeBruijn -> expr_DeBruijn -> Prop :=
  | E_App1 : forall e1 e1' e2,
    (expr_lc e2) ->
    e1 -->eval e1' ->
    [{ e1 e2 }] -->eval [{ e1' e2 }]

  | E_App2 : forall v1 e2 e2',
    (is_value v1) ->
    e2 -->eval e2' ->
    [{ v1 e2 }] -->eval [{ v1 e2' }]

  | E_TypApp : forall e e',
    expr_lc [{ Λ e }] ->
    e -->eval e' ->
    [{ Λ e }] -->eval [{ Λ e' }]

  | E_TypAppLam : forall e τ,
    expr_lc [{ (Λ e ) }] ->
    [{ (Λ e) # τ }] -->eval (expr_open_type τ e)

  | E_AppLam : forall τ e a,
    expr_lc [{ (λ τ ↦ e) }] ->
    [{ (λ τ ↦ e) a }] -->eval (expr_open a e)

  | E_AppMorph : forall τ e a,
    expr_lc [{ (λ τ ↦morph e) }] ->
    [{ (λ τ ↦morph e) a }] -->eval (expr_open a e)

  | E_Let : forall e a,
    expr_lc [{ let a in e }] ->
    [{ let a in e }] -->eval (expr_open a e)

  | E_Ascend : forall τ e e',
    (e -->eval e') ->
    [{ e as τ }] -->eval [{ e' as τ }]

  | E_AscendCollapse : forall τ' τ e,
    [{ (e as τ) as τ' }] -->eval [{ e as (τ'~τ) }]

  | E_DescendCollapse : forall τ' τ e,
    (τ':<=τ) ->
    [{ (e des τ') des τ }] -->eval [{ e des τ }]

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