Lemma gt0_ltr_powR (#1027)

* Lemma gt0_ltr_powR * add powR_injective --------- Co-authored-by: Reynald Affeldt <[email protected]>
math-comp · Oct 2, 2023 · 3381621 · 3381621
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diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md
@@ -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
@@ -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
 
@@ -101,6 +107,9 @@
   + `lee_opp` -> `leeN2`
   + `lte_opp` -> `lteN2`
 
+- in `exp.v`:
+  + `gt0_ler_powR` -> `ge0_ler_powR`
+
 ### Generalized
 
 ### Deprecated

diff --git a/theories/exp.v b/theories/exp.v
@@ -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.
@@ -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}.