ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- BoxIntegral.Prepartition.IsPartition.existsUnique
protected theorem existsUnique (h : π.IsPartition) (hx : x ∈ I) :
    ∃! J ∈ π, x ∈ J 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π : BoxIntegral.Prepartition I} {x : ι → ℝ} (h : π.IsPartition) (hx : x ∈ I)
Goal: ∃! J, J ∈ π ∧ x ∈ J
-- BoxIntegral.Prepartition.IsPartition.nonempty_boxes
theorem nonempty_boxes (h : π.IsPartition) : π.boxes.Nonempty 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π : BoxIntegral.Prepartition I} (h : π.IsPartition)
Goal: π.boxes.Nonempty
-- BoxIntegral.Prepartition.IsPartition.eq_of_boxes_subset
theorem eq_of_boxes_subset (h₁ : π₁.IsPartition) (h₂ : π₁.boxes ⊆ π₂.boxes) : π₁ = π₂ 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.Prepartition I} (h₁ : π₁.IsPartition) (h₂ : π₁.boxes ⊆ π₂.boxes)
Goal: π₁ = π₂
-- BoxIntegral.Prepartition.IsPartition.le_iff
theorem le_iff (h : π₂.IsPartition) :
    π₁ ≤ π₂ ↔ ∀ J ∈ π₁, ∀ J' ∈ π₂, (J ∩ J' : Set (ι → ℝ)).Nonempty → J ≤ J' 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.Prepartition I} (h : π₂.IsPartition)
Goal: π₁ ≤ π₂ ↔ ∀ J ∈ π₁, ∀ J' ∈ π₂, (↑J ∩ ↑J').Nonempty → J ≤ J'
-- BoxIntegral.Prepartition.IsPartition.biUnion
protected theorem biUnion (h : IsPartition π) (hi : ∀ J ∈ π, IsPartition (πi J)) :
    IsPartition (π.biUnion πi) 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π : BoxIntegral.Prepartition I} {πi : (J : BoxIntegral.Box ι) → BoxIntegral.Prepartition J} (h : π.IsPartition) (hi : ∀ J ∈ π, (πi J).IsPartition)
Goal: (π.biUnion πi).IsPartition
-- BoxIntegral.Prepartition.IsPartition.restrict
protected theorem restrict (h : IsPartition π) (hJ : J ≤ I) : IsPartition (π.restrict J) 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {π : BoxIntegral.Prepartition I} (h : π.IsPartition) (hJ : J ≤ I)
Goal: (π.restrict J).IsPartition
-- BoxIntegral.Prepartition.IsPartition.inf
protected theorem inf (h₁ : IsPartition π₁) (h₂ : IsPartition π₂) : IsPartition (π₁ ⊓ π₂) 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.Prepartition I} (h₁ : π₁.IsPartition) (h₂ : π₂.IsPartition)
Goal: (π₁ ⊓ π₂).IsPartition
-- BoxIntegral.Prepartition.iUnion_biUnion_partition
theorem iUnion_biUnion_partition (h : ∀ J ∈ π, (πi J).IsPartition) :
    (π.biUnion πi).iUnion = π.iUnion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) {πi : (J : BoxIntegral.Box ι) → BoxIntegral.Prepartition J} (h : ∀ J ∈ π, (πi J).IsPartition)
Goal: (π.biUnion πi).iUnion = π.iUnion
-- BoxIntegral.Prepartition.isPartitionDisjUnionOfEqDiff
theorem isPartitionDisjUnionOfEqDiff (h : π₂.iUnion = ↑I \ π₁.iUnion) :
    IsPartition (π₁.disjUnion π₂ <| h.symm ▸ disjoint_sdiff_self_right) 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.Prepartition I} (h : π₂.iUnion = ↑I \ π₁.iUnion)
Goal: (π₁.disjUnion π₂ ⋯).IsPartition
-- fderivInnerCLM_apply
@[simp]
theorem fderivInnerCLM_apply (p x : E × E) : fderivInnerCLM 𝕜 p x = ⟪p.1, x.2⟫ + ⟪x.1, p.2⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] (p x : E × E)
Goal: (fderivInnerCLM 𝕜 p) x = ⟪p.1, x.2⟫_𝕜 + ⟪x.1, p.2⟫_𝕜
-- contDiff_inner
theorem contDiff_inner {n} : ContDiff ℝ n fun p : E × E => ⟪p.1, p.2⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {n : ℕ∞}
Goal: ContDiff ℝ n fun p => ⟪p.1, p.2⟫_𝕜
-- contDiffAt_inner
theorem contDiffAt_inner {p : E × E} {n} : ContDiffAt ℝ n (fun p : E × E => ⟪p.1, p.2⟫) p 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {p : E × E} {n : ℕ∞}
Goal: ContDiffAt ℝ n (fun p => ⟪p.1, p.2⟫_𝕜) p
-- differentiable_inner
theorem differentiable_inner : Differentiable ℝ fun p : E × E => ⟪p.1, p.2⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E]
Goal: Differentiable ℝ fun p => ⟪p.1, p.2⟫_𝕜
-- ContDiffWithinAt.inner
theorem ContDiffWithinAt.inner (hf : ContDiffWithinAt ℝ n f s x) (hg : ContDiffWithinAt ℝ n g s x) :
    ContDiffWithinAt ℝ n (fun x => ⟪f x, g x⟫) s x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {s : Set G} {x : G} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) (hg : ContDiffWithinAt ℝ n g s x)
Goal: ContDiffWithinAt ℝ n (fun x => ⟪f x, g x⟫_𝕜) s x
-- ContDiffAt.inner
nonrec theorem ContDiffAt.inner (hf : ContDiffAt ℝ n f x) (hg : ContDiffAt ℝ n g x) :
    ContDiffAt ℝ n (fun x => ⟪f x, g x⟫) x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {x : G} {n : ℕ∞} (hf : ContDiffAt ℝ n f x) (hg : ContDiffAt ℝ n g x)
Goal: ContDiffAt ℝ n (fun x => ⟪f x, g x⟫_𝕜) x
-- ContDiffOn.inner
theorem ContDiffOn.inner (hf : ContDiffOn ℝ n f s) (hg : ContDiffOn ℝ n g s) :
    ContDiffOn ℝ n (fun x => ⟪f x, g x⟫) s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {s : Set G} {n : ℕ∞} (hf : ContDiffOn ℝ n f s) (hg : ContDiffOn ℝ n g s)
Goal: ContDiffOn ℝ n (fun x => ⟪f x, g x⟫_𝕜) s
-- ContDiff.inner
theorem ContDiff.inner (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g) :
    ContDiff ℝ n fun x => ⟪f x, g x⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {n : ℕ∞} (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g)
Goal: ContDiff ℝ n fun x => ⟪f x, g x⟫_𝕜
-- HasFDerivWithinAt.inner
theorem HasFDerivWithinAt.inner (hf : HasFDerivWithinAt f f' s x)
    (hg : HasFDerivWithinAt g g' s x) :
    HasFDerivWithinAt (fun t => ⟪f t, g t⟫) ((fderivInnerCLM 𝕜 (f x, g x)).comp <| f'.prod g') s
      x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {f' g' : G →L[ℝ] E} {s : Set G} {x : G} (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt g g' s x)
Goal: HasFDerivWithinAt (fun t => ⟪f t, g t⟫_𝕜) ((fderivInnerCLM 𝕜 (f x, g x)).comp (f'.prod g')) s x
-- HasStrictFDerivAt.inner
theorem HasStrictFDerivAt.inner (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) :
    HasStrictFDerivAt (fun t => ⟪f t, g t⟫) ((fderivInnerCLM 𝕜 (f x, g x)).comp <| f'.prod g') x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {f' g' : G →L[ℝ] E} {x : G} (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x)
Goal: HasStrictFDerivAt (fun t => ⟪f t, g t⟫_𝕜) ((fderivInnerCLM 𝕜 (f x, g x)).comp (f'.prod g')) x
-- HasFDerivAt.inner
theorem HasFDerivAt.inner (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x) :
    HasFDerivAt (fun t => ⟪f t, g t⟫) ((fderivInnerCLM 𝕜 (f x, g x)).comp <| f'.prod g') x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {f' g' : G →L[ℝ] E} {x : G} (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x)
Goal: HasFDerivAt (fun t => ⟪f t, g t⟫_𝕜) ((fderivInnerCLM 𝕜 (f x, g x)).comp (f'.prod g')) x
-- HasDerivWithinAt.inner
theorem HasDerivWithinAt.inner {f g : ℝ → E} {f' g' : E} {s : Set ℝ} {x : ℝ}
    (hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x) :
    HasDerivWithinAt (fun t => ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) s x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {f g : ℝ → E} {f' g' : E} {s : Set ℝ} {x : ℝ} (hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x)
Goal: HasDerivWithinAt (fun t => ⟪f t, g t⟫_𝕜) (⟪f x, g'⟫_𝕜 + ⟪f', g x⟫_𝕜) s x
-- HasDerivAt.inner
theorem HasDerivAt.inner {f g : ℝ → E} {f' g' : E} {x : ℝ} :
    HasDerivAt f f' x → HasDerivAt g g' x →
      HasDerivAt (fun t => ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {f g : ℝ → E} {f' g' : E} {x : ℝ}
Goal: HasDerivAt f f' x → HasDerivAt g g' x → HasDerivAt (fun t => ⟪f t, g t⟫_𝕜) (⟪f x, g'⟫_𝕜 + ⟪f', g x⟫_𝕜) x
-- DifferentiableWithinAt.inner
theorem DifferentiableWithinAt.inner (hf : DifferentiableWithinAt ℝ f s x)
    (hg : DifferentiableWithinAt ℝ g s x) : DifferentiableWithinAt ℝ (fun x => ⟪f x, g x⟫) s x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {s : Set G} {x : G} (hf : DifferentiableWithinAt ℝ f s x) (hg : DifferentiableWithinAt ℝ g s x)
Goal: DifferentiableWithinAt ℝ (fun x => ⟪f x, g x⟫_𝕜) s x
-- DifferentiableAt.inner
theorem DifferentiableAt.inner (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) :
    DifferentiableAt ℝ (fun x => ⟪f x, g x⟫) x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {x : G} (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x)
Goal: DifferentiableAt ℝ (fun x => ⟪f x, g x⟫_𝕜) x
-- DifferentiableOn.inner
theorem DifferentiableOn.inner (hf : DifferentiableOn ℝ f s) (hg : DifferentiableOn ℝ g s) :
    DifferentiableOn ℝ (fun x => ⟪f x, g x⟫) s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {s : Set G} (hf : DifferentiableOn ℝ f s) (hg : DifferentiableOn ℝ g s)
Goal: DifferentiableOn ℝ (fun x => ⟪f x, g x⟫_𝕜) s
-- Differentiable.inner
theorem Differentiable.inner (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) :
    Differentiable ℝ fun x => ⟪f x, g x⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} (hf : Differentiable ℝ f) (hg : Differentiable ℝ g)
Goal: Differentiable ℝ fun x => ⟪f x, g x⟫_𝕜
-- fderiv_inner_apply
theorem fderiv_inner_apply (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) (y : G) :
    fderiv ℝ (fun t => ⟪f t, g t⟫) x y = ⟪f x, fderiv ℝ g x y⟫ + ⟪fderiv ℝ f x y, g x⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {x : G} (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) (y : G)
Goal: (fderiv ℝ (fun t => ⟪f t, g t⟫_𝕜) x) y = ⟪f x, (fderiv ℝ g x) y⟫_𝕜 + ⟪(fderiv ℝ f x) y, g x⟫_𝕜
-- deriv_inner_apply
theorem deriv_inner_apply {f g : ℝ → E} {x : ℝ} (hf : DifferentiableAt ℝ f x)
    (hg : DifferentiableAt ℝ g x) :
    deriv (fun t => ⟪f t, g t⟫) x = ⟪f x, deriv g x⟫ + ⟪deriv f x, g x⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {f g : ℝ → E} {x : ℝ} (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x)
Goal: deriv (fun t => ⟪f t, g t⟫_𝕜) x = ⟪f x, deriv g x⟫_𝕜 + ⟪deriv f x, g x⟫_𝕜
-- contDiff_norm_sq
theorem contDiff_norm_sq : ContDiff ℝ n fun x : E => ‖x‖ ^ 2 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {n : ℕ∞}
Goal: ContDiff ℝ n fun x => ‖x‖ ^ 2
-- ContDiff.norm_sq
theorem ContDiff.norm_sq (hf : ContDiff ℝ n f) : ContDiff ℝ n fun x => ‖f x‖ ^ 2 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {n : ℕ∞} (hf : ContDiff ℝ n f)
Goal: ContDiff ℝ n fun x => ‖f x‖ ^ 2
-- ContDiffWithinAt.norm_sq
theorem ContDiffWithinAt.norm_sq (hf : ContDiffWithinAt ℝ n f s x) :
    ContDiffWithinAt ℝ n (fun y => ‖f y‖ ^ 2) s x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {s : Set G} {x : G} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x)
Goal: ContDiffWithinAt ℝ n (fun y => ‖f y‖ ^ 2) s x
-- ContDiffAt.norm_sq
nonrec theorem ContDiffAt.norm_sq (hf : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (‖f ·‖ ^ 2) x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {x : G} {n : ℕ∞} (hf : ContDiffAt ℝ n f x)
Goal: ContDiffAt ℝ n (fun x => ‖f x‖ ^ 2) x
-- contDiffAt_norm
theorem contDiffAt_norm {x : E} (hx : x ≠ 0) : ContDiffAt ℝ n norm x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {n : ℕ∞} {x : E} (hx : x ≠ 0)
Goal: ContDiffAt ℝ n norm x
-- ContDiffAt.norm
theorem ContDiffAt.norm (hf : ContDiffAt ℝ n f x) (h0 : f x ≠ 0) :
    ContDiffAt ℝ n (fun y => ‖f y‖) x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {x : G} {n : ℕ∞} (hf : ContDiffAt ℝ n f x) (h0 : f x ≠ 0)
Goal: ContDiffAt ℝ n (fun y => ‖f y‖) x
-- ContDiffAt.dist
theorem ContDiffAt.dist (hf : ContDiffAt ℝ n f x) (hg : ContDiffAt ℝ n g x) (hne : f x ≠ g x) :
    ContDiffAt ℝ n (fun y => dist (f y) (g y)) x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {x : G} {n : ℕ∞} (hf : ContDiffAt ℝ n f x) (hg : ContDiffAt ℝ n g x) (hne : f x ≠ g x)
Goal: ContDiffAt ℝ n (fun y => dist (f y) (g y)) x
-- ContDiffWithinAt.norm
theorem ContDiffWithinAt.norm (hf : ContDiffWithinAt ℝ n f s x) (h0 : f x ≠ 0) :
    ContDiffWithinAt ℝ n (fun y => ‖f y‖) s x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {s : Set G} {x : G} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) (h0 : f x ≠ 0)
Goal: ContDiffWithinAt ℝ n (fun y => ‖f y‖) s x
-- ContDiffWithinAt.dist
theorem ContDiffWithinAt.dist (hf : ContDiffWithinAt ℝ n f s x) (hg : ContDiffWithinAt ℝ n g s x)
    (hne : f x ≠ g x) : ContDiffWithinAt ℝ n (fun y => dist (f y) (g y)) s x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {s : Set G} {x : G} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) (hg : ContDiffWithinAt ℝ n g s x) (hne : f x ≠ g x)
Goal: ContDiffWithinAt ℝ n (fun y => dist (f y) (g y)) s x
-- ContDiffOn.norm_sq
theorem ContDiffOn.norm_sq (hf : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun y => ‖f y‖ ^ 2) s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {s : Set G} {n : ℕ∞} (hf : ContDiffOn ℝ n f s)
Goal: ContDiffOn ℝ n (fun y => ‖f y‖ ^ 2) s
-- ContDiffOn.norm
theorem ContDiffOn.norm (hf : ContDiffOn ℝ n f s) (h0 : ∀ x ∈ s, f x ≠ 0) :
    ContDiffOn ℝ n (fun y => ‖f y‖) s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {s : Set G} {n : ℕ∞} (hf : ContDiffOn ℝ n f s) (h0 : ∀ x ∈ s, f x ≠ 0)
Goal: ContDiffOn ℝ n (fun y => ‖f y‖) s
-- ContDiffOn.dist
theorem ContDiffOn.dist (hf : ContDiffOn ℝ n f s) (hg : ContDiffOn ℝ n g s)
    (hne : ∀ x ∈ s, f x ≠ g x) : ContDiffOn ℝ n (fun y => dist (f y) (g y)) s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {s : Set G} {n : ℕ∞} (hf : ContDiffOn ℝ n f s) (hg : ContDiffOn ℝ n g s) (hne : ∀ x ∈ s, f x ≠ g x)
Goal: ContDiffOn ℝ n (fun y => dist (f y) (g y)) s
-- ContDiff.norm
theorem ContDiff.norm (hf : ContDiff ℝ n f) (h0 : ∀ x, f x ≠ 0) : ContDiff ℝ n fun y => ‖f y‖ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {n : ℕ∞} (hf : ContDiff ℝ n f) (h0 : ∀ (x : G), f x ≠ 0)
Goal: ContDiff ℝ n fun y => ‖f y‖
-- ContDiff.dist
theorem ContDiff.dist (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g) (hne : ∀ x, f x ≠ g x) :
    ContDiff ℝ n fun y => dist (f y) (g y) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {n : ℕ∞} (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g) (hne : ∀ (x : G), f x ≠ g x)
Goal: ContDiff ℝ n fun y => dist (f y) (g y)
-- hasStrictFDerivAt_norm_sq
theorem hasStrictFDerivAt_norm_sq (x : F) :
    HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • (innerSL ℝ x)) x 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x : F)
Goal: HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • (innerSL ℝ) x) x
-- HasFDerivAt.norm_sq
theorem HasFDerivAt.norm_sq {f : G → F} {f' : G →L[ℝ] F} (hf : HasFDerivAt f f' x) :
    HasFDerivAt (‖f ·‖ ^ 2) (2 • (innerSL ℝ (f x)).comp f') x 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {x : G} {f : G → F} {f' : G →L[ℝ] F} (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => ‖f x‖ ^ 2) (2 • ((innerSL ℝ) (f x)).comp f') x
-- HasDerivAt.norm_sq
theorem HasDerivAt.norm_sq {f : ℝ → F} {f' : F} {x : ℝ} (hf : HasDerivAt f f' x) :
    HasDerivAt (‖f ·‖ ^ 2) (2 * Inner.inner (f x) f') x 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {f : ℝ → F} {f' : F} {x : ℝ} (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => ‖f x‖ ^ 2) (2 * ⟪f x, f'⟫_ℝ) x
-- HasFDerivWithinAt.norm_sq
theorem HasFDerivWithinAt.norm_sq {f : G → F} {f' : G →L[ℝ] F} (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (‖f ·‖ ^ 2) (2 • (innerSL ℝ (f x)).comp f') s x 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {s : Set G} {x : G} {f : G → F} {f' : G →L[ℝ] F} (hf : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (fun x => ‖f x‖ ^ 2) (2 • ((innerSL ℝ) (f x)).comp f') s x
-- HasDerivWithinAt.norm_sq
theorem HasDerivWithinAt.norm_sq {f : ℝ → F} {f' : F} {s : Set ℝ} {x : ℝ}
    (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (‖f ·‖ ^ 2) (2 * Inner.inner (f x) f') s x 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {f : ℝ → F} {f' : F} {s : Set ℝ} {x : ℝ} (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => ‖f x‖ ^ 2) (2 * ⟪f x, f'⟫_ℝ) s x
-- DifferentiableAt.norm_sq
theorem DifferentiableAt.norm_sq (hf : DifferentiableAt ℝ f x) :
    DifferentiableAt ℝ (fun y => ‖f y‖ ^ 2) x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {x : G} (hf : DifferentiableAt ℝ f x)
Goal: DifferentiableAt ℝ (fun y => ‖f y‖ ^ 2) x
-- DifferentiableAt.norm
theorem DifferentiableAt.norm (hf : DifferentiableAt ℝ f x) (h0 : f x ≠ 0) :
    DifferentiableAt ℝ (fun y => ‖f y‖) x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {x : G} (hf : DifferentiableAt ℝ f x) (h0 : f x ≠ 0)
Goal: DifferentiableAt ℝ (fun y => ‖f y‖) x
-- DifferentiableAt.dist
theorem DifferentiableAt.dist (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x)
    (hne : f x ≠ g x) : DifferentiableAt ℝ (fun y => dist (f y) (g y)) x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {x : G} (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) (hne : f x ≠ g x)
Goal: DifferentiableAt ℝ (fun y => dist (f y) (g y)) x
-- Differentiable.norm_sq
theorem Differentiable.norm_sq (hf : Differentiable ℝ f) : Differentiable ℝ fun y => ‖f y‖ ^ 2 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} (hf : Differentiable ℝ f)
Goal: Differentiable ℝ fun y => ‖f y‖ ^ 2
-- Differentiable.norm
theorem Differentiable.norm (hf : Differentiable ℝ f) (h0 : ∀ x, f x ≠ 0) :
    Differentiable ℝ fun y => ‖f y‖ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} (hf : Differentiable ℝ f) (h0 : ∀ (x : G), f x ≠ 0)
Goal: Differentiable ℝ fun y => ‖f y‖
-- Differentiable.dist
theorem Differentiable.dist (hf : Differentiable ℝ f) (hg : Differentiable ℝ g)
    (hne : ∀ x, f x ≠ g x) : Differentiable ℝ fun y => dist (f y) (g y) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) (hne : ∀ (x : G), f x ≠ g x)
Goal: Differentiable ℝ fun y => dist (f y) (g y)
-- DifferentiableWithinAt.norm_sq
theorem DifferentiableWithinAt.norm_sq (hf : DifferentiableWithinAt ℝ f s x) :
    DifferentiableWithinAt ℝ (fun y => ‖f y‖ ^ 2) s x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {s : Set G} {x : G} (hf : DifferentiableWithinAt ℝ f s x)
Goal: DifferentiableWithinAt ℝ (fun y => ‖f y‖ ^ 2) s x
-- DifferentiableWithinAt.norm
theorem DifferentiableWithinAt.norm (hf : DifferentiableWithinAt ℝ f s x) (h0 : f x ≠ 0) :
    DifferentiableWithinAt ℝ (fun y => ‖f y‖) s x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {s : Set G} {x : G} (hf : DifferentiableWithinAt ℝ f s x) (h0 : f x ≠ 0)
Goal: DifferentiableWithinAt ℝ (fun y => ‖f y‖) s x
-- DifferentiableWithinAt.dist
theorem DifferentiableWithinAt.dist (hf : DifferentiableWithinAt ℝ f s x)
    (hg : DifferentiableWithinAt ℝ g s x) (hne : f x ≠ g x) :
    DifferentiableWithinAt ℝ (fun y => dist (f y) (g y)) s x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {s : Set G} {x : G} (hf : DifferentiableWithinAt ℝ f s x) (hg : DifferentiableWithinAt ℝ g s x) (hne : f x ≠ g x)
Goal: DifferentiableWithinAt ℝ (fun y => dist (f y) (g y)) s x
-- DifferentiableOn.norm_sq
theorem DifferentiableOn.norm_sq (hf : DifferentiableOn ℝ f s) :
    DifferentiableOn ℝ (fun y => ‖f y‖ ^ 2) s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {s : Set G} (hf : DifferentiableOn ℝ f s)
Goal: DifferentiableOn ℝ (fun y => ‖f y‖ ^ 2) s
-- DifferentiableOn.norm
theorem DifferentiableOn.norm (hf : DifferentiableOn ℝ f s) (h0 : ∀ x ∈ s, f x ≠ 0) :
    DifferentiableOn ℝ (fun y => ‖f y‖) s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f : G → E} {s : Set G} (hf : DifferentiableOn ℝ f s) (h0 : ∀ x ∈ s, f x ≠ 0)
Goal: DifferentiableOn ℝ (fun y => ‖f y‖) s