Eliminating cofix in Coq proof
(* I post my answer as a Coq file. In it I show that supercoooldave's
   definition of a universal tree is not what he intended. His isUniv
   means "the tree has an infinite branch". I provide the correct
   definition, show that the universal tree is universal according to
   the new definition, and I provide counter-examples to
   supercooldave's definition. I also point out that the universal
   tree of branching type A has an infinite path iff A is inhabited.
   *)

Set Implicit Arguments.

CoInductive Tree (A : Set): Set := Node : (A -&gt; option (Tree A)) -&gt; Tree A.

Definition child (A : Set) (t : Tree A) (a : A) :=
  match t with
    Node f =&gt; f a
  end.

(* We consider two trees, one is the universal tree on A (always
   branches out fully), and the other is a binary tree which always
   branches to one side and not to the other, so it is like an
   infinite path with branches of length 1 shooting off at each node.  *)

CoFixpoint Univ (A : Set) : Tree A := Node (fun _ =&gt; Some (Univ A)).

CoFixpoint Thread : Tree (bool) :=
  Node (fun (b : bool) =&gt; if b then Some Thread else None).

(* The original definition of supercooldave should be called "has an
   infinite path", so we rename it to "hasInfinitePath". *)
CoInductive hasInfinitePath (A : Set) : Tree A -&gt; Prop :=
  haspath : forall (f : A -&gt; option (Tree A)) (a : A) (t : Tree A),
    f a = Some t -&gt;
    hasInfinitePath t -&gt; 
    hasInfinitePath (Node f).

(* The correct definition of universal tree. *)
CoInductive isUniv (A : Set) : Tree A -&gt; Prop :=
  isuniv : forall (f : A -&gt; option (Tree A)),
    (forall  a, exists t, f a = Some t /\ isUniv t) -&gt; 
    isUniv (Node f).

(* Technicalities that allow us to get coinductive proofs done. *)
Definition TT (A : Set) (t : Tree A) :=
  match t with
    | Node o =&gt; Node o
  end.

Lemma TTid (A : Set) : forall t: Tree A, t = TT t.
  intros A t.
  destruct t.
  reflexivity.
  Qed.

(* Thread has an infinite path. *)
Lemma ThreadHasInfinitePath : hasInfinitePath Thread.
Proof.
  cofix H.
  rewrite TTid.
  unfold TT.
  unfold Thread.
  (* there is a path down the "true" branch leading to Thread. *)
  apply haspath with (a := true) (t := Thread).
  auto.
  auto.
Qed.

(* Auxiliary lemma *)
Lemma univChildNotNone (A : Set) (t : Tree A) (a : A):
  isUniv t -&gt; (child t a &lt;&gt; None).
Proof.
  intros A t a [f H].
  destruct (H a) as [u [G _]].
  unfold child.
  rewrite G.
  discriminate.
Qed.

(* Thread is not universal. *)
Lemma ThreadNotUniversal: ~ (isUniv Thread).
Proof.
  unfold not.
  intro H.
  eapply univChildNotNone with (t := Thread) (a := false).
  auto.
  unfold Thread, child.
  auto.
Qed.

(* Now let us show that Univ is universal. *)
Lemma univIsuniv (A : Set): isUniv (Univ A).
Proof.
  intro A.
  cofix H.
  rewrite TTid.
  unfold TT.
  unfold Univ.
  apply isuniv.
  intro a.
  exists (Univ A).
  auto.
Qed.

(* By the way, it need not be the case that a universal tree has
   an infinite path! In fact, the universal tree of branching type
   A has an infinite path iff A is inhabited. *)

Lemma whenUnivHasInfiniteBranch (A : Set):
  hasInfinitePath (Univ A) &lt;-&gt; exists a : A, True.
Proof.
  intro A.
  split.
  intro H.
  destruct H as [f a t _].
  exists a.
  trivial.
  intros [a _].
  cofix H.
  rewrite TTid.
  unfold TT.
  unfold Univ.
  apply haspath with (t := Univ A); auto.
Qed.