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Lemma gt0_ltr_powR #1027

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Oct 2, 2023
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Lemma gt0_ltr_powR #1027

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7819b72
Lemma gt0_ltr_powR
hoheinzollern Sep 23, 2023
932a46a
add powR_injective
affeldt-aist Sep 28, 2023
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9 changes: 9 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Original file line number Diff line number Diff line change
Expand Up @@ -70,6 +70,10 @@
- in `measure.v`:
+ definition `ess_sup`, lemma `ess_sup_ge0`

- in `exp.v`:
+ lemma `gt0_ltr_powR`
+ lemma `powR_injective`

### Changed

- `mnormalize` moved from `kernel.v` to `measure.v` and generalized
Expand All @@ -91,6 +95,8 @@

- in `measure.v`:
+ implicits of `measurable_fst` and `measurable_snd`
- in `exp.v`:
+ `gt0_ler_powR` now uses `Num.nneg`

### Renamed

Expand All @@ -101,6 +107,9 @@
+ `lee_opp` -> `leeN2`
+ `lte_opp` -> `lteN2`

- in `exp.v`:
+ `gt0_ler_powR` -> `ge0_ler_powR`

### Generalized

### Deprecated
Expand Down
26 changes: 23 additions & 3 deletions theories/exp.v
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Expand Up @@ -664,18 +664,35 @@ move=> a1 x y xy.
by rewrite /powR gt_eqF ?(lt_le_trans _ a1)// ler_expR ler_wpmul2r ?ln_ge0.
Qed.

Lemma gt0_ler_powR (r : R) : 0 <= r ->
{in `[0, +oo[ &, {homo powR ^~ r : x y / x <= y >-> x <= y}}.
Lemma powR_injective r : 0 < r -> {in Num.nneg &, injective (powR ^~ r)}.
Proof.
move=> r0 x y x0 y0; rewrite /powR; case: ifPn => [/eqP ->|xneq0].
by case: ifPn => [/eqP ->//|_ /eqP]; rewrite (gt_eqF r0) eq_sym expR_eq0.
case: ifPn => [/eqP -> /eqP|yneq0]; first by rewrite (gt_eqF r0) expR_eq0.
by move/expR_inj/mulfI => /(_ (negbT (gt_eqF r0))); apply: ln_inj;
rewrite posrE lt_neqAle eq_sym (xneq0,yneq0).
Qed.

Lemma ge0_ler_powR (r : R) : 0 <= r ->
{in Num.nneg &, {homo powR ^~ r : x y / x <= y >-> x <= y}}.
Proof.
rewrite le_eqVlt => /predU1P[<- x y _ _ _|]; first by rewrite !powRr0.
move=> a0 x y; rewrite !in_itv/= !andbT !le_eqVlt => /predU1P[<-|x0].
move=> a0 x y; rewrite 2!nnegrE !le_eqVlt => /predU1P[<-|x0].
move=> /predU1P[<- _|y0 _]; first by rewrite eqxx.
by rewrite !powR0 ?(gt_eqF a0)// powR_gt0 ?orbT.
move=> /predU1P[<-|y0]; first by rewrite gt_eqF//= ltNge (ltW x0).
move=> /predU1P[->//|xy]; first by rewrite eqxx.
by apply/orP; right; rewrite /powR !gt_eqF// ltr_expR ltr_pmul2l// ltr_ln.
Qed.

Lemma gt0_ltr_powR (r : R) : 0 < r ->
{in Num.nneg &, {homo powR ^~ r : x y / x < y >-> x < y}}.
Proof.
move=> r0 x y x0 y0 xy; have := ge0_ler_powR (ltW r0) x0 y0 (ltW xy).
rewrite le_eqVlt => /orP[/eqP/(powR_injective r0 x0 y0)/eqP|//].
by rewrite lt_eqF.
Qed.

Lemma powRM x y r : 0 <= x -> 0 <= y -> (x * y) `^ r = x `^ r * y `^ r.
Proof.
rewrite /powR mulf_eq0.
Expand Down Expand Up @@ -786,6 +803,9 @@ Qed.
End PowR.
Notation "a `^ x" := (powR a x) : ring_scope.

#[deprecated(since="mathcomp-analysis 0.6.5", note="renamed `ge0_ler_powR`")]
Notation gt0_ler_powR := ge0_ler_powR.

Section poweR.
Local Open Scope ereal_scope.
Context {R : realType}.
Expand Down
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