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fix: forgotten lemma about derivability of sqrt #1909

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fix: forgotten lemma about derivability of sqrt #1909

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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -4,6 +4,9 @@

### Added

- in `realfun.v`:
+ lemma `derivable_sqrt`

### Changed

### Renamed
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5 changes: 4 additions & 1 deletion theories/realfun.v
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Expand Up @@ -1943,9 +1943,12 @@ rewrite -[x]sqrtK//; apply: (@is_derive_inverse _ (fun x => x ^+ 2)).
- by rewrite mulf_neq0// gt_eqF// sqrtr_gt0 exprn_gt0// sqrtr_gt0.
Unshelve. all: by end_near. Qed.

Lemma derivable_sqrt {K : realType} (x : K) : 0 < x -> derivable Num.sqrt x 1.
Proof. by move=> x0; exact/ex_derive/is_derive1_sqrt. Qed.

Lemma derive_sqrt {K : realType} (r : K) : 0 < r ->
'D_1 Num.sqrt r = (2 * Num.sqrt r)^-1.
Proof. by move=> r0; apply: derive_val; exact: is_derive1_sqrt. Qed.
Proof. by move=> r0; exact/derive_val/is_derive1_sqrt. Qed.

#[global] Hint Extern 0 (is_derive _ _ (fun _ => (_ _)^-1) _) =>
(eapply is_deriveV; first by []) : typeclass_instances.
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