From Coq Require Import Lists.List.
Import ListNotations.
Require Import Atom.
Require Import terms_debruijn.
Require Import subtype_debruijn.
Require Import context_debruijn.
Require Import morph_debruijn.

Open Scope ladder_type_scope.
Open Scope ladder_expr_scope.

Reserved Notation "Γ '|-' x '\is' X"  (at level 101).
Inductive typing : context -> expr_DeBruijn -> type_DeBruijn -> Prop :=
  | T_Var : forall Γ x τ,
    (In (x, τ) Γ) ->
    (Γ |- [{ $x }] \is τ)

  | T_Let : forall Γ s σ t τ x,
    (Γ |- s \is σ) ->
    (((x σ) :: Γ) |- t \is τ) ->
    (Γ |- [{ let s in t }] \is τ)

  | T_TypeAbs : forall Γ e τ,
    (Γ |- e \is τ) ->
    (Γ |- [{ Λ e }] \is [< ∀ τ >])

  | T_TypeApp : forall Γ e σ τ,
    (Γ |- e \is [< ∀ τ >]) ->
    (Γ |- [{ e # σ }] \is (type_open σ τ))

  | T_Abs : forall Γ x σ t τ,
    (((x σ) :: Γ) |- t \is τ) ->
    (Γ |- [{ λ σ ↦ t }] \is [< σ -> τ >])

  | T_MorphAbs : forall Γ x σ t τ,
    (((x σ) :: Γ) |- t \is τ) ->
    (Γ |- [{ λ σ ↦morph t }] \is [< σ ->morph τ >])

  | T_App : forall Γ f a σ' σ τ,
    (Γ |- f \is [< σ -> τ >]) ->
    (Γ |- a \is σ') ->
    (Γ |- σ' ~~> σ) ->
    (Γ |- [{ f a }] \is τ)

  | T_MorphFun : forall Γ f σ τ,
    (Γ |- f \is [< σ ->morph τ >]) ->
    (Γ |- f \is [< σ -> τ >])

  | T_Ascend : forall Γ e τ τ',
    (Γ |- e \is τ) ->
    (Γ |- [{ e as τ' }] \is [< τ' ~ τ >])

  | T_DescendImplicit : forall Γ x τ τ',
    (Γ |- x \is τ) ->
    (τ :<= τ') ->
    (Γ |- x \is τ')

  | T_Descend : forall Γ x τ τ',
    (Γ |- x \is τ) ->
    (τ :<= τ') ->
    (Γ |- [{ x des τ' }] \is τ')

where "Γ '|-' x '\is' τ" := (typing Γ x τ).