Minkowski (#1000)

* Minkowski's inequality and accessory lemmas --------- Co-authored-by: Alessandro Bruni <[email protected]> Co-authored-by: Reynald Affeldt <[email protected]>
math-comp · Nov 15, 2023 · f485ccc · f485ccc
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diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md
@@ -88,6 +88,15 @@
   + lemmas `expeR_eq0`, `expeRD`, `expeR_ge1Dx`
   + lemmas `ltr_expeR`, `ler_expeR`, `expeR_inj`, `expeR_total`
 
+- in `exp.v`:
+  + lemmas `mulr_powRB1`, `fin_num_poweR`, `poweRN`, `poweR_lty`, `lty_poweRy`, `gt0_ler_poweR`
+
+- in `mathcomp_extra.v`:
+  + lemma `onemV`
+
+- in `hoelder.v`:
+  + lemmas `powR_Lnorm`, `minkowski`
+
 ### Changed
 
 - in `hoelder.v`:

diff --git a/classical/mathcomp_extra.v b/classical/mathcomp_extra.v
@@ -469,6 +469,8 @@ Arguments big_rmcond_in {R idx op I r} P.
 (* MathComp > 1.15.0 additions *)
 (*******************************)
 
+Reserved Notation "`1- x" (format "`1- x", at level 2).
+
 Section onem.
 Variable R : numDomainType.
 Implicit Types r : R.
@@ -518,6 +520,9 @@ Qed.
 End onem.
 Notation "`1- r" := (onem r) : ring_scope.
 
+Lemma onemV (F : numFieldType) (x : F) : x != 0 -> `1-(x^-1) = (x - 1) / x.
+Proof. by move=> ?; rewrite mulrDl divff// mulN1r. Qed.
+
 Lemma lez_abs2 (a b : int) : 0 <= a -> a <= b -> (`|a| <= `|b|)%N.
 Proof. by case: a => //= n _; case: b. Qed.
 

diff --git a/theories/exp.v b/theories/exp.v
@@ -694,7 +694,7 @@ End Ln.
 
 Section PowR.
 Variable R : realType.
-Implicit Types a x : R.
+Implicit Types a x y z r : R.
 
 Definition powR a x := if a == 0 then (x == 0)%:R else expR (x * ln a).
 
@@ -781,7 +781,7 @@ move=> /andP[a0 a1] r1.
 by rewrite (le_trans (ger_powR _ r1)) ?powRr1 ?a0// ltW.
 Qed.
 
-Lemma ge0_ler_powR (r : R) : 0 <= r ->
+Lemma ge0_ler_powR 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.
@@ -793,7 +793,7 @@ move=> /predU1P[->//|xy]; first by rewrite eqxx.
 by apply/orP; right; rewrite /powR !gt_eqF// ltr_expR ltr_pM2l// ltr_ln.
 Qed.
 
-Lemma gt0_ltr_powR (r : R) : 0 < r ->
+Lemma gt0_ltr_powR 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).
@@ -823,7 +823,7 @@ move=> x1 y0 r1.
 by rewrite (powRM _ (le_trans _ x1))// ler_wpM2r ?powR_ge0// le1r_powR// x0.
 Qed.
 
-Lemma powRrM (x y z : R) : x `^ (y * z) = (x `^ y) `^ z.
+Lemma powRrM x y z : x `^ (y * z) = (x `^ y) `^ z.
 Proof.
 rewrite /powR mulf_eq0; have [_|xN0] := eqVneq x 0.
   by case: (y == 0); rewrite ?eqxx//= oner_eq0 ln1 mulr0 expR0.
@@ -841,6 +841,12 @@ have [->|] := eqVneq r 0; first by rewrite mul1r add0r.
 by rewrite implybF mul0r => _ /negPf ->.
 Qed.
 
+Lemma mulr_powRB1 x p : 0 <= x -> 0 < p -> x * x `^ (p - 1) = x `^ p.
+Proof.
+rewrite le_eqVlt => /predU1P[<- p0|x0 p0]; first by rewrite mul0r powR0 ?gt_eqF.
+by rewrite -{1}(powRr1 (ltW x0))// -powRD addrCA subrr addr0// gt_eqF.
+Qed.
+
 Lemma powRN x r : x `^ (- r) = (x `^ r)^-1.
 Proof.
 have [r0|r0] := eqVneq r 0%R; first by rewrite r0 oppr0 powRr0 invr1.
@@ -956,6 +962,9 @@ Proof.
 by move: x => [x'| |]//= x0; rewrite ?powRr1// (negbTE (oner_neq0 _)).
 Qed.
 
+Lemma poweRN x r : x \is a fin_num -> x `^ (- r) = (fine x `^ r)^-1%:E.
+Proof. by case: x => // x xf; rewrite poweR_EFin powRN. Qed.
+
 Lemma poweRNyr r : r != 0%R -> -oo `^ r = 0.
 Proof. by move=> r0 /=; rewrite (negbTE r0). Qed.
 
@@ -965,6 +974,16 @@ Proof. by case: x => [x| |] //=; case: ifP. Qed.
 Lemma eqy_poweR x r : (0 < r)%R -> x = +oo -> x `^ r = +oo.
 Proof. by move: x => [| |]//= r0 _; rewrite gt_eqF. Qed.
 
+Lemma poweR_lty x r : x < +oo -> x `^ r < +oo.
+Proof.
+by move: x => [x| |]//=; rewrite ?ltry//; case: ifPn => // _; rewrite ltry.
+Qed.
+
+Lemma lty_poweRy x r : r != 0%R -> x `^ r < +oo -> x < +oo.
+Proof.
+by move=> r0; move: x => [x| | _]//=; rewrite ?ltry// (negbTE r0).
+Qed.
+
 Lemma poweR0r r : r != 0%R -> 0 `^ r = 0.
 Proof. by move=> r0; rewrite poweR_EFin powR0. Qed.
 
@@ -998,6 +1017,21 @@ Qed.
 Lemma poweR_eq0_eq0 x r : 0 <= x -> x `^ r = 0 -> x = 0.
 Proof. by move=> + /eqP => /poweR_eq0-> /andP[/eqP]. Qed.
 
+Lemma gt0_ler_poweR r : (0 <= r)%R ->
+  {in `[0, +oo] &, {homo poweR ^~ r : x y / x <= y >-> x <= y}}.
+Proof.
+move=> r0 + y; case=> //= [x /[1!in_itv]/= /andP[xint _]| _ _].
+- case: y => //= [y /[1!in_itv]/= /andP[yint _] xy| _ _].
+  + by rewrite !lee_fin ge0_ler_powR.
+  + by case: eqP => [->|]; rewrite ?powRr0 ?leey.
+- by rewrite leye_eq => /eqP ->.
+Qed.
+
+Lemma fin_num_poweR x r : x \is a fin_num -> x `^ r \is a fin_num.
+Proof.
+by move=> xfin; rewrite ge0_fin_numE ?poweR_lty ?ltey_eq ?xfin// poweR_ge0.
+Qed.
+
 Lemma poweRM x y r : 0 <= x -> 0 <= y -> (x * y) `^ r = x `^ r * y `^ r.
 Proof.
 have [->|rN0] := eqVneq r 0%R; first by rewrite !poweRe0 mule1.

diff --git a/theories/hoelder.v b/theories/hoelder.v
@@ -35,17 +35,15 @@ Reserved Notation "'N_ p [ F ]"
 
 Declare Scope Lnorm_scope.
 
-Local Open Scope ereal_scope.
-
 HB.lock Definition Lnorm {d} {T : measurableType d} {R : realType}
     (mu : {measure set T -> \bar R}) (p : \bar R) (f : T -> R) :=
   match p with
-  | p%:E => if p == 0%R then
+  | p%:E => (if p == 0%R then
               mu (f @^-1` (setT `\ 0%R))
             else
-              (\int[mu]_x (`|f x| `^ p)%:E) `^ p^-1
-  | +oo => if mu [set: T] > 0 then ess_sup mu (normr \o f) else 0
-  | -oo => 0
+              (\int[mu]_x (`|f x| `^ p)%:E) `^ p^-1)%E
+  | +oo%E => (if mu [set: T] > 0 then ess_sup mu (normr \o f) else 0)%E
+  | -oo%E => 0%E
   end.
 Canonical locked_Lnorm := Unlockable Lnorm.unlock.
 Arguments Lnorm {d T R} mu p f.
@@ -87,7 +85,15 @@ under eq_integral => x _ do rewrite ger0_norm ?powR_ge0//.
 by rewrite fp//; apply: integral_ge0 => t _; rewrite lee_fin powR_ge0.
 Qed.
 
+Lemma powR_Lnorm f r : r != 0%R ->
+  'N_r%:E[f] `^ r = \int[mu]_x (`| f x | `^ r)%:E.
+Proof.
+move=> r0; rewrite unlock (negbTE r0) -poweRrM mulVf// poweRe1//.
+by apply: integral_ge0 => x _; rewrite lee_fin// powR_ge0.
+Qed.
+
 End Lnorm_properties.
+
 #[global]
 Hint Extern 0 (0 <= Lnorm _ _ _) => solve [apply: Lnorm_ge0] : core.
 
@@ -96,7 +102,7 @@ Notation "'N[ mu ]_ p [ f ]" := (Lnorm mu p f).
 Section lnorm.
 (* l-norm is just L-norm applied to counting *)
 Context d {T : measurableType d} {R : realType}.
-
+Local Open Scope ereal_scope.
 Local Notation "'N_ p [ f ]" := (Lnorm [the measure _ _ of counting] p f).
 
 Lemma Lnorm_counting p (f : R^nat) : (0 < p)%R ->
@@ -186,8 +192,8 @@ have [f0|f0] := eqVneq 'N_p%:E[f] 0%E; first exact: hoelder0.
 have [g0|g0] := eqVneq 'N_q%:E[g] 0%E.
   rewrite muleC; apply: le_trans; last by apply: hoelder0 => //; rewrite addrC.
   by under eq_Lnorm do rewrite /= mulrC.
-have {f0}fpos : 0 < 'N_p%:E[f] by rewrite lt_neqAle eq_sym f0// Lnorm_ge0.
-have {g0}gpos : 0 < 'N_q%:E[g] by rewrite lt_neqAle eq_sym g0// Lnorm_ge0.
+have {f0}fpos : 0 < 'N_p%:E[f] by rewrite lt0e f0 Lnorm_ge0.
+have {g0}gpos : 0 < 'N_q%:E[g] by rewrite lt0e g0 Lnorm_ge0.
 have [foo|foo] := eqVneq 'N_p%:E[f] +oo%E; first by rewrite foo gt0_mulye ?leey.
 have [goo|goo] := eqVneq 'N_q%:E[g] +oo%E; first by rewrite goo gt0_muley ?leey.
 pose F := normalized p f; pose G := normalized q g.
@@ -296,8 +302,7 @@ have [->|w10] := eqVneq w1 0.
     by rewrite !mul0r powR0// gt_eqF.
   by rewrite ge1r_powRZ// /w2 lt_neqAle eq_sym w20/=; apply/andP.
 have [->|w20] := eqVneq w2 0.
-  rewrite !mul0r !addr0 ge1r_powRZ// onem_le1// andbT.
-  by rewrite lt_neqAle eq_sym onem_ge0// andbT.
+  by rewrite !mul0r !addr0 ge1r_powRZ// onem_le1// andbT lt0r w10 onem_ge0.
 have [->|p_neq1] := eqVneq p 1.
   by rewrite !powRr1// addr_ge0// mulr_ge0// /w2 ?onem_ge0.
 have {p_neq1} {}p1 : 1 < p by rewrite lt_neqAle eq_sym p_neq1.
@@ -330,3 +335,173 @@ by rewrite {2}/w1 {2}/w2 subrK powR1 mulr1.
 Qed.
 
 End convex_powR.
+
+Section minkowski.
+Context d (T : measurableType d) (R : realType).
+Variable mu : {measure set T -> \bar R}.
+Implicit Types (f g : T -> R) (p : R).
+
+Let convex_powR_abs_half f g p x : 1 <= p ->
+  `| 2^-1 * f x + 2^-1 * g x | `^ p <=
+   2^-1 * `| f x | `^ p + 2^-1 * `| g x | `^ p.
+Proof.
+move=> p1; rewrite (@le_trans _ _ ((2^-1 * `| f x | + 2^-1 * `| g x |) `^ p))//.
+  rewrite ge0_ler_powR ?nnegrE ?(le_trans _ p1)//.
+  by rewrite (le_trans (ler_normD _ _))// 2!normrM ger0_norm.
+rewrite {1 3}(_ : 2^-1 = 1 - 2^-1); last by rewrite {2}(splitr 1) div1r addrK.
+rewrite (@convex_powR _ _ p1 (@Itv.mk _ _ _ _)) ?inE/= ?in_itv/= ?normr_ge0//.
+by rewrite /Itv.itv_cond/= in_itv/= invr_ge0 ler0n invf_le1 ?ler1n.
+Qed.
+
+Let measurableT_comp_powR f p :
+  measurable_fun setT f -> measurable_fun setT (fun x => f x `^ p)%R.
+Proof. exact: (@measurableT_comp _ _ _ _ _ _ (@powR R ^~ p)). Qed.
+
+Local Notation "'N_ p [ f ]" := (Lnorm mu p f).
+Local Open Scope ereal_scope.
+
+Let minkowski1 f g p : measurable_fun setT f -> measurable_fun setT g ->
+  'N_1[(f \+ g)%R] <= 'N_1[f] + 'N_1[g].
+Proof.
+move=> mf mg.
+rewrite !Lnorm1 -ge0_integralD//; [|by do 2 apply: measurableT_comp..].
+rewrite ge0_le_integral//.
+- by do 2 apply: measurableT_comp => //; exact: measurable_funD.
+- by move=> x _; rewrite lee_fin.
+- by apply/measurableT_comp/measurable_funD; exact/measurableT_comp.
+- by move=> x _; rewrite lee_fin ler_normD.
+Qed.
+
+Let minkowski_lty f g p :
+  measurable_fun setT f -> measurable_fun setT g -> (1 <= p)%R ->
+  'N_p%:E[f] < +oo -> 'N_p%:E[g] < +oo -> 'N_p%:E[(f \+ g)%R] < +oo.
+Proof.
+move=> mf mg p1 Nfoo Ngoo.
+have p0 : p != 0%R by rewrite gt_eqF// (lt_le_trans _ p1).
+have h x : (`| f x + g x | `^ p <=
+            2 `^ (p - 1) * (`| f x | `^ p + `| g x | `^ p))%R.
+  have := convex_powR_abs_half (fun x => 2 * f x)%R (fun x => 2 * g x)%R x p1.
+  rewrite !normrM (@ger0_norm _ 2)// !mulrA mulVf// !mul1r => /le_trans; apply.
+  rewrite !powRM// !mulrA -powR_inv1// -powRD ?pnatr_eq0 ?implybT//.
+  by rewrite (addrC _ p) -mulrDr.
+rewrite unlock (gt_eqF (lt_le_trans _ p1))// poweR_lty//.
+pose x := \int[mu]_x (2 `^ (p - 1) * (`|f x| `^ p + `|g x| `^ p))%:E.
+apply: (@le_lt_trans _ _ x).
+  rewrite ge0_le_integral//=.
+  - by move=> t _; rewrite lee_fin// powR_ge0.
+  - apply/EFin_measurable_fun/measurableT_comp_powR/measurableT_comp => //.
+    exact: measurable_funD.
+  - by move=> t _; rewrite lee_fin mulr_ge0 ?addr_ge0 ?powR_ge0.
+  - by apply/EFin_measurable_fun/measurable_funM/measurable_funD => //;
+      exact/measurableT_comp_powR/measurableT_comp.
+  - by move=> ? _; rewrite lee_fin.
+rewrite {}/x; under eq_integral do rewrite EFinM.
+rewrite ge0_integralZl_EFin ?powR_ge0//; last 2 first.
+  - by move=> x _; rewrite lee_fin addr_ge0// powR_ge0.
+  - by apply/EFin_measurable_fun/measurable_funD => //;
+      exact/measurableT_comp_powR/measurableT_comp.
+rewrite lte_mul_pinfty ?lee_fin ?powR_ge0//.
+under eq_integral do rewrite EFinD.
+rewrite ge0_integralD//; last 4 first.
+  - by move=> x _; rewrite lee_fin powR_ge0.
+  - exact/EFin_measurable_fun/measurableT_comp_powR/measurableT_comp.
+  - by move=> x _; rewrite lee_fin powR_ge0.
+  - exact/EFin_measurable_fun/measurableT_comp_powR/measurableT_comp.
+by rewrite lte_add_pinfty// -powR_Lnorm ?(gt_eqF (lt_trans _ p1))// poweR_lty.
+Qed.
+
+Lemma minkowski f g p :
+  measurable_fun setT f -> measurable_fun setT g -> (1 <= p)%R ->
+  'N_p%:E[(f \+ g)%R] <= 'N_p%:E[f] + 'N_p%:E[g].
+Proof.
+move=> mf mg; rewrite le_eqVlt => /predU1P[<-|p1]; first exact: minkowski1.
+have [->|Nfoo] := eqVneq 'N_p%:E[f] +oo.
+  by rewrite addye ?leey// -ltNye (lt_le_trans _ (Lnorm_ge0 _ _ _)).
+have [->|Ngoo] := eqVneq 'N_p%:E[g] +oo.
+  by rewrite addey ?leey// -ltNye (lt_le_trans _ (Lnorm_ge0 _ _ _)).
+have Nfgoo : 'N_p%:E[(f \+ g)%R] < +oo.
+  by rewrite minkowski_lty// ?ltW// ltey; [exact: Nfoo|exact: Ngoo].
+suff : 'N_p%:E[(f \+ g)%R] `^ p <= ('N_p%:E[f] + 'N_p%:E[g]) *
+    'N_p%:E[(f \+ g)%R] `^ p * (fine 'N_p%:E[(f \+ g)%R])^-1%:E.
+  have [-> _|Nfg0] := eqVneq 'N_p%:E[(f \+ g)%R] 0.
+    by rewrite adde_ge0 ?Lnorm_ge0.
+  rewrite lee_pdivl_mulr ?fine_gt0// ?lt0e ?Nfg0 ?Lnorm_ge0//.
+  rewrite -{1}(@fineK _ ('N_p%:E[(f \+ g)%R] `^ p)); last first.
+    by rewrite fin_num_poweR// ge0_fin_numE// Lnorm_ge0.
+  rewrite -(invrK (fine _)) lee_pdivr_mull; last first.
+    rewrite invr_gt0 fine_gt0// (poweR_lty _ Nfgoo) andbT poweR_gt0//.
+    by rewrite lt0e Nfg0 Lnorm_ge0.
+  rewrite fineK ?ge0_fin_numE ?Lnorm_ge0// => /le_trans; apply.
+  rewrite lee_pdivr_mull; last first.
+    by rewrite fine_gt0// poweR_lty// andbT poweR_gt0// lt0e Nfg0 Lnorm_ge0.
+  by rewrite fineK// 1?muleC// fin_num_poweR// ge0_fin_numE ?Lnorm_ge0.
+have p0 : (0 < p)%R by exact: (lt_trans _ p1).
+rewrite powR_Lnorm ?gt_eqF//.
+under eq_integral => x _ do rewrite -mulr_powRB1//.
+apply: (@le_trans _ _
+    (\int[mu]_x ((`|f x| + `|g x|) * `|f x + g x| `^ (p - 1))%:E)).
+  rewrite ge0_le_integral//.
+  - by move=> ? _; rewrite lee_fin mulr_ge0// powR_ge0.
+  - apply: measurableT_comp => //; apply: measurable_funM.
+      exact/measurableT_comp/measurable_funD.
+    exact/measurableT_comp_powR/measurableT_comp/measurable_funD.
+  - by move=> ? _; rewrite lee_fin mulr_ge0// powR_ge0.
+  - apply/measurableT_comp => //; apply: measurable_funM.
+      by apply/measurable_funD => //; exact: measurableT_comp.
+    exact/measurableT_comp_powR/measurableT_comp/measurable_funD.
+  - by move=> ? _; rewrite lee_fin ler_wpM2r// ?powR_ge0// ler_normD.
+under eq_integral=> ? _ do rewrite mulrDl EFinD.
+rewrite ge0_integralD//; last 4 first.
+  - by move=> x _; rewrite lee_fin mulr_ge0// powR_ge0.
+  - apply: measurableT_comp => //; apply: measurable_funM.
+      exact: measurableT_comp.
+    exact/measurableT_comp_powR/measurableT_comp/measurable_funD.
+  - by move=> x _; rewrite lee_fin mulr_ge0// powR_ge0.
+  - apply: measurableT_comp => //; apply: measurable_funM.
+      exact: measurableT_comp.
+    exact/measurableT_comp_powR/measurableT_comp/measurable_funD.
+rewrite [leRHS](_ : _ = ('N_p%:E[f] + 'N_p%:E[g]) *
+    (\int[mu]_x (`|f x + g x| `^ p)%:E) `^ `1-(p^-1)).
+  rewrite muleDl; last 2 first.
+    - rewrite fin_num_poweR// -powR_Lnorm ?gt_eqF// fin_num_poweR//.
+      by rewrite ge0_fin_numE ?Lnorm_ge0.
+    - by rewrite ge0_adde_def// inE Lnorm_ge0.
+  apply: lee_add.
+  - pose h := (@powR R ^~ (p - 1) \o normr \o (f \+ g))%R; pose i := (f \* h)%R.
+    rewrite [leLHS](_ : _ = 'N_1[i]%R); last first.
+      rewrite Lnorm1; apply: eq_integral => x _.
+      by rewrite normrM (ger0_norm (powR_ge0 _ _)).
+    rewrite [X in _ * X](_ : _ = 'N_(p / (p - 1))%:E[h]); last first.
+      rewrite unlock mulf_eq0 gt_eqF//= invr_eq0 subr_eq0 (gt_eqF p1).
+      rewrite onemV ?gt_eqF// invf_div; apply: congr2; last by [].
+      apply: eq_integral => x _; congr EFin.
+      rewrite norm_powR// normr_id -powRrM mulrCA divff ?mulr1//.
+      by rewrite subr_eq0 gt_eqF.
+    apply: (@hoelder _ _ _ _ _ _ p (p / (p - 1))) => //.
+    + exact/measurableT_comp_powR/measurableT_comp/measurable_funD.
+    + by rewrite divr_gt0// subr_gt0.
+    + by rewrite invf_div -onemV ?gt_eqF// addrCA subrr addr0.
+  - pose h := (fun x => `|f x + g x| `^ (p - 1))%R; pose i := (g \* h)%R.
+    rewrite [leLHS](_ : _ = 'N_1[i]); last first.
+      rewrite Lnorm1; apply: eq_integral => x _ .
+      by rewrite normrM norm_powR// normr_id.
+    rewrite [X in _ * X](_ : _ = 'N_((1 - p^-1)^-1)%:E[h])//; last first.
+      rewrite unlock invrK invr_eq0 subr_eq0 eq_sym invr_eq1 (gt_eqF p1).
+      apply: congr2; last by [].
+      apply: eq_integral => x _; congr EFin.
+      rewrite -/(onem p^-1) onemV ?gt_eqF// norm_powR// normr_id -powRrM.
+      by rewrite invf_div mulrCA divff ?subr_eq0 ?gt_eqF// ?mulr1.
+    apply: (le_trans (@hoelder _ _ _ _ _ _ p (1 - p^-1)^-1 _ _ _ _ _)) => //.
+    + exact/measurableT_comp_powR/measurableT_comp/measurable_funD.
+    + by rewrite invr_gt0 onem_gt0// invf_lt1.
+    + by rewrite invrK addrCA subrr addr0.
+rewrite -muleA; congr (_ * _).
+under [X in X * _]eq_integral=> x _ do rewrite mulr_powRB1 ?subr_gt0//.
+rewrite poweRD; last by rewrite poweRD_defE gt_eqF ?implyFb// subr_gt0 invf_lt1.
+rewrite poweRe1; last by apply: integral_ge0 => x _; rewrite lee_fin powR_ge0.
+congr (_ * _); rewrite poweRN.
+- by rewrite unlock gt_eqF// fine_poweR.
+- by rewrite -powR_Lnorm ?gt_eqF// fin_num_poweR// ge0_fin_numE ?Lnorm_ge0.
+Qed.
+
+End minkowski.
diff --git a/theories/probability.v b/theories/probability.v
@@ -541,7 +541,7 @@ rewrite (le_trans _ (markov _ (expR_gt0 (r * a)) _ _ _))//; last first.
   exact: (monoW_in (@ger0_le_norm _)).
 rewrite ger0_norm ?expR_ge0// muleC lee_pmul2l// ?lte_fin ?expR_gt0//.
 rewrite [X in _ <= P X](_ : _ = [set x | a <= X x]%R)//; apply: eq_set => t/=.
-by rewrite ger0_norm ?expR_ge0// lee_fin ler_expR  mulrC ler_pmul2r.
+by rewrite ger0_norm ?expR_ge0// lee_fin ler_expR  mulrC ler_pM2r.
 Qed.
 
 Lemma chebyshev (X : {RV P >-> R}) (eps : R) : (0 < eps)%R ->