natural logarithm for extended real numbers

CHANGELOG_UNRELEASED.md

@@ -9,14 +9,16 @@
 
 - in `realfun.v`:
   + instance `is_derive1_sqrt`
+
 - in `mathcomp_extra.v`:
   + lemmas `subrKC`, `sumr_le0`, `card_fset_sum1`
 
 - in `functions.v`:
   + lemmas `fct_prodE`, `prodrfctE`
 
-- in `exp.v:
+- in `exp.v`:
   + lemma `norm_expR`
+
 - in `hoelder.v`
   + lemma `hoelder_conj_ge1`
 
@@ -35,17 +37,39 @@
   + lemmas `closed_Fsigma`, `Gdelta_measurable`, `Gdelta_subspace_open`,
     `irrational_Gdelta`, `not_rational_Gdelta`
 
+- in `constructive_ereal.v`:
+  + lemma `expe0`, `mule0n`, `muleS`
+
+- in `exp.v`:
+  + lemmas `expeR_eqy`
+  + lemmas `lt0_powR1`, `powR_eq1`
+  + definition `lne`
+  + lemmas `le0_lneNy`, `lne_EFin`, `expeRK`, `lneK`, `lneK_eq`, `lne1`, `lneM`, 
+    `lne_inj`, `lneV`, `lne_div`, `lte_lne`, `lee_lne`, `lneXn`, `le_lne1Dx`, 
+    `lne_sublinear`, `lne_ge0`, `lne_lt0`, `lne_gt0`, `lne_le0`
+  + lemma `lne_eq0`
+
+- in `charge.v`:
+  + definition `copp`, lemma `cscaleN1`
+
 ### Changed
 
 - moved from `pi_irrational.v` to `reals.v` and changed
   + definition `rational`
 
+- in `constructive_ereal.v`:
+  + lemma `mulN1e`
+
 ### Renamed
 
 - in `lebesgue_stieltjes_measure.v`:
   + `cumulativeNy0` -> `cumulativeNy`
   + `cumulativey1` -> `cumulativey`
 
+- in `exp.v`:
+  + `ltr_expeR` -> `lte_expeR`
+  + `ler_expeR` -> `lee_expeR`
+
 ### Generalized
 
 - in `functions.v`

reals/constructive_ereal.v

@@ -726,8 +726,12 @@ Proof. by elim: n => //= n ->. Qed.
 Lemma enatmul_ninfty n : -oo *+ n.+1 = -oo :> \bar R.
 Proof. by elim: n => //= n ->. Qed.
 
+Lemma mule0n x : x *+ 0 = 0. Proof. by []. Qed.
+
 Lemma mule2n x : x *+ 2 = x + x. Proof. by []. Qed.
 
+Lemma expe0 x : x ^+ 0 = 1. Proof. by []. Qed.
+
 Lemma expe2 x : x ^+ 2 = x * x. Proof. by []. Qed.
 
 Lemma leeN2 : {mono @oppe R : x y /~ x <= y}.
@@ -862,6 +866,9 @@ Lemma addeC : commutative (S := \bar R) +%E. Proof. exact: addrC. Qed.
 
 Lemma adde0 : right_id (0 : \bar R) +%E. Proof. exact: addr0. Qed.
 
+Lemma muleS x n : x *+ n.+1 = x + x *+ n.
+Proof. by case: n => //=; rewrite adde0. Qed.
+
 Lemma add0e : left_id (0 : \bar R) +%E. Proof. exact: add0r. Qed.
 
 Lemma addeA : associative (S := \bar R) +%E. Proof. exact: addrA. Qed.
@@ -1266,13 +1273,13 @@ Proof. by move=> rreal; rewrite muleC real_mulrNy. Qed.
 
 Definition real_mulr_infty := (real_mulry, real_mulyr, real_mulrNy, real_mulNyr).
 
-Lemma mulN1e x : - 1%E * x = - x.
+Lemma mulN1e x : (- 1%E) * x = - x.
 Proof.
-rewrite -EFinN /mule/=; case: x => [x||];
-  do ?[by rewrite mulN1r|by rewrite eqe oppr_eq0 oner_eq0 lte_fin ltr0N1].
+by case: x => [r| |]/=;
+  rewrite /mule ?mulN1r// eqe oppr_eq0 oner_eq0/= lte_fin oppr_gt0 ltr10.
 Qed.
 
-Lemma muleN1 x : x * - 1%E = - x. Proof. by rewrite muleC mulN1e. Qed.
+Lemma muleN1 x : x * (- 1%E) = - x. Proof. by rewrite muleC mulN1e. Qed.
 
 Lemma mule_neq0 x y : x != 0 -> y != 0 -> x * y != 0.
 Proof.
@@ -1284,7 +1291,7 @@ Qed.
 Lemma mule_eq0 x y : (x * y == 0) = (x == 0) || (y == 0).
 Proof.
 apply/idP/idP => [|/orP[] /eqP->]; rewrite ?(mule0, mul0e)//.
-by apply: contraTT => /norP[]; apply: mule_neq0.
+by apply: contraTT => /norP[]; exact: mule_neq0.
 Qed.
 
 Lemma mule_ge0 x y : 0 <= x -> 0 <= y -> 0 <= x * y.

theories/charge.v

@@ -43,7 +43,9 @@ From mathcomp Require Import lebesgue_measure lebesgue_integral.
 (*                              non-measurable sets                           *)
 (*                     czero == zero charge                                   *)
 (*               cscale r nu == charge nu scaled by a factor r : R            *)
-(*          charge_add n1 n2 == the charge corresponding to the sum of        *)
+(*                   copp nu == the charge corresponding to the opposite of   *)
+(*                              the charges nu                                *)
+(*                cadd n1 n2 == the charge corresponding to the sum of        *)
 (*                              charges n1 and n2                             *)
 (* charge_of_finite_measure mu == charge corresponding to a finite measure mu *)
 (* ```                                                                        *)
@@ -440,6 +442,38 @@ move=> /negbTE c0 munu E mE /eqP; rewrite /cscale mule_eq0 eqe c0/=.
 by move=> /eqP/munu; exact.
 Qed.
 
+Section charge_opp.
+Local Open Scope ereal_scope.
+Context d (T : measurableType d) (R : realType).
+Variables (nu : {charge set T -> \bar R}).
+
+Definition copp := \- nu.
+
+Let copp0 : copp set0 = 0.
+Proof. by rewrite /copp charge0 oppe0. Qed.
+
+Let copp_finite A : measurable A -> copp A \is a fin_num.
+Proof. by move=> mA; rewrite fin_numN fin_num_measure. Qed.
+
+Let copp_sigma_additive : semi_sigma_additive copp.
+Proof.
+move=> F mF tF mUF; rewrite /copp; under eq_fun.
+  move=> n; rewrite sumeN; last first.
+    by move=> p q _ _; rewrite fin_num_adde_defl// fin_num_measure.
+  over.
+exact/cvgeN/charge_semi_sigma_additive.
+Qed.
+
+HB.instance Definition _ := isCharge.Build _ _ _ copp
+  copp0 copp_finite copp_sigma_additive.
+
+End charge_opp.
+
+Lemma cscaleN1 {d} {T : ringOfSetsType d} {R : realFieldType}
+    (nu : {charge set T -> \bar R}) :
+  cscale (-1) nu = \- nu.
+Proof. by rewrite /cscale/=; apply/funext => x; rewrite mulN1e. Qed.
+
 Section charge_add.
 Local Open Scope ereal_scope.
 Context d (T : measurableType d) (R : realType).
@@ -944,7 +978,7 @@ Local Definition cjordan_neg : {charge set T -> \bar R} :=
   cscale (-1) (crestr0 nu mN).
 
 Lemma cjordan_negE A : cjordan_neg A = - crestr0 nu mN A.
-Proof. by rewrite /= /cscale/= EFinN mulN1e. Qed.
+Proof. by rewrite /= cscaleN1. Qed.
 
 Let positive_set_cjordan_neg E : 0 <= cjordan_neg E.
 Proof.
@@ -970,7 +1004,7 @@ Lemma jordan_decomp (A : set T) : measurable A ->
   nu A = cadd jordan_pos (cscale (-1) jordan_neg) A.
 Proof.
 move=> mA.
-rewrite /cadd cjordan_posE /= /cscale EFinN mulN1e cjordan_negE oppeK.
+rewrite /cadd cjordan_posE/= cscaleN1 cjordan_negE oppeK.
 rewrite /crestr0 mem_set// -[in LHS](setIT A).
 case: nuPN => _ _ <- PN0; rewrite setIUr chargeU//.
 - exact: measurableI.
@@ -1045,7 +1079,7 @@ Lemma abse_charge_variation d (T : measurableType d) (R : realType)
   measurable A -> `|nu A| <= charge_variation PN A.
 Proof.
 move=> mA.
-rewrite (jordan_decomp PN mA) /cadd/= /cscale/= mulN1e /charge_variation.
+rewrite (jordan_decomp PN mA) /cadd/= cscaleN1 /charge_variation.
 by rewrite (le_trans (lee_abs_sub _ _))// !gee0_abs.
 Qed.
 
@@ -2034,8 +2068,7 @@ exists (fp \- fn); split; first by move=> x; rewrite fin_numB// fpfin fnfin.
   exact: integrableB.
 move=> E mE; rewrite [LHS](jordan_decomp nuPN mE)// integralB//;
   [|exact: (integrableS measurableT)..].
-rewrite -fpE ?inE// -fnE ?inE//= /cadd/= jordan_posE jordan_negE.
-by rewrite /cscale EFinN mulN1e.
+by rewrite -fpE ?inE// -fnE ?inE//= /cadd/= cscaleN1.
 Qed.
 
 Definition Radon_Nikodym : T -> \bar R :=

theories/ereal.v

@@ -1066,7 +1066,7 @@ Definition lt_expandRL := monoRL_in
 Lemma contract_eq0 x : (contract x == 0%R) = (x == 0).
 Proof. by rewrite -(can_eq contractK) contract0. Qed.
 
-Lemma contract_eqN1 x : (contract x == -1) = (x == -oo).
+Lemma contract_eqN1 x : (contract x == (- 1)%R) = (x == -oo).
 Proof. by rewrite -(can_eq contractK). Qed.
 
 Lemma contract_eq1 x : (contract x == 1%R) = (x == +oo).
@@ -1352,7 +1352,7 @@ move: reN1; rewrite eq_sym neq_lt => /orP[reN1|reN1].
       rewrite (@lt_le_trans _ _ 1%R) // ?lerDr//.
       by rewrite (le_lt_trans (ler_norm _))// contract_lt1.
     have ? : (`|contract r%:E + e%:num| < 1)%R.
-      rewrite ltr_norml re1 andbT -(addr0 (-1)) ler_ltD //.
+      rewrite ltr_norml re1 andbT -(addr0 (- 1)%R) ler_ltD //.
       by move: (contract_le1 r%:E); rewrite ler_norml => /andP[].
     pose e' : R := Num.min
       (r - fine (expand (contract r%:E - e%:num)))%R