ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- summable_of_isEquivalent
theorem summable_of_isEquivalent {ι E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [FiniteDimensional ℝ E] {f : ι → E} {g : ι → E} (hg : Summable g) (h : f ~[cofinite] g) :
    Summable f 
Arguments: {ι : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {f g : ι → E} (hg : Summable g) (h : f ~[Filter.cofinite] g)
Goal: Summable f
-- summable_of_isEquivalent_nat
theorem summable_of_isEquivalent_nat {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [FiniteDimensional ℝ E] {f : ℕ → E} {g : ℕ → E} (hg : Summable g) (h : f ~[atTop] g) :
    Summable f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {f g : ℕ → E} (hg : Summable g) (h : f ~[Filter.atTop] g)
Goal: Summable f
-- IsEquivalent.summable_iff
theorem IsEquivalent.summable_iff {ι E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [FiniteDimensional ℝ E] {f : ι → E} {g : ι → E} (h : f ~[cofinite] g) :
    Summable f ↔ Summable g 
Arguments: {ι : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {f g : ι → E} (h : f ~[Filter.cofinite] g)
Goal: Summable f ↔ Summable g
-- IsEquivalent.summable_iff_nat
theorem IsEquivalent.summable_iff_nat {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [FiniteDimensional ℝ E] {f : ℕ → E} {g : ℕ → E} (h : f ~[atTop] g) : Summable f ↔ Summable g 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {f g : ℕ → E} (h : f ~[Filter.atTop] g)
Goal: Summable f ↔ Summable g
-- exists_hasDerivWithinAt_eq_of_gt_of_lt
theorem exists_hasDerivWithinAt_eq_of_gt_of_lt (hab : a ≤ b)
    (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) {m : ℝ} (hma : f' a < m)
    (hmb : m < f' b) : m ∈ f' '' Ioo a b 
Arguments: {a b : ℝ} {f f' : ℝ → ℝ} (hab : a ≤ b) (hf : ∀ x ∈ Set.Icc a b, HasDerivWithinAt f (f' x) (Set.Icc a b) x) {m : ℝ} (hma : f' a < m) (hmb : m < f' b)
Goal: m ∈ f' '' Set.Ioo a b
-- exists_hasDerivWithinAt_eq_of_lt_of_gt
theorem exists_hasDerivWithinAt_eq_of_lt_of_gt (hab : a ≤ b)
    (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) {m : ℝ} (hma : m < f' a)
    (hmb : f' b < m) : m ∈ f' '' Ioo a b 
Arguments: {a b : ℝ} {f f' : ℝ → ℝ} (hab : a ≤ b) (hf : ∀ x ∈ Set.Icc a b, HasDerivWithinAt f (f' x) (Set.Icc a b) x) {m : ℝ} (hma : m < f' a) (hmb : f' b < m)
Goal: m ∈ f' '' Set.Ioo a b
-- Set.OrdConnected.image_hasDerivWithinAt
theorem Set.OrdConnected.image_hasDerivWithinAt {s : Set ℝ} (hs : OrdConnected s)
    (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) : OrdConnected (f' '' s) 
Arguments: {f f' : ℝ → ℝ} {s : Set ℝ} (hs : s.OrdConnected) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x)
Goal: (f' '' s).OrdConnected
-- Set.OrdConnected.image_derivWithin
theorem Set.OrdConnected.image_derivWithin {s : Set ℝ} (hs : OrdConnected s)
    (hf : DifferentiableOn ℝ f s) : OrdConnected (derivWithin f s '' s) 
Arguments: {f : ℝ → ℝ} {s : Set ℝ} (hs : s.OrdConnected) (hf : DifferentiableOn ℝ f s)
Goal: (derivWithin f s '' s).OrdConnected
-- Set.OrdConnected.image_deriv
theorem Set.OrdConnected.image_deriv {s : Set ℝ} (hs : OrdConnected s)
    (hf : ∀ x ∈ s, DifferentiableAt ℝ f x) : OrdConnected (deriv f '' s) 
Arguments: {f : ℝ → ℝ} {s : Set ℝ} (hs : s.OrdConnected) (hf : ∀ x ∈ s, DifferentiableAt ℝ f x)
Goal: (deriv f '' s).OrdConnected
-- Convex.image_hasDerivWithinAt
theorem Convex.image_hasDerivWithinAt {s : Set ℝ} (hs : Convex ℝ s)
    (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) : Convex ℝ (f' '' s) 
Arguments: {f f' : ℝ → ℝ} {s : Set ℝ} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x)
Goal: Convex ℝ (f' '' s)
-- Convex.image_derivWithin
theorem Convex.image_derivWithin {s : Set ℝ} (hs : Convex ℝ s) (hf : DifferentiableOn ℝ f s) :
    Convex ℝ (derivWithin f s '' s) 
Arguments: {f : ℝ → ℝ} {s : Set ℝ} (hs : Convex ℝ s) (hf : DifferentiableOn ℝ f s)
Goal: Convex ℝ (derivWithin f s '' s)
-- Convex.image_deriv
theorem Convex.image_deriv {s : Set ℝ} (hs : Convex ℝ s) (hf : ∀ x ∈ s, DifferentiableAt ℝ f x) :
    Convex ℝ (deriv f '' s) 
Arguments: {f : ℝ → ℝ} {s : Set ℝ} (hs : Convex ℝ s) (hf : ∀ x ∈ s, DifferentiableAt ℝ f x)
Goal: Convex ℝ (deriv f '' s)
-- exists_hasDerivWithinAt_eq_of_ge_of_le
theorem exists_hasDerivWithinAt_eq_of_ge_of_le (hab : a ≤ b)
    (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) {m : ℝ} (hma : f' a ≤ m)
    (hmb : m ≤ f' b) : m ∈ f' '' Icc a b 
Arguments: {a b : ℝ} {f f' : ℝ → ℝ} (hab : a ≤ b) (hf : ∀ x ∈ Set.Icc a b, HasDerivWithinAt f (f' x) (Set.Icc a b) x) {m : ℝ} (hma : f' a ≤ m) (hmb : m ≤ f' b)
Goal: m ∈ f' '' Set.Icc a b
-- exists_hasDerivWithinAt_eq_of_le_of_ge
theorem exists_hasDerivWithinAt_eq_of_le_of_ge (hab : a ≤ b)
    (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) {m : ℝ} (hma : f' a ≤ m)
    (hmb : m ≤ f' b) : m ∈ f' '' Icc a b 
Arguments: {a b : ℝ} {f f' : ℝ → ℝ} (hab : a ≤ b) (hf : ∀ x ∈ Set.Icc a b, HasDerivWithinAt f (f' x) (Set.Icc a b) x) {m : ℝ} (hma : f' a ≤ m) (hmb : m ≤ f' b)
Goal: m ∈ f' '' Set.Icc a b
-- hasDerivWithinAt_forall_lt_or_forall_gt_of_forall_ne
theorem hasDerivWithinAt_forall_lt_or_forall_gt_of_forall_ne {s : Set ℝ} (hs : Convex ℝ s)
    (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) {m : ℝ} (hf' : ∀ x ∈ s, f' x ≠ m) :
    (∀ x ∈ s, f' x < m) ∨ ∀ x ∈ s, m < f' x 
Arguments: {f f' : ℝ → ℝ} {s : Set ℝ} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) {m : ℝ} (hf' : ∀ x ∈ s, f' x ≠ m)
Goal: (∀ x ∈ s, f' x < m) ∨ ∀ x ∈ s, m < f' x
-- support_deriv_subset
theorem support_deriv_subset : support (deriv f) ⊆ tsupport f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E}
Goal: Function.support (deriv f) ⊆ tsupport f
-- HasCompactSupport.deriv
protected theorem HasCompactSupport.deriv (hf : HasCompactSupport f) :
    HasCompactSupport (deriv f) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} (hf : HasCompactSupport f)
Goal: HasCompactSupport (deriv f)
-- Quaternion.inner_self
theorem inner_self (a : ℍ) : ⟪a, a⟫ = normSq a 
Arguments: (a : Quaternion ℝ)
Goal: ⟪a, a⟫_ℝ = Quaternion.normSq a
-- Quaternion.inner_def
theorem inner_def (a b : ℍ) : ⟪a, b⟫ = (a * star b).re 
Arguments: (a b : Quaternion ℝ)
Goal: ⟪a, b⟫_ℝ = (a * star b).re
-- Quaternion.normSq_eq_norm_mul_self
theorem normSq_eq_norm_mul_self (a : ℍ) : normSq a = ‖a‖ * ‖a‖ 
Arguments: (a : Quaternion ℝ)
Goal: Quaternion.normSq a = ‖a‖ * ‖a‖
-- Quaternion.norm_coe
@[simp, norm_cast]
theorem norm_coe (a : ℝ) : ‖(a : ℍ)‖ = ‖a‖ 
Arguments: (a : ℝ)
Goal: ‖↑a‖ = ‖a‖
-- Quaternion.nnnorm_coe
@[simp, norm_cast]
theorem nnnorm_coe (a : ℝ) : ‖(a : ℍ)‖₊ = ‖a‖₊ 
Arguments: (a : ℝ)
Goal: ‖↑a‖₊ = ‖a‖₊
-- Quaternion.norm_star
@[simp, nolint simpNF] -- Porting note (#10959): simp cannot prove this
theorem norm_star (a : ℍ) : ‖star a‖ = ‖a‖ 
Arguments: (a : Quaternion ℝ)
Goal: ‖star a‖ = ‖a‖
-- Quaternion.nnnorm_star
@[simp, nolint simpNF] -- Porting note (#10959): simp cannot prove this
theorem nnnorm_star (a : ℍ) : ‖star a‖₊ = ‖a‖₊ 
Arguments: (a : Quaternion ℝ)
Goal: ‖star a‖₊ = ‖a‖₊
-- Quaternion.coeComplex_re
@[simp, norm_cast]
theorem coeComplex_re (z : ℂ) : (z : ℍ).re = z.re 
Arguments: (z : ℂ)
Goal: (↑z).re = z.re
-- Quaternion.coeComplex_imI
@[simp, norm_cast]
theorem coeComplex_imI (z : ℂ) : (z : ℍ).imI = z.im 
Arguments: (z : ℂ)
Goal: (↑z).imI = z.im
-- Quaternion.coeComplex_imJ
@[simp, norm_cast]
theorem coeComplex_imJ (z : ℂ) : (z : ℍ).imJ = 0 
Arguments: (z : ℂ)
Goal: (↑z).imJ = 0
-- Quaternion.coeComplex_imK
@[simp, norm_cast]
theorem coeComplex_imK (z : ℂ) : (z : ℍ).imK = 0 
Arguments: (z : ℂ)
Goal: (↑z).imK = 0
-- Quaternion.coeComplex_add
@[simp, norm_cast]
theorem coeComplex_add (z w : ℂ) : ↑(z + w) = (z + w : ℍ) 
Arguments: (z w : ℂ)
Goal: ↑(z + w) = ↑z + ↑w
-- Quaternion.coeComplex_mul
@[simp, norm_cast]
theorem coeComplex_mul (z w : ℂ) : ↑(z * w) = (z * w : ℍ) 
Arguments: (z w : ℂ)
Goal: ↑(z * w) = ↑z * ↑w
-- Quaternion.coeComplex_zero
@[simp, norm_cast]
theorem coeComplex_zero : ((0 : ℂ) : ℍ) = 0 
Goal: ↑0 = 0
-- Quaternion.coeComplex_one
@[simp, norm_cast]
theorem coeComplex_one : ((1 : ℂ) : ℍ) = 1 
Goal: ↑1 = 1
-- Quaternion.coe_real_complex_mul
@[simp, norm_cast]
theorem coe_real_complex_mul (r : ℝ) (z : ℂ) : (r • z : ℍ) = ↑r * ↑z 
Arguments: (r : ℝ) (z : ℂ)
Goal: r • ↑z = ↑r * ↑z
-- Quaternion.coeComplex_coe
@[simp, norm_cast]
theorem coeComplex_coe (r : ℝ) : ((r : ℂ) : ℍ) = r 
Arguments: (r : ℝ)
Goal: ↑↑r = ↑r
-- Quaternion.coe_ofComplex
@[simp]
theorem coe_ofComplex : ⇑ofComplex = coeComplex 
Goal: ⇑Quaternion.ofComplex = Quaternion.coeComplex
-- Quaternion.norm_piLp_equiv_symm_equivTuple
theorem norm_piLp_equiv_symm_equivTuple (x : ℍ) :
    ‖(WithLp.equiv 2 (Fin 4 → _)).symm (equivTuple ℝ x)‖ = ‖x‖ 
Arguments: (x : Quaternion ℝ)
Goal: ‖(WithLp.equiv 2 (Fin 4 → ℝ)).symm ((Quaternion.equivTuple ℝ) x)‖ = ‖x‖
-- Quaternion.linearIsometryEquivTuple_apply
@[simps apply symm_apply]
noncomputable def linearIsometryEquivTuple : ℍ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin 4) 
Arguments: (a : Quaternion ℝ)
Goal: Quaternion.linearIsometryEquivTuple a = (WithLp.equiv 2 (Fin 4 → ℝ)).symm ![a.re, a.imI, a.imJ, a.imK]
-- Quaternion.linearIsometryEquivTuple_symm_apply
@[simps apply symm_apply]
noncomputable def linearIsometryEquivTuple : ℍ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin 4) 
Arguments: (a : EuclideanSpace ℝ (Fin 4))
Goal: Quaternion.linearIsometryEquivTuple.symm a = { re := a 0, imI := a 1, imJ := a 2, imK := a 3 }
-- Quaternion.continuous_coe
@[continuity]
theorem continuous_coe : Continuous (coe : ℝ → ℍ) 
Goal: Continuous Quaternion.coe
-- Quaternion.continuous_normSq
@[continuity]
theorem continuous_normSq : Continuous (normSq : ℍ → ℝ) 
Goal: Continuous ⇑Quaternion.normSq
-- Quaternion.continuous_re
@[continuity]
theorem continuous_re : Continuous fun q : ℍ => q.re 
Goal: Continuous fun q => q.re
-- Quaternion.continuous_imI
@[continuity]
theorem continuous_imI : Continuous fun q : ℍ => q.imI 
Goal: Continuous fun q => q.imI
-- Quaternion.continuous_imJ
@[continuity]
theorem continuous_imJ : Continuous fun q : ℍ => q.imJ 
Goal: Continuous fun q => q.imJ
-- Quaternion.continuous_imK
@[continuity]
theorem continuous_imK : Continuous fun q : ℍ => q.imK 
Goal: Continuous fun q => q.imK
-- Quaternion.continuous_im
@[continuity]
theorem continuous_im : Continuous fun q : ℍ => q.im 
Goal: Continuous fun q => q.im
-- Quaternion.hasSum_coe
@[simp, norm_cast]
theorem hasSum_coe {f : α → ℝ} {r : ℝ} : HasSum (fun a => (f a : ℍ)) (↑r : ℍ) ↔ HasSum f r 
Arguments: {α : Type u_1} {f : α → ℝ} {r : ℝ}
Goal: HasSum (fun a => ↑(f a)) ↑r ↔ HasSum f r
-- Quaternion.summable_coe
@[simp, norm_cast]
theorem summable_coe {f : α → ℝ} : (Summable fun a => (f a : ℍ)) ↔ Summable f 
Arguments: {α : Type u_1} {f : α → ℝ}
Goal: (Summable fun a => ↑(f a)) ↔ Summable f
-- Quaternion.tsum_coe
@[norm_cast]
theorem tsum_coe (f : α → ℝ) : (∑' a, (f a : ℍ)) = ↑(∑' a, f a) 
Arguments: {α : Type u_1} (f : α → ℝ)
Goal: ∑' (a : α), ↑(f a) = ↑(∑' (a : α), f a)
-- HasStrictDerivAt.hasStrictFDerivAt_equiv
theorem HasStrictDerivAt.hasStrictFDerivAt_equiv {f : 𝕜 → 𝕜} {f' x : 𝕜}
    (hf : HasStrictDerivAt f f' x) (hf' : f' ≠ 0) :
    HasStrictFDerivAt f (ContinuousLinearEquiv.unitsEquivAut 𝕜 (Units.mk0 f' hf') : 𝕜 →L[𝕜] 𝕜) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {f : 𝕜 → 𝕜} {f' x : 𝕜} (hf : HasStrictDerivAt f f' x) (hf' : f' ≠ 0)
Goal: HasStrictFDerivAt f (↑((ContinuousLinearEquiv.unitsEquivAut 𝕜) (Units.mk0 f' hf'))) x
-- HasDerivAt.hasFDerivAt_equiv
theorem HasDerivAt.hasFDerivAt_equiv {f : 𝕜 → 𝕜} {f' x : 𝕜} (hf : HasDerivAt f f' x)
    (hf' : f' ≠ 0) :
    HasFDerivAt f (ContinuousLinearEquiv.unitsEquivAut 𝕜 (Units.mk0 f' hf') : 𝕜 →L[𝕜] 𝕜) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {f : 𝕜 → 𝕜} {f' x : 𝕜} (hf : HasDerivAt f f' x) (hf' : f' ≠ 0)
Goal: HasFDerivAt f (↑((ContinuousLinearEquiv.unitsEquivAut 𝕜) (Units.mk0 f' hf'))) x
-- HasStrictDerivAt.of_local_left_inverse
theorem HasStrictDerivAt.of_local_left_inverse {f g : 𝕜 → 𝕜} {f' a : 𝕜} (hg : ContinuousAt g a)
    (hf : HasStrictDerivAt f f' (g a)) (hf' : f' ≠ 0) (hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) :
    HasStrictDerivAt g f'⁻¹ a 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {f g : 𝕜 → 𝕜} {f' a : 𝕜} (hg : ContinuousAt g a) (hf : HasStrictDerivAt f f' (g a)) (hf' : f' ≠ 0) (hfg : ∀ᶠ (y : 𝕜) in 𝓝 a, f (g y) = y)
Goal: HasStrictDerivAt g f'⁻¹ a
-- PartialHomeomorph.hasStrictDerivAt_symm
theorem PartialHomeomorph.hasStrictDerivAt_symm (f : PartialHomeomorph 𝕜 𝕜) {a f' : 𝕜}
    (ha : a ∈ f.target) (hf' : f' ≠ 0) (htff' : HasStrictDerivAt f f' (f.symm a)) :
    HasStrictDerivAt f.symm f'⁻¹ a 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (f : PartialHomeomorph 𝕜 𝕜) {a f' : 𝕜} (ha : a ∈ f.target) (hf' : f' ≠ 0) (htff' : HasStrictDerivAt (↑f) f' (↑f.symm a))
Goal: HasStrictDerivAt (↑f.symm) f'⁻¹ a
-- HasDerivAt.of_local_left_inverse
theorem HasDerivAt.of_local_left_inverse {f g : 𝕜 → 𝕜} {f' a : 𝕜} (hg : ContinuousAt g a)
    (hf : HasDerivAt f f' (g a)) (hf' : f' ≠ 0) (hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) :
    HasDerivAt g f'⁻¹ a 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {f g : 𝕜 → 𝕜} {f' a : 𝕜} (hg : ContinuousAt g a) (hf : HasDerivAt f f' (g a)) (hf' : f' ≠ 0) (hfg : ∀ᶠ (y : 𝕜) in 𝓝 a, f (g y) = y)
Goal: HasDerivAt g f'⁻¹ a
-- PartialHomeomorph.hasDerivAt_symm
theorem PartialHomeomorph.hasDerivAt_symm (f : PartialHomeomorph 𝕜 𝕜) {a f' : 𝕜} (ha : a ∈ f.target)
    (hf' : f' ≠ 0) (htff' : HasDerivAt f f' (f.symm a)) : HasDerivAt f.symm f'⁻¹ a 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (f : PartialHomeomorph 𝕜 𝕜) {a f' : 𝕜} (ha : a ∈ f.target) (hf' : f' ≠ 0) (htff' : HasDerivAt (↑f) f' (↑f.symm a))
Goal: HasDerivAt (↑f.symm) f'⁻¹ a
-- HasDerivAt.eventually_ne
theorem HasDerivAt.eventually_ne (h : HasDerivAt f f' x) (hf' : f' ≠ 0) :
    ∀ᶠ z in 𝓝[≠] x, f z ≠ f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (h : HasDerivAt f f' x) (hf' : f' ≠ 0)
Goal: ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z ≠ f x
-- HasDerivAt.tendsto_punctured_nhds
theorem HasDerivAt.tendsto_punctured_nhds (h : HasDerivAt f f' x) (hf' : f' ≠ 0) :
    Tendsto f (𝓝[≠] x) (𝓝[≠] f x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (h : HasDerivAt f f' x) (hf' : f' ≠ 0)
Goal: Filter.Tendsto f (𝓝[≠] x) (𝓝[≠] f x)
-- not_differentiableWithinAt_of_local_left_inverse_hasDerivWithinAt_zero
theorem not_differentiableWithinAt_of_local_left_inverse_hasDerivWithinAt_zero {f g : 𝕜 → 𝕜} {a : 𝕜}
    {s t : Set 𝕜} (ha : a ∈ s) (hsu : UniqueDiffWithinAt 𝕜 s a) (hf : HasDerivWithinAt f 0 t (g a))
    (hst : MapsTo g s t) (hfg : f ∘ g =ᶠ[𝓝[s] a] id) : ¬DifferentiableWithinAt 𝕜 g s a 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {f g : 𝕜 → 𝕜} {a : 𝕜} {s t : Set 𝕜} (ha : a ∈ s) (hsu : UniqueDiffWithinAt 𝕜 s a) (hf : HasDerivWithinAt f 0 t (g a)) (hst : Set.MapsTo g s t) (hfg : f ∘ g =ᶠ[𝓝[s] a] id)
Goal: ¬DifferentiableWithinAt 𝕜 g s a
-- not_differentiableAt_of_local_left_inverse_hasDerivAt_zero
theorem not_differentiableAt_of_local_left_inverse_hasDerivAt_zero {f g : 𝕜 → 𝕜} {a : 𝕜}
    (hf : HasDerivAt f 0 (g a)) (hfg : f ∘ g =ᶠ[𝓝 a] id) : ¬DifferentiableAt 𝕜 g a 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {f g : 𝕜 → 𝕜} {a : 𝕜} (hf : HasDerivAt f 0 (g a)) (hfg : f ∘ g =ᶠ[𝓝 a] id)
Goal: ¬DifferentiableAt 𝕜 g a
-- HasFTaylorSeriesUpToOn.hasStrictFDerivAt
theorem HasFTaylorSeriesUpToOn.hasStrictFDerivAt {s : Set E'} {f : E' → F'} {x : E'}
    {p : E' → FormalMultilinearSeries 𝕂 E' F'} (hf : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
    (hs : s ∈ 𝓝 x) : HasStrictFDerivAt f ((continuousMultilinearCurryFin1 𝕂 E' F') (p x 1)) x 
Arguments: {n : ℕ∞} {𝕂 : Type u_1} [RCLike 𝕂] {E' : Type u_2} [NormedAddCommGroup E'] [NormedSpace 𝕂 E'] {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace 𝕂 F'] {s : Set E'} {f : E' → F'} {x : E'} {p : E' → FormalMultilinearSeries 𝕂 E' F'} (hf : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n) (hs : s ∈ 𝓝 x)
Goal: HasStrictFDerivAt f ((continuousMultilinearCurryFin1 𝕂 E' F') (p x 1)) x
-- ContDiffAt.hasStrictFDerivAt'
theorem ContDiffAt.hasStrictFDerivAt' {f : E' → F'} {f' : E' →L[𝕂] F'} {x : E'}
    (hf : ContDiffAt 𝕂 n f x) (hf' : HasFDerivAt f f' x) (hn : 1 ≤ n) :
    HasStrictFDerivAt f f' x 
Arguments: {n : ℕ∞} {𝕂 : Type u_1} [RCLike 𝕂] {E' : Type u_2} [NormedAddCommGroup E'] [NormedSpace 𝕂 E'] {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace 𝕂 F'] {f : E' → F'} {f' : E' →L[𝕂] F'} {x : E'} (hf : ContDiffAt 𝕂 n f x) (hf' : HasFDerivAt f f' x) (hn : 1 ≤ n)
Goal: HasStrictFDerivAt f f' x
-- ContDiffAt.hasStrictDerivAt'
theorem ContDiffAt.hasStrictDerivAt' {f : 𝕂 → F'} {f' : F'} {x : 𝕂} (hf : ContDiffAt 𝕂 n f x)
    (hf' : HasDerivAt f f' x) (hn : 1 ≤ n) : HasStrictDerivAt f f' x 
Arguments: {n : ℕ∞} {𝕂 : Type u_1} [RCLike 𝕂] {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace 𝕂 F'] {f : 𝕂 → F'} {f' : F'} {x : 𝕂} (hf : ContDiffAt 𝕂 n f x) (hf' : HasDerivAt f f' x) (hn : 1 ≤ n)
Goal: HasStrictDerivAt f f' x
-- ContDiffAt.hasStrictFDerivAt
theorem ContDiffAt.hasStrictFDerivAt {f : E' → F'} {x : E'} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) :
    HasStrictFDerivAt f (fderiv 𝕂 f x) x 
Arguments: {n : ℕ∞} {𝕂 : Type u_1} [RCLike 𝕂] {E' : Type u_2} [NormedAddCommGroup E'] [NormedSpace 𝕂 E'] {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace 𝕂 F'] {f : E' → F'} {x : E'} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n)
Goal: HasStrictFDerivAt f (fderiv 𝕂 f x) x
-- ContDiffAt.hasStrictDerivAt
theorem ContDiffAt.hasStrictDerivAt {f : 𝕂 → F'} {x : 𝕂} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) :
    HasStrictDerivAt f (deriv f x) x 
Arguments: {n : ℕ∞} {𝕂 : Type u_1} [RCLike 𝕂] {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace 𝕂 F'] {f : 𝕂 → F'} {x : 𝕂} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n)
Goal: HasStrictDerivAt f (deriv f x) x
-- ContDiff.hasStrictFDerivAt
theorem ContDiff.hasStrictFDerivAt {f : E' → F'} {x : E'} (hf : ContDiff 𝕂 n f) (hn : 1 ≤ n) :
    HasStrictFDerivAt f (fderiv 𝕂 f x) x 
Arguments: {n : ℕ∞} {𝕂 : Type u_1} [RCLike 𝕂] {E' : Type u_2} [NormedAddCommGroup E'] [NormedSpace 𝕂 E'] {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace 𝕂 F'] {f : E' → F'} {x : E'} (hf : ContDiff 𝕂 n f) (hn : 1 ≤ n)
Goal: HasStrictFDerivAt f (fderiv 𝕂 f x) x
-- ContDiff.hasStrictDerivAt
theorem ContDiff.hasStrictDerivAt {f : 𝕂 → F'} {x : 𝕂} (hf : ContDiff 𝕂 n f) (hn : 1 ≤ n) :
    HasStrictDerivAt f (deriv f x) x 
Arguments: {n : ℕ∞} {𝕂 : Type u_1} [RCLike 𝕂] {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace 𝕂 F'] {f : 𝕂 → F'} {x : 𝕂} (hf : ContDiff 𝕂 n f) (hn : 1 ≤ n)
Goal: HasStrictDerivAt f (deriv f x) x
-- HasFTaylorSeriesUpToOn.exists_lipschitzOnWith_of_nnnorm_lt
theorem HasFTaylorSeriesUpToOn.exists_lipschitzOnWith_of_nnnorm_lt {E F : Type*}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F}
    {p : E → FormalMultilinearSeries ℝ E F} {s : Set E} {x : E}
    (hf : HasFTaylorSeriesUpToOn 1 f p (insert x s)) (hs : Convex ℝ s) (K : ℝ≥0)
    (hK : ‖p x 1‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {p : E → FormalMultilinearSeries ℝ E F} {s : Set E} {x : E} (hf : HasFTaylorSeriesUpToOn 1 f p (insert x s)) (hs : Convex ℝ s) (K : ℝ≥0) (hK : ‖p x 1‖₊ < K)
Goal: ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t
-- HasFTaylorSeriesUpToOn.exists_lipschitzOnWith
theorem HasFTaylorSeriesUpToOn.exists_lipschitzOnWith {E F : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F}
    {p : E → FormalMultilinearSeries ℝ E F} {s : Set E} {x : E}
    (hf : HasFTaylorSeriesUpToOn 1 f p (insert x s)) (hs : Convex ℝ s) :
    ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {p : E → FormalMultilinearSeries ℝ E F} {s : Set E} {x : E} (hf : HasFTaylorSeriesUpToOn 1 f p (insert x s)) (hs : Convex ℝ s)
Goal: ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t
-- ContDiffWithinAt.exists_lipschitzOnWith
theorem ContDiffWithinAt.exists_lipschitzOnWith {E F : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {s : Set E} {x : E}
    (hf : ContDiffWithinAt ℝ 1 f s x) (hs : Convex ℝ s) :
    ∃ K : ℝ≥0, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {s : Set E} {x : E} (hf : ContDiffWithinAt ℝ 1 f s x) (hs : Convex ℝ s)
Goal: ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t
-- ContDiffAt.exists_lipschitzOnWith_of_nnnorm_lt
theorem ContDiffAt.exists_lipschitzOnWith_of_nnnorm_lt {f : E' → F'} {x : E'}
    (hf : ContDiffAt 𝕂 1 f x) (K : ℝ≥0) (hK : ‖fderiv 𝕂 f x‖₊ < K) :
    ∃ t ∈ 𝓝 x, LipschitzOnWith K f t 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂] {E' : Type u_2} [NormedAddCommGroup E'] [NormedSpace 𝕂 E'] {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace 𝕂 F'] {f : E' → F'} {x : E'} (hf : ContDiffAt 𝕂 1 f x) (K : ℝ≥0) (hK : ‖fderiv 𝕂 f x‖₊ < K)
Goal: ∃ t ∈ 𝓝 x, LipschitzOnWith K f t
-- ContDiffAt.exists_lipschitzOnWith
theorem ContDiffAt.exists_lipschitzOnWith {f : E' → F'} {x : E'} (hf : ContDiffAt 𝕂 1 f x) :
    ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂] {E' : Type u_2} [NormedAddCommGroup E'] [NormedSpace 𝕂 E'] {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace 𝕂 F'] {f : E' → F'} {x : E'} (hf : ContDiffAt 𝕂 1 f x)
Goal: ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t
-- ContDiff.locallyLipschitz
lemma ContDiff.locallyLipschitz {f : E' → F'} (hf : ContDiff 𝕂 1 f) : LocallyLipschitz f 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂] {E' : Type u_2} [NormedAddCommGroup E'] [NormedSpace 𝕂 E'] {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace 𝕂 F'] {f : E' → F'} (hf : ContDiff 𝕂 1 f)
Goal: LocallyLipschitz f
-- ContDiff.lipschitzWith_of_hasCompactSupport
theorem ContDiff.lipschitzWith_of_hasCompactSupport {f : E' → F'} {n : ℕ∞}
    (hf : HasCompactSupport f) (h'f : ContDiff 𝕂 n f) (hn : 1 ≤ n) :
    ∃ C, LipschitzWith C f 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂] {E' : Type u_2} [NormedAddCommGroup E'] [NormedSpace 𝕂 E'] {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace 𝕂 F'] {f : E' → F'} {n : ℕ∞} (hf : HasCompactSupport f) (h'f : ContDiff 𝕂 n f) (hn : 1 ≤ n)
Goal: ∃ C, LipschitzWith C f
-- map_one_ne_zero
theorem map_one_ne_zero (f_ne_zero : f ≠ 0) (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) : f 1 ≠ 0 
Arguments: {R : Type u_1} [CommRing R] {f : R → ℝ} {c : ℝ} (f_ne_zero : f ≠ 0) (f_nonneg : 0 ≤ f) (f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y)
Goal: f 1 ≠ 0
-- map_pow_ne_zero
theorem map_pow_ne_zero (f_nonneg : 0 ≤ f) {x : R} (hx : IsUnit x) (hfx : f x ≠ 0) (n : ℕ)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) : f (x ^ n) ≠ 0 
Arguments: {R : Type u_1} [CommRing R] {f : R → ℝ} {c : ℝ} (f_nonneg : 0 ≤ f) {x : R} (hx : IsUnit x) (hfx : f x ≠ 0) (n : ℕ) (f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y)
Goal: f (x ^ n) ≠ 0
-- map_mul_zero_of_map_zero
theorem map_mul_zero_of_map_zero (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) {x : R} (hx : f x = 0)
    (y : R) : f (x * y) = 0 
Arguments: {R : Type u_1} [CommRing R] {f : R → ℝ} {c : ℝ} (f_nonneg : 0 ≤ f) (f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y) {x : R} (hx : f x = 0) (y : R)
Goal: f (x * y) = 0
-- seminormFromBounded_zero
theorem seminormFromBounded_zero (f_zero : f 0 = 0) : seminormFromBounded' f (0 : R) = 0 
Arguments: {R : Type u_1} [CommRing R] {f : R → ℝ} (f_zero : f 0 = 0)
Goal: seminormFromBounded' f 0 = 0
-- seminormFromBounded_aux
theorem seminormFromBounded_aux (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) (x : R) : 0 ≤ c * f x 
Arguments: {R : Type u_1} [CommRing R] {f : R → ℝ} {c : ℝ} (f_nonneg : 0 ≤ f) (f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y) (x : R)
Goal: 0 ≤ c * f x