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reversed parameterized integral #2091

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reversed parameterized integral #2091

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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -55,6 +55,9 @@
- in `Rstruct_topology.v`:
+ lemmas `RcosE`, `Rtrigo_PIE`, `RsinE`

- in `ftc.v`:
+ lemmas `parameterized_integralN`, `parameterized_integralN_continuous`

### Changed

- in `derive.v`:
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34 changes: 34 additions & 0 deletions theories/ftc.v
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Expand Up @@ -1765,6 +1765,40 @@ Qed.

End integration_by_substitution.

Lemma parameterized_integralN {R : realType} x b (f : R -> R) : (x <= b) ->
{within `[x, b], continuous f} ->
parameterized_integral lebesgue_measure x b f =
parameterized_integral lebesgue_measure (- b) (- x) (f \o -%R).
Proof.
move=> xb cf.
rewrite /parameterized_integral /Rintegral.
rewrite -(@integration_by_substitution_oppr _ f (- b) (- x)) ?opprK//.
by rewrite lerN2.
Qed.

Section parameterized_integralN_continuous.
Context {R : realType}.
Notation mu := (@lebesgue_measure R).
Let int := (parameterized_integral mu).

Lemma parameterized_integralN_continuous a b (f : R -> R) : a <= b ->
{within `[a, b], continuous f} ->

@holgerthies holgerthies Aug 26, 2026 •
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Could the assumption {within [a, b], continuous f} be weakened to mu.-integrable [a, b] (EFin \o f) ?

@affeldt-aist affeldt-aist Aug 27, 2026 •
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Yes, it could be (like parameterized_integral_continuous). (And it should maybe be done before merging.)

@affeldt-aist affeldt-aist Aug 30, 2026

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Let's postpone since it might take some time.

{within `[a, b], continuous (fun x => int x b f)}.
Proof.
move=> ab abf; suff: {within `[a, b], continuous
((fun x => parameterized_integral mu (- b) x (f \o -%R)) \o -%R)}.
apply: subspace_eq_continuous => /= x /[!inE] xab/=.
rewrite /from_subspace/= -parameterized_integralN ?(itvP xab)//.
apply: continuous_subspaceW abf.
by apply: subset_itvr; rewrite bnd_simp (itvP xab).
apply: within_continuous_compN.
apply: parameterized_integral_continuous; first by rewrite lerN2.
apply: continuous_compact_integrable => //; first exact: segment_compact.
by apply: within_continuous_compN; rewrite !opprK.
Qed.

End parameterized_integralN_continuous.

Section ge0_integration_by_substitution_shift.
Context {R : realType}.
Notation mu := (@lebesgue_measure R).
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