ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Complex.cderiv_sub
theorem cderiv_sub (hr : 0 < r) (hf : ContinuousOn f (sphere z r))
    (hg : ContinuousOn g (sphere z r)) : cderiv r (f - g) z = cderiv r f z - cderiv r g z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {z : ℂ} {r : ℝ} {f g : ℂ → E} (hr : 0 < r) (hf : ContinuousOn f (Metric.sphere z r)) (hg : ContinuousOn g (Metric.sphere z r))
Goal: Complex.cderiv r (f - g) z = Complex.cderiv r f z - Complex.cderiv r g z
-- Complex.norm_cderiv_lt
theorem norm_cderiv_lt (hr : 0 < r) (hfM : ∀ w ∈ sphere z r, ‖f w‖ < M)
    (hf : ContinuousOn f (sphere z r)) : ‖cderiv r f z‖ < M / r 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {z : ℂ} {M r : ℝ} {f : ℂ → E} (hr : 0 < r) (hfM : ∀ w ∈ Metric.sphere z r, ‖f w‖ < M) (hf : ContinuousOn f (Metric.sphere z r))
Goal: ‖Complex.cderiv r f z‖ < M / r
-- Complex.norm_cderiv_sub_lt
theorem norm_cderiv_sub_lt (hr : 0 < r) (hfg : ∀ w ∈ sphere z r, ‖f w - g w‖ < M)
    (hf : ContinuousOn f (sphere z r)) (hg : ContinuousOn g (sphere z r)) :
    ‖cderiv r f z - cderiv r g z‖ < M / r 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {z : ℂ} {M r : ℝ} {f g : ℂ → E} (hr : 0 < r) (hfg : ∀ w ∈ Metric.sphere z r, ‖f w - g w‖ < M) (hf : ContinuousOn f (Metric.sphere z r)) (hg : ContinuousOn g (Metric.sphere z r))
Goal: ‖Complex.cderiv r f z - Complex.cderiv r g z‖ < M / r
-- TendstoUniformlyOn.cderiv
theorem _root_.TendstoUniformlyOn.cderiv (hF : TendstoUniformlyOn F f φ (cthickening δ K))
    (hδ : 0 < δ) (hFn : ∀ᶠ n in φ, ContinuousOn (F n) (cthickening δ K)) :
    TendstoUniformlyOn (cderiv δ ∘ F) (cderiv δ f) φ K 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {K : Set ℂ} {δ : ℝ} {φ : Filter ι} {F : ι → ℂ → E} {f : ℂ → E} (hF : TendstoUniformlyOn F f φ (Metric.cthickening δ K)) (hδ : 0 < δ) (hFn : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (Metric.cthickening δ K))
Goal: TendstoUniformlyOn (Complex.cderiv δ ∘ F) (Complex.cderiv δ f) φ K
-- Complex.tendstoUniformlyOn_deriv_of_cthickening_subset
theorem tendstoUniformlyOn_deriv_of_cthickening_subset (hf : TendstoLocallyUniformlyOn F f φ U)
    (hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U) {δ : ℝ} (hδ : 0 < δ) (hK : IsCompact K)
    (hU : IsOpen U) (hKU : cthickening δ K ⊆ U) :
    TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {U K : Set ℂ} {φ : Filter ι} {F : ι → ℂ → E} {f : ℂ → E} (hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U) {δ : ℝ} (hδ : 0 < δ) (hK : IsCompact K) (hU : IsOpen U) (hKU : Metric.cthickening δ K ⊆ U)
Goal: TendstoUniformlyOn (deriv ∘ F) (Complex.cderiv δ f) φ K
-- Complex.exists_cthickening_tendstoUniformlyOn
theorem exists_cthickening_tendstoUniformlyOn (hf : TendstoLocallyUniformlyOn F f φ U)
    (hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U) (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
    ∃ δ > 0, cthickening δ K ⊆ U ∧ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {U K : Set ℂ} {φ : Filter ι} {F : ι → ℂ → E} {f : ℂ → E} (hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U) (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
Goal: ∃ δ > 0, Metric.cthickening δ K ⊆ U ∧ TendstoUniformlyOn (deriv ∘ F) (Complex.cderiv δ f) φ K
-- TendstoLocallyUniformlyOn.differentiableOn
theorem _root_.TendstoLocallyUniformlyOn.differentiableOn [φ.NeBot]
    (hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U)
    (hU : IsOpen U) : DifferentiableOn ℂ f U 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {U : Set ℂ} {φ : Filter ι} {F : ι → ℂ → E} {f : ℂ → E} [φ.NeBot] (hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U) (hU : IsOpen U)
Goal: DifferentiableOn ℂ f U
-- TendstoLocallyUniformlyOn.deriv
theorem _root_.TendstoLocallyUniformlyOn.deriv (hf : TendstoLocallyUniformlyOn F f φ U)
    (hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U) (hU : IsOpen U) :
    TendstoLocallyUniformlyOn (deriv ∘ F) (deriv f) φ U 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {U : Set ℂ} {φ : Filter ι} {F : ι → ℂ → E} {f : ℂ → E} (hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U) (hU : IsOpen U)
Goal: TendstoLocallyUniformlyOn (deriv ∘ F) (deriv f) φ U
-- Complex.differentiableOn_tsum_of_summable_norm
theorem differentiableOn_tsum_of_summable_norm {u : ι → ℝ} (hu : Summable u)
    (hf : ∀ i : ι, DifferentiableOn ℂ (F i) U) (hU : IsOpen U)
    (hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i) :
    DifferentiableOn ℂ (fun w : ℂ => ∑' i : ι, F i w) U 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {U : Set ℂ} {F : ι → ℂ → E} {u : ι → ℝ} (hu : Summable u) (hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U) (hU : IsOpen U) (hF_le : ∀ (i : ι), ∀ w ∈ U, ‖F i w‖ ≤ u i)
Goal: DifferentiableOn ℂ (fun w => ∑' (i : ι), F i w) U
-- Complex.hasSum_deriv_of_summable_norm
theorem hasSum_deriv_of_summable_norm {u : ι → ℝ} (hu : Summable u)
    (hf : ∀ i : ι, DifferentiableOn ℂ (F i) U) (hU : IsOpen U)
    (hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i) (hz : z ∈ U) :
    HasSum (fun i : ι => deriv (F i) z) (deriv (fun w : ℂ => ∑' i : ι, F i w) z) 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {U : Set ℂ} {z : ℂ} {F : ι → ℂ → E} {u : ι → ℝ} (hu : Summable u) (hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U) (hU : IsOpen U) (hF_le : ∀ (i : ι), ∀ w ∈ U, ‖F i w‖ ≤ u i) (hz : z ∈ U)
Goal: HasSum (fun i => deriv (F i) z) (deriv (fun w => ∑' (i : ι), F i w) z)
-- Complex.logDeriv_tendsto
theorem logDeriv_tendsto {ι : Type*} {p : Filter ι} (f : ι → ℂ → ℂ) (g : ℂ → ℂ)
    {s : Set ℂ} (hs : IsOpen s) (x : s) (hF : TendstoLocallyUniformlyOn f g p s)
    (hf : ∀ᶠ n : ι in p, DifferentiableOn ℂ (f n) s) (hg : g x ≠ 0) :
    Tendsto (fun n : ι => logDeriv (f n) x) p (𝓝 ((logDeriv g) x)) 
Arguments: {ι : Type u_3} {p : Filter ι} (f : ι → ℂ → ℂ) (g : ℂ → ℂ) {s : Set ℂ} (hs : IsOpen s) (x : ↑s) (hF : TendstoLocallyUniformlyOn f g p s) (hf : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (f n) s) (hg : g ↑x ≠ 0)
Goal: Filter.Tendsto (fun n => logDeriv (f n) ↑x) p (𝓝 (logDeriv g ↑x))
-- isConformalMap_conj
theorem isConformalMap_conj : IsConformalMap (conjLIE : ℂ →L[ℝ] ℂ) 
Goal: IsConformalMap ↑{ toLinearEquiv := Complex.conjLIE.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }
-- isConformalMap_complex_linear
theorem isConformalMap_complex_linear {map : ℂ →L[ℂ] E} (nonzero : map ≠ 0) :
    IsConformalMap (map.restrictScalars ℝ) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace ℂ E] {map : ℂ →L[ℂ] E} (nonzero : map ≠ 0)
Goal: IsConformalMap (ContinuousLinearMap.restrictScalars ℝ map)
-- isConformalMap_complex_linear_conj
theorem isConformalMap_complex_linear_conj {map : ℂ →L[ℂ] E} (nonzero : map ≠ 0) :
    IsConformalMap ((map.restrictScalars ℝ).comp (conjCLE : ℂ →L[ℝ] ℂ)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace ℂ E] {map : ℂ →L[ℂ] E} (nonzero : map ≠ 0)
Goal: IsConformalMap ((ContinuousLinearMap.restrictScalars ℝ map).comp ↑Complex.conjCLE)
-- IsConformalMap.is_complex_or_conj_linear
theorem IsConformalMap.is_complex_or_conj_linear (h : IsConformalMap g) :
    (∃ map : ℂ →L[ℂ] ℂ, map.restrictScalars ℝ = g) ∨
      ∃ map : ℂ →L[ℂ] ℂ, map.restrictScalars ℝ = g ∘L ↑conjCLE 
Arguments: {g : ℂ →L[ℝ] ℂ} (h : IsConformalMap g)
Goal: (∃ map, ContinuousLinearMap.restrictScalars ℝ map = g) ∨
  ∃ map, ContinuousLinearMap.restrictScalars ℝ map = g.comp ↑Complex.conjCLE
-- isConformalMap_iff_is_complex_or_conj_linear
theorem isConformalMap_iff_is_complex_or_conj_linear :
    IsConformalMap g ↔
      ((∃ map : ℂ →L[ℂ] ℂ, map.restrictScalars ℝ = g) ∨
          ∃ map : ℂ →L[ℂ] ℂ, map.restrictScalars ℝ = g ∘L ↑conjCLE) ∧
        g ≠ 0 
Arguments: {g : ℂ →L[ℝ] ℂ}
Goal: IsConformalMap g ↔
  ((∃ map, ContinuousLinearMap.restrictScalars ℝ map = g) ∨
      ∃ map, ContinuousLinearMap.restrictScalars ℝ map = g.comp ↑Complex.conjCLE) ∧
    g ≠ 0
-- image_le_of_liminf_slope_right_lt_deriv_boundary'
theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b))
    -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
    (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
    {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x 
Arguments: {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Set.Icc a b)) (hB' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt B (B' x) (Set.Ici x) x) (bound : ∀ x ∈ Set.Ico a b, f x = B x → f' x < B' x) ⦃x : ℝ⦄
Goal: x ∈ Set.Icc a b → f x ≤ B x
-- image_le_of_liminf_slope_right_lt_deriv_boundary
theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b))
    -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
    (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
    {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x)
    (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x 
Arguments: {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ (x : ℝ), HasDerivAt B (B' x) x) (bound : ∀ x ∈ Set.Ico a b, f x = B x → f' x < B' x) ⦃x : ℝ⦄
Goal: x ∈ Set.Icc a b → f x ≤ B x
-- image_le_of_liminf_slope_right_le_deriv_boundary
theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x`
    (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) :
    ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x 
Arguments: {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Set.Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Set.Icc a b)) (hB' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt B (B' x) (Set.Ici x) x) (bound : ∀ x ∈ Set.Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r) ⦃x : ℝ⦄
Goal: x ∈ Set.Icc a b → f x ≤ B x
-- image_le_of_deriv_right_lt_deriv_boundary'
theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
    {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x 
Arguments: {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt f (f' x) (Set.Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Set.Icc a b)) (hB' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt B (B' x) (Set.Ici x) x) (bound : ∀ x ∈ Set.Ico a b, f x = B x → f' x < B' x) ⦃x : ℝ⦄
Goal: x ∈ Set.Icc a b → f x ≤ B x
-- image_le_of_deriv_right_lt_deriv_boundary
theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
    {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x)
    (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x 
Arguments: {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt f (f' x) (Set.Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ (x : ℝ), HasDerivAt B (B' x) x) (bound : ∀ x ∈ Set.Ico a b, f x = B x → f' x < B' x) ⦃x : ℝ⦄
Goal: x ∈ Set.Icc a b → f x ≤ B x
-- image_le_of_deriv_right_le_deriv_boundary
theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
    {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x 
Arguments: {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt f (f' x) (Set.Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Set.Icc a b)) (hB' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt B (B' x) (Set.Ici x) x) (bound : ∀ x ∈ Set.Ico a b, f' x ≤ B' x) ⦃x : ℝ⦄
Goal: x ∈ Set.Icc a b → f x ≤ B x
-- image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary
theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*}
    [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b))
    -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x`
    (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r)
    {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x 
Arguments: {a b : ℝ} {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Set.Icc a b)) (hB' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt B (B' x) (Set.Ici x) x) (bound : ∀ x ∈ Set.Ico a b, ‖f x‖ = B x → f' x < B' x) ⦃x : ℝ⦄
Goal: x ∈ Set.Icc a b → ‖f x‖ ≤ B x
-- image_norm_le_of_norm_deriv_right_lt_deriv_boundary'
theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E}
    (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
    {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {f' : ℝ → E} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt f (f' x) (Set.Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Set.Icc a b)) (hB' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt B (B' x) (Set.Ici x) x) (bound : ∀ x ∈ Set.Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) ⦃x : ℝ⦄
Goal: x ∈ Set.Icc a b → ‖f x‖ ≤ B x
-- image_norm_le_of_norm_deriv_right_lt_deriv_boundary
theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E}
    (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
    {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x)
    (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {f' : ℝ → E} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt f (f' x) (Set.Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ (x : ℝ), HasDerivAt B (B' x) x) (bound : ∀ x ∈ Set.Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) ⦃x : ℝ⦄
Goal: x ∈ Set.Icc a b → ‖f x‖ ≤ B x
-- image_norm_le_of_norm_deriv_right_le_deriv_boundary'
theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E}
    (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
    {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {f' : ℝ → E} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt f (f' x) (Set.Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Set.Icc a b)) (hB' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt B (B' x) (Set.Ici x) x) (bound : ∀ x ∈ Set.Ico a b, ‖f' x‖ ≤ B' x) ⦃x : ℝ⦄
Goal: x ∈ Set.Icc a b → ‖f x‖ ≤ B x
-- image_norm_le_of_norm_deriv_right_le_deriv_boundary
theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E}
    (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
    {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x)
    (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {f' : ℝ → E} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt f (f' x) (Set.Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ (x : ℝ), HasDerivAt B (B' x) x) (bound : ∀ x ∈ Set.Ico a b, ‖f' x‖ ≤ B' x) ⦃x : ℝ⦄
Goal: x ∈ Set.Icc a b → ‖f x‖ ≤ B x
-- norm_image_sub_le_of_norm_deriv_right_le_segment
theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ}
    (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt f (f' x) (Set.Ici x) x) (bound : ∀ x ∈ Set.Ico a b, ‖f' x‖ ≤ C) (x : ℝ)
Goal: x ∈ Set.Icc a b → ‖f x - f a‖ ≤ C * (x - a)
-- norm_image_sub_le_of_norm_deriv_le_segment'
theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ}
    (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x)
    (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Set.Icc a b, HasDerivWithinAt f (f' x) (Set.Icc a b) x) (bound : ∀ x ∈ Set.Ico a b, ‖f' x‖ ≤ C) (x : ℝ)
Goal: x ∈ Set.Icc a b → ‖f x - f a‖ ≤ C * (x - a)
-- norm_image_sub_le_of_norm_deriv_le_segment
theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b))
    (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) :
    ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b C : ℝ} (hf : DifferentiableOn ℝ f (Set.Icc a b)) (bound : ∀ x ∈ Set.Ico a b, ‖derivWithin f (Set.Icc a b) x‖ ≤ C) (x : ℝ)
Goal: x ∈ Set.Icc a b → ‖f x - f a‖ ≤ C * (x - a)
-- norm_image_sub_le_of_norm_deriv_le_segment_01'
theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ}
    (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x)
    (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Set.Icc 0 1, HasDerivWithinAt f (f' x) (Set.Icc 0 1) x) (bound : ∀ x ∈ Set.Ico 0 1, ‖f' x‖ ≤ C)
Goal: ‖f 1 - f 0‖ ≤ C
-- norm_image_sub_le_of_norm_deriv_le_segment_01
theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ}
    (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1))
    (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {C : ℝ} (hf : DifferentiableOn ℝ f (Set.Icc 0 1)) (bound : ∀ x ∈ Set.Ico 0 1, ‖derivWithin f (Set.Icc 0 1) x‖ ≤ C)
Goal: ‖f 1 - f 0‖ ≤ C
-- constant_of_has_deriv_right_zero
theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b))
    (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} (hcont : ContinuousOn f (Set.Icc a b)) (hderiv : ∀ x ∈ Set.Ico a b, HasDerivWithinAt f 0 (Set.Ici x) x) (x : ℝ)
Goal: x ∈ Set.Icc a b → f x = f a
-- constant_of_derivWithin_zero
theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b))
    (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} (hdiff : DifferentiableOn ℝ f (Set.Icc a b)) (hderiv : ∀ x ∈ Set.Ico a b, derivWithin f (Set.Icc a b) x = 0) (x : ℝ)
Goal: x ∈ Set.Icc a b → f x = f a
-- eq_of_has_deriv_right_eq
theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
    (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b))
    (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {f' g : ℝ → E} (derivf : ∀ x ∈ Set.Ico a b, HasDerivWithinAt f (f' x) (Set.Ici x) x) (derivg : ∀ x ∈ Set.Ico a b, HasDerivWithinAt g (f' x) (Set.Ici x) x) (fcont : ContinuousOn f (Set.Icc a b)) (gcont : ContinuousOn g (Set.Icc a b)) (hi : f a = g a) (y : ℝ)
Goal: y ∈ Set.Icc a b → f y = g y
-- eq_of_derivWithin_eq
theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b))
    (gdiff : DifferentiableOn ℝ g (Icc a b))
    (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) :
    ∀ y ∈ Icc a b, f y = g y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {g : ℝ → E} (fdiff : DifferentiableOn ℝ f (Set.Icc a b)) (gdiff : DifferentiableOn ℝ g (Set.Icc a b)) (hderiv : Set.EqOn (derivWithin f (Set.Icc a b)) (derivWithin g (Set.Icc a b)) (Set.Ico a b)) (hi : f a = g a) (y : ℝ)
Goal: y ∈ Set.Icc a b → f y = g y
-- Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le
theorem norm_image_sub_le_of_norm_hasFDerivWithin_le
    (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s)
    (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → G} {C : ℝ} {s : Set E} {x y : E} {f' : E → E →L[𝕜] G} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s)
Goal: ‖f y - f x‖ ≤ C * ‖y - x‖
-- Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0}
    (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C)
    (hs : Convex ℝ s) : LipschitzOnWith C f s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → G} {s : Set E} {f' : E → E →L[𝕜] G} {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s)
Goal: LipschitzOnWith C f s
-- Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt
theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s)
    {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y)
    (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) :
    ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {s : Set E} {x : E} {f' : E → E →L[𝕜] G} (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ (y : E) in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K)
Goal: ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t