Merge pull request #891 from affeldt-aist/powere_pos

power function for extended real numbers

CHANGELOG_UNRELEASED.md

@@ -69,6 +69,13 @@
   + definition `sqrte`
   + lemmas `sqrte0`, `sqrte_ge0`, `lee_sqrt`, `sqrteM`, `sqr_sqrte`,
     `sqrte_sqr`, `sqrte_fin_num`
+- in `exp.v`:
+  + lemma `ln_power_pos`
+  + definition `powere_pos`, notation ``` _ `^ _ ``` in `ereal_scope`
+  + lemmas `powere_pos_EFin`, `powere_posyr`, `powere_pose0`,
+    `powere_pose1`, `powere_posNyr` `powere_pos0r`, `powere_pos1r`,
+    `powere_posNyr`, `fine_powere_pos`, `powere_pos_ge0`,
+    `powere_pos_gt0`, `powere_pos_eq0`, `powere_posM`, `powere12_sqrt`
 
 ### Changed
 
@@ -133,6 +140,8 @@
 - in `lebesgue_measure.v`:
   + lemmas `emeasurable_itv_bnd_pinfty`, `emeasurable_itv_ninfty_bnd`
     (use `emeasurable_itv` instead)
+- in `measure.v`:
+  + lemma `measurable_fun_ext`
 
 ### Removed
 

theories/exp.v

@@ -19,7 +19,8 @@ Require Import signed topology normedtype landau sequences derive realfun.
 (*   pseries_diffs f i == (i + 1) * f (i + 1)                                 *)
 (*                                                                            *)
 (*                ln x == the natural logarithm                               *)
-(*              a `^ x == power function (assumes a >= 0)                     *)
+(*              s `^ r == power function, in ring_scope (assumes s >= 0)      *)
+(*              e `^ r == power function, in ereal_scope (assumes e >= 0)     *)
 (*          riemannR a == sequence n |-> 1 / (n.+1) `^ a where a has a type   *)
 (*                        of type realType                                    *)
 (*                                                                            *)
@@ -686,6 +687,9 @@ rewrite -(opprK z) (_ : - z = `|z|%N); last by rewrite ltz0_abs.
 by rewrite -exprnN -power_pos_inv// nmulrn.
 Qed.
 
+Lemma ln_power_pos s r : s != 0 -> ln (s `^ r) = r * ln s.
+Proof. by move=> s0; rewrite /power_pos (negbTE s0) expK. Qed.
+
 Lemma power12_sqrt a : 0 <= a -> a `^ (2^-1) = Num.sqrt a.
 Proof.
 rewrite le_eqVlt => /predU1P[<-|a0].
@@ -699,7 +703,96 @@ by rewrite h oppr_gt0 ltNge sqrtr_ge0.
 Qed.
 
 End PowerPos.
-Notation "a `^ x" := (power_pos a x) : real_scope.
+Notation "a `^ x" := (power_pos a x) : ring_scope.
+
+Section powere_pos.
+Local Open Scope ereal_scope.
+Context {R : realType}.
+Implicit Types (r : R) (x y : \bar R).
+
+Definition powere_pos x r :=
+  match x with
+  | x'%:E => (x' `^ r)%:E
+  | +oo => if r == 0%R then 1%E else +oo
+  | -oo => if r == 0%R then 1%E else 0%E
+  end.
+
+Local Notation "x `^ r" := (powere_pos x r).
+
+Lemma powere_pos_EFin (s : R) r : s%:E `^ r = (s `^ r)%:E.
+Proof. by []. Qed.
+
+Lemma powere_posyr r : r != 0%R -> +oo `^ r = +oo.
+Proof. by move/negbTE => /= ->. Qed.
+
+Lemma powere_pose0 x : x `^ 0 = 1.
+Proof. by move: x => [x'| |]/=; rewrite ?power_posr0// eqxx. Qed.
+
+Lemma powere_pose1 x : 0 <= x -> x `^ 1 = x.
+Proof.
+by move: x => [x'| |]//= x0; rewrite ?power_posr1// (negbTE (oner_neq0 _)).
+Qed.
+
+Lemma powere_posNyr r : r != 0%R -> -oo `^ r = 0.
+Proof.
+by move => xne0; rewrite /powere_pos ifF //; apply/eqP; move: xne0 => /eqP.
+Qed.
+
+Lemma powere_pos0r r : 0 `^ r = (r == 0)%:R%:E.
+Proof. by rewrite powere_pos_EFin power_pos0. Qed.
+
+Lemma powere_pos1r r : 1 `^ r = 1.
+Proof. by rewrite powere_pos_EFin power_pos1. Qed.
+
+Lemma fine_powere_pos x r : fine (x `^ r) = ((fine x) `^ r)%R.
+Proof. by move: x => [x| |]//=; rewrite power_pos0; case: ifPn. Qed.
+
+Lemma powere_pos_ge0 x r : 0 <= x `^ r.
+Proof.
+by move: x => [x| |];
+  rewrite ?powere_pos_EFin ?lee_fin ?power_pos_ge0// /powere_pos; case: ifPn.
+Qed.
+
+Lemma powere_pos_gt0 x r : 0 < x -> 0 < x `^ r.
+Proof.
+move: x => [x|_|//]; rewrite ?lte_fin; first exact: power_pos_gt0.
+by rewrite /powere_pos; case: ifPn.
+Qed.
+
+Lemma powere_pos_eq0 x r : -oo < x -> x `^ r = 0 -> x = 0.
+Proof.
+move: x => [x _|_/=|//].
+- by rewrite powere_pos_EFin => -[] /power_pos_eq0 ->.
+- by case: ifPn => // _ /eqP; rewrite onee_eq0.
+Qed.
+
+Lemma powere_posM x y r : 0 <= x -> 0 <= y -> (x * y) `^ r = x `^ r * y `^ r.
+Proof.
+move: x y => [x| |] [y| |]//=.
+- by move=> x0 y0; rewrite -EFinM power_posM.
+- move=> x0 _; case: ifPn => /= [/eqP ->|r0].
+  + by rewrite mule1 power_posr0 powere_pose0.
+  + move: x0; rewrite le_eqVlt => /predU1P[[]<-|/[1!(@lte_fin R)] x0].
+    * by rewrite mul0e powere_pos0r power_pos0 (negbTE r0)/= mul0e.
+    * by rewrite mulry [RHS]mulry !gtr0_sg ?power_pos_gt0// !mul1e powere_posyr.
+- move=> _ y0; case: ifPn => /= [/eqP ->|r0].
+  + by rewrite power_posr0 powere_pose0 mule1.
+  + move: y0; rewrite le_eqVlt => /predU1P[[]<-|/[1!(@lte_fin R)] u0].
+    by rewrite mule0 powere_pos0r power_pos0 (negbTE r0) mule0.
+  + by rewrite 2!mulyr !gtr0_sg ?power_pos_gt0// mul1e powere_posyr.
+- move=> _ _; case: ifPn => /= [/eqP ->|r0].
+  + by rewrite powere_pose0 mule1.
+  + by rewrite mulyy powere_posyr.
+Qed.
+
+Lemma powere12_sqrt x : 0 <= x -> x `^ 2^-1 = sqrte x.
+Proof.
+move: x => [x|_|//]; last by rewrite powere_posyr.
+by rewrite lee_fin => x0 /=; rewrite power12_sqrt.
+Qed.
+
+End powere_pos.
+Notation "a `^ x" := (powere_pos a x) : ereal_scope.
 
 Section riemannR_series.
 Variable R : realType.

theories/lebesgue_measure.v

@@ -1726,7 +1726,7 @@ apply: measurable_fun_if => //.
 - rewrite setTI; apply: (@measurable_fun_comp _ _ _ _ _ _ setT) => //.
     by apply: continuous_measurable_fun; exact: continuous_expR.
   rewrite (_ : _ @^-1` _ = [set~ 0]); last first.
-    by apply/seteqP; split => [x [/negP/negP/eqP]|x x0]//=; exact/negbTE/eqP.
+    by apply/seteqP; split => [x /negP/negP/eqP|x x0]//=; exact/negbTE/eqP.
   by apply: measurable_funrM; exact: measurable_fun_ln.
 Qed.
 
@@ -1890,7 +1890,7 @@ Lemma emeasurable_fun_cvg D (f_ : (T -> \bar R)^nat) (f : T -> \bar R) :
 Proof.
 move=> mf_ f_f; have fE x : D x -> f x = lim_esup (f_^~ x).
   by move=> Dx; have /cvg_lim  <-// := @cvg_esups _ (f_^~x) (f x) (f_f x Dx).
-apply: (measurable_fun_ext (fun x => lim_esup (f_ ^~ x))) => //.
+apply: (eq_measurable_fun (fun x => lim_esup (f_ ^~ x))) => //.
   by move=> x; rewrite inE => Dx; rewrite fE.
 exact: measurable_fun_lim_esup.
 Qed.

theories/measure.v

@@ -1023,10 +1023,8 @@ Proof. exact: measurable_fun_comp. Qed.
 Lemma eq_measurable_fun D (f g : T1 -> T2) :
   {in D, f =1 g} -> measurable_fun D f -> measurable_fun D g.
 Proof.
-move=> Dfg Df mD A mA; rewrite (_ : D `&` _ = D `&` f @^-1` A); first exact: Df.
-apply/seteqP; rewrite /preimage; split => [x /= [Dx Agx]|x /= [Dx Afx]].
-  by split=> //; rewrite Dfg// inE.
-by split=> //; rewrite -Dfg// inE.
+by move=> fg mf mD A mA; rewrite [X in measurable X](_ : _ = D `&` f @^-1` A);
+  [exact: mf|exact/esym/eq_preimage].
 Qed.
 
 Lemma measurable_fun_cst D (r : T2) : measurable_fun D (cst r : T1 -> _).
@@ -1062,13 +1060,6 @@ Lemma measurable_funTS D (f : T1 -> T2) :
   measurable_fun setT f -> measurable_fun D f.
 Proof. exact: measurable_funS. Qed.
 
-Lemma measurable_fun_ext D (f g : T1 -> T2) :
-  {in D, f =1 g} -> measurable_fun D f -> measurable_fun D g.
-Proof.
-by move=> fg mf mD A mA; rewrite [X in measurable X](_ : _ = D `&` f @^-1` A);
-  [exact: mf|exact/esym/eq_preimage].
-Qed.
-
 Lemma measurable_restrict D E (f : T1 -> T2) :
   measurable D -> measurable E -> D `<=` E ->
   measurable_fun D f <-> measurable_fun E (f \_ D).
@@ -1127,7 +1118,9 @@ have [-> _|-> _|-> _ |-> _] := subset_set2 YT.
 Qed.
 
 End measurable_fun.
-Arguments measurable_fun_ext {d1 d2 T1 T2 D} f {g}.
+Arguments eq_measurable_fun {d1 d2 T1 T2 D} f {g}.
+#[deprecated(since="mathcomp-analysis 0.6.2", note="renamed `eq_measurable_fun`")]
+Notation measurable_fun_ext := eq_measurable_fun.
 Arguments measurable_fun_bool {d1 T1 D f} b.
 
 Section measurability.