From Coq Require Import Lists.List.
Require Import Atom.
Require Import Environment.
Require Import Metatheory.
Require Import debruijn.
Require Import subtype.
Require Import env.
Require Import morph.
Require Import subst_lemmas.
Require Import typing.


Lemma typing_inv_tabs : forall Γ t τ,
  (Γ |- [{ Λ t }] \is [< ∀ τ >]) ->
  forall L x, x `notin` L ->
  (Γ |- (expr_open_type (ty_fvar x) t) \is τ)
.
Proof.
Admitted.


Lemma typing_inv_abs : forall Γ σ t τ,
  (Γ |- [{ λ σ ↦ t }] \is [< σ -> τ >]) ->
  forall L x, x `notin` L ->
  ((x,σ)::Γ |- (expr_open (ex_fvar x) t) \is τ)
.
Proof.
Admitted.


Lemma typing_inv_morph : forall Γ σ t τ,
  (Γ |- [{ λ σ ↦morph t }] \is [< σ ->morph τ >]) ->
  forall L x, x `notin` L ->
  ((x,σ)::Γ |- (expr_open (ex_fvar x) t) \is τ)
.
Proof.
  intros.
  
  inversion H.
  subst.
  
Admitted.


Lemma typing_inv_let : forall Γ s σ t τ,
  (Γ |- s \is σ) ->
  (Γ |- [{ let s in t }] \is [< τ >]) ->
  forall L x, x `notin` L ->
  ((x,σ)::Γ |- (expr_open (ex_fvar x) t) \is τ)
.
Proof.
Admitted.