Define `wf_guard` in terms of Coq stdlib's `Acc_intro_generator`.

theories/relations.v

@@ -194,33 +194,8 @@ End subrel.
 
 (** * Theorems on well founded relations *)
 Notation wf := well_founded.
-
-Section wf.
-  Context `{R : relation A}.
-
-  Lemma Acc_def x : Acc R x ↔ (∀ y : A, R y x → Acc R y).
-  Proof. split. by destruct 1. by constructor. Qed.
-
-  (** A trick by Thomas Braibant to compute with well-founded recursions:
-  it lazily adds [2^n] [Acc_intro] constructors in front of a well foundedness
-  proof, so that the actual proof is never reached in practise. *)
-  Fixpoint wf_guard (n : nat) (wfR : wf R) : wf R :=
-    match n with
-    | 0 => wfR
-    | S n => λ x, Acc_intro x (λ y _, wf_guard n (wf_guard n wfR) y)
-    end.
-
-  Lemma wf_projected `(R2 : relation B) (f : A → B) :
-    (∀ x y, R x y → R2 (f x) (f y)) →
-    wf R2 → wf R.
-  Proof.
-    intros Hf Hwf.
-    cut (∀ y, Acc R2 y → ∀ x, y = f x → Acc R x).
-    { intros aux x. apply (aux (f x)); auto. }
-    induction 1 as [y _ IH]. intros x ?. subst.
-    constructor. intros. apply (IH (f y)); auto.
-  Qed.
-End wf.
+Definition wf_guard `{R : relation A} (n : nat) (wfR : wf R) : wf R :=
+  Acc_intro_generator n wfR.
 
 (* Generally we do not want [wf_guard] to be expanded (neither by tactics,
 nor by conversion tests in the kernel), but in some cases we do need it for
@@ -228,6 +203,17 @@ computation (that is, we cannot make it opaque). We use the [Strategy]
 command to make its expanding behavior less eager. *)
 Strategy 100 [wf_guard].
 
+Lemma wf_projected `{R1 : relation A} `(R2 : relation B) (f : A → B) :
+  (∀ x y, R1 x y → R2 (f x) (f y)) →
+  wf R2 → wf R1.
+Proof.
+  intros Hf Hwf.
+  cut (∀ y, Acc R2 y → ∀ x, y = f x → Acc R1 x).
+  { intros aux x. apply (aux (f x)); auto. }
+  induction 1 as [y _ IH]. intros x ?. subst.
+  constructor. intros. apply (IH (f y)); auto.
+Qed.
+
 Lemma Fix_F_proper `{R : relation A} (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),