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Jun 5, 2025
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is_deriveX: is that a generalization? #1637

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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -134,6 +134,9 @@
- in `convex.v`
+ parameter `R` of `convType` from `realDomainType` to `numDomainType`

- in `derive.v`:
+ lemmas `is_deriveX`, `deriveX`, `exp_derive1`

### Deprecated

### Removed
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12 changes: 7 additions & 5 deletions theories/derive.v
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Expand Up @@ -1322,9 +1322,11 @@ by rewrite deriveM // !derive_val.
Qed.

Global Instance is_deriveX f n x v (df : R) :
is_derive x v f df -> is_derive x v (f ^+ n.+1) ((n.+1%:R * f x ^+n) *: df).
is_derive x v f df -> is_derive x v (f ^+ n) ((n%:R * f x ^+ n.-1) *: df).
Proof.
move=> dfx; elim: n => [|n ihn]; first by rewrite expr1 expr0 mulr1 scale1r.
case: n => [fdf|n dfx].
by rewrite expr0 mul0r scale0r; exact: is_derive_cst.
elim: n => [|n ihn]; first by rewrite expr1 expr0 mulr1 scale1r.
rewrite exprS; apply: is_derive_eq.
rewrite scalerA -scalerDl mulrCA -[f x * _]exprS.
by rewrite [in LHS]mulr_natl exprfctE -mulrSr mulr_natl.
Expand All @@ -1334,7 +1336,7 @@ Lemma derivableX f n x v : derivable f x v -> derivable (f ^+ n) x v.
Proof. by case: n => [_|n /derivableP]; [rewrite expr0|]. Qed.

Lemma deriveX f n x v : derivable f x v ->
'D_v (f ^+ n.+1) x = (n.+1%:R * f x ^+ n) *: 'D_v f x.
'D_v (f ^+ n) x = (n%:R * f x ^+ n.-1) *: 'D_v f x.
Proof. by move=> /derivableP df; rewrite derive_val. Qed.

Fact der_inv f x v : f x != 0 -> derivable f x v ->
Expand Down Expand Up @@ -1413,14 +1415,14 @@ by apply: derivableM; first exact: derivable_id.
Qed.

Lemma exp_derive {R : numFieldType} n x v :
'D_v (@GRing.exp R ^~ n.+1) x = n.+1%:R *: x ^+ n *: v.
'D_v (@GRing.exp R ^~ n) x = n%:R *: x ^+ n.-1 *: v.
Proof.
have /= := @deriveX R R id n x v (@derivable_id _ _ _ _).
by rewrite fctE => ->; rewrite derive_id.
Qed.

Lemma exp_derive1 {R : numFieldType} n x :
(@GRing.exp R ^~ n.+1)^`() x = n.+1%:R *: x ^+ n.
(@GRing.exp R ^~ n)^`() x = n%:R *: x ^+ n.-1.
Proof. by rewrite derive1E exp_derive [LHS]mulr1. Qed.

Lemma EVT_max (R : realType) (f : R -> R) (a b : R) : (* TODO : Filter not infered *)
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