ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousMultilinearMap.norm_compContinuous_linearIsometry_le
theorem norm_compContinuous_linearIsometry_le (g : ContinuousMultilinearMap 𝕜 E₁ G)
    (f : ∀ i, E i →ₗᵢ[𝕜] E₁ i) :
    ‖g.compContinuousLinearMap fun i => (f i).toContinuousLinearMap‖ ≤ ‖g‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i →ₗᵢ[𝕜] E₁ i)
Goal: ‖g.compContinuousLinearMap fun i => (f i).toContinuousLinearMap‖ ≤ ‖g‖
-- ContinuousMultilinearMap.norm_compContinuous_linearIsometryEquiv
theorem norm_compContinuous_linearIsometryEquiv (g : ContinuousMultilinearMap 𝕜 E₁ G)
    (f : ∀ i, E i ≃ₗᵢ[𝕜] E₁ i) :
    ‖g.compContinuousLinearMap fun i => (f i : E i →L[𝕜] E₁ i)‖ = ‖g‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i)
Goal: ‖g.compContinuousLinearMap fun i =>
      ↑{ toLinearEquiv := (f i).toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }‖ =
  ‖g‖
-- ContinuousMultilinearMap.compContinuousLinearMapL_apply
@[simp]
theorem compContinuousLinearMapL_apply (g : ContinuousMultilinearMap 𝕜 E₁ G)
    (f : ∀ i, E i →L[𝕜] E₁ i) : compContinuousLinearMapL f g = g.compContinuousLinearMap f 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i →L[𝕜] E₁ i)
Goal: (ContinuousMultilinearMap.compContinuousLinearMapL f) g = g.compContinuousLinearMap f
-- ContinuousMultilinearMap.norm_compContinuousLinearMapL_le
theorem norm_compContinuousLinearMapL_le (f : ∀ i, E i →L[𝕜] E₁ i) :
    ‖compContinuousLinearMapL (G := G) f‖ ≤ ∏ i, ‖f i‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : (i : ι) → E i →L[𝕜] E₁ i)
Goal: ‖ContinuousMultilinearMap.compContinuousLinearMapL f‖ ≤ ∏ i : ι, ‖f i‖
-- ContinuousMultilinearMap.compContinuousLinearMapLRight_apply
@[simp]
theorem compContinuousLinearMapLRight_apply (g : ContinuousMultilinearMap 𝕜 E₁ G)
    (f : ∀ i, E i →L[𝕜] E₁ i) : compContinuousLinearMapLRight g f = g.compContinuousLinearMap f 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i →L[𝕜] E₁ i)
Goal: g.compContinuousLinearMapLRight f = g.compContinuousLinearMap f
-- ContinuousMultilinearMap.norm_compContinuousLinearMapLRight_le
theorem norm_compContinuousLinearMapLRight_le (g : ContinuousMultilinearMap 𝕜 E₁ G) :
    ‖compContinuousLinearMapLRight (E := E) g‖ ≤ ‖g‖ 
Arguments: {𝕜 : Type u} {ι : Type v} (E : ι → Type wE) {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G)
Goal: ‖g.compContinuousLinearMapLRight‖ ≤ ‖g‖
-- ContinuousMultilinearMap.compContinuousLinearMapEquivL_symm
@[simp]
theorem compContinuousLinearMapEquivL_symm (f : ∀ i, E i ≃L[𝕜] E₁ i) :
    (compContinuousLinearMapEquivL G f).symm =
      compContinuousLinearMapEquivL G fun i : ι => (f i).symm 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : (i : ι) → E i ≃L[𝕜] E₁ i)
Goal: (ContinuousMultilinearMap.compContinuousLinearMapEquivL G f).symm =
  ContinuousMultilinearMap.compContinuousLinearMapEquivL G fun i => (f i).symm
-- ContinuousMultilinearMap.compContinuousLinearMapEquivL_apply
@[simp]
theorem compContinuousLinearMapEquivL_apply (g : ContinuousMultilinearMap 𝕜 E₁ G)
    (f : ∀ i, E i ≃L[𝕜] E₁ i) :
    compContinuousLinearMapEquivL G f g =
      g.compContinuousLinearMap fun i => (f i : E i →L[𝕜] E₁ i) 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i ≃L[𝕜] E₁ i)
Goal: (ContinuousMultilinearMap.compContinuousLinearMapEquivL G f) g = g.compContinuousLinearMap fun i => ↑(f i)
-- ContinuousMultilinearMap.iteratedFDerivComponent_apply
@[simp] lemma iteratedFDerivComponent_apply {α : Type*} [Fintype α]
    (f : ContinuousMultilinearMap 𝕜 E₁ G) {s : Set ι} (e : α ≃ s) [DecidablePred (· ∈ s)]
    (v : ∀ i : {a : ι // a ∉ s}, E₁ i) (w : α → (∀ i, E₁ i)) :
    f.iteratedFDerivComponent e v w =
      f (fun j ↦ if h : j ∈ s then w (e.symm ⟨j, h⟩) j else v ⟨j, h⟩) 
Arguments: {𝕜 : Type u} {ι : Type v} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {α : Type u_1} [Fintype α] (f : ContinuousMultilinearMap 𝕜 E₁ G) {s : Set ι} (e : α ≃ ↑s) [DecidablePred fun x => x ∈ s] (v : (i : { a // a ∉ s }) → E₁ ↑i) (w : α → (i : ι) → E₁ i)
Goal: ((f.iteratedFDerivComponent e) v) w = f fun j => if h : j ∈ s then w (e.symm ⟨j, h⟩) j else v ⟨j, h⟩
-- ContinuousMultilinearMap.norm_iteratedFDerivComponent_le
lemma norm_iteratedFDerivComponent_le {α : Type*} [Fintype α]
    (f : ContinuousMultilinearMap 𝕜 E₁ G) {s : Set ι} (e : α ≃ s) [DecidablePred (· ∈ s)]
    (x : (i : ι) → E₁ i) :
    ‖f.iteratedFDerivComponent e (x ·)‖ ≤ ‖f‖ * ‖x‖ ^ (Fintype.card ι - Fintype.card α) 
Arguments: {𝕜 : Type u} {ι : Type v} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {α : Type u_1} [Fintype α] (f : ContinuousMultilinearMap 𝕜 E₁ G) {s : Set ι} (e : α ≃ ↑s) [DecidablePred fun x => x ∈ s] (x : (i : ι) → E₁ i)
Goal: ‖(f.iteratedFDerivComponent e) fun x_1 => x ↑x_1‖ ≤ ‖f‖ * ‖x‖ ^ (Fintype.card ι - Fintype.card α)
-- ContinuousMultilinearMap.norm_iteratedFDeriv_le'
lemma norm_iteratedFDeriv_le' (f : ContinuousMultilinearMap 𝕜 E₁ G) (k : ℕ) (x : (i : ι) → E₁ i) :
    ‖f.iteratedFDeriv k x‖
      ≤ Nat.descFactorial (Fintype.card ι) k * ‖f‖ * ‖x‖ ^ (Fintype.card ι - k) 
Arguments: {𝕜 : Type u} {ι : Type v} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E₁ G) (k : ℕ) (x : (i : ι) → E₁ i)
Goal: ‖f.iteratedFDeriv k x‖ ≤ ↑((Fintype.card ι).descFactorial k) * ‖f‖ * ‖x‖ ^ (Fintype.card ι - k)
-- ContinuousMultilinearMap.opNorm_zero_iff
theorem opNorm_zero_iff : ‖f‖ = 0 ↔ f = 0 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G)
Goal: ‖f‖ = 0 ↔ f = 0
-- ContinuousMultilinearMap.norm_ofSubsingleton_id
theorem norm_ofSubsingleton_id [Subsingleton ι] [Nontrivial G] (i : ι) :
    ‖ofSubsingleton 𝕜 G G i (.id _ _)‖ = 1 
Arguments: (𝕜 : Type u) {ι : Type v} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [Subsingleton ι] [Nontrivial G] (i : ι)
Goal: ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖ = 1
-- ContinuousMultilinearMap.nnnorm_ofSubsingleton_id
theorem nnnorm_ofSubsingleton_id [Subsingleton ι] [Nontrivial G] (i : ι) :
    ‖ofSubsingleton 𝕜 G G i (.id _ _)‖₊ = 1 
Arguments: (𝕜 : Type u) {ι : Type v} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [Subsingleton ι] [Nontrivial G] (i : ι)
Goal: ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖₊ = 1
-- MultilinearMap.bound_of_shell
theorem bound_of_shell {ε : ι → ℝ} {C : ℝ} (hε : ∀ i, 0 < ε i) {c : ι → 𝕜} (hc : ∀ i, 1 < ‖c i‖)
    (hf : ∀ m : ∀ i, E i, (∀ i, ε i / ‖c i‖ ≤ ‖m i‖) → (∀ i, ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖)
    (m : ∀ i, E i) : ‖f m‖ ≤ C * ∏ i, ‖m i‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) {ε : ι → ℝ} {C : ℝ} (hε : ∀ (i : ι), 0 < ε i) {c : ι → 𝕜} (hc : ∀ (i : ι), 1 < ‖c i‖) (hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) (m : (i : ι) → E i)
Goal: ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
-- Polynomial.hasStrictDerivAt
protected theorem hasStrictDerivAt (x : 𝕜) :
    HasStrictDerivAt (fun x => p.eval x) (p.derivative.eval x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (p : 𝕜[X]) (x : 𝕜)
Goal: HasStrictDerivAt (fun x => Polynomial.eval x p) (Polynomial.eval x (Polynomial.derivative p)) x
-- Polynomial.hasStrictDerivAt_aeval
protected theorem hasStrictDerivAt_aeval (x : 𝕜) :
    HasStrictDerivAt (fun x => aeval x q) (aeval x (derivative q)) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X]) (x : 𝕜)
Goal: HasStrictDerivAt (fun x => (Polynomial.aeval x) q) ((Polynomial.aeval x) (Polynomial.derivative q)) x
-- Polynomial.hasDerivAt
protected theorem hasDerivAt (x : 𝕜) : HasDerivAt (fun x => p.eval x) (p.derivative.eval x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (p : 𝕜[X]) (x : 𝕜)
Goal: HasDerivAt (fun x => Polynomial.eval x p) (Polynomial.eval x (Polynomial.derivative p)) x
-- Polynomial.hasDerivAt_aeval
protected theorem hasDerivAt_aeval (x : 𝕜) :
    HasDerivAt (fun x => aeval x q) (aeval x (derivative q)) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X]) (x : 𝕜)
Goal: HasDerivAt (fun x => (Polynomial.aeval x) q) ((Polynomial.aeval x) (Polynomial.derivative q)) x
-- Polynomial.hasDerivWithinAt
protected theorem hasDerivWithinAt (x : 𝕜) (s : Set 𝕜) :
    HasDerivWithinAt (fun x => p.eval x) (p.derivative.eval x) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (p : 𝕜[X]) (x : 𝕜) (s : Set 𝕜)
Goal: HasDerivWithinAt (fun x => Polynomial.eval x p) (Polynomial.eval x (Polynomial.derivative p)) s x
-- Polynomial.hasDerivWithinAt_aeval
protected theorem hasDerivWithinAt_aeval (x : 𝕜) (s : Set 𝕜) :
    HasDerivWithinAt (fun x => aeval x q) (aeval x (derivative q)) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X]) (x : 𝕜) (s : Set 𝕜)
Goal: HasDerivWithinAt (fun x => (Polynomial.aeval x) q) ((Polynomial.aeval x) (Polynomial.derivative q)) s x
-- Polynomial.differentiableAt
protected theorem differentiableAt : DifferentiableAt 𝕜 (fun x => p.eval x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} (p : 𝕜[X])
Goal: DifferentiableAt 𝕜 (fun x => Polynomial.eval x p) x
-- Polynomial.differentiableAt_aeval
protected theorem differentiableAt_aeval : DifferentiableAt 𝕜 (fun x => aeval x q) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X])
Goal: DifferentiableAt 𝕜 (fun x => (Polynomial.aeval x) q) x
-- Polynomial.differentiableWithinAt
protected theorem differentiableWithinAt : DifferentiableWithinAt 𝕜 (fun x => p.eval x) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} (p : 𝕜[X])
Goal: DifferentiableWithinAt 𝕜 (fun x => Polynomial.eval x p) s x
-- Polynomial.differentiableWithinAt_aeval
protected theorem differentiableWithinAt_aeval :
    DifferentiableWithinAt 𝕜 (fun x => aeval x q) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X])
Goal: DifferentiableWithinAt 𝕜 (fun x => (Polynomial.aeval x) q) s x
-- Polynomial.differentiable
protected theorem differentiable : Differentiable 𝕜 fun x => p.eval x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (p : 𝕜[X])
Goal: Differentiable 𝕜 fun x => Polynomial.eval x p
-- Polynomial.differentiable_aeval
protected theorem differentiable_aeval : Differentiable 𝕜 fun x : 𝕜 => aeval x q 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X])
Goal: Differentiable 𝕜 fun x => (Polynomial.aeval x) q
-- Polynomial.differentiableOn
protected theorem differentiableOn : DifferentiableOn 𝕜 (fun x => p.eval x) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {s : Set 𝕜} (p : 𝕜[X])
Goal: DifferentiableOn 𝕜 (fun x => Polynomial.eval x p) s
-- Polynomial.differentiableOn_aeval
protected theorem differentiableOn_aeval : DifferentiableOn 𝕜 (fun x => aeval x q) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {s : Set 𝕜} {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X])
Goal: DifferentiableOn 𝕜 (fun x => (Polynomial.aeval x) q) s
-- Polynomial.deriv
@[simp]
protected theorem deriv : deriv (fun x => p.eval x) x = p.derivative.eval x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} (p : 𝕜[X])
Goal: deriv (fun x => Polynomial.eval x p) x = Polynomial.eval x (Polynomial.derivative p)
-- Polynomial.deriv_aeval
@[simp]
protected theorem deriv_aeval : deriv (fun x => aeval x q) x = aeval x (derivative q) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X])
Goal: deriv (fun x => (Polynomial.aeval x) q) x = (Polynomial.aeval x) (Polynomial.derivative q)
-- Polynomial.derivWithin
protected theorem derivWithin (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => p.eval x) s x = p.derivative.eval x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} (p : 𝕜[X]) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun x => Polynomial.eval x p) s x = Polynomial.eval x (Polynomial.derivative p)
-- Polynomial.derivWithin_aeval
protected theorem derivWithin_aeval (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => aeval x q) s x = aeval x (derivative q) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X]) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun x => (Polynomial.aeval x) q) s x = (Polynomial.aeval x) (Polynomial.derivative q)
-- Polynomial.hasFDerivAt
protected theorem hasFDerivAt (x : 𝕜) :
    HasFDerivAt (fun x => p.eval x) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x)) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (p : 𝕜[X]) (x : 𝕜)
Goal: HasFDerivAt (fun x => Polynomial.eval x p)
  (ContinuousLinearMap.smulRight 1 (Polynomial.eval x (Polynomial.derivative p))) x
-- Polynomial.hasFDerivAt_aeval
protected theorem hasFDerivAt_aeval (x : 𝕜) :
    HasFDerivAt (fun x => aeval x q) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (aeval x (derivative q))) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X]) (x : 𝕜)
Goal: HasFDerivAt (fun x => (Polynomial.aeval x) q)
  (ContinuousLinearMap.smulRight 1 ((Polynomial.aeval x) (Polynomial.derivative q))) x
-- Polynomial.hasFDerivWithinAt
protected theorem hasFDerivWithinAt (x : 𝕜) :
    HasFDerivWithinAt (fun x => p.eval x) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x)) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {s : Set 𝕜} (p : 𝕜[X]) (x : 𝕜)
Goal: HasFDerivWithinAt (fun x => Polynomial.eval x p)
  (ContinuousLinearMap.smulRight 1 (Polynomial.eval x (Polynomial.derivative p))) s x
-- Polynomial.hasFDerivWithinAt_aeval
protected theorem hasFDerivWithinAt_aeval (x : 𝕜) :
    HasFDerivWithinAt (fun x => aeval x q) (smulRight (1 : 𝕜 →L[𝕜] 𝕜)
      (aeval x (derivative q))) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {s : Set 𝕜} {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X]) (x : 𝕜)
Goal: HasFDerivWithinAt (fun x => (Polynomial.aeval x) q)
  (ContinuousLinearMap.smulRight 1 ((Polynomial.aeval x) (Polynomial.derivative q))) s x
-- Polynomial.fderiv
@[simp]
protected theorem fderiv :
    fderiv 𝕜 (fun x => p.eval x) x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} (p : 𝕜[X])
Goal: fderiv 𝕜 (fun x => Polynomial.eval x p) x =
  ContinuousLinearMap.smulRight 1 (Polynomial.eval x (Polynomial.derivative p))
-- Polynomial.fderiv_aeval
@[simp]
protected theorem fderiv_aeval :
    fderiv 𝕜 (fun x => aeval x q) x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (aeval x (derivative q)) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X])
Goal: fderiv 𝕜 (fun x => (Polynomial.aeval x) q) x =
  ContinuousLinearMap.smulRight 1 ((Polynomial.aeval x) (Polynomial.derivative q))
-- Polynomial.fderivWithin
protected theorem fderivWithin (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun x => p.eval x) s x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} (p : 𝕜[X]) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (fun x => Polynomial.eval x p) s x =
  ContinuousLinearMap.smulRight 1 (Polynomial.eval x (Polynomial.derivative p))
-- Polynomial.fderivWithin_aeval
protected theorem fderivWithin_aeval (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun x => aeval x q) s x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (aeval x (derivative q)) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {R : Type u_1} [CommSemiring R] [Algebra R 𝕜] (q : R[X]) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (fun x => (Polynomial.aeval x) q) s x =
  ContinuousLinearMap.smulRight 1 ((Polynomial.aeval x) (Polynomial.derivative q))
-- ContinuousLinearMap.NonlinearRightInverse.bound'
structure NonlinearRightInverse where
  toFun : F → E
  nnnorm : ℝ≥0
  bound' : ∀ y, ‖toFun y‖ ≤ nnnorm * ‖y‖
  right_inv' : ∀ y, f (toFun y) = y
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] {f : E →SL[σ] F} (self : f.NonlinearRightInverse) (y : F)
Goal: ‖self.toFun y‖ ≤ ↑self.nnnorm * ‖y‖
-- ContinuousLinearMap.NonlinearRightInverse.right_inv'
structure NonlinearRightInverse where
  toFun : F → E
  nnnorm : ℝ≥0
  bound' : ∀ y, ‖toFun y‖ ≤ nnnorm * ‖y‖
  right_inv' : ∀ y, f (toFun y) = y
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] {f : E →SL[σ] F} (self : f.NonlinearRightInverse) (y : F)
Goal: f (self.toFun y) = y
-- ContinuousLinearMap.NonlinearRightInverse.right_inv
@[simp]
theorem NonlinearRightInverse.right_inv {f : E →SL[σ] F} (fsymm : NonlinearRightInverse f) (y : F) :
    f (fsymm y) = y 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] {f : E →SL[σ] F} (fsymm : f.NonlinearRightInverse) (y : F)
Goal: f (fsymm.toFun y) = y
-- ContinuousLinearMap.NonlinearRightInverse.bound
theorem NonlinearRightInverse.bound {f : E →SL[σ] F} (fsymm : NonlinearRightInverse f) (y : F) :
    ‖fsymm y‖ ≤ fsymm.nnnorm * ‖y‖ 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] {f : E →SL[σ] F} (fsymm : f.NonlinearRightInverse) (y : F)
Goal: ‖fsymm.toFun y‖ ≤ ↑fsymm.nnnorm * ‖y‖
-- ContinuousLinearMap.exists_approx_preimage_norm_le
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] (f : E →SL[σ] F) [CompleteSpace F] (surj : Function.Surjective ⇑f)
Goal: ∃ C ≥ 0, ∀ (y : F), ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖
-- ContinuousLinearMap.exists_preimage_norm_le
theorem exists_preimage_norm_le (surj : Surjective f) :
    ∃ C > 0, ∀ y, ∃ x, f x = y ∧ ‖x‖ ≤ C * ‖y‖ 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] (f : E →SL[σ] F) [CompleteSpace F] [CompleteSpace E] (surj : Function.Surjective ⇑f)
Goal: ∃ C > 0, ∀ (y : F), ∃ x, f x = y ∧ ‖x‖ ≤ C * ‖y‖
-- ContinuousLinearMap.isOpenMap
protected theorem isOpenMap (surj : Surjective f) : IsOpenMap f 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] (f : E →SL[σ] F) [CompleteSpace F] [CompleteSpace E] (surj : Function.Surjective ⇑f)
Goal: IsOpenMap ⇑f
-- ContinuousLinearMap.quotientMap
protected theorem quotientMap (surj : Surjective f) : QuotientMap f 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] [RingHomIsometric σ] [RingHomIsometric σ'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F] (f : E →SL[σ] F) [CompleteSpace F] [CompleteSpace E] (surj : Function.Surjective ⇑f)
Goal: QuotientMap ⇑f
-- AffineMap.isOpenMap
theorem _root_.AffineMap.isOpenMap {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
    [CompleteSpace F] {P Q : Type*} [MetricSpace P] [NormedAddTorsor E P] [MetricSpace Q]
    [NormedAddTorsor F Q] (f : P →ᵃ[𝕜] Q) (hf : Continuous f) (surj : Surjective f) :
    IsOpenMap f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {P : Type u_6} {Q : Type u_7} [MetricSpace P] [NormedAddTorsor E P] [MetricSpace Q] [NormedAddTorsor F Q] (f : P →ᵃ[𝕜] Q) (hf : Continuous ⇑f) (surj : Function.Surjective ⇑f)
Goal: IsOpenMap ⇑f