ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LipschitzWith.ae_lineDeriv_sum_eq
theorem ae_lineDeriv_sum_eq
    (hf : LipschitzWith C f) {ι : Type*} (s : Finset ι) (a : ι → ℝ) (v : ι → E) :
    ∀ᵐ x ∂μ, lineDeriv ℝ f x (∑ i ∈ s, a i • v i) = ∑ i ∈ s, a i • lineDeriv ℝ f x (v i) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {C : ℝ≥0} {f : E → ℝ} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] (hf : LipschitzWith C f) {ι : Type u_3} (s : Finset ι) (a : ι → ℝ) (v : ι → E)
Goal: ∀ᵐ (x : E) ∂μ, lineDeriv ℝ f x (∑ i ∈ s, a i • v i) = ∑ i ∈ s, a i • lineDeriv ℝ f x (v i)
-- LipschitzWith.ae_exists_fderiv_of_countable
theorem ae_exists_fderiv_of_countable
    (hf : LipschitzWith C f) {s : Set E} (hs : s.Countable) :
    ∀ᵐ x ∂μ, ∃ (L : E →L[ℝ] ℝ), ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {C : ℝ≥0} {f : E → ℝ} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] (hf : LipschitzWith C f) {s : Set E} (hs : s.Countable)
Goal: ∀ᵐ (x : E) ∂μ, ∃ L, ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
-- LipschitzWith.hasFderivAt_of_hasLineDerivAt_of_closure
theorem hasFderivAt_of_hasLineDerivAt_of_closure {f : E → F}
    (hf : LipschitzWith C f) {s : Set E} (hs : sphere 0 1 ⊆ closure s)
    {L : E →L[ℝ] F} {x : E} (hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v) :
    HasFDerivAt f L x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {C : ℝ≥0} {f : E → F} (hf : LipschitzWith C f) {s : Set E} (hs : Metric.sphere 0 1 ⊆ closure s) {L : E →L[ℝ] F} {x : E} (hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v)
Goal: HasFDerivAt f L x
-- LipschitzWith.ae_differentiableAt_of_real
theorem ae_differentiableAt_of_real (hf : LipschitzWith C f) :
    ∀ᵐ x ∂μ, DifferentiableAt ℝ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {C : ℝ≥0} {f : E → ℝ} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] (hf : LipschitzWith C f)
Goal: ∀ᵐ (x : E) ∂μ, DifferentiableAt ℝ f x
-- LipschitzOnWith.ae_differentiableWithinAt_of_mem_of_real
theorem ae_differentiableWithinAt_of_mem_of_real (hf : LipschitzOnWith C f s) :
    ∀ᵐ x ∂μ, x ∈ s → DifferentiableWithinAt ℝ f s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {C : ℝ≥0} {f : E → ℝ} {s : Set E} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] (hf : LipschitzOnWith C f s)
Goal: ∀ᵐ (x : E) ∂μ, x ∈ s → DifferentiableWithinAt ℝ f s x
-- LipschitzOnWith.ae_differentiableWithinAt_of_mem_pi
theorem ae_differentiableWithinAt_of_mem_pi
    {ι : Type*} [Fintype ι] {f : E → ι → ℝ} {s : Set E}
    (hf : LipschitzOnWith C f s) : ∀ᵐ x ∂μ, x ∈ s → DifferentiableWithinAt ℝ f s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {C : ℝ≥0} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] {ι : Type u_3} [Fintype ι] {f : E → ι → ℝ} {s : Set E} (hf : LipschitzOnWith C f s)
Goal: ∀ᵐ (x : E) ∂μ, x ∈ s → DifferentiableWithinAt ℝ f s x
-- LipschitzOnWith.ae_differentiableWithinAt_of_mem
theorem ae_differentiableWithinAt_of_mem {f : E → F} (hf : LipschitzOnWith C f s) :
    ∀ᵐ x ∂μ, x ∈ s → DifferentiableWithinAt ℝ f s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {C : ℝ≥0} {s : Set E} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] [FiniteDimensional ℝ F] {f : E → F} (hf : LipschitzOnWith C f s)
Goal: ∀ᵐ (x : E) ∂μ, x ∈ s → DifferentiableWithinAt ℝ f s x
-- LipschitzOnWith.ae_differentiableWithinAt
theorem ae_differentiableWithinAt {f : E → F} (hf : LipschitzOnWith C f s)
    (hs : MeasurableSet s) :
    ∀ᵐ x ∂(μ.restrict s), DifferentiableWithinAt ℝ f s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {C : ℝ≥0} {s : Set E} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] [FiniteDimensional ℝ F] {f : E → F} (hf : LipschitzOnWith C f s) (hs : MeasurableSet s)
Goal: ∀ᵐ (x : E) ∂μ.restrict s, DifferentiableWithinAt ℝ f s x
-- LipschitzWith.ae_differentiableAt
theorem LipschitzWith.ae_differentiableAt {f : E → F} (h : LipschitzWith C f) :
    ∀ᵐ x ∂μ, DifferentiableAt ℝ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {C : ℝ≥0} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] [FiniteDimensional ℝ F] {f : E → F} (h : LipschitzWith C f)
Goal: ∀ᵐ (x : E) ∂μ, DifferentiableAt ℝ f x
-- Complex.det_conjLIE
@[simp]
theorem det_conjLIE : LinearMap.det (conjLIE.toLinearEquiv : ℂ →ₗ[ℝ] ℂ) = -1 
Goal: LinearMap.det ↑Complex.conjLIE.toLinearEquiv = -1
-- Complex.linearEquiv_det_conjLIE
@[simp]
theorem linearEquiv_det_conjLIE : LinearEquiv.det conjLIE.toLinearEquiv = -1 
Goal: LinearEquiv.det Complex.conjLIE.toLinearEquiv = -1
-- Complex.reCLM_norm
@[simp]
theorem reCLM_norm : ‖reCLM‖ = 1 
Goal: ‖Complex.reCLM‖ = 1
-- Complex.reCLM_nnnorm
@[simp]
theorem reCLM_nnnorm : ‖reCLM‖₊ = 1 
Goal: ‖Complex.reCLM‖₊ = 1
-- Complex.imCLM_norm
@[simp]
theorem imCLM_norm : ‖imCLM‖ = 1 
Goal: ‖Complex.imCLM‖ = 1
-- Complex.imCLM_nnnorm
@[simp]
theorem imCLM_nnnorm : ‖imCLM‖₊ = 1 
Goal: ‖Complex.imCLM‖₊ = 1
-- Complex.conjCLE_norm
@[simp]
theorem conjCLE_norm : ‖(conjCLE : ℂ →L[ℝ] ℂ)‖ = 1 
Goal: ‖↑Complex.conjCLE‖ = 1
-- Complex.conjCLE_nnorm
@[simp]
theorem conjCLE_nnorm : ‖(conjCLE : ℂ →L[ℝ] ℂ)‖₊ = 1 
Goal: ‖↑Complex.conjCLE‖₊ = 1
-- Complex.ofRealCLM_norm
@[simp]
theorem ofRealCLM_norm : ‖ofRealCLM‖ = 1 
Goal: ‖Complex.ofRealCLM‖ = 1
-- Complex.ofRealCLM_nnnorm
@[simp]
theorem ofRealCLM_nnnorm : ‖ofRealCLM‖₊ = 1 
Goal: ‖Complex.ofRealCLM‖₊ = 1
-- uniqueDiffWithinAt_iff
@[mk_iff]
structure UniqueDiffWithinAt (s : Set E) (x : E) : Prop where
  dense_tangentCone : Dense (Submodule.span 𝕜 (tangentConeAt 𝕜 s x) : Set E)
  mem_closure : x ∈ closure s
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (s : Set E) (x : E)
Goal: UniqueDiffWithinAt 𝕜 s x ↔ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure s
-- UniqueDiffWithinAt.mem_closure
@[mk_iff]
structure UniqueDiffWithinAt (s : Set E) (x : E) : Prop where
  dense_tangentCone : Dense (Submodule.span 𝕜 (tangentConeAt 𝕜 s x) : Set E)
  mem_closure : x ∈ closure s
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] {s : Set E} {x : E} (self : UniqueDiffWithinAt 𝕜 s x)
Goal: x ∈ closure s
-- UniqueDiffWithinAt.dense_tangentCone
@[mk_iff]
structure UniqueDiffWithinAt (s : Set E) (x : E) : Prop where
  dense_tangentCone : Dense (Submodule.span 𝕜 (tangentConeAt 𝕜 s x) : Set E)
  mem_closure : x ∈ closure s
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] {s : Set E} {x : E} (self : UniqueDiffWithinAt 𝕜 s x)
Goal: Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x))
-- mem_tangentConeAt_of_pow_smul
theorem mem_tangentConeAt_of_pow_smul {r : 𝕜} (hr₀ : r ≠ 0) (hr : ‖r‖ < 1)
    (hs : ∀ᶠ n : ℕ in atTop, x + r ^ n • y ∈ s) : y ∈ tangentConeAt 𝕜 s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x y : E} {s : Set E} {r : 𝕜} (hr₀ : r ≠ 0) (hr : ‖r‖ < 1) (hs : ∀ᶠ (n : ℕ) in Filter.atTop, x + r ^ n • y ∈ s)
Goal: y ∈ tangentConeAt 𝕜 s x
-- tangentCone_univ
theorem tangentCone_univ : tangentConeAt 𝕜 univ x = univ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E}
Goal: tangentConeAt 𝕜 Set.univ x = Set.univ
-- tangentCone_mono
theorem tangentCone_mono (h : s ⊆ t) : tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 t x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s t : Set E} (h : s ⊆ t)
Goal: tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 t x
-- tangentConeAt.lim_zero
theorem tangentConeAt.lim_zero {α : Type*} (l : Filter α) {c : α → 𝕜} {d : α → E}
    (hc : Tendsto (fun n => ‖c n‖) l atTop) (hd : Tendsto (fun n => c n • d n) l (𝓝 y)) :
    Tendsto d l (𝓝 0) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {y : E} {α : Type u_5} (l : Filter α) {c : α → 𝕜} {d : α → E} (hc : Filter.Tendsto (fun n => ‖c n‖) l Filter.atTop) (hd : Filter.Tendsto (fun n => c n • d n) l (𝓝 y))
Goal: Filter.Tendsto d l (𝓝 0)
-- tangentCone_mono_nhds
theorem tangentCone_mono_nhds (h : 𝓝[s] x ≤ 𝓝[t] x) :
    tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 t x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s t : Set E} (h : 𝓝[s] x ≤ 𝓝[t] x)
Goal: tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 t x
-- tangentCone_congr
theorem tangentCone_congr (h : 𝓝[s] x = 𝓝[t] x) : tangentConeAt 𝕜 s x = tangentConeAt 𝕜 t x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s t : Set E} (h : 𝓝[s] x = 𝓝[t] x)
Goal: tangentConeAt 𝕜 s x = tangentConeAt 𝕜 t x
-- tangentCone_inter_nhds
theorem tangentCone_inter_nhds (ht : t ∈ 𝓝 x) : tangentConeAt 𝕜 (s ∩ t) x = tangentConeAt 𝕜 s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s t : Set E} (ht : t ∈ 𝓝 x)
Goal: tangentConeAt 𝕜 (s ∩ t) x = tangentConeAt 𝕜 s x
-- subset_tangentCone_prod_left
theorem subset_tangentCone_prod_left {t : Set F} {y : F} (ht : y ∈ closure t) :
    LinearMap.inl 𝕜 E F '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {s : Set E} {t : Set F} {y : F} (ht : y ∈ closure t)
Goal: ⇑(LinearMap.inl 𝕜 E F) '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y)
-- subset_tangentCone_prod_right
theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
    LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {s : Set E} {t : Set F} {y : F} (hs : x ∈ closure s)
Goal: ⇑(LinearMap.inr 𝕜 E F) '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y)
-- mapsTo_tangentCone_pi
theorem mapsTo_tangentCone_pi {ι : Type*} [DecidableEq ι] {E : ι → Type*}
    [∀ i, NormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)] {s : ∀ i, Set (E i)} {x : ∀ i, E i}
    {i : ι} (hi : ∀ j ≠ i, x j ∈ closure (s j)) :
    MapsTo (LinearMap.single i : E i →ₗ[𝕜] ∀ j, E j) (tangentConeAt 𝕜 (s i) (x i))
      (tangentConeAt 𝕜 (Set.pi univ s) x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_5} [DecidableEq ι] {E : ι → Type u_6} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {s : (i : ι) → Set (E i)} {x : (i : ι) → E i} {i : ι} (hi : ∀ (j : ι), j ≠ i → x j ∈ closure (s j))
Goal: Set.MapsTo (⇑(LinearMap.single i)) (tangentConeAt 𝕜 (s i) (x i)) (tangentConeAt 𝕜 (Set.univ.pi s) x)
-- mem_tangentCone_of_openSegment_subset
theorem mem_tangentCone_of_openSegment_subset {s : Set G} {x y : G} (h : openSegment ℝ x y ⊆ s) :
    y - x ∈ tangentConeAt ℝ s x 
Arguments: {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {s : Set G} {x y : G} (h : openSegment ℝ x y ⊆ s)
Goal: y - x ∈ tangentConeAt ℝ s x
-- mem_tangentCone_of_segment_subset
theorem mem_tangentCone_of_segment_subset {s : Set G} {x y : G} (h : segment ℝ x y ⊆ s) :
    y - x ∈ tangentConeAt ℝ s x 
Arguments: {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {s : Set G} {x y : G} (h : segment ℝ x y ⊆ s)
Goal: y - x ∈ tangentConeAt ℝ s x
-- UniqueDiffOn.uniqueDiffWithinAt
theorem UniqueDiffOn.uniqueDiffWithinAt {s : Set E} {x} (hs : UniqueDiffOn 𝕜 s) (h : x ∈ s) :
    UniqueDiffWithinAt 𝕜 s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {x : E} (hs : UniqueDiffOn 𝕜 s) (h : x ∈ s)
Goal: UniqueDiffWithinAt 𝕜 s x
-- uniqueDiffWithinAt_univ
theorem uniqueDiffWithinAt_univ : UniqueDiffWithinAt 𝕜 univ x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E}
Goal: UniqueDiffWithinAt 𝕜 Set.univ x
-- uniqueDiffOn_univ
theorem uniqueDiffOn_univ : UniqueDiffOn 𝕜 (univ : Set E) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: UniqueDiffOn 𝕜 Set.univ
-- uniqueDiffOn_empty
theorem uniqueDiffOn_empty : UniqueDiffOn 𝕜 (∅ : Set E) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: UniqueDiffOn 𝕜 ∅
-- UniqueDiffWithinAt.congr_pt
theorem UniqueDiffWithinAt.congr_pt (h : UniqueDiffWithinAt 𝕜 s x) (hy : x = y) :
    UniqueDiffWithinAt 𝕜 s y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x y : E} {s : Set E} (h : UniqueDiffWithinAt 𝕜 s x) (hy : x = y)
Goal: UniqueDiffWithinAt 𝕜 s y
-- UniqueDiffWithinAt.mono_nhds
theorem UniqueDiffWithinAt.mono_nhds (h : UniqueDiffWithinAt 𝕜 s x) (st : 𝓝[s] x ≤ 𝓝[t] x) :
    UniqueDiffWithinAt 𝕜 t x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s t : Set E} (h : UniqueDiffWithinAt 𝕜 s x) (st : 𝓝[s] x ≤ 𝓝[t] x)
Goal: UniqueDiffWithinAt 𝕜 t x
-- UniqueDiffWithinAt.mono
theorem UniqueDiffWithinAt.mono (h : UniqueDiffWithinAt 𝕜 s x) (st : s ⊆ t) :
    UniqueDiffWithinAt 𝕜 t x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s t : Set E} (h : UniqueDiffWithinAt 𝕜 s x) (st : s ⊆ t)
Goal: UniqueDiffWithinAt 𝕜 t x
-- uniqueDiffWithinAt_congr
theorem uniqueDiffWithinAt_congr (st : 𝓝[s] x = 𝓝[t] x) :
    UniqueDiffWithinAt 𝕜 s x ↔ UniqueDiffWithinAt 𝕜 t x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s t : Set E} (st : 𝓝[s] x = 𝓝[t] x)
Goal: UniqueDiffWithinAt 𝕜 s x ↔ UniqueDiffWithinAt 𝕜 t x
-- uniqueDiffWithinAt_inter
theorem uniqueDiffWithinAt_inter (ht : t ∈ 𝓝 x) :
    UniqueDiffWithinAt 𝕜 (s ∩ t) x ↔ UniqueDiffWithinAt 𝕜 s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s t : Set E} (ht : t ∈ 𝓝 x)
Goal: UniqueDiffWithinAt 𝕜 (s ∩ t) x ↔ UniqueDiffWithinAt 𝕜 s x
-- UniqueDiffWithinAt.inter
theorem UniqueDiffWithinAt.inter (hs : UniqueDiffWithinAt 𝕜 s x) (ht : t ∈ 𝓝 x) :
    UniqueDiffWithinAt 𝕜 (s ∩ t) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s t : Set E} (hs : UniqueDiffWithinAt 𝕜 s x) (ht : t ∈ 𝓝 x)
Goal: UniqueDiffWithinAt 𝕜 (s ∩ t) x
-- uniqueDiffWithinAt_inter'
theorem uniqueDiffWithinAt_inter' (ht : t ∈ 𝓝[s] x) :
    UniqueDiffWithinAt 𝕜 (s ∩ t) x ↔ UniqueDiffWithinAt 𝕜 s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s t : Set E} (ht : t ∈ 𝓝[s] x)
Goal: UniqueDiffWithinAt 𝕜 (s ∩ t) x ↔ UniqueDiffWithinAt 𝕜 s x
-- UniqueDiffWithinAt.inter'
theorem UniqueDiffWithinAt.inter' (hs : UniqueDiffWithinAt 𝕜 s x) (ht : t ∈ 𝓝[s] x) :
    UniqueDiffWithinAt 𝕜 (s ∩ t) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s t : Set E} (hs : UniqueDiffWithinAt 𝕜 s x) (ht : t ∈ 𝓝[s] x)
Goal: UniqueDiffWithinAt 𝕜 (s ∩ t) x
-- uniqueDiffWithinAt_of_mem_nhds
theorem uniqueDiffWithinAt_of_mem_nhds (h : s ∈ 𝓝 x) : UniqueDiffWithinAt 𝕜 s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} (h : s ∈ 𝓝 x)
Goal: UniqueDiffWithinAt 𝕜 s x
-- IsOpen.uniqueDiffWithinAt
theorem IsOpen.uniqueDiffWithinAt (hs : IsOpen s) (xs : x ∈ s) : UniqueDiffWithinAt 𝕜 s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} (hs : IsOpen s) (xs : x ∈ s)
Goal: UniqueDiffWithinAt 𝕜 s x
-- UniqueDiffOn.inter
theorem UniqueDiffOn.inter (hs : UniqueDiffOn 𝕜 s) (ht : IsOpen t) : UniqueDiffOn 𝕜 (s ∩ t) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s t : Set E} (hs : UniqueDiffOn 𝕜 s) (ht : IsOpen t)
Goal: UniqueDiffOn 𝕜 (s ∩ t)
-- IsOpen.uniqueDiffOn
theorem IsOpen.uniqueDiffOn (hs : IsOpen s) : UniqueDiffOn 𝕜 s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} (hs : IsOpen s)
Goal: UniqueDiffOn 𝕜 s
-- UniqueDiffWithinAt.prod
theorem UniqueDiffWithinAt.prod {t : Set F} {y : F} (hs : UniqueDiffWithinAt 𝕜 s x)
    (ht : UniqueDiffWithinAt 𝕜 t y) : UniqueDiffWithinAt 𝕜 (s ×ˢ t) (x, y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {s : Set E} {t : Set F} {y : F} (hs : UniqueDiffWithinAt 𝕜 s x) (ht : UniqueDiffWithinAt 𝕜 t y)
Goal: UniqueDiffWithinAt 𝕜 (s ×ˢ t) (x, y)
-- UniqueDiffWithinAt.univ_pi
theorem UniqueDiffWithinAt.univ_pi (ι : Type*) [Finite ι] (E : ι → Type*)
    [∀ i, NormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)] (s : ∀ i, Set (E i)) (x : ∀ i, E i)
    (h : ∀ i, UniqueDiffWithinAt 𝕜 (s i) (x i)) : UniqueDiffWithinAt 𝕜 (Set.pi univ s) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (ι : Type u_5) [Finite ι] (E : ι → Type u_6) [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (s : (i : ι) → Set (E i)) (x : (i : ι) → E i) (h : ∀ (i : ι), UniqueDiffWithinAt 𝕜 (s i) (x i))
Goal: UniqueDiffWithinAt 𝕜 (Set.univ.pi s) x
-- UniqueDiffWithinAt.pi
theorem UniqueDiffWithinAt.pi (ι : Type*) [Finite ι] (E : ι → Type*)
    [∀ i, NormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)] (s : ∀ i, Set (E i)) (x : ∀ i, E i)
    (I : Set ι) (h : ∀ i ∈ I, UniqueDiffWithinAt 𝕜 (s i) (x i)) :
    UniqueDiffWithinAt 𝕜 (Set.pi I s) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (ι : Type u_5) [Finite ι] (E : ι → Type u_6) [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (s : (i : ι) → Set (E i)) (x : (i : ι) → E i) (I : Set ι) (h : ∀ i ∈ I, UniqueDiffWithinAt 𝕜 (s i) (x i))
Goal: UniqueDiffWithinAt 𝕜 (I.pi s) x
-- UniqueDiffOn.prod
theorem UniqueDiffOn.prod {t : Set F} (hs : UniqueDiffOn 𝕜 s) (ht : UniqueDiffOn 𝕜 t) :
    UniqueDiffOn 𝕜 (s ×ˢ t) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {t : Set F} (hs : UniqueDiffOn 𝕜 s) (ht : UniqueDiffOn 𝕜 t)
Goal: UniqueDiffOn 𝕜 (s ×ˢ t)
-- UniqueDiffOn.pi
theorem UniqueDiffOn.pi (ι : Type*) [Finite ι] (E : ι → Type*) [∀ i, NormedAddCommGroup (E i)]
    [∀ i, NormedSpace 𝕜 (E i)] (s : ∀ i, Set (E i)) (I : Set ι)
    (h : ∀ i ∈ I, UniqueDiffOn 𝕜 (s i)) : UniqueDiffOn 𝕜 (Set.pi I s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (ι : Type u_5) [Finite ι] (E : ι → Type u_6) [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (s : (i : ι) → Set (E i)) (I : Set ι) (h : ∀ i ∈ I, UniqueDiffOn 𝕜 (s i))
Goal: UniqueDiffOn 𝕜 (I.pi s)
-- UniqueDiffOn.univ_pi
theorem UniqueDiffOn.univ_pi (ι : Type*) [Finite ι] (E : ι → Type*)
    [∀ i, NormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)] (s : ∀ i, Set (E i))
    (h : ∀ i, UniqueDiffOn 𝕜 (s i)) : UniqueDiffOn 𝕜 (Set.pi univ s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (ι : Type u_5) [Finite ι] (E : ι → Type u_6) [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (s : (i : ι) → Set (E i)) (h : ∀ (i : ι), UniqueDiffOn 𝕜 (s i))
Goal: UniqueDiffOn 𝕜 (Set.univ.pi s)
-- uniqueDiffWithinAt_convex
theorem uniqueDiffWithinAt_convex {s : Set G} (conv : Convex ℝ s) (hs : (interior s).Nonempty)
    {x : G} (hx : x ∈ closure s) : UniqueDiffWithinAt ℝ s x 
Arguments: {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {s : Set G} (conv : Convex ℝ s) (hs : (interior s).Nonempty) {x : G} (hx : x ∈ closure s)
Goal: UniqueDiffWithinAt ℝ s x
-- uniqueDiffOn_convex
theorem uniqueDiffOn_convex {s : Set G} (conv : Convex ℝ s) (hs : (interior s).Nonempty) :
    UniqueDiffOn ℝ s 
Arguments: {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {s : Set G} (conv : Convex ℝ s) (hs : (interior s).Nonempty)
Goal: UniqueDiffOn ℝ s
-- uniqueDiffOn_Ici
theorem uniqueDiffOn_Ici (a : ℝ) : UniqueDiffOn ℝ (Ici a) 
Arguments: (a : ℝ)
Goal: UniqueDiffOn ℝ (Set.Ici a)
-- uniqueDiffOn_Iic
theorem uniqueDiffOn_Iic (a : ℝ) : UniqueDiffOn ℝ (Iic a) 
Arguments: (a : ℝ)
Goal: UniqueDiffOn ℝ (Set.Iic a)
-- uniqueDiffOn_Ioi
theorem uniqueDiffOn_Ioi (a : ℝ) : UniqueDiffOn ℝ (Ioi a) 
Arguments: (a : ℝ)
Goal: UniqueDiffOn ℝ (Set.Ioi a)
-- uniqueDiffOn_Iio
theorem uniqueDiffOn_Iio (a : ℝ) : UniqueDiffOn ℝ (Iio a) 
Arguments: (a : ℝ)
Goal: UniqueDiffOn ℝ (Set.Iio a)
-- uniqueDiffOn_Icc
theorem uniqueDiffOn_Icc {a b : ℝ} (hab : a < b) : UniqueDiffOn ℝ (Icc a b) 
Arguments: {a b : ℝ} (hab : a < b)
Goal: UniqueDiffOn ℝ (Set.Icc a b)
-- uniqueDiffOn_Ico
theorem uniqueDiffOn_Ico (a b : ℝ) : UniqueDiffOn ℝ (Ico a b) 
Arguments: (a b : ℝ)
Goal: UniqueDiffOn ℝ (Set.Ico a b)
-- uniqueDiffOn_Ioc
theorem uniqueDiffOn_Ioc (a b : ℝ) : UniqueDiffOn ℝ (Ioc a b) 
Arguments: (a b : ℝ)
Goal: UniqueDiffOn ℝ (Set.Ioc a b)
-- uniqueDiffOn_Ioo
theorem uniqueDiffOn_Ioo (a b : ℝ) : UniqueDiffOn ℝ (Ioo a b) 
Arguments: (a b : ℝ)
Goal: UniqueDiffOn ℝ (Set.Ioo a b)
-- uniqueDiffOn_Icc_zero_one
theorem uniqueDiffOn_Icc_zero_one : UniqueDiffOn ℝ (Icc (0 : ℝ) 1) 
Goal: UniqueDiffOn ℝ (Set.Icc 0 1)
-- uniqueDiffWithinAt_Ioo
theorem uniqueDiffWithinAt_Ioo {a b t : ℝ} (ht : t ∈ Set.Ioo a b) :
    UniqueDiffWithinAt ℝ (Set.Ioo a b) t 
Arguments: {a b t : ℝ} (ht : t ∈ Set.Ioo a b)
Goal: UniqueDiffWithinAt ℝ (Set.Ioo a b) t
-- uniqueDiffWithinAt_Ioi
theorem uniqueDiffWithinAt_Ioi (a : ℝ) : UniqueDiffWithinAt ℝ (Ioi a) a 
Arguments: (a : ℝ)
Goal: UniqueDiffWithinAt ℝ (Set.Ioi a) a
-- uniqueDiffWithinAt_Iio
theorem uniqueDiffWithinAt_Iio (a : ℝ) : UniqueDiffWithinAt ℝ (Iio a) a 
Arguments: (a : ℝ)
Goal: UniqueDiffWithinAt ℝ (Set.Iio a) a
-- hasGradientWithinAt_iff_hasFDerivWithinAt
theorem hasGradientWithinAt_iff_hasFDerivWithinAt {s : Set F} :
    HasGradientWithinAt f f' s x ↔ HasFDerivWithinAt f (toDual 𝕜 F f') s x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s : Set F}
Goal: HasGradientWithinAt f f' s x ↔ HasFDerivWithinAt f ((InnerProductSpace.toDual 𝕜 F) f') s x
-- hasFDerivWithinAt_iff_hasGradientWithinAt
theorem hasFDerivWithinAt_iff_hasGradientWithinAt {frechet : F →L[𝕜] 𝕜} {s : Set F} :
    HasFDerivWithinAt f frechet s x ↔ HasGradientWithinAt f ((toDual 𝕜 F).symm frechet) s x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {x : F} {frechet : F →L[𝕜] 𝕜} {s : Set F}
Goal: HasFDerivWithinAt f frechet s x ↔ HasGradientWithinAt f ((InnerProductSpace.toDual 𝕜 F).symm frechet) s x
-- hasGradientAt_iff_hasFDerivAt
theorem hasGradientAt_iff_hasFDerivAt :
    HasGradientAt f f' x ↔ HasFDerivAt f (toDual 𝕜 F f') x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F}
Goal: HasGradientAt f f' x ↔ HasFDerivAt f ((InnerProductSpace.toDual 𝕜 F) f') x