diff --git a/theories/binders.v b/theories/binders.v
index 72c0f7e8929fd984cf8951e152416af74adc9757..da5dea024d51570fa3c21c96eefd023537d7b739 100644
--- a/theories/binders.v
+++ b/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.