Proofs about binders

Now that POPL deadline is over, here are some more lemmata. As always let me know if I should change anything.

theories/binders.v

@@ -21,29 +21,50 @@ Instance binder_inhabited : Inhabited binder := populate BAnon.
 Instance binder_countable : Countable binder.
 Proof.
  refine (inj_countable'
-   (λ mx, match mx with BAnon => None | BNamed x => Some x end)
-   (λ mx, match mx with None => BAnon | Some x => BNamed x end) _); by intros [].
+   (λ b, match b with BAnon => None | BNamed s => Some s end)
+   (λ b, match b with None => BAnon | Some s => BNamed s end) _); by intros [].
 Qed.
 
-(** The functions [cons_binder mx X] and [app_binder mxs X] are typically used
-to collect the free variables of an expression. Here [X] is the current list of
-free variables, and [mx], respectively [mxs], are the binders that are being
+(** The functions [cons_binder b ss] and [app_binder bs ss] are typically used
+to collect the free variables of an expression. Here [ss] is the current list of
+free variables, and [b], respectively [bs], are the binders that are being
 added. *)
-Definition cons_binder (mx : binder) (X : list string) : list string :=
-  match mx with BAnon => X | BNamed x => x :: X end.
+Definition cons_binder (b : binder) (ss : list string) : list string :=
+  match b with BAnon => ss | BNamed s => s :: ss end.
 Infix ":b:" := cons_binder (at level 60, right associativity).
-Fixpoint app_binder (mxs : list binder) (X : list string) : list string :=
-  match mxs with [] => X | b :: mxs => b :b: app_binder mxs X end.
+Fixpoint app_binder (bs : list binder) (ss : list string) : list string :=
+  match bs with [] => ss | b :: bs => b :b: app_binder bs ss end.
 Infix "+b+" := app_binder (at level 60, right associativity).
 
-Instance set_unfold_cons_binder x mx X P :
-  SetUnfoldElemOf x X P → SetUnfoldElemOf x (mx :b: X) (BNamed x = mx ∨ P).
+Instance set_unfold_cons_binder s b ss P :
+  SetUnfoldElemOf s ss P → SetUnfoldElemOf s (b :b: ss) (BNamed s = b ∨ P).
 Proof.
-  constructor. rewrite <-(set_unfold (x ∈ X) P).
-  destruct mx; simpl; rewrite ?elem_of_cons; naive_solver.
+  constructor. rewrite <-(set_unfold (s ∈ ss) P).
+  destruct b; simpl; rewrite ?elem_of_cons; naive_solver.
 Qed.
-Instance set_unfold_app_binder x mxl X P :
-  SetUnfoldElemOf x X P → SetUnfoldElemOf x (mxl +b+ X) (BNamed x ∈ mxl ∨ P).
+Instance set_unfold_app_binder s bs ss P Q :
+  SetUnfoldElemOf (BNamed s) bs P → SetUnfoldElemOf s ss Q →
+  SetUnfoldElemOf s (bs +b+ ss) (P ∨ Q).
 Proof.
-  constructor. rewrite <-(set_unfold (x ∈ X) P). induction mxl; set_solver.
+  intros HinP HinQ.
+  constructor. rewrite <-(set_unfold (s ∈ ss) Q), <-(set_unfold (BNamed s ∈ bs) P).
+  clear HinP HinQ.
+  induction bs; set_solver.
+Qed.
+
+Lemma app_binder_named ss1 ss2 : (BNamed <$> ss1) +b+ ss2 = ss1 ++ ss2.
+Proof. induction ss1; by f_equal/=. Qed.
+
+Lemma app_binder_snoc bs s ss : bs +b+ (s :: ss) = (bs ++ [BNamed s]) +b+ ss.
+Proof. induction bs; by f_equal/=. Qed.
+
+Instance cons_binder_Permutation b : Proper ((≡ₚ) ==> (≡ₚ)) (cons_binder b).
+Proof. intros ss1 ss2 Hss. destruct b; csimpl; by rewrite Hss. Qed.
+
+Instance app_binder_Permutation : Proper ((≡ₚ) ==> (≡ₚ) ==> (≡ₚ)) app_binder.
+Proof.
+  assert (∀ bs, Proper ((≡ₚ) ==> (≡ₚ)) (app_binder bs)).
+  { induction bs as [|[]]; intros ss1 ss2; simpl; by intros ->. }
+  induction 1 as [|[]|[] []|]; intros ss1 ss2 Hss; simpl;
+    first [by eauto using perm_trans|by rewrite 1?perm_swap, Hss].
 Qed.