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Alternative Proof of Gauss Integral with improper integral #1646

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Alternative Proof of Gauss Integral with improper integral #1646

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13 changes: 13 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -86,6 +86,19 @@
`le0_nondecreasing_set_nonincreasing_integral`,
`le0_nondecreasing_set_cvg_integral`

- in new file `gauss_integral_unbounded`
+ add lemmas `integral0y_gauss_fin_num`,
`integral0y_u0`,
`integrable0y_u`,
`max_y_ge0`,
`u_dominates`,
`int0yu_fin_num`,
`cvgy_int0yu0`,
`derive1_int0yuE`,
`derivable_int0yu`,
`rcvg0_int0yu`,
`gauss_integration`

### Changed

- in `convex.v`:
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1 change: 1 addition & 0 deletions _CoqProject
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Expand Up @@ -120,5 +120,6 @@ theories/kernel.v
theories/pi_irrational.v
theories/gauss_integral.v
theories/showcase/summability.v
theories/showcase/gauss_integral_unbounded.v
analysis_stdlib/Rstruct_topology.v
analysis_stdlib/showcase/uniform_bigO.v
1 change: 1 addition & 0 deletions theories/Make
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Expand Up @@ -86,3 +86,4 @@ pi_irrational.v
gauss_integral.v
all_analysis.v
showcase/summability.v
showcase/gauss_integral_unbounded.v
8 changes: 4 additions & 4 deletions theories/gauss_integral.v
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Expand Up @@ -62,7 +62,7 @@ Let mu : {measure set _ -> \bar R} := @lebesgue_measure R.

Definition integral0y_gauss := \int[mu]_(x in `[0%R, +oo[) gauss_fun x.

Let integral0y_gauss_ge0 : 0 <= integral0y_gauss.
Lemma integral0y_gauss_ge0 : 0 <= integral0y_gauss.
Proof. by apply: Rintegral_ge0 => //= x _; rewrite gauss_fun_ge0. Qed.

Definition integral0_gauss x := \int[mu]_(t in `[0, x]) gauss_fun t.
Expand All @@ -81,10 +81,10 @@ Qed.

Definition u x t := expR (- x ^+ 2 * oneDsqr t) / oneDsqr t.

Let u_ge0 x t : 0 <= u x t.
Lemma u_ge0 x t : 0 <= u x t.
Proof. by rewrite /u divr_ge0// ?expR_ge0. Qed.

Let measurable_u x : measurable_fun setT (u x).
Lemma measurable_u x : measurable_fun setT (u x).
Proof.
apply: measurable_funM => //=.
apply: measurableT_comp => //=; apply: measurable_funM => //=.
Expand All @@ -95,7 +95,7 @@ Qed.

Local Notation "'d1 f" := (partial1of2 f).

Let partial1_u x t : ('d1 u) x t = - 2 * x * gauss_fun x * gauss_fun (t * x).
Lemma partial1_u x t : ('d1 u) x t = - 2 * x * gauss_fun x * gauss_fun (t * x).
Proof.
rewrite partial1of2E /u /= deriveMr//= -derive1E.
rewrite derive1_comp// [in X in _ * (_ * X)]derive1Mr//.
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