ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DifferentiableAt.restrictScalars
@[fun_prop]
theorem DifferentiableAt.restrictScalars (h : DifferentiableAt 𝕜' f x) : DifferentiableAt 𝕜 f x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {x : E} (h : DifferentiableAt 𝕜' f x)
Goal: DifferentiableAt 𝕜 f x
-- DifferentiableWithinAt.restrictScalars
@[fun_prop]
theorem DifferentiableWithinAt.restrictScalars (h : DifferentiableWithinAt 𝕜' f s x) :
    DifferentiableWithinAt 𝕜 f s x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {s : Set E} {x : E} (h : DifferentiableWithinAt 𝕜' f s x)
Goal: DifferentiableWithinAt 𝕜 f s x
-- DifferentiableOn.restrictScalars
@[fun_prop]
theorem DifferentiableOn.restrictScalars (h : DifferentiableOn 𝕜' f s) : DifferentiableOn 𝕜 f s 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {s : Set E} (h : DifferentiableOn 𝕜' f s)
Goal: DifferentiableOn 𝕜 f s
-- Differentiable.restrictScalars
@[fun_prop]
theorem Differentiable.restrictScalars (h : Differentiable 𝕜' f) : Differentiable 𝕜 f 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} (h : Differentiable 𝕜' f)
Goal: Differentiable 𝕜 f
-- HasFDerivWithinAt.of_restrictScalars
@[fun_prop]
theorem HasFDerivWithinAt.of_restrictScalars {g' : E →L[𝕜] F} (h : HasFDerivWithinAt f g' s x)
    (H : f'.restrictScalars 𝕜 = g') : HasFDerivWithinAt f f' s x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {f' : E →L[𝕜'] F} {s : Set E} {x : E} {g' : E →L[𝕜] F} (h : HasFDerivWithinAt f g' s x) (H : ContinuousLinearMap.restrictScalars 𝕜 f' = g')
Goal: HasFDerivWithinAt f f' s x
-- hasFDerivAt_of_restrictScalars
@[fun_prop]
theorem hasFDerivAt_of_restrictScalars {g' : E →L[𝕜] F} (h : HasFDerivAt f g' x)
    (H : f'.restrictScalars 𝕜 = g') : HasFDerivAt f f' x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {f' : E →L[𝕜'] F} {x : E} {g' : E →L[𝕜] F} (h : HasFDerivAt f g' x) (H : ContinuousLinearMap.restrictScalars 𝕜 f' = g')
Goal: HasFDerivAt f f' x
-- DifferentiableAt.fderiv_restrictScalars
theorem DifferentiableAt.fderiv_restrictScalars (h : DifferentiableAt 𝕜' f x) :
    fderiv 𝕜 f x = (fderiv 𝕜' f x).restrictScalars 𝕜 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {x : E} (h : DifferentiableAt 𝕜' f x)
Goal: fderiv 𝕜 f x = ContinuousLinearMap.restrictScalars 𝕜 (fderiv 𝕜' f x)
-- differentiableWithinAt_iff_restrictScalars
theorem differentiableWithinAt_iff_restrictScalars (hf : DifferentiableWithinAt 𝕜 f s x)
    (hs : UniqueDiffWithinAt 𝕜 s x) : DifferentiableWithinAt 𝕜' f s x ↔
      ∃ g' : E →L[𝕜'] F, g'.restrictScalars 𝕜 = fderivWithin 𝕜 f s x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {s : Set E} {x : E} (hf : DifferentiableWithinAt 𝕜 f s x) (hs : UniqueDiffWithinAt 𝕜 s x)
Goal: DifferentiableWithinAt 𝕜' f s x ↔ ∃ g', ContinuousLinearMap.restrictScalars 𝕜 g' = fderivWithin 𝕜 f s x
-- differentiableAt_iff_restrictScalars
theorem differentiableAt_iff_restrictScalars (hf : DifferentiableAt 𝕜 f x) :
    DifferentiableAt 𝕜' f x ↔ ∃ g' : E →L[𝕜'] F, g'.restrictScalars 𝕜 = fderiv 𝕜 f x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {x : E} (hf : DifferentiableAt 𝕜 f x)
Goal: DifferentiableAt 𝕜' f x ↔ ∃ g', ContinuousLinearMap.restrictScalars 𝕜 g' = fderiv 𝕜 f x
-- hasFDerivAt_update
@[fun_prop]
theorem hasFDerivAt_update (x : ∀ i, E i) {i : ι} (y : E i) :
    HasFDerivAt (Function.update x i) (.pi (Pi.single i (.id 𝕜 (E i)))) y 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} [DecidableEq ι] [Fintype ι] [NontriviallyNormedField 𝕜] {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (x : (i : ι) → E i) {i : ι} (y : E i)
Goal: HasFDerivAt (Function.update x i) (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id 𝕜 (E i)))) y
-- hasFDerivAt_single
@[fun_prop]
theorem hasFDerivAt_single {i : ι} (y : E i) :
    HasFDerivAt (Pi.single i) (.pi (Pi.single i (.id 𝕜 (E i)))) y 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} [DecidableEq ι] [Fintype ι] [NontriviallyNormedField 𝕜] {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {i : ι} (y : E i)
Goal: HasFDerivAt (Pi.single i) (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id 𝕜 (E i)))) y
-- fderiv_update
theorem fderiv_update (x : ∀ i, E i) {i : ι} (y : E i) :
    fderiv 𝕜 (Function.update x i) y = .pi (Pi.single i (.id 𝕜 (E i))) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} [DecidableEq ι] [Fintype ι] [NontriviallyNormedField 𝕜] {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (x : (i : ι) → E i) {i : ι} (y : E i)
Goal: fderiv 𝕜 (Function.update x i) y = ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id 𝕜 (E i)))
-- fderiv_single
theorem fderiv_single {i : ι} (y : E i) :
    fderiv 𝕜 (Pi.single i) y = .pi (Pi.single i (.id 𝕜 (E i))) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} [DecidableEq ι] [Fintype ι] [NontriviallyNormedField 𝕜] {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {i : ι} (y : E i)
Goal: fderiv 𝕜 (Pi.single i) y = ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id 𝕜 (E i)))
-- posTangentConeAt_mono
theorem posTangentConeAt_mono : Monotone fun s => posTangentConeAt s a 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {a : E}
Goal: Monotone fun s => posTangentConeAt s a
-- mem_posTangentConeAt_of_frequently_mem
theorem mem_posTangentConeAt_of_frequently_mem (h : ∃ᶠ t : ℝ in 𝓝[>] 0, x + t • y ∈ s) :
    y ∈ posTangentConeAt s x 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {x y : E} (h : ∃ᶠ (t : ℝ) in 𝓝[>] 0, x + t • y ∈ s)
Goal: y ∈ posTangentConeAt s x
-- mem_posTangentConeAt_of_segment_subset
theorem mem_posTangentConeAt_of_segment_subset (h : [x -[ℝ] x + y] ⊆ s) :
    y ∈ posTangentConeAt s x 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {x y : E} (h : [x-[ℝ]x + y] ⊆ s)
Goal: y ∈ posTangentConeAt s x
-- sub_mem_posTangentConeAt_of_segment_subset
theorem sub_mem_posTangentConeAt_of_segment_subset (h : segment ℝ x y ⊆ s) :
    y - x ∈ posTangentConeAt s x 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {x y : E} (h : [x-[ℝ]y] ⊆ s)
Goal: y - x ∈ posTangentConeAt s x
-- posTangentConeAt_univ
@[simp]
theorem posTangentConeAt_univ : posTangentConeAt univ a = univ 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {a : E}
Goal: posTangentConeAt Set.univ a = Set.univ
-- IsLocalMaxOn.hasFDerivWithinAt_nonpos
theorem IsLocalMaxOn.hasFDerivWithinAt_nonpos (h : IsLocalMaxOn f s a)
    (hf : HasFDerivWithinAt f f' s a) (hy : y ∈ posTangentConeAt s a) : f' y ≤ 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {s : Set E} {a y : E} (h : IsLocalMaxOn f s a) (hf : HasFDerivWithinAt f f' s a) (hy : y ∈ posTangentConeAt s a)
Goal: f' y ≤ 0
-- IsLocalMaxOn.fderivWithin_nonpos
theorem IsLocalMaxOn.fderivWithin_nonpos (h : IsLocalMaxOn f s a)
    (hy : y ∈ posTangentConeAt s a) : (fderivWithin ℝ f s a : E → ℝ) y ≤ 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {a y : E} (h : IsLocalMaxOn f s a) (hy : y ∈ posTangentConeAt s a)
Goal: (fderivWithin ℝ f s a) y ≤ 0
-- IsLocalMaxOn.hasFDerivWithinAt_eq_zero
theorem IsLocalMaxOn.hasFDerivWithinAt_eq_zero (h : IsLocalMaxOn f s a)
    (hf : HasFDerivWithinAt f f' s a) (hy : y ∈ posTangentConeAt s a)
    (hy' : -y ∈ posTangentConeAt s a) : f' y = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {s : Set E} {a y : E} (h : IsLocalMaxOn f s a) (hf : HasFDerivWithinAt f f' s a) (hy : y ∈ posTangentConeAt s a) (hy' : -y ∈ posTangentConeAt s a)
Goal: f' y = 0
-- IsLocalMaxOn.fderivWithin_eq_zero
theorem IsLocalMaxOn.fderivWithin_eq_zero (h : IsLocalMaxOn f s a)
    (hy : y ∈ posTangentConeAt s a) (hy' : -y ∈ posTangentConeAt s a) :
    (fderivWithin ℝ f s a : E → ℝ) y = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {a y : E} (h : IsLocalMaxOn f s a) (hy : y ∈ posTangentConeAt s a) (hy' : -y ∈ posTangentConeAt s a)
Goal: (fderivWithin ℝ f s a) y = 0
-- IsLocalMinOn.hasFDerivWithinAt_nonneg
theorem IsLocalMinOn.hasFDerivWithinAt_nonneg (h : IsLocalMinOn f s a)
    (hf : HasFDerivWithinAt f f' s a) (hy : y ∈ posTangentConeAt s a) : 0 ≤ f' y 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {s : Set E} {a y : E} (h : IsLocalMinOn f s a) (hf : HasFDerivWithinAt f f' s a) (hy : y ∈ posTangentConeAt s a)
Goal: 0 ≤ f' y
-- IsLocalMinOn.fderivWithin_nonneg
theorem IsLocalMinOn.fderivWithin_nonneg (h : IsLocalMinOn f s a)
    (hy : y ∈ posTangentConeAt s a) : (0 : ℝ) ≤ (fderivWithin ℝ f s a : E → ℝ) y 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {a y : E} (h : IsLocalMinOn f s a) (hy : y ∈ posTangentConeAt s a)
Goal: 0 ≤ (fderivWithin ℝ f s a) y
-- IsLocalMinOn.hasFDerivWithinAt_eq_zero
theorem IsLocalMinOn.hasFDerivWithinAt_eq_zero (h : IsLocalMinOn f s a)
    (hf : HasFDerivWithinAt f f' s a) (hy : y ∈ posTangentConeAt s a)
    (hy' : -y ∈ posTangentConeAt s a) : f' y = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {s : Set E} {a y : E} (h : IsLocalMinOn f s a) (hf : HasFDerivWithinAt f f' s a) (hy : y ∈ posTangentConeAt s a) (hy' : -y ∈ posTangentConeAt s a)
Goal: f' y = 0
-- IsLocalMinOn.fderivWithin_eq_zero
theorem IsLocalMinOn.fderivWithin_eq_zero (h : IsLocalMinOn f s a)
    (hy : y ∈ posTangentConeAt s a) (hy' : -y ∈ posTangentConeAt s a) :
    (fderivWithin ℝ f s a : E → ℝ) y = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {a y : E} (h : IsLocalMinOn f s a) (hy : y ∈ posTangentConeAt s a) (hy' : -y ∈ posTangentConeAt s a)
Goal: (fderivWithin ℝ f s a) y = 0
-- IsLocalMin.hasFDerivAt_eq_zero
theorem IsLocalMin.hasFDerivAt_eq_zero (h : IsLocalMin f a) (hf : HasFDerivAt f f' a) : f' = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {a : E} (h : IsLocalMin f a) (hf : HasFDerivAt f f' a)
Goal: f' = 0
-- IsLocalMin.fderiv_eq_zero
theorem IsLocalMin.fderiv_eq_zero (h : IsLocalMin f a) : fderiv ℝ f a = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {a : E} (h : IsLocalMin f a)
Goal: fderiv ℝ f a = 0
-- IsLocalMax.hasFDerivAt_eq_zero
theorem IsLocalMax.hasFDerivAt_eq_zero (h : IsLocalMax f a) (hf : HasFDerivAt f f' a) : f' = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {a : E} (h : IsLocalMax f a) (hf : HasFDerivAt f f' a)
Goal: f' = 0
-- IsLocalMax.fderiv_eq_zero
theorem IsLocalMax.fderiv_eq_zero (h : IsLocalMax f a) : fderiv ℝ f a = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {a : E} (h : IsLocalMax f a)
Goal: fderiv ℝ f a = 0
-- IsLocalExtr.hasFDerivAt_eq_zero
theorem IsLocalExtr.hasFDerivAt_eq_zero (h : IsLocalExtr f a) : HasFDerivAt f f' a → f' = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {a : E} (h : IsLocalExtr f a)
Goal: HasFDerivAt f f' a → f' = 0
-- IsLocalExtr.fderiv_eq_zero
theorem IsLocalExtr.fderiv_eq_zero (h : IsLocalExtr f a) : fderiv ℝ f a = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {a : E} (h : IsLocalExtr f a)
Goal: fderiv ℝ f a = 0
-- one_mem_posTangentConeAt_iff_mem_closure
lemma one_mem_posTangentConeAt_iff_mem_closure :
    1 ∈ posTangentConeAt s a ↔ a ∈ closure (Ioi a ∩ s) 
Arguments: {s : Set ℝ} {a : ℝ}
Goal: 1 ∈ posTangentConeAt s a ↔ a ∈ closure (Set.Ioi a ∩ s)
-- one_mem_posTangentConeAt_iff_frequently
lemma one_mem_posTangentConeAt_iff_frequently :
    1 ∈ posTangentConeAt s a ↔ ∃ᶠ x in 𝓝[>] a, x ∈ s 
Arguments: {s : Set ℝ} {a : ℝ}
Goal: 1 ∈ posTangentConeAt s a ↔ ∃ᶠ (x : ℝ) in 𝓝[>] a, x ∈ s
-- IsLocalMin.hasDerivAt_eq_zero
theorem IsLocalMin.hasDerivAt_eq_zero (h : IsLocalMin f a) (hf : HasDerivAt f f' a) : f' = 0 
Arguments: {f : ℝ → ℝ} {f' a : ℝ} (h : IsLocalMin f a) (hf : HasDerivAt f f' a)
Goal: f' = 0
-- IsLocalMin.deriv_eq_zero
theorem IsLocalMin.deriv_eq_zero (h : IsLocalMin f a) : deriv f a = 0 
Arguments: {f : ℝ → ℝ} {a : ℝ} (h : IsLocalMin f a)
Goal: deriv f a = 0
-- IsLocalMax.hasDerivAt_eq_zero
theorem IsLocalMax.hasDerivAt_eq_zero (h : IsLocalMax f a) (hf : HasDerivAt f f' a) : f' = 0 
Arguments: {f : ℝ → ℝ} {f' a : ℝ} (h : IsLocalMax f a) (hf : HasDerivAt f f' a)
Goal: f' = 0
-- IsLocalMax.deriv_eq_zero
theorem IsLocalMax.deriv_eq_zero (h : IsLocalMax f a) : deriv f a = 0 
Arguments: {f : ℝ → ℝ} {a : ℝ} (h : IsLocalMax f a)
Goal: deriv f a = 0
-- IsLocalExtr.hasDerivAt_eq_zero
theorem IsLocalExtr.hasDerivAt_eq_zero (h : IsLocalExtr f a) : HasDerivAt f f' a → f' = 0 
Arguments: {f : ℝ → ℝ} {f' a : ℝ} (h : IsLocalExtr f a)
Goal: HasDerivAt f f' a → f' = 0
-- IsLocalExtr.deriv_eq_zero
theorem IsLocalExtr.deriv_eq_zero (h : IsLocalExtr f a) : deriv f a = 0 
Arguments: {f : ℝ → ℝ} {a : ℝ} (h : IsLocalExtr f a)
Goal: deriv f a = 0
-- MeasureTheory.GridLines.T_univ
@[simp] lemma T_univ (f : (∀ i, A i) → ℝ≥0∞) (x : ∀ i, A i) :
    T μ p f univ x =
    ∫⁻ (x : ∀ i, A i), (f x ^ (1 - (#ι - 1 : ℝ) * p)
    * ∏ i : ι, (∫⁻ t : A i, f (update x i t) ∂(μ i)) ^ p) ∂(.pi μ) 
Arguments: {ι : Type u_1} [Fintype ι] {A : ι → Type u_2} [(i : ι) → MeasurableSpace (A i)] (μ : (i : ι) → MeasureTheory.Measure (A i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [DecidableEq ι] {p : ℝ} (f : ((i : ι) → A i) → ℝ≥0∞) (x : (i : ι) → A i)
Goal: MeasureTheory.GridLines.T μ p f Finset.univ x =
  ∫⁻ (x : (i : ι) → A i),
    f x ^ (1 - (↑#ι - 1) * p) * ∏ i : ι, (∫⁻ (t : A i), f (Function.update x i t) ∂μ i) ^ p ∂MeasureTheory.Measure.pi μ
-- MeasureTheory.GridLines.T_empty
@[simp] lemma T_empty (f : (∀ i, A i) → ℝ≥0∞) (x : ∀ i, A i) :
    T μ p f ∅ x = f x ^ (1 + p) 
Arguments: {ι : Type u_1} {A : ι → Type u_2} [(i : ι) → MeasurableSpace (A i)] (μ : (i : ι) → MeasureTheory.Measure (A i)) [DecidableEq ι] {p : ℝ} (f : ((i : ι) → A i) → ℝ≥0∞) (x : (i : ι) → A i)
Goal: MeasureTheory.GridLines.T μ p f ∅ x = f x ^ (1 + p)
-- MeasureTheory.GridLines.T_insert_le_T_lmarginal_singleton
theorem T_insert_le_T_lmarginal_singleton (hp₀ : 0 ≤ p) (s : Finset ι)
    (hp : (s.card : ℝ) * p ≤ 1)
    (i : ι) (hi : i ∉ s) {f : (∀ i, A i) → ℝ≥0∞} (hf : Measurable f) :
    T μ p f (insert i s) ≤ T μ p (∫⋯∫⁻_{i}, f ∂μ) s 
Arguments: {ι : Type u_1} {A : ι → Type u_2} [(i : ι) → MeasurableSpace (A i)] (μ : (i : ι) → MeasureTheory.Measure (A i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [DecidableEq ι] {p : ℝ} (hp₀ : 0 ≤ p) (s : Finset ι) (hp : ↑s.card * p ≤ 1) (i : ι) (hi : i ∉ s) {f : ((i : ι) → A i) → ℝ≥0∞} (hf : Measurable f)
Goal: MeasureTheory.GridLines.T μ p f (insert i s) ≤ MeasureTheory.GridLines.T μ p (∫⋯∫⁻_{i}, f ∂μ) s
-- MeasureTheory.GridLines.T_lmarginal_antitone
theorem T_lmarginal_antitone (hp₀ : 0 ≤ p) (hp : (#ι - 1 : ℝ) * p ≤ 1)
    {f : (∀ i, A i) → ℝ≥0∞} (hf : Measurable f) :
    Antitone (fun s ↦ T μ p (∫⋯∫⁻_sᶜ, f ∂μ) s) 
Arguments: {ι : Type u_1} [Fintype ι] {A : ι → Type u_2} [(i : ι) → MeasurableSpace (A i)] (μ : (i : ι) → MeasureTheory.Measure (A i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [DecidableEq ι] {p : ℝ} (hp₀ : 0 ≤ p) (hp : (↑#ι - 1) * p ≤ 1) {f : ((i : ι) → A i) → ℝ≥0∞} (hf : Measurable f)
Goal: Antitone fun s => MeasureTheory.GridLines.T μ p (∫⋯∫⁻_sᶜ, f ∂μ) s
-- MeasureTheory.lintegral_mul_prod_lintegral_pow_le
theorem lintegral_mul_prod_lintegral_pow_le {p : ℝ} (hp₀ : 0 ≤ p)
    (hp : (#ι - 1 : ℝ) * p ≤ 1) {f : (∀ i : ι, A i) → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ x, f x ^ (1 - (#ι - 1 : ℝ) * p) * ∏ i, (∫⁻ xᵢ, f (update x i xᵢ) ∂μ i) ^ p ∂.pi μ
    ≤ (∫⁻ x, f x ∂.pi μ) ^ (1 + p) 
Arguments: {ι : Type u_1} [Fintype ι] {A : ι → Type u_2} [(i : ι) → MeasurableSpace (A i)] (μ : (i : ι) → MeasureTheory.Measure (A i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [DecidableEq ι] {p : ℝ} (hp₀ : 0 ≤ p) (hp : (↑#ι - 1) * p ≤ 1) {f : ((i : ι) → A i) → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (x : (i : ι) → A i),
    f x ^ (1 - (↑#ι - 1) * p) *
      ∏ i : ι, (∫⁻ (xᵢ : A i), f (Function.update x i xᵢ) ∂μ i) ^ p ∂MeasureTheory.Measure.pi μ ≤
  (∫⁻ (x : (i : ι) → A i), f x ∂MeasureTheory.Measure.pi μ) ^ (1 + p)
-- MeasureTheory.lintegral_prod_lintegral_pow_le
theorem lintegral_prod_lintegral_pow_le
    {p : ℝ} (hp : Real.IsConjExponent #ι p)
    {f} (hf : Measurable f) :
    ∫⁻ x, ∏ i, (∫⁻ xᵢ, f (update x i xᵢ) ∂μ i) ^ ((1 : ℝ) / (#ι - 1 : ℝ)) ∂.pi μ
    ≤ (∫⁻ x, f x ∂.pi μ) ^ p 
Arguments: {ι : Type u_1} [Fintype ι] {A : ι → Type u_2} [(i : ι) → MeasurableSpace (A i)] (μ : (i : ι) → MeasureTheory.Measure (A i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [DecidableEq ι] {p : ℝ} (hp : (↑#ι).IsConjExponent p) {f : ((a : ι) → A a) → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (x : (i : ι) → A i),
    ∏ i : ι, (∫⁻ (xᵢ : A i), f (Function.update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂MeasureTheory.Measure.pi μ ≤
  (∫⁻ (x : (i : ι) → A i), f x ∂MeasureTheory.Measure.pi μ) ^ p
-- MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv_aux
theorem lintegral_pow_le_pow_lintegral_fderiv_aux
    {p : ℝ} (hp : Real.IsConjExponent #ι p)
    {u : (ι → ℝ) → F} (hu : ContDiff ℝ 1 u)
    (h2u : HasCompactSupport u) :
    ∫⁻ x, (‖u x‖₊ : ℝ≥0∞) ^ p ≤ (∫⁻ x, ‖fderiv ℝ u x‖₊) ^ p 
Arguments: {ι : Type u_1} [Fintype ι] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {p : ℝ} (hp : (↑#ι).IsConjExponent p) {u : (ι → ℝ) → F} (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
Goal: ∫⁻ (x : ι → ℝ), ↑‖u x‖₊ ^ p ≤ (∫⁻ (x : ι → ℝ), ↑‖fderiv ℝ u x‖₊) ^ p
-- MeasureTheory.lintegralPowLePowLIntegralFDerivConst_def
irreducible_def lintegralPowLePowLIntegralFDerivConst (p : ℝ) : ℝ≥0 
Arguments: {E : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (p : ℝ)
Goal: MeasureTheory.lintegralPowLePowLIntegralFDerivConst μ p =
  let ι := Fin (FiniteDimensional.finrank ℝ E);
  let_fun this := ⋯;
  let e := ContinuousLinearEquiv.ofFinrankEq this;
  let c := μ.addHaarScalarFactor (MeasureTheory.Measure.map (⇑e.symm) MeasureTheory.volume);
  c * ‖↑e.symm‖₊ ^ p * (c ^ p)⁻¹
-- MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv
theorem lintegral_pow_le_pow_lintegral_fderiv {u : E → F}
    (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
    {p : ℝ} (hp : Real.IsConjExponent (finrank ℝ E) p) :
    ∫⁻ x, (‖u x‖₊ : ℝ≥0∞) ^ p ∂μ ≤
    lintegralPowLePowLIntegralFDerivConst μ p * (∫⁻ x, ‖fderiv ℝ u x‖₊ ∂μ) ^ p 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {u : E → F} (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u) {p : ℝ} (hp : (↑(FiniteDimensional.finrank ℝ E)).IsConjExponent p)
Goal: ∫⁻ (x : E), ↑‖u x‖₊ ^ p ∂μ ≤
  ↑(MeasureTheory.lintegralPowLePowLIntegralFDerivConst μ p) * (∫⁻ (x : E), ↑‖fderiv ℝ u x‖₊ ∂μ) ^ p
-- MeasureTheory.eLpNormLESNormFDerivOneConst_def
irreducible_def eLpNormLESNormFDerivOneConst (p : ℝ) : ℝ≥0 
Arguments: {E : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (p : ℝ)
Goal: MeasureTheory.eLpNormLESNormFDerivOneConst μ p = MeasureTheory.lintegralPowLePowLIntegralFDerivConst μ p ^ p⁻¹
-- MeasureTheory.eLpNorm_le_eLpNorm_fderiv_one
theorem eLpNorm_le_eLpNorm_fderiv_one  {u : E → F} (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
    {p : ℝ≥0} (hp : NNReal.IsConjExponent (finrank ℝ E) p) :
    eLpNorm u p μ ≤ eLpNormLESNormFDerivOneConst μ p * eLpNorm (fderiv ℝ u) 1 μ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {u : E → F} (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u) {p : ℝ≥0} (hp : (↑(FiniteDimensional.finrank ℝ E)).IsConjExponent p)
Goal: MeasureTheory.eLpNorm u (↑p) μ ≤
  ↑(MeasureTheory.eLpNormLESNormFDerivOneConst μ ↑p) * MeasureTheory.eLpNorm (fderiv ℝ u) 1 μ
-- MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_eq_inner
theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner  {u : E → F'}
    (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
    {p p' : ℝ≥0} (hp : 1 ≤ p) (hn : 0 < finrank ℝ E)
    (hp' : (p' : ℝ)⁻¹ = p⁻¹ - (finrank ℝ E : ℝ)⁻¹) :
    eLpNorm u p' μ ≤ eLpNormLESNormFDerivOfEqInnerConst μ p * eLpNorm (fderiv ℝ u) p μ 
Arguments: {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {F' : Type u_5} [NormedAddCommGroup F'] [InnerProductSpace ℝ F'] [CompleteSpace F'] {u : E → F'} (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u) {p p' : ℝ≥0} (hp : 1 ≤ p) (hn : 0 < FiniteDimensional.finrank ℝ E) (hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(FiniteDimensional.finrank ℝ E))⁻¹)
Goal: MeasureTheory.eLpNorm u (↑p') μ ≤
  ↑(MeasureTheory.eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * MeasureTheory.eLpNorm (fderiv ℝ u) (↑p) μ
-- MeasureTheory.SNormLESNormFDerivOfEqConst_def
irreducible_def SNormLESNormFDerivOfEqConst [FiniteDimensional ℝ F] (p : ℝ) : ℝ≥0 
Arguments: (F : Type u_6) [NormedAddCommGroup F] [NormedSpace ℝ F] {E : Type u_7} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] [FiniteDimensional ℝ F] (p : ℝ)
Goal: MeasureTheory.SNormLESNormFDerivOfEqConst F μ p =
  let F' := EuclideanSpace ℝ (Fin (FiniteDimensional.finrank ℝ F));
  let e := toEuclidean;
  ‖↑e.symm‖₊ * MeasureTheory.eLpNormLESNormFDerivOfEqInnerConst μ p * ‖↑e‖₊