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[Merged by Bors] - feat: add a version of FTC-2 with an integral over the unit interval #8615

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[Merged by Bors] - feat: add a version of FTC-2 with an integral over the unit interval #8615

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feat: add a version of FTC-2 with an integral over the unit interval
MichaelStollBayreuth Nov 24, 2023
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use IsROrC
MichaelStollBayreuth Nov 30, 2023
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MichaelStollBayreuth Dec 7, 2023
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23 changes: 22 additions & 1 deletion Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
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Expand Up @@ -90,7 +90,7 @@ We use FTC-1 to prove several versions of FTC-2 for the Lebesgue measure, using
scheme as for the versions of FTC-1. They include:
* `intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le` - most general version, for functions
with a right derivative
* `intervalIntegral.integral_eq_sub_of_hasDerivAt'` - version for functions with a derivative on
* `intervalIntegral.integral_eq_sub_of_hasDerivAt` - version for functions with a derivative on
an open set
* `intervalIntegral.integral_deriv_eq_sub'` - version that is easiest to use when computing the
integral of a specific function
Expand Down Expand Up @@ -1243,6 +1243,27 @@ theorem integral_deriv_eq_sub' (f) (hderiv : deriv f = f')
exact hcont.intervalIntegrable
#align interval_integral.integral_deriv_eq_sub' intervalIntegral.integral_deriv_eq_sub'

/-- A variant of `intervalIntegral.integral_deriv_eq_sub`, the Fundamental theorem
of calculus, involving integrating over the unit interval. -/
lemma integral_unitInterval_deriv_eq_sub [IsROrC 𝕜] [NormedSpace 𝕜 E] [IsScalarTower ℝ 𝕜 E]
{f f' : 𝕜 → E} {z₀ z₁ : 𝕜}
(hcont : ContinuousOn (fun t : ℝ ↦ f' (z₀ + t • z₁)) (Set.Icc 0 1))
(hderiv : ∀ t ∈ Set.Icc (0 : ℝ) 1, HasDerivAt f (f' (z₀ + t • z₁)) (z₀ + t • z₁)) :
z₁ • ∫ t in (0 : ℝ)..1, f' (z₀ + t • z₁) = f (z₀ + z₁) - f z₀ := by
let γ (t : ℝ) : 𝕜 := z₀ + t • z₁
have hint : IntervalIntegrable (z₁ • (f' ∘ γ)) MeasureTheory.volume 0 1 :=
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(ContinuousOn.const_smul hcont z₁).intervalIntegrable_of_Icc zero_le_one
have hderiv' : ∀ t ∈ Set.uIcc (0 : ℝ) 1, HasDerivAt (f ∘ γ) (z₁ • (f' ∘ γ) t) t
· intro t ht
refine (hderiv t <| (Set.uIcc_of_le (α := ℝ) zero_le_one).symm ▸ ht).scomp t ?_
have : HasDerivAt (fun t : ℝ ↦ t • z₁) z₁ t
· convert (hasDerivAt_id t).smul_const (F := 𝕜) _ using 1
simp only [one_smul]
exact this.const_add z₀
convert (integral_eq_sub_of_hasDerivAt hderiv' hint) using 1
· simp_rw [← integral_smul, Function.comp_apply]
· simp only [Function.comp_apply, one_smul, zero_smul, add_zero]

/-!
### Automatic integrability for nonnegative derivatives
-/
Expand Down