File: Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean

full_name: integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux1
code: lemma integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux1 [SigmaFinite μ]
    {f f' : E × ℝ → F} {g g' : E × ℝ → G} {B : F →L[ℝ] G →L[ℝ] W}
    (hf'g : Integrable (fun x ↦ B (f' x) (g x)) (μ.prod volume))
    (hfg' : Integrable (fun x ↦ B (f x) (g' x)) (μ.prod volume))
    (hfg : Integrable (fun x ↦ B (f x) (g x)) (μ.prod volume))
    (hf : ∀ x, HasLineDerivAt ℝ f (f' x) x (0, 1)) (hg : ∀ x, HasLineDerivAt ℝ g (g' x) x (0, 1)) :
    ∫ x, B (f x) (g' x) ∂(μ.prod volume) = - ∫ x, B (f' x) (g x) ∂(μ.prod volume) 
goal: ∫ (x : E × ℝ), (B (f x)) (g' x) ∂μ.prod MeasureTheory.volume =
  -∫ (x : E × ℝ), (B (f' x)) (g x) ∂μ.prod MeasureTheory.volume

full_name: integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2
code: lemma integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2
    [FiniteDimensional ℝ E] {μ : Measure (E × ℝ)} [IsAddHaarMeasure μ]
    {f f' : E × ℝ → F} {g g' : E × ℝ → G} {B : F →L[ℝ] G →L[ℝ] W}
    (hf'g : Integrable (fun x ↦ B (f' x) (g x)) μ)
    (hfg' : Integrable (fun x ↦ B (f x) (g' x)) μ)
    (hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
    (hf : ∀ x, HasLineDerivAt ℝ f (f' x) x (0, 1)) (hg : ∀ x, HasLineDerivAt ℝ g (g' x) x (0, 1)) :
    ∫ x, B (f x) (g' x) ∂μ = - ∫ x, B (f' x) (g x) ∂μ 
goal: ∫ (x : E × ℝ), (B (f x)) (g' x) ∂μ = -∫ (x : E × ℝ), (B (f' x)) (g x) ∂μ

full_name: integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable
code: /-- **Integration by parts for line derivatives**
Version with a general bilinear form `B`.
If `B f g` is integrable, as well as `B f' g` and `B f g'` where `f'` and `g'` are derivatives
of `f` and `g` in a given direction `v`, then `∫ B f g' = - ∫ B f' g`. -/
theorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable
    {f f' : E → F} {g g' : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W}
    (hf'g : Integrable (fun x ↦ B (f' x) (g x)) μ) (hfg' : Integrable (fun x ↦ B (f x) (g' x)) μ)
    (hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
    (hf : ∀ x, HasLineDerivAt ℝ f (f' x) x v) (hg : ∀ x, HasLineDerivAt ℝ g (g' x) x v) :
    ∫ x, B (f x) (g' x) ∂μ = - ∫ x, B (f' x) (g x) ∂μ 
goal: ∫ (x : E), (B (f x)) (g' x) ∂μ = -∫ (x : E), (B (f' x)) (g x) ∂μ

full_name: integral_bilinear_hasFDerivAt_right_eq_neg_left_of_integrable
code: /-- **Integration by parts for Fréchet derivatives**
Version with a general bilinear form `B`.
If `B f g` is integrable, as well as `B f' g` and `B f g'` where `f'` and `g'` are derivatives
of `f` and `g` in a given direction `v`, then `∫ B f g' = - ∫ B f' g`. -/
theorem integral_bilinear_hasFDerivAt_right_eq_neg_left_of_integrable
    {f : E → F} {f' : E → (E →L[ℝ] F)}
    {g : E → G} {g' : E → (E →L[ℝ] G)} {v : E} {B : F →L[ℝ] G →L[ℝ] W}
    (hf'g : Integrable (fun x ↦ B (f' x v) (g x)) μ)
    (hfg' : Integrable (fun x ↦ B (f x) (g' x v)) μ)
    (hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
    (hf : ∀ x, HasFDerivAt f (f' x) x) (hg : ∀ x, HasFDerivAt g (g' x) x) :
    ∫ x, B (f x) (g' x v) ∂μ = - ∫ x, B (f' x v) (g x) ∂μ 
goal: ∫ (x : E), (B (f x)) ((g' x) v) ∂μ = -∫ (x : E), (B ((f' x) v)) (g x) ∂μ

full_name: integral_bilinear_fderiv_right_eq_neg_left_of_integrable
code: /-- **Integration by parts for Fréchet derivatives**
Version with a general bilinear form `B`.
If `B f g` is integrable, as well as `B f' g` and `B f g'` where `f'` and `g'` are the derivatives
of `f` and `g` in a given direction `v`, then `∫ B f g' = - ∫ B f' g`. -/
theorem integral_bilinear_fderiv_right_eq_neg_left_of_integrable
    {f : E → F} {g : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W}
    (hf'g : Integrable (fun x ↦ B (fderiv ℝ f x v) (g x)) μ)
    (hfg' : Integrable (fun x ↦ B (f x) (fderiv ℝ g x v)) μ)
    (hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
    (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) :
    ∫ x, B (f x) (fderiv ℝ g x v) ∂μ = - ∫ x, B (fderiv ℝ f x v) (g x) ∂μ 
goal: ∫ (x : E), (B (f x)) ((fderiv ℝ g x) v) ∂μ = -∫ (x : E), (B ((fderiv ℝ f x) v)) (g x) ∂μ

full_name: integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable
code: /-- **Integration by parts for Fréchet derivatives**
Version with a scalar function: `∫ f • g' = - ∫ f' • g` when `f • g'` and `f' • g` and `f • g`
are integrable, where `f'` and `g'` are the derivatives of `f` and `g` in a given direction `v`. -/
theorem integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable
    {f : E → 𝕜} {g : E → G} {v : E}
    (hf'g : Integrable (fun x ↦ fderiv ℝ f x v • g x) μ)
    (hfg' : Integrable (fun x ↦ f x • fderiv ℝ g x v) μ)
    (hfg : Integrable (fun x ↦ f x • g x) μ)
    (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) :
    ∫ x, f x • fderiv ℝ g x v ∂μ = - ∫ x, fderiv ℝ f x v • g x ∂μ 
goal: ∫ (x : E), f x • (fderiv ℝ g x) v ∂μ = -∫ (x : E), (fderiv ℝ f x) v • g x ∂μ

full_name: integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable
code: /-- **Integration by parts for Fréchet derivatives**
Version with two scalar functions: `∫ f * g' = - ∫ f' * g` when `f * g'` and `f' * g` and `f * g`
are integrable, where `f'` and `g'` are the derivatives of `f` and `g` in a given direction `v`. -/
theorem integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable
    {f : E → 𝕜} {g : E → 𝕜} {v : E}
    (hf'g : Integrable (fun x ↦ fderiv ℝ f x v * g x) μ)
    (hfg' : Integrable (fun x ↦ f x * fderiv ℝ g x v) μ)
    (hfg : Integrable (fun x ↦ f x * g x) μ)
    (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) :
    ∫ x, f x * fderiv ℝ g x v ∂μ = - ∫ x, fderiv ℝ f x v * g x ∂μ 
goal: ∫ (x : E), f x * (fderiv ℝ g x) v ∂μ = -∫ (x : E), (fderiv ℝ f x) v * g x ∂μ