ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- measurableSet_lineDifferentiableAt
theorem measurableSet_lineDifferentiableAt (hf : Continuous f) :
    MeasurableSet {x : E | LineDifferentiableAt 𝕜 f x v} 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {v : E} (hf : Continuous f)
Goal: MeasurableSet {x | LineDifferentiableAt 𝕜 f x v}
-- measurable_lineDeriv
theorem measurable_lineDeriv [MeasurableSpace F] [BorelSpace F]
    (hf : Continuous f) : Measurable (fun x ↦ lineDeriv 𝕜 f x v) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {v : E} [MeasurableSpace F] [BorelSpace F] (hf : Continuous f)
Goal: Measurable fun x => lineDeriv 𝕜 f x v
-- stronglyMeasurable_lineDeriv
theorem stronglyMeasurable_lineDeriv [SecondCountableTopologyEither E F] (hf : Continuous f) :
    StronglyMeasurable (fun x ↦ lineDeriv 𝕜 f x v) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {v : E} [SecondCountableTopologyEither E F] (hf : Continuous f)
Goal: MeasureTheory.StronglyMeasurable fun x => lineDeriv 𝕜 f x v
-- aemeasurable_lineDeriv
theorem aemeasurable_lineDeriv [MeasurableSpace F] [BorelSpace F]
    (hf : Continuous f) (μ : Measure E) :
    AEMeasurable (fun x ↦ lineDeriv 𝕜 f x v) μ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {v : E} [MeasurableSpace F] [BorelSpace F] (hf : Continuous f) (μ : MeasureTheory.Measure E)
Goal: AEMeasurable (fun x => lineDeriv 𝕜 f x v) μ
-- aestronglyMeasurable_lineDeriv
theorem aestronglyMeasurable_lineDeriv [SecondCountableTopologyEither E F]
    (hf : Continuous f) (μ : Measure E) :
    AEStronglyMeasurable (fun x ↦ lineDeriv 𝕜 f x v) μ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {v : E} [SecondCountableTopologyEither E F] (hf : Continuous f) (μ : MeasureTheory.Measure E)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => lineDeriv 𝕜 f x v) μ
-- measurableSet_lineDifferentiableAt_uncurry
theorem measurableSet_lineDifferentiableAt_uncurry (hf : Continuous f) :
    MeasurableSet {p : E × E | LineDifferentiableAt 𝕜 f p.1 p.2} 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} [SecondCountableTopology E] (hf : Continuous f)
Goal: MeasurableSet {p | LineDifferentiableAt 𝕜 f p.1 p.2}
-- measurable_lineDeriv_uncurry
theorem measurable_lineDeriv_uncurry [MeasurableSpace F] [BorelSpace F]
    (hf : Continuous f) : Measurable (fun (p : E × E) ↦ lineDeriv 𝕜 f p.1 p.2) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} [SecondCountableTopology E] [MeasurableSpace F] [BorelSpace F] (hf : Continuous f)
Goal: Measurable fun p => lineDeriv 𝕜 f p.1 p.2
-- stronglyMeasurable_lineDeriv_uncurry
theorem stronglyMeasurable_lineDeriv_uncurry (hf : Continuous f) :
    StronglyMeasurable (fun (p : E × E) ↦ lineDeriv 𝕜 f p.1 p.2) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} [SecondCountableTopology E] (hf : Continuous f)
Goal: MeasureTheory.StronglyMeasurable fun p => lineDeriv 𝕜 f p.1 p.2
-- aemeasurable_lineDeriv_uncurry
theorem aemeasurable_lineDeriv_uncurry [MeasurableSpace F] [BorelSpace F]
    (hf : Continuous f) (μ : Measure (E × E)) :
    AEMeasurable (fun (p : E × E) ↦ lineDeriv 𝕜 f p.1 p.2) μ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} [SecondCountableTopology E] [MeasurableSpace F] [BorelSpace F] (hf : Continuous f) (μ : MeasureTheory.Measure (E × E))
Goal: AEMeasurable (fun p => lineDeriv 𝕜 f p.1 p.2) μ
-- aestronglyMeasurable_lineDeriv_uncurry
theorem aestronglyMeasurable_lineDeriv_uncurry (hf : Continuous f) (μ : Measure (E × E)) :
    AEStronglyMeasurable (fun (p : E × E) ↦ lineDeriv 𝕜 f p.1 p.2) μ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} [SecondCountableTopology E] (hf : Continuous f) (μ : MeasureTheory.Measure (E × E))
Goal: MeasureTheory.AEStronglyMeasurable (fun p => lineDeriv 𝕜 f p.1 p.2) μ
-- exists_hasDerivAt_eq_zero
theorem exists_hasDerivAt_eq_zero (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b)
    (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) : ∃ c ∈ Ioo a b, f' c = 0 
Arguments: {f f' : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Set.Icc a b)) (hfI : f a = f b) (hff' : ∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x)
Goal: ∃ c ∈ Set.Ioo a b, f' c = 0
-- exists_deriv_eq_zero
theorem exists_deriv_eq_zero (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) :
    ∃ c ∈ Ioo a b, deriv f c = 0 
Arguments: {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Set.Icc a b)) (hfI : f a = f b)
Goal: ∃ c ∈ Set.Ioo a b, deriv f c = 0
-- exists_hasDerivAt_eq_zero'
theorem exists_hasDerivAt_eq_zero' (hab : a < b) (hfa : Tendsto f (𝓝[>] a) (𝓝 l))
    (hfb : Tendsto f (𝓝[<] b) (𝓝 l)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) :
    ∃ c ∈ Ioo a b, f' c = 0 
Arguments: {f f' : ℝ → ℝ} {a b l : ℝ} (hab : a < b) (hfa : Filter.Tendsto f (𝓝[>] a) (𝓝 l)) (hfb : Filter.Tendsto f (𝓝[<] b) (𝓝 l)) (hff' : ∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x)
Goal: ∃ c ∈ Set.Ioo a b, f' c = 0
-- exists_deriv_eq_zero'
theorem exists_deriv_eq_zero' (hab : a < b) (hfa : Tendsto f (𝓝[>] a) (𝓝 l))
    (hfb : Tendsto f (𝓝[<] b) (𝓝 l)) : ∃ c ∈ Ioo a b, deriv f c = 0 
Arguments: {f : ℝ → ℝ} {a b l : ℝ} (hab : a < b) (hfa : Filter.Tendsto f (𝓝[>] a) (𝓝 l)) (hfb : Filter.Tendsto f (𝓝[<] b) (𝓝 l))
Goal: ∃ c ∈ Set.Ioo a b, deriv f c = 0
-- Polynomial.card_roots_toFinset_le_card_roots_derivative_diff_roots_succ
theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
    p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 
Arguments: (p : Polynomial ℝ)
Goal: p.roots.toFinset.card ≤ ((Polynomial.derivative p).roots.toFinset \ p.roots.toFinset).card + 1
-- Polynomial.card_roots_toFinset_le_derivative
theorem card_roots_toFinset_le_derivative (p : ℝ[X]) :
    p.roots.toFinset.card ≤ p.derivative.roots.toFinset.card + 1 
Arguments: (p : Polynomial ℝ)
Goal: p.roots.toFinset.card ≤ (Polynomial.derivative p).roots.toFinset.card + 1
-- Polynomial.card_roots_le_derivative
theorem card_roots_le_derivative (p : ℝ[X]) :
    Multiset.card p.roots ≤ Multiset.card (derivative p).roots + 1 
Arguments: (p : Polynomial ℝ)
Goal: Multiset.card p.roots ≤ Multiset.card (Polynomial.derivative p).roots + 1
-- Polynomial.card_rootSet_le_derivative
theorem card_rootSet_le_derivative {F : Type*} [CommRing F] [Algebra F ℝ] (p : F[X]) :
    Fintype.card (p.rootSet ℝ) ≤ Fintype.card (p.derivative.rootSet ℝ) + 1 
Arguments: {F : Type u_1} [CommRing F] [Algebra F ℝ] (p : Polynomial F)
Goal: Fintype.card ↑(p.rootSet ℝ) ≤ Fintype.card ↑((Polynomial.derivative p).rootSet ℝ) + 1
-- conformalAt_iff'
theorem conformalAt_iff' {f : E → F} {x : E} : ConformalAt f x ↔
    ∃ c : ℝ, 0 < c ∧ ∀ u v : E, ⟪fderiv ℝ f x u, fderiv ℝ f x v⟫ = c * ⟪u, v⟫ 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace ℝ E] [InnerProductSpace ℝ F] {f : E → F} {x : E}
Goal: ConformalAt f x ↔ ∃ c, 0 < c ∧ ∀ (u v : E), ⟪(fderiv ℝ f x) u, (fderiv ℝ f x) v⟫_ℝ = c * ⟪u, v⟫_ℝ
-- conformalAt_iff
theorem conformalAt_iff {f : E → F} {x : E} {f' : E →L[ℝ] F} (h : HasFDerivAt f f' x) :
    ConformalAt f x ↔ ∃ c : ℝ, 0 < c ∧ ∀ u v : E, ⟪f' u, f' v⟫ = c * ⟪u, v⟫ 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace ℝ E] [InnerProductSpace ℝ F] {f : E → F} {x : E} {f' : E →L[ℝ] F} (h : HasFDerivAt f f' x)
Goal: ConformalAt f x ↔ ∃ c, 0 < c ∧ ∀ (u v : E), ⟪f' u, f' v⟫_ℝ = c * ⟪u, v⟫_ℝ
-- conformalFactorAt_pos
theorem conformalFactorAt_pos {f : E → F} {x : E} (h : ConformalAt f x) : 0 < conformalFactorAt h 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace ℝ E] [InnerProductSpace ℝ F] {f : E → F} {x : E} (h : ConformalAt f x)
Goal: 0 < conformalFactorAt h
-- conformalFactorAt_inner_eq_mul_inner'
theorem conformalFactorAt_inner_eq_mul_inner' {f : E → F} {x : E} (h : ConformalAt f x) (u v : E) :
    ⟪(fderiv ℝ f x) u, (fderiv ℝ f x) v⟫ = (conformalFactorAt h : ℝ) * ⟪u, v⟫ 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace ℝ E] [InnerProductSpace ℝ F] {f : E → F} {x : E} (h : ConformalAt f x) (u v : E)
Goal: ⟪(fderiv ℝ f x) u, (fderiv ℝ f x) v⟫_ℝ = conformalFactorAt h * ⟪u, v⟫_ℝ
-- conformalFactorAt_inner_eq_mul_inner
theorem conformalFactorAt_inner_eq_mul_inner {f : E → F} {x : E} {f' : E →L[ℝ] F}
    (h : HasFDerivAt f f' x) (H : ConformalAt f x) (u v : E) :
    ⟪f' u, f' v⟫ = (conformalFactorAt H : ℝ) * ⟪u, v⟫ 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace ℝ E] [InnerProductSpace ℝ F] {f : E → F} {x : E} {f' : E →L[ℝ] F} (h : HasFDerivAt f f' x) (H : ConformalAt f x) (u v : E)
Goal: ⟪f' u, f' v⟫_ℝ = conformalFactorAt H * ⟪u, v⟫_ℝ
-- ImplicitFunctionData.left_has_deriv
-- Porting note(#5171): linter not yet ported @[nolint has_nonempty_instance]
structure ImplicitFunctionData (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type*)
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] (F : Type*) [NormedAddCommGroup F]
    [NormedSpace 𝕜 F] [CompleteSpace F] (G : Type*) [NormedAddCommGroup G] [NormedSpace 𝕜 G]
    [CompleteSpace G] where
  leftFun : E → F
  leftDeriv : E →L[𝕜] F
  rightFun : E → G
  rightDeriv : E →L[𝕜] G
  pt : E
  left_has_deriv : HasStrictFDerivAt leftFun leftDeriv pt
  right_has_deriv : HasStrictFDerivAt rightFun rightDeriv pt
  left_range : range leftDeriv = ⊤
  right_range : range rightDeriv = ⊤
  isCompl_ker : IsCompl (ker leftDeriv) (ker rightDeriv)
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (self : ImplicitFunctionData 𝕜 E F G)
Goal: HasStrictFDerivAt self.leftFun self.leftDeriv self.pt
-- ImplicitFunctionData.isCompl_ker
-- Porting note(#5171): linter not yet ported @[nolint has_nonempty_instance]
structure ImplicitFunctionData (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type*)
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] (F : Type*) [NormedAddCommGroup F]
    [NormedSpace 𝕜 F] [CompleteSpace F] (G : Type*) [NormedAddCommGroup G] [NormedSpace 𝕜 G]
    [CompleteSpace G] where
  leftFun : E → F
  leftDeriv : E →L[𝕜] F
  rightFun : E → G
  rightDeriv : E →L[𝕜] G
  pt : E
  left_has_deriv : HasStrictFDerivAt leftFun leftDeriv pt
  right_has_deriv : HasStrictFDerivAt rightFun rightDeriv pt
  left_range : range leftDeriv = ⊤
  right_range : range rightDeriv = ⊤
  isCompl_ker : IsCompl (ker leftDeriv) (ker rightDeriv)
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (self : ImplicitFunctionData 𝕜 E F G)
Goal: IsCompl (LinearMap.ker self.leftDeriv) (LinearMap.ker self.rightDeriv)
-- ImplicitFunctionData.right_range
-- Porting note(#5171): linter not yet ported @[nolint has_nonempty_instance]
structure ImplicitFunctionData (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type*)
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] (F : Type*) [NormedAddCommGroup F]
    [NormedSpace 𝕜 F] [CompleteSpace F] (G : Type*) [NormedAddCommGroup G] [NormedSpace 𝕜 G]
    [CompleteSpace G] where
  leftFun : E → F
  leftDeriv : E →L[𝕜] F
  rightFun : E → G
  rightDeriv : E →L[𝕜] G
  pt : E
  left_has_deriv : HasStrictFDerivAt leftFun leftDeriv pt
  right_has_deriv : HasStrictFDerivAt rightFun rightDeriv pt
  left_range : range leftDeriv = ⊤
  right_range : range rightDeriv = ⊤
  isCompl_ker : IsCompl (ker leftDeriv) (ker rightDeriv)
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (self : ImplicitFunctionData 𝕜 E F G)
Goal: LinearMap.range self.rightDeriv = ⊤
-- ImplicitFunctionData.right_has_deriv
-- Porting note(#5171): linter not yet ported @[nolint has_nonempty_instance]
structure ImplicitFunctionData (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type*)
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] (F : Type*) [NormedAddCommGroup F]
    [NormedSpace 𝕜 F] [CompleteSpace F] (G : Type*) [NormedAddCommGroup G] [NormedSpace 𝕜 G]
    [CompleteSpace G] where
  leftFun : E → F
  leftDeriv : E →L[𝕜] F
  rightFun : E → G
  rightDeriv : E →L[𝕜] G
  pt : E
  left_has_deriv : HasStrictFDerivAt leftFun leftDeriv pt
  right_has_deriv : HasStrictFDerivAt rightFun rightDeriv pt
  left_range : range leftDeriv = ⊤
  right_range : range rightDeriv = ⊤
  isCompl_ker : IsCompl (ker leftDeriv) (ker rightDeriv)
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (self : ImplicitFunctionData 𝕜 E F G)
Goal: HasStrictFDerivAt self.rightFun self.rightDeriv self.pt
-- ImplicitFunctionData.left_range
-- Porting note(#5171): linter not yet ported @[nolint has_nonempty_instance]
structure ImplicitFunctionData (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type*)
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] (F : Type*) [NormedAddCommGroup F]
    [NormedSpace 𝕜 F] [CompleteSpace F] (G : Type*) [NormedAddCommGroup G] [NormedSpace 𝕜 G]
    [CompleteSpace G] where
  leftFun : E → F
  leftDeriv : E →L[𝕜] F
  rightFun : E → G
  rightDeriv : E →L[𝕜] G
  pt : E
  left_has_deriv : HasStrictFDerivAt leftFun leftDeriv pt
  right_has_deriv : HasStrictFDerivAt rightFun rightDeriv pt
  left_range : range leftDeriv = ⊤
  right_range : range rightDeriv = ⊤
  isCompl_ker : IsCompl (ker leftDeriv) (ker rightDeriv)
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (self : ImplicitFunctionData 𝕜 E F G)
Goal: LinearMap.range self.leftDeriv = ⊤
-- ImplicitFunctionData.prodFun_apply
@[simp]
theorem prodFun_apply (x : E) : φ.prodFun x = (φ.leftFun x, φ.rightFun x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G) (x : E)
Goal: φ.prodFun x = (φ.leftFun x, φ.rightFun x)
-- ImplicitFunctionData.hasStrictFDerivAt
protected theorem hasStrictFDerivAt :
    HasStrictFDerivAt φ.prodFun
      (φ.leftDeriv.equivProdOfSurjectiveOfIsCompl φ.rightDeriv φ.left_range φ.right_range
          φ.isCompl_ker :
        E →L[𝕜] F × G)
      φ.pt 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G)
Goal: HasStrictFDerivAt φ.prodFun (↑(φ.leftDeriv.equivProdOfSurjectiveOfIsCompl φ.rightDeriv ⋯ ⋯ ⋯)) φ.pt
-- ImplicitFunctionData.toPartialHomeomorph_coe
@[simp]
theorem toPartialHomeomorph_coe : ⇑φ.toPartialHomeomorph = φ.prodFun 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G)
Goal: ↑φ.toPartialHomeomorph = φ.prodFun
-- ImplicitFunctionData.toPartialHomeomorph_apply
theorem toPartialHomeomorph_apply (x : E) : φ.toPartialHomeomorph x = (φ.leftFun x, φ.rightFun x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G) (x : E)
Goal: ↑φ.toPartialHomeomorph x = (φ.leftFun x, φ.rightFun x)
-- ImplicitFunctionData.pt_mem_toPartialHomeomorph_source
theorem pt_mem_toPartialHomeomorph_source : φ.pt ∈ φ.toPartialHomeomorph.source 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G)
Goal: φ.pt ∈ φ.toPartialHomeomorph.source
-- ImplicitFunctionData.map_pt_mem_toPartialHomeomorph_target
theorem map_pt_mem_toPartialHomeomorph_target :
    (φ.leftFun φ.pt, φ.rightFun φ.pt) ∈ φ.toPartialHomeomorph.target 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G)
Goal: (φ.leftFun φ.pt, φ.rightFun φ.pt) ∈ φ.toPartialHomeomorph.target
-- ImplicitFunctionData.prod_map_implicitFunction
theorem prod_map_implicitFunction :
    ∀ᶠ p : F × G in 𝓝 (φ.prodFun φ.pt), φ.prodFun (φ.implicitFunction p.1 p.2) = p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G)
Goal: ∀ᶠ (p : F × G) in 𝓝 (φ.prodFun φ.pt), φ.prodFun (φ.implicitFunction p.1 p.2) = p
-- ImplicitFunctionData.left_map_implicitFunction
theorem left_map_implicitFunction :
    ∀ᶠ p : F × G in 𝓝 (φ.prodFun φ.pt), φ.leftFun (φ.implicitFunction p.1 p.2) = p.1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G)
Goal: ∀ᶠ (p : F × G) in 𝓝 (φ.prodFun φ.pt), φ.leftFun (φ.implicitFunction p.1 p.2) = p.1
-- ImplicitFunctionData.right_map_implicitFunction
theorem right_map_implicitFunction :
    ∀ᶠ p : F × G in 𝓝 (φ.prodFun φ.pt), φ.rightFun (φ.implicitFunction p.1 p.2) = p.2 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G)
Goal: ∀ᶠ (p : F × G) in 𝓝 (φ.prodFun φ.pt), φ.rightFun (φ.implicitFunction p.1 p.2) = p.2
-- ImplicitFunctionData.implicitFunction_apply_image
theorem implicitFunction_apply_image :
    ∀ᶠ x in 𝓝 φ.pt, φ.implicitFunction (φ.leftFun x) (φ.rightFun x) = x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G)
Goal: ∀ᶠ (x : E) in 𝓝 φ.pt, φ.implicitFunction (φ.leftFun x) (φ.rightFun x) = x
-- ImplicitFunctionData.map_nhds_eq
theorem map_nhds_eq : map φ.leftFun (𝓝 φ.pt) = 𝓝 (φ.leftFun φ.pt) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G)
Goal: Filter.map φ.leftFun (𝓝 φ.pt) = 𝓝 (φ.leftFun φ.pt)
-- ImplicitFunctionData.implicitFunction_hasStrictFDerivAt
theorem implicitFunction_hasStrictFDerivAt (g'inv : G →L[𝕜] E)
    (hg'inv : φ.rightDeriv.comp g'inv = ContinuousLinearMap.id 𝕜 G)
    (hg'invf : φ.leftDeriv.comp g'inv = 0) :
    HasStrictFDerivAt (φ.implicitFunction (φ.leftFun φ.pt)) g'inv (φ.rightFun φ.pt) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] (φ : ImplicitFunctionData 𝕜 E F G) (g'inv : G →L[𝕜] E) (hg'inv : φ.rightDeriv.comp g'inv = ContinuousLinearMap.id 𝕜 G) (hg'invf : φ.leftDeriv.comp g'inv = 0)
Goal: HasStrictFDerivAt (φ.implicitFunction (φ.leftFun φ.pt)) g'inv (φ.rightFun φ.pt)
-- HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_fst
@[simp]
theorem implicitToPartialHomeomorphOfComplemented_fst (hf : HasStrictFDerivAt f f' a)
    (hf' : range f' = ⊤) (hker : (ker f').ClosedComplemented) (x : E) :
    (hf.implicitToPartialHomeomorphOfComplemented f f' hf' hker x).fst = f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) (hf' : LinearMap.range f' = ⊤) (hker : (LinearMap.ker f').ClosedComplemented) (x : E)
Goal: (↑(HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented f f' hf hf' hker) x).1 = f x