Lspace

.nix/config.nix

@@ -50,12 +50,16 @@ in
   ## will be created per bundle
   bundles."8.19".coqPackages = common-bundle // {
     coq.override.version = "8.19";
-    mathcomp.override.version = "2.2.0";
+    mathcomp.override.version = "CohenCyril:seminorm";
+    mathcomp-bigenough.override.version = "master";
+    mathcomp-finmap.override.version = "master";
   };
 
   bundles."8.20".coqPackages = common-bundle // {
     coq.override.version = "8.20";
-    mathcomp.override.version = "2.2.0";
+    mathcomp.override.version = "CohenCyril:seminorm";
+    mathcomp-bigenough.override.version = "master";
+    mathcomp-finmap.override.version = "master";
     ssprove.job = false;
   };
 
@@ -65,7 +69,7 @@ in
     coq-elpi.override.version = "master";
     coq-elpi.override.elpi-version = "2.0.7";
     hierarchy-builder.override.version = "master";
-    mathcomp.override.version = "master";
+    mathcomp.override.version = "CohenCyril:seminorm";
     mathcomp-bigenough.override.version = "master";
     mathcomp-finmap.override.version = "master";
     ssprove.job = false;

_CoqProject

@@ -94,3 +94,4 @@ theories/pi_irrational.v
 theories/showcase/summability.v
 analysis_stdlib/Rstruct_topology.v
 analysis_stdlib/showcase/uniform_bigO.v
+theories/lspace.v

theories/charge.v

@@ -1867,7 +1867,7 @@ have nuf A : d.-measurable A -> nu A = \int[mu]_(x in A) f x.
 move=> A mA; rewrite nuf ?inE//; apply: ae_eq_integral => //.
 - exact/measurable_funTS.
 - exact/measurable_funTS.
-- exact: ae_eq_subset ff'.
+- exact: (@ae_eq_subset _ _ _ _ mu setT A f f' (@subsetT _ A)).
 Qed.
 
 End radon_nikodym_sigma_finite.
@@ -2093,6 +2093,10 @@ move=> mE; apply: integral_ae_eq => //.
   by rewrite -Radon_Nikodym_SigmaFinite.f_integral.
 Qed.
 
+(* TODO: move back to measure.v, current version incompatible *)
+Lemma ae_eq_mul2l (f g h : T -> \bar R) D : f = g %[ae mu in D] -> (h \* f) = (h \* g) %[ae mu in D].
+Proof. by apply: filterS => x /= /[apply] ->. Qed.
+
 Lemma Radon_Nikodym_change_of_variables f E : measurable E ->
     nu.-integrable E f ->
   \int[mu]_(x in E) (f x * ('d (charge_of_finite_measure nu) '/d mu) x) =

theories/hoelder.v

@@ -10,10 +10,10 @@ From mathcomp Require Import numfun exp convex itv.
 (**md**************************************************************************)
 (* # Hoelder's Inequality                                                     *)
 (*                                                                            *)
-(* This file provides Hoelder's inequality.                                   *)
+(* This file provides Hoelder's inequality and its consequences, most notably *)
+(* Minkowski's inequality and the convexity of the power function.            *)
 (* ```                                                                        *)
-(*           'N[mu]_p[f] := (\int[mu]_x (`|f x| `^ p)%:E) `^ p^-1             *)
-(*                          The corresponding definition is Lnorm.            *)
+(*           'N[mu]_p[f] == the p-norm of f with measure mu                   *)
 (* ```                                                                        *)
 (*                                                                            *)
 (******************************************************************************)
@@ -40,7 +40,7 @@ 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
-              mu (f @^-1` (setT `\ 0%R))
+              (mu (f @^-1` (setT `\ 0%R))) `^ p^-1
             else
               (\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
@@ -57,6 +57,20 @@ Implicit Types (p : \bar R) (f g : T -> R) (r : R).
 
 Local Notation "'N_ p [ f ]" := (Lnorm mu p f).
 
+Lemma Lnorm0 p : 1 <= p -> 'N_p[cst 0%R] = 0.
+Proof.
+rewrite unlock /Lnorm.
+case: p => [r||//].
+- rewrite lee_fin => r1.
+  have r0: r != 0%R by rewrite gt_eqF// (lt_le_trans _ r1).
+  rewrite gt_eqF ?(lt_le_trans _ r1)//.
+  under eq_integral => x _ do rewrite /= normr0 powR0//.
+  by rewrite integral0 poweR0r// invr_neq0.
+case: ifPn => //mu0 _.
+rewrite (_ : normr \o _ = 0%R); last by apply: funext => x/=; rewrite normr0.
+exact: ess_sup_cst.
+Qed.
+
 Lemma Lnorm1 f : 'N_1[f] = \int[mu]_x `|f x|%:E.
 Proof.
 rewrite unlock oner_eq0 invr1// poweRe1//.
@@ -74,16 +88,36 @@ Qed.
 Lemma eq_Lnorm p f g : f =1 g -> 'N_p[f] = 'N_p[g].
 Proof. by move=> fg; congr Lnorm; exact/funext. Qed.
 
-Lemma Lnorm_eq0_eq0 r f : (0 < r)%R -> measurable_fun setT f ->
-  'N_r%:E[f] = 0 -> ae_eq mu [set: T] (fun t => (`|f t| `^ r)%:E) (cst 0).
+Lemma Lnorm_eq0_eq0 (f : T -> R) p :
+  measurable_fun setT f -> (0 < p)%E -> 'N_p[f] = 0 -> f = 0%R %[ae mu].
 Proof.
-move=> r0 mf; rewrite unlock (gt_eqF r0) => /poweR_eq0_eq0 fp.
-apply/ae_eq_integral_abs => //=.
-  apply: measurableT_comp => //.
-  apply: (@measurableT_comp _ _ _ _ _ _ (@powR R ^~ r)) => //.
-  exact: measurableT_comp.
-under eq_integral => x _ do rewrite ger0_norm ?powR_ge0//.
-by rewrite fp//; apply: integral_ge0 => t _; rewrite lee_fin powR_ge0.
+rewrite unlock /Lnorm => mf.
+case: p => [r r0||].
+- case: ifPn => _.
+    rewrite preimage_setI preimage_setT setTI -preimage_setC.
+    move=> /poweR_eq0_eq0 /negligibleP.
+    move/(_ (measurableC _)); rewrite -[X in d.-measurable X]setTI.
+    move/(_ (mf _ _ _)).
+    by case=> // A [mA muA0 fA]; exists A; split => // x/= ?; exact: fA.
+  move=> /poweR_eq0_eq0.
+  move=> /(_ (integral_ge0 _ _)) h.
+  have: (\int[mu]_x (`|f x| `^ r)%:E)%E = 0 by apply: h => x _; rewrite lee_fin powR_ge0.
+  under eq_integral => x _ do rewrite -[_%:E]gee0_abs ?lee_fin ?powR_ge0//.
+  have mp: measurable_fun [set: T] (fun x : T => (`|f x| `^ r)%:E).
+    apply: measurableT_comp => //.
+    apply (measurableT_comp (measurable_powR _)) => //.
+    exact: measurableT_comp.
+  move/(ae_eq_integral_abs _ measurableT mp).
+  apply: filterS => x/= /[apply].
+  by case=> /powR_eq0_eq0 /eqP; rewrite normr_eq0 => /eqP.
+- case: ifPn => [mu0 _|].
+    exact: ess_sup_eq0.
+  rewrite ltNge => /negbNE mu0 _ _.
+  suffices mueq0: mu setT = 0 by exact: ae_eq0.
+  move: mu0 (measure_ge0 mu setT) => mu0 mu1.
+  suffices: (mu setT <= 0 <= mu setT)%E by move/le_anti.
+  by rewrite mu0 mu1.
+by [].
 Qed.
 
 Lemma powR_Lnorm f r : r != 0%R ->
@@ -93,9 +127,39 @@ move=> r0; rewrite unlock (negbTE r0) -poweRrM mulVf// poweRe1//.
 by apply: integral_ge0 => x _; rewrite lee_fin// powR_ge0.
 Qed.
 
+Lemma oppr_Lnorm f p :
+  'N_p[\- f]%R = 'N_p[f].
+Proof.
+rewrite unlock /Lnorm.
+case: p => /= [r||//].
+  case: eqP => _. congr ((mu _) `^ _).
+    rewrite !preimage_setI.
+    congr (_ `&` _).
+    rewrite -!preimage_setC.
+    congr (~` _).
+    rewrite /preimage.
+    apply: funext => x/=.
+    rewrite -{1}oppr0.
+    apply: propext. split; last by move=> ->.
+    by move/oppr_inj.
+  by under eq_integral => x _ do rewrite normrN.
+rewrite compA (_ : normr \o -%R = normr)//.
+apply: funext => x/=; exact: normrN.
+Qed.
+
+Lemma Lnorm_cst1 r : ('N_r%:E[cst 1%R] = (mu setT)`^(r^-1)).
+Proof.
+rewrite unlock /Lnorm.
+case: ifPn => [_|].
+  by rewrite preimage_cst ifT// inE/=; split => //; apply/eqP; rewrite oner_neq0.
+under eq_integral => x _ do rewrite normr1 powR1 (_ : 1 = cst 1 x)%R// -indicT.
+by rewrite integral_indic// setTI.
+Qed.
+
 End Lnorm_properties.
 
 #[global]
+
 Hint Extern 0 (0 <= Lnorm _ _ _) => solve [apply: Lnorm_ge0] : core.
 
 Notation "'N[ mu ]_ p [ f ]" := (Lnorm mu p f).
@@ -144,11 +208,12 @@ Let hoelder0 f g p q : measurable_fun setT f -> measurable_fun setT g ->
     (0 < p)%R -> (0 < q)%R -> (p^-1 + q^-1 = 1)%R ->
   'N_p%:E[f] = 0 -> 'N_1[(f \* g)%R]  <= 'N_p%:E[f] * 'N_q%:E[g].
 Proof.
+rewrite -lte_fin.
 move=> mf mg p0 q0 pq f0; rewrite f0 mul0e Lnorm1 [leLHS](_ : _ = 0)//.
 rewrite (ae_eq_integral (cst 0)) => [|//||//|]; first by rewrite integral0.
 - by do 2 apply: measurableT_comp => //; exact: measurable_funM.
-- apply: filterS (Lnorm_eq0_eq0 p0 mf f0) => x /(_ I)[] /powR_eq0_eq0 + _.
-  by rewrite normrM => ->; rewrite mul0r.
+- apply: filterS (Lnorm_eq0_eq0 mf p0 f0) => x /(_ I)[] + _.
+  by rewrite normrM => ->; rewrite normr0 mul0r.
 Qed.
 
 Let normalized p f x := `|f x| / fine 'N_p%:E[f].
@@ -502,4 +567,20 @@ congr (_ * _); rewrite poweRN.
 - by rewrite -powR_Lnorm ?gt_eqF// fin_num_poweR// ge0_fin_numE ?Lnorm_ge0.
 Qed.
 
+Lemma minkowski' f g p :
+  measurable_fun setT f -> measurable_fun setT g -> (1 <= p)%R ->
+  'N_p%:E[f] <= 'N_p%:E[f \+ g] + 'N_p%:E[g].
+Proof.
+move=> mf mg p1.
+rewrite (_ : f = ((f \+ g) \+ (-%R \o g))%R); last first.
+  by apply: funext => x /=; rewrite -addrA subrr addr0.
+rewrite [X in _ <= 'N__[X] + _](_ : ((f \+ g \- g) \+ g)%R = (f \+ g)%R); last first.
+  by apply: funext => x /=; rewrite -addrA [X in _ + _ + X]addrC subrr addr0.
+rewrite (_ : 'N__[g] = 'N_p%:E[-%R \o g]); last first.
+  by rewrite oppr_Lnorm.
+apply: minkowski => //.
+  apply: measurable_funD => //.
+apply: measurableT_comp => //.
+Qed.
+
 End minkowski.

theories/lebesgue_integral.v

@@ -272,6 +272,8 @@ Lemma mfunN f : - f = \- f :> (_ -> _). Proof. by []. Qed.
 Lemma mfunD f g : f + g = f \+ g :> (_ -> _). Proof. by []. Qed.
 Lemma mfunB f g : f - g = f \- g :> (_ -> _). Proof. by []. Qed.
 Lemma mfunM f g : f * g = f \* g :> (_ -> _). Proof. by []. Qed.
+Lemma mfunMn f n : (f *+ n) = (fun x => f x *+ n) :> (_ -> _).
+Proof. by apply/funext=> x; elim: n => //= n; rewrite !mulrS !mfunD /= => ->. Qed.
 Lemma mfun_sum I r (P : {pred I}) (f : I -> {mfun aT >-> rT}) (x : aT) :
   (\sum_(i <- r | P i) f i) x = \sum_(i <- r | P i) f i x.
 Proof. by elim/big_rec2: _ => //= i y ? Pi <-. Qed.
@@ -285,6 +287,7 @@ HB.instance Definition _ f g := MeasurableFun.copy (f \+ g) (f + g).
 HB.instance Definition _ f g := MeasurableFun.copy (\- f) (- f).
 HB.instance Definition _ f g := MeasurableFun.copy (f \- g) (f - g).
 HB.instance Definition _ f g := MeasurableFun.copy (f \* g) (f * g).
+(* TODO: fix this: HB.instance Definition _ f (n : nat) := MeasurableFun.copy (fun x => f x *+ n) (f *+ n). *)
 
 Definition mindic (D : set aT) of measurable D : aT -> rT := \1_D.
 
@@ -3726,11 +3729,9 @@ Local Open Scope ereal_scope.
 Context d (T : measurableType d) (R : realType)
         (mu : {measure set T -> \bar R}).
 
-Local Notation ae_eq := (ae_eq mu).
-
 Let ae_eq_integral_abs_bounded (D : set T) (mD : measurable D) (f : T -> \bar R)
     M : measurable_fun D f -> (forall x, D x -> `|f x| <= M%:E) ->
-  ae_eq D f (cst 0) -> \int[mu]_(x in D) `|f x|%E  = 0.
+  (\forall x \ae mu, D x -> f x = 0) -> \int[mu]_(x in D) `|f x|%E  = 0.
 Proof.
 move=> mf fM [N [mA mN0 Df0N]].
 pose Df_neq0 := D `&` [set x | f x != 0].
@@ -3759,7 +3760,8 @@ by rewrite mule0 -eq_le => /eqP.
 Qed.
 
 Lemma ae_eq_integral_abs (D : set T) (mD : measurable D) (f : T -> \bar R) :
-  measurable_fun D f -> \int[mu]_(x in D) `|f x| = 0 <-> ae_eq D f (cst 0).
+    measurable_fun D f ->
+  \int[mu]_(x in D) `|f x| = 0 <-> (\forall x \ae mu, D x -> f x = 0).
 Proof.
 move=> mf; split=> [iDf0|Df0].
   exists (D `&` [set x | f x != 0]); split;
@@ -3813,7 +3815,7 @@ transitivity (limn (fun n => \int[mu]_(x in D) (f_ n x) )).
     have [ftm|ftm] := leP `|f t|%E m%:R%:E.
       by rewrite lexx /= (le_trans ftm)// lee_fin ler_nat.
     by rewrite (ltW ftm) /= lee_fin ler_nat.
-have ae_eq_f_ n : ae_eq D (f_ n) (cst 0).
+have ae_eq_f_ n : (f_ n) = (cst 0) %[ae mu in D].
   case: Df0 => N [mN muN0 DfN].
   exists N; split => // t /= /not_implyP[Dt fnt0].
   apply: DfN => /=; apply/not_implyP; split => //.
@@ -3922,7 +3924,7 @@ Qed.
 Lemma ge0_ae_eq_integral (D : set T) (f g : T -> \bar R) :
   measurable D -> measurable_fun D f -> measurable_fun D g ->
   (forall x, D x -> 0 <= f x) -> (forall x, D x -> 0 <= g x) ->
-  ae_eq D f g -> \int[mu]_(x in D) (f x) = \int[mu]_(x in D) (g x).
+  f = g %[ae mu in D] -> \int[mu]_(x in D) (f x) = \int[mu]_(x in D) (g x).
 Proof.
 move=> mD mf mg f0 g0 [N [mN N0 subN]].
 rewrite integralEpatch// [RHS]integralEpatch//.
@@ -3940,7 +3942,7 @@ Qed.
 
 Lemma ae_eq_integral (D : set T) (g f : T -> \bar R) :
   measurable D -> measurable_fun D f -> measurable_fun D g ->
-  ae_eq D f g -> integral mu D f = integral mu D g.
+  f = g %[ae mu in D] -> integral mu D f = integral mu D g.
 Proof.
 move=> mD mf mg /ae_eq_funeposneg[Dfgp Dfgn].
 rewrite integralE// [in RHS]integralE//; congr (_ - _).