Add syntax.v

repl25-notes/syntax.v

@@ -0,0 +1,117 @@
+Inductive Term : Set :=
+| Var : nat -> Term
+| App : Term -> Term -> Term
+| Abs : Term -> Term.
+
+(* Renamings *)
+(* nat -> nat *)
+Definition id (x : nat) : nat := x.
+Definition shift (x : nat) : nat := S x.
+Definition comp (f g : nat -> nat) (x : nat) :=
+  f (g x).
+Definition ext (f : nat -> nat) (x : nat)
+  : nat -> nat :=
+  fun y =>
+    match y with
+    | 0 => x
+    | S z => f z
+    end.
+Definition lift (f : nat -> nat) :=
+  ext (comp shift f) 0.
+
+Compute (id 0).
+Compute (id 1).
+Compute (id 2).
+Lemma id_is_id : forall x,  id x = x.
+  reflexivity.
+Qed.
+
+Compute (ext id 2) 0.
+Compute (ext id 2) 1.
+Compute (ext id 2) 2.
+
+Lemma ext_zero : forall f x,  (ext f x) 0 = x.
+  reflexivity.
+Qed.
+
+Lemma ext_succ : forall f x y,  (ext f x) (S y) = f y.
+Proof.
+  reflexivity.
+Qed.
+
+Lemma lift_zero : forall f, (lift f) 0 = 0.
+Proof.
+  reflexivity.
+Qed.
+
+Lemma lift_succ : forall f x,
+    (lift f) (S x) = S (f x).
+  reflexivity.
+Qed.
+
+Fixpoint ren (ξ : nat -> nat) (a : Term)
+  : Term :=
+  match a with
+  | Var n => Var (ξ n)
+  | App b0 b1 => App (ren ξ b0) (ren ξ b1)
+  | Abs a => Abs (ren (lift ξ) a)
+  end.
+
+Require Import Logic.FunctionalExtensionality.
+
+Lemma lift_id_is_id_helper : forall x, lift id x = id x.
+Proof.
+  now destruct x.
+Qed.
+
+Lemma lift_id_is_id : lift id = id.
+Proof.
+  apply functional_extensionality.
+  apply lift_id_is_id_helper.
+Qed.
+
+Lemma ren_id : forall (a : Term), ren id a = a.
+Proof.
+  intros a.
+  induction a.
+  - easy.
+  - simpl.
+    rewrite IHa1.
+    rewrite IHa2.
+    easy.
+  - simpl.
+    rewrite lift_id_is_id.
+    rewrite IHa.
+    easy.
+Qed.
+
+Lemma lift_comp ξ0 ξ1 :
+  lift (comp ξ0 ξ1) = comp (lift ξ0) (lift ξ1).
+Proof.
+  apply functional_extensionality.
+  now destruct x.
+Qed.
+
+Lemma ren_comp : forall (a : Term) (ξ0 ξ1 : nat -> nat), ren (comp ξ0 ξ1) a = ren ξ0 (ren ξ1 a).
+Proof.
+  induction a.
+  - easy.
+  - intros ξ0 ξ1.
+    simpl.
+    rewrite IHa1.
+    rewrite IHa2.
+    easy.
+  - intros ξ0 ξ1.
+    simpl.
+    rewrite lift_comp.
+    rewrite IHa.
+    easy.
+Qed.
+
+
+Lemma ren_comp' : forall (ξ0 ξ1 : nat -> nat) a, ren (comp ξ0 ξ1) a = ren ξ0 (ren ξ1 a).
+Proof.
+  intros ξ0 ξ1 a.
+  revert a ξ0 ξ1.
+  apply ren_comp.
+Qed.