beta distribution

Author: affeldt-aist

CHANGELOG_UNRELEASED.md

@@ -57,6 +57,19 @@
   + `measure_extension.v`
   + `measurable_function.v`
   + `measure.v`
+- in `lebesgue_integral_theory/lebesgue_integral_nonneg.v`:
+  + lemmas `ge0_nondecreasing_set_nondecreasing_integral`,
+           `ge0_nondecreasing_set_cvg_integral`,
+           `le0_nondecreasing_set_nonincreasing_integral`,
+	   `le0_nondecreasing_set_cvg_integral`
+- in `pseudometric_normed_Zmodule.v`:
+  + lemma `continuous_comp_cvg`
+
+- in `derive.v`:
+  + lemma `derive1_onem`
+
+- in `ftc.v`:
+  + lemmas `integration_by_substitution_onem`, `Rintegration_by_substitution_onem`
 
 ### Changed
 
@@ -305,6 +318,27 @@
 
 ### Renamed
 
+- in `measure.v`
+  + definition `ess_sup` moved to `ess_sup_inf.v`
+
+- in `convex.v`
+  + lemma `conv_gt0` to `convR_gt0`
+- in `tvs.v`
+  + HB class `TopologicalNmodule` moved to `PreTopologicalNmodule`
+  + HB class `TopologicalZmodule` moved to `PreTopologicalZmodule`
+  + HB class `TopologicalLmodule` moved to `PreTopologicalLmodule`
+  + structure `topologicalLmodule` moved to `preTopologicalLmodule`
+  + HB class `UniformNmodule` moved to `PreUniformNmodule`
+  + HB class `UniformZmodule` moved to `PreUniformZmodule`
+  + HB class `UniformLmodule` moved to `PreUniformLmodule`
+
+- in `probability.v`:
+  + `bernoulli` -> `bernoulli_prob`
+  + `bernoulli_probE` -> `bernoulliE`
+  + `integral_bernoulli_prob` -> `integral_bernoulli_prob`
+  + `measurable_bernoulli` -> `measurable_bernoulli_prob`
+  + `measurable_bernoulli2` -> `measurable_bernoulli_prob2`
+
 ### Generalized
 
 - in `pseudometric_normed_Zmodule.v`:

theories/derive.v

@@ -1267,6 +1267,12 @@ Proof. by have /derivableP := @derivable_id x v; rewrite derive_val. Qed.
 
 End derive_id.
 
+Lemma derive1_onem {R : numFieldType} :
+  (fun x => 1 - x : R^o)^`()%classic = cst (-1).
+Proof.
+by apply/funext => x; rewrite derive1E deriveB// derive_id derive_cst sub0r.
+Qed.
+
 Section Derive_lemmasVR.
 Variables (R : numFieldType) (V : normedModType R).
 Implicit Types (f g : V -> R) (x v : V).
@@ -1332,13 +1338,26 @@ rewrite scalerA -scalerDl mulrCA -[f x * _]exprS.
 by rewrite [in LHS]mulr_natl exprfctE -mulrSr mulr_natl.
 Qed.
 
+(*Global Instance is_deriveX f n x v (df : R) :
+  is_derive x v f df -> is_derive x v (f ^+ n.+1) ((n.+1%:R * f x ^+n) *: df).
+Proof.
+move=> dfx; elim: n => [|n ihn]; first by rewrite expr1 expr0 mulr1 scale1r.
+rewrite exprS; apply: is_derive_eq.
+rewrite scalerA -scalerDl mulrCA -[f x * _]exprS.
+by rewrite [in LHS]mulr_natl exprfctE -mulrSr mulr_natl.
+Qed.*)
+
 Lemma derivableX f n x v : derivable f x v -> derivable (f ^+ n) x v.
 Proof. by case: n => [_|n /derivableP]; [rewrite expr0|]. Qed.
 
 Lemma deriveX f n x v : derivable f x v ->
   'D_v (f ^+ n) x = (n%:R * f x ^+ n.-1) *: 'D_v f x.
 Proof. by move=> /derivableP df; rewrite derive_val. Qed.
 
+(*Lemma deriveX f n x v : derivable f x v ->
+  'D_v (f ^+ n.+1) x = (n.+1%:R * f x ^+ n) *: 'D_v f x.
+Proof. by move=> /derivableP df; rewrite derive_val. Qed.*)
+
 Fact der_inv f x v : f x != 0 -> derivable f x v ->
   (fun h => h^-1 *: (((fun y => (f y)^-1) \o shift x) (h *: v) - (f x)^-1)) @
   0^' --> - (f x) ^-2 *: 'D_v f x.

theories/ftc.v

@@ -1793,6 +1793,46 @@ Qed.
 
 End integration_by_substitution.
 
+Section integration_by_substitution_onem.
+Context {R : realType}.
+Let mu := (@lebesgue_measure R).
+Local Open Scope ereal_scope.
+
+Lemma integration_by_substitution_onem (G : R -> R) (r : R) :
+  (0 < r <= 1)%R ->
+  {within `[0%R, r], continuous G} ->
+  (\int[mu]_(x in `[0%R, r]) (G x)%:E =
+  \int[mu]_(x in `[(1 - r)%R, 1%R]) (G (1 - x))%:E).
+Proof.
+move=> r01 cG.
+have := @integration_by_substitution_decreasing R (fun x => 1 - x)%R G (1 - r) 1.
+rewrite subKr subrr => -> //.
+- by apply: eq_integral => x xr; rewrite !fctE derive1_onem opprK mulr1.
+- by rewrite ltrBlDl ltrDr; case/andP : r01.
+- by move=> x y _ _ xy; rewrite ler_ltB.
+- by rewrite derive1_onem; move=> ? ?; exact: cvg_cst.
+- by rewrite derive1_onem; exact: is_cvg_cst.
+- by rewrite derive1_onem; exact: is_cvg_cst.
+- split => /=.
+  + by move=> x xr1; exact: derivableB.
+  + apply: cvg_at_right_filter; rewrite subKr.
+    apply: (@continuous_comp_cvg _ R^o R^o _ (fun x => 1 - x)%R)=> //=.
+      by move=> x; apply: (@continuousB _ R^o)  => //; exact: cvg_cst.
+    by under eq_fun do rewrite subKr; exact: cvg_id.
+  + by apply: cvg_at_left_filter; apply: (@cvgB _ R^o) => //; exact: cvg_cst.
+Qed.
+
+Lemma Rintegration_by_substitution_onem (G : R -> R) (r : R) :
+  (0 < r <= 1)%R ->
+  {within `[0%R, r], continuous G} ->
+  (\int[mu]_(x in `[0%R, r]) (G x) =
+  \int[mu]_(x in `[(1 - r)%R, 1%R]) (G (1 - x)))%R.
+Proof.
+by move=> r01 cG; rewrite [in LHS]/Rintegral integration_by_substitution_onem.
+Qed.
+
+End integration_by_substitution_onem.
+
 Section ge0_integralT_even.
 Context {R : realType}.
 Let mu := @lebesgue_measure R.

theories/normedtype_theory/pseudometric_normed_Zmodule.v

@@ -817,6 +817,20 @@ Arguments cvgr_neq0 {R V T F FF f}.
 #[global] Hint Extern 0 (ProperFilter _^'+) =>
   (apply: at_right_proper_filter) : typeclass_instances.
 
+Lemma continuous_comp_cvg {R : numFieldType} (U V : pseudoMetricNormedZmodType R)
+  (f : U -> V) (g : R -> U) (r : R) (l : V) : continuous f ->
+  (f \o g) x @[x --> r] --> l -> f x @[x --> g r] --> l.
+Proof.
+move=> cf fgl; apply/(@cvgrPdist_le _ V) => /= e e0.
+have e20 : 0 < e / 2 by rewrite divr_gt0.
+move/(@cvgrPdist_le _ V) : fgl => /(_ _ e20) fgl.
+have /(@cvgrPdist_le _ V) /(_ _ e20) fgf := cf (g r).
+rewrite !near_simpl/=; near=> t.
+rewrite -(@subrK _ (f (g r)) l) -(addrA (_ + _)) (le_trans (ler_normD _ _))//.
+rewrite (splitr e) lerD//; last by near: t.
+by case: fgl => d /= d0; apply; rewrite /ball_/= subrr normr0.
+Unshelve. all: by end_near. Qed.
+
 Section closure_left_right_open.
 Variable R : realFieldType.
 Implicit Types z : R.