Hi, I'm self-learning Coq and arrived here. So, given these:

```
Lemma leb_spec : forall (n m : nat),
leb n m = true \/ (leb n m = false /\ leb m n = true).
Lemma leb_nm__leb_mn : forall (n m : nat),
leb n m = false -> leb m n = true.
```

the author(s) claims that `leb_nm__leb_mn`

is a specialisation of `leb_spec`

, and that confuses me.

Problems:

- I can prove
`leb_nm__leb_mn`

without using`leb_spec`

. Still, I couldn't see how these two are related. - What does it mean for a theorem A to be a specialisation of another theorem B?

I guess it means theorem B is a more general theorem, and that theorem A can be proven simply by proving a special case. But from my POV, that is not the case with`leb_nm__leb_mn`

and`leb_spec`

.

`leb_nm__leb_mn`

is indeed a specialization of `leb_spec`

. You can prove `leb_nm__leb_mn`

without ever looking at the properties of `leb`

. In fact, you could just as well prove the following:

```
Lemma foo : forall p q : bool,
(p = true \/ (p = false /\ q = true)) ->
(p = false -> q = true).
```

Note that you can also prove the converse implication. So you could also say the second lemma is not a specialization / corollary of the first one, but just an equivalent formulation. Both lemmas are just as strong as the other.

Thanks, now it makes sense to me. As for the converse implication, it'll involve proving this:

```
Lemma implies_or : forall p q : Prop,
(p -> q) <-> (~ p \/ q).
```

Can this be proven in Coq?

Not for arbitrary inhabitants of `Prop`

. But if you replace your atoms by `p = true`

and `q = true`

, then it becomes provable.

(Actually, you only need to make `p`

a boolean; `q`

can stay in `Prop`

.)

If `forall p q`

is really needed, I suppose classical logic could be used, but then you lose the ability to do extraction (heard something like that recently from here :smile: ).

Classical logic in Prop is perfectly fine w.r.t. extraction. More generally any consistent set of axioms in Prop are fine, that's the whole point of the Prop-Type separation.

Extraction is possible even with `p \/ ~p`

added as axiom?

Oh is it that Prop is erased away during extraction?

Is there any disadvantage in using classical logic, though?

Guillaume Melquiond said:

Not for arbitrary inhabitants of

`Prop`

. But if you replace your atoms by`p = true`

and`q = true`

, then it becomes provable.(Actually, you only need to make

`p`

a boolean;`q`

can stay in`Prop`

.)

Make sense, thanks!

Jiahong Lee has marked this topic as resolved.

Julin S said:

Is there any disadvantage in using classical logic, though?

Ermmm, I think it deserves as a new thread? Thanks!

Maybe yeah. Still, I had thought the extraction part was what would get in trouble because if excluded middle is there, we can't say which clause was true to make the whole thing true.

You could state excluded middle in Type (forall T : Type, {T} + {~T}), and then you'd definitely have extraction trouble.

Jiahong Lee said:

Hi, I'm self-learning Coq and arrived here. So, given these:

`Lemma leb_spec : forall (n m : nat), leb n m = true \/ (leb n m = false /\ leb m n = true). Lemma leb_nm__leb_mn : forall (n m : nat), leb n m = false -> leb m n = true.`

the author(s) claims that

`leb_nm__leb_mn`

is a specialisation of`leb_spec`

, and that confuses me.

Revisiting the problem, I can complete the prove simply by introducing the theorem into the context.

```
Lemma leb_nm__leb_mn' : forall (n m : nat),
leb n m = false -> leb m n = true.
Proof.
intros n m H. destruct (leb_spec n m) as [H1 | [H2 H3]].
- rewrite H1 in H. discriminate H.
- apply H3.
Qed.
```

Cheers.

Last updated: Sep 30 2023 at 07:01 UTC