(** _Note_: One might want to eliminate the use of [pattern] in the proof script by replacing the corresponding line with the following:  [rewrite sunf_eq at 1; simpl.]  However, doing so will result in an invalid proof (rejected at [Qed]-time).  The reason is that the proof term created by [rewrite] ... [at 1] involves some lemmas from the [Setoid] library.  Like most lemmas, these are opaque, so guardedness checking cannot inspect their proofs and thus fails.  This is a great example of two of the previously mentioned problems with the [cofix] approach, namely lack of compositionality and poor interaction with standard tactics. *)(* *)
(** *** A proof using our [pcofix]  We will now do the same proof, but using the [paco] library.  We first simply show the new (but equivalent) definition of stream equality and the new proof of the example, and then explain what is going on behind the scenes.  In both, we use [paco] constructs with suffix "2" because we are dealing with predicates of arity 2 here.  _Note_: [Paco] supports predicates of arity up to 8.  Also, it supports up to three mutually coinductive predicates (see the last example of this tutorial).  In either case, extending this is just a matter of copy and paste.  We also have to prove monotonicity of the generating function [seq_gen], which can be discharged by the tactic [pmonauto], and register it in the Hint databse [paco].  _Remark_: Unlike [CoInductive], [paco] does not care whether the generating function is given in a _strictly positive_ syntactic form; all that matters is that the function is monotone. More specifically, [paco2 f] is well defined for an arbitrary generating function [f] regardless of whether it is monotone or not. However, in order to ensure that [paco2 f bot2] is the greatest fixed point of [f], we need monotonicity of [f]. *)
Hint Unfold seq'.

Lemma seq_gen_mon: monotone2 seq_gen. Proof. pmonauto. Qed.
Hint Resolve seq_gen_mon : paco.