Unfolding lemma for Fix in setoids.

This generalizes Fix_unfold to a setoid setting. In particular,
we can use this to unfold multi-argument fixpoints without
requiring functional extensionality.

theories/relations.v

@@ -235,3 +235,16 @@ Lemma Fix_F_proper `{R : relation A} (B : A → Type) (E : ∀ x, relation (B x)
     (x : A) (acc1 acc2 : Acc R x) :
   E _ (Fix_F B F acc1) (Fix_F B F acc2).
 Proof. revert x acc1 acc2. fix 2. intros x [acc1] [acc2]; simpl; auto. Qed.
+
+Lemma Fix_unfold_rel `{R: relation A} (wfR: wf R) (B: A → Type) (E: ∀ x, relation (B x))
+      (F: ∀ x, (∀ y, R y x → B y) → B x)
+      (HF: ∀ (x: A) (f g: ∀ y, R y x → B y),
+             (∀ y Hy Hy', E _ (f y Hy) (g y Hy')) → E _ (F x f) (F x g))
+      (x: A):
+  E _ (Fix wfR B F x) (F x (λ y _, Fix wfR B F y)).
+Proof.
+  unfold Fix.
+  destruct (wfR x); cbn.
+  apply HF; intros.
+  apply Fix_F_proper; auto.
+Qed.