How do I rewrite negb true to false in Coq?

Question

How do I rewrite negb true to false in Coq?

I've been searching in the libraries for a simply way to rewrite negb true to false .

However, I haven't found anything useful.

I know of simpl. , but I prefer the more basic syntax.

Answer 1

When you search for lemmas the search engine looks in the imported modules only.

That's why you first need to

Require Import Bool.

Then

Search (negb _ = false).

reveals the lemma

negb_false_iff: forall b : bool, negb b = false <-> b = true

You can use the lemma for rewriting if you Require Import Setoid .

Goal negb true = false.
  rewrite negb_false_iff. reflexivity. Qed.

You, probably, don't want to use simpl because you have negb true in some context and simpl messes up your goal/hypotheses, because it unfolds too much and creates large unreadable terms. You can narrow the context in which simpl applies by using it this way:

simpl (negb true).       (* rewrites `negb true` to `false` in the goal *)

or

simpl (negb true) in *.  (* rewrites `negb true` in the goal and hypotheses *)

Answer 2

You already know that simpl will indeed reduce negb true to false . This is because the two are definitionaly equivalent.

With this knowledge you could first prove the statement that you need, then rewrite with it:

assert (H: negb true = false) by reflexivity.
rewrite H.

But even better, you can use change to do all of this in just one line:

change (negb true) with false.