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 transl_inv_let : forall Γ s s' σ t t' τ,
  (Γ |- s \is σ) ->
  (Γ |- [[ [{ let s in t }] \is τ ]] = [{ let s' in t' }]) ->
  forall L x, x `notin` L ->
  ((x, σ) :: Γ |- [[expr_open [{$ x}] t \is τ]] = expr_open [{$ x}] t')
.
Proof.
Admitted.

Lemma transl_inv_abs : forall Γ σ e e' τ,
  (Γ |- [[ [{ λ σ ↦ e }] \is τ ]] = [{ λ σ ↦ e' }]) ->
  forall L x, x `notin` L ->
  ((x, σ) :: Γ |- [[expr_open [{$ x}] e \is τ]] = expr_open [{$ x}] e')
.
Proof.
Admitted.

Lemma transl_inv_morph : forall Γ σ e e' τ,
  (Γ |- [[ [{ λ σ ↦morph e }] \is τ ]] = [{ λ σ ↦morph e' }]) ->
  forall L x, x `notin` L ->
  ((x, σ) :: Γ |- [[expr_open [{$ x}] e \is τ]] = expr_open [{$ x}] e')
.
Proof.
Admitted.

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