add preconditions of expr_lc in eval; use coinductive quantification in T_TypeAbs

coq/lemmas/typing_regular.v

@@ -0,0 +1,46 @@
+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 typing.
+
+
+Lemma type_lc_sub : forall τ1 τ2,
+  type_lc τ1 ->
+  τ1 :<= τ2 ->
+  type_lc τ2
+.
+Proof.
+  intros.
+Admitted.
+
+Lemma type_morph_lc_inv : forall τ1 τ2,
+  type_lc (ty_morph τ1 τ2) ->
+  type_lc τ1 /\ type_lc τ2
+.
+Proof.
+  intros.
+  inversion H.
+  auto.
+Qed.
+
+Lemma typing_regular_expr_lc : forall Γ e τ,
+  (Γ |- e \is τ) ->
+  expr_lc e.
+Proof.
+  intros Γ e τ H.
+  induction H; eauto.
+  - apply Elc_Var.
+  - apply Elc_Let with (L:=L).
+    apply IHtyping.
+Admitted.
+
+Lemma typing_regular_type_lc : forall Γ e τ,
+  (Γ |- e \is τ) ->
+  type_lc τ.
+Proof.
+Admitted.