add expr_open_lc and expr_subst_open lemmas

coq/lemmas/subst_lemmas.v

@@ -1,3 +1,5 @@
+From Coq Require Import Lists.List.
+Import ListNotations.
 Require Import Atom.
 Require Import Metatheory.
 Require Import FSetNotin.
@@ -103,7 +105,7 @@ Lemma type_open_lc_core : forall (τ:type_DeBruijn) i (σ1:type_DeBruijn) j (σ2
   {i ~tt~> σ1} τ = {j ~tt~> σ2} ({i ~tt~> σ1} τ) ->
   ({j ~tt~> σ2} τ) = τ
 .
-Proof with eauto*.
+Proof.
   induction τ;
   intros i σ1 j σ2 Neq H;
   simpl in *;
@@ -181,3 +183,95 @@ Proof.
     reflexivity.
     trivial.
 Qed.
+
+
+
+
+
+
+
+
+Lemma expr_open_lc_core : forall (t:expr_DeBruijn) i (s1:expr_DeBruijn) j (s2:expr_DeBruijn),
+  i <> j ->
+  {i ~ee~> s1} t = {j ~ee~> s2} ({i ~ee~> s1} t) ->
+  ({j ~ee~> s2} t) = t
+.
+Proof.
+  induction t;
+  intros i s1 j s2 Neq H;
+  simpl in *;
+  inversion H;
+  f_equal; eauto.
+
+  (* bound var *)
+  - simpl in *.
+    destruct (j === n).
+    destruct (i === n).
+    3:reflexivity.
+
+    rewrite e,e0 in Neq.
+    contradiction Neq.
+    reflexivity.
+
+    rewrite H,e.
+    simpl.
+    destruct (n===n).
+    reflexivity.
+    contradict n1.
+    reflexivity.
+Qed.
+
+
+(*
+ * opening is idempotent on locally closed expressions
+ *)
+Lemma expr_open_lc : forall k s t,
+  expr_lc t ->
+  ({ k ~ee~> s } t) = t
+.
+Proof.
+  intros.
+  generalize dependent k.
+  induction H; eauto; simpl in *; intro k; f_equal; eauto.
+
+  - unfold expr_open in *.
+    pick fresh x for L.
+    apply expr_open_lc_core with (i:=0) (s1:=(ex_fvar x)) (j:=S k) (s2:=s).
+    discriminate.
+    apply eq_sym, H1.
+    assumption.
+
+  - unfold expr_open in *.
+    pick fresh x for L.
+    apply expr_open_lc_core with (i:=0) (s1:=(ex_fvar x)) (j:=S k) (s2:=s).
+    discriminate.
+    apply eq_sym, H1.
+    assumption.
+Qed.
+
+(*
+ * type substitution distributes over opening
+ *)
+Lemma expr_subst_open : forall t s1 s2 x k,
+  expr_lc s2 ->
+
+  [x ~ee~> s2] ({k ~ee~> s1} t)
+  =
+  {k ~ee~> [x ~ee~> s2] s1} ([x ~ee~> s2] t).
+Proof.
+  induction t;
+  intros; simpl; f_equal; auto.
+
+  (* free var *)
+  - destruct (x == a).
+    subst.
+    apply eq_sym, expr_open_lc.
+    assumption.
+    trivial.
+
+  (* bound var *)
+  - destruct (k === n).
+    reflexivity.
+    trivial.
+Qed.
+