ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- seminormFromBounded_bddAbove_range
theorem seminormFromBounded_bddAbove_range (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) (x : R) :
    BddAbove (Set.range fun y ↦ f (x * y) / f y) 
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: BddAbove (Set.range fun y => f (x * y) / f y)
-- seminormFromBounded_le
theorem seminormFromBounded_le (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) (x : R) :
    seminormFromBounded' f x ≤ 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: seminormFromBounded' f x ≤ c * f x
-- seminormFromBounded_ge
theorem seminormFromBounded_ge (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) (x : R) :
    f x ≤ f 1 * seminormFromBounded' 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: f x ≤ f 1 * seminormFromBounded' f x
-- seminormFromBounded_nonneg
theorem seminormFromBounded_nonneg (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y)  :
    0 ≤ seminormFromBounded' f 
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)
Goal: 0 ≤ seminormFromBounded' f
-- seminormFromBounded_eq_zero_iff
theorem seminormFromBounded_eq_zero_iff (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) (x : R) :
    seminormFromBounded' f x = 0 ↔ f x = 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)
Goal: seminormFromBounded' f x = 0 ↔ f x = 0
-- seminormFromBounded_neg
theorem seminormFromBounded_neg (f_neg : ∀ x : R, f (-x) = f x) (x : R) :
    seminormFromBounded' f (-x) = seminormFromBounded' f x 
Arguments: {R : Type u_1} [CommRing R] {f : R → ℝ} (f_neg : ∀ (x : R), f (-x) = f x) (x : R)
Goal: seminormFromBounded' f (-x) = seminormFromBounded' f x
-- seminormFromBounded_mul
theorem seminormFromBounded_mul (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) (x y : R) :
    seminormFromBounded' f (x * y) ≤ seminormFromBounded' f x * seminormFromBounded' f y 
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 y : R)
Goal: seminormFromBounded' f (x * y) ≤ seminormFromBounded' f x * seminormFromBounded' f y
-- seminormFromBounded_one
theorem seminormFromBounded_one (f_ne_zero : f ≠ 0) (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) :
    seminormFromBounded' f 1 = 1 
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: seminormFromBounded' f 1 = 1
-- seminormFromBounded_one_le
theorem seminormFromBounded_one_le (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) :
    seminormFromBounded' f 1 ≤ 1 
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)
Goal: seminormFromBounded' f 1 ≤ 1
-- seminormFromBounded_add
theorem seminormFromBounded_add (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y)
    (f_add : ∀ a b, f (a + b) ≤ f a + f b) (x y : R) :
    seminormFromBounded' f (x + y) ≤ seminormFromBounded' f x + seminormFromBounded' f y 
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) (f_add : ∀ (a b : R), f (a + b) ≤ f a + f b) (x y : R)
Goal: seminormFromBounded' f (x + y) ≤ seminormFromBounded' f x + seminormFromBounded' f y
-- seminormFromBounded_isNonarchimedean
theorem seminormFromBounded_isNonarchimedean (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y)
    (hna : IsNonarchimedean f) : IsNonarchimedean (seminormFromBounded' f) 
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) (hna : IsNonarchimedean f)
Goal: IsNonarchimedean (seminormFromBounded' f)
-- seminormFromBounded_of_mul_apply
theorem seminormFromBounded_of_mul_apply (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) {x : R}
    (hx : ∀ y : R, f (x * y) = f x * f y) : seminormFromBounded' f x = 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} (hx : ∀ (y : R), f (x * y) = f x * f y)
Goal: seminormFromBounded' f x = f x
-- seminormFromBounded_of_mul_le
theorem seminormFromBounded_of_mul_le (f_nonneg : 0 ≤ f) {x : R}
    (hx : ∀ y : R, f (x * y) ≤ f x * f y) (h_one : f 1 ≤ 1) : seminormFromBounded' f x = f x 
Arguments: {R : Type u_1} [CommRing R] {f : R → ℝ} (f_nonneg : 0 ≤ f) {x : R} (hx : ∀ (y : R), f (x * y) ≤ f x * f y) (h_one : f 1 ≤ 1)
Goal: seminormFromBounded' f x = f x
-- seminormFromBounded_nonzero
theorem seminormFromBounded_nonzero (f_ne_zero : f ≠ 0) (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) :
    seminormFromBounded' f ≠ 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: seminormFromBounded' f ≠ 0
-- seminormFromBounded_ker
theorem seminormFromBounded_ker (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) :
    seminormFromBounded' f ⁻¹' {0} = f ⁻¹' {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)
Goal: seminormFromBounded' f ⁻¹' {0} = f ⁻¹' {0}
-- seminormFromBounded_is_norm_iff
theorem seminormFromBounded_is_norm_iff (f_zero : f 0 = 0) (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y)
    (f_add : ∀ a b, f (a + b) ≤ f a + f b) (f_neg : ∀ x : R, f (-x) = f x) :
    (∀ x : R, (seminormFromBounded f_zero f_nonneg f_mul f_add f_neg).toFun x = 0 → x = 0) ↔
      f ⁻¹' {0} = {0} 
Arguments: {R : Type u_1} [CommRing R] {f : R → ℝ} {c : ℝ} (f_zero : f 0 = 0) (f_nonneg : 0 ≤ f) (f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y) (f_add : ∀ (a b : R), f (a + b) ≤ f a + f b) (f_neg : ∀ (x : R), f (-x) = f x)
Goal: (∀ (x : R), (seminormFromBounded f_zero f_nonneg f_mul f_add f_neg).toFun x = 0 → x = 0) ↔ f ⁻¹' {0} = {0}
-- seminormFromBounded_of_mul_is_mul
theorem seminormFromBounded_of_mul_is_mul (f_nonneg : 0 ≤ f)
    (f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) {x : R}
    (hx : ∀ y : R, f (x * y) = f x * f y) (y : R) :
    seminormFromBounded' f (x * y) = seminormFromBounded' f x * seminormFromBounded' f y 
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 : ∀ (y : R), f (x * y) = f x * f y) (y : R)
Goal: seminormFromBounded' f (x * y) = seminormFromBounded' f x * seminormFromBounded' f y
-- _private.Mathlib.Analysis.MellinInversion.0.rexp_neg_deriv_aux
private theorem rexp_neg_deriv_aux :
    ∀ x ∈ univ, HasDerivWithinAt (rexp ∘ Neg.neg) (-rexp (-x)) univ x 
Arguments: (x : ℝ)
Goal: x ∈ Set.univ → HasDerivWithinAt (rexp ∘ Neg.neg) (-rexp (-x)) Set.univ x
-- _private.Mathlib.Analysis.MellinInversion.0.rexp_neg_image_aux
private theorem rexp_neg_image_aux : rexp ∘ Neg.neg '' univ = Ioi 0 
Goal: rexp ∘ Neg.neg '' Set.univ = Set.Ioi 0
-- _private.Mathlib.Analysis.MellinInversion.0.rexp_neg_injOn_aux
private theorem rexp_neg_injOn_aux : univ.InjOn (rexp ∘ Neg.neg) 
Goal: Set.InjOn (rexp ∘ Neg.neg) Set.univ
-- _private.Mathlib.Analysis.MellinInversion.0.rexp_cexp_aux
private theorem rexp_cexp_aux (x : ℝ) (s : ℂ) (f : E) :
    rexp (-x) • cexp (-↑x) ^ (s - 1) • f = cexp (-s * ↑x) • f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (x : ℝ) (s : ℂ) (f : E)
Goal: rexp (-x) • cexp (-↑x) ^ (s - 1) • f = cexp (-s * ↑x) • f
-- mellin_eq_fourierIntegral
theorem mellin_eq_fourierIntegral (f : ℝ → E) {s : ℂ} :
    mellin f s = 𝓕 (fun (u : ℝ) ↦ (Real.exp (-s.re * u) • f (Real.exp (-u)))) (s.im / (2 * π)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E) {s : ℂ}
Goal: mellin f s = 𝓕 (fun u => rexp (-s.re * u) • f (rexp (-u))) (s.im / (2 * π))
-- mellinInv_eq_fourierIntegralInv
theorem mellinInv_eq_fourierIntegralInv (σ : ℝ) (f : ℂ → E) {x : ℝ} (hx : 0 < x) :
    mellinInv σ f x =
    (x : ℂ) ^ (-σ : ℂ) • 𝓕⁻ (fun (y : ℝ) ↦ f (σ + 2 * π * y * I)) (-Real.log x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (σ : ℝ) (f : ℂ → E) {x : ℝ} (hx : 0 < x)
Goal: mellinInv σ f x = ↑x ^ (-↑σ) • 𝓕⁻ (fun y => f (↑σ + 2 * ↑π * ↑y * Complex.I)) (-Real.log x)
-- mellin_inversion
theorem mellin_inversion (σ : ℝ) (f : ℝ → E) {x : ℝ} (hx : 0 < x) (hf : MellinConvergent f σ)
    (hFf : VerticalIntegrable (mellin f) σ) (hfx : ContinuousAt f x) :
    mellinInv σ (mellin f) x = f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] (σ : ℝ) (f : ℝ → E) {x : ℝ} (hx : 0 < x) (hf : MellinConvergent f ↑σ) (hFf : Complex.VerticalIntegrable (mellin f) σ MeasureTheory.volume) (hfx : ContinuousAt f x)
Goal: mellinInv σ (mellin f) x = f x
-- HasConstantSpeedOnWith.hasLocallyBoundedVariationOn
theorem HasConstantSpeedOnWith.hasLocallyBoundedVariationOn (h : HasConstantSpeedOnWith f s l) :
    LocallyBoundedVariationOn f s 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {s : Set ℝ} {l : ℝ≥0} (h : HasConstantSpeedOnWith f s l)
Goal: LocallyBoundedVariationOn f s
-- hasConstantSpeedOnWith_of_subsingleton
theorem hasConstantSpeedOnWith_of_subsingleton (f : ℝ → E) {s : Set ℝ} (hs : s.Subsingleton)
    (l : ℝ≥0) : HasConstantSpeedOnWith f s l 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] (f : ℝ → E) {s : Set ℝ} (hs : s.Subsingleton) (l : ℝ≥0)
Goal: HasConstantSpeedOnWith f s l
-- hasConstantSpeedOnWith_iff_ordered
theorem hasConstantSpeedOnWith_iff_ordered :
    HasConstantSpeedOnWith f s l ↔ ∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s),
      x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (l * (y - x)) 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {s : Set ℝ} {l : ℝ≥0}
Goal: HasConstantSpeedOnWith f s l ↔
  ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Set.Icc x y) = ENNReal.ofReal (↑l * (y - x))
-- hasConstantSpeedOnWith_iff_variationOnFromTo_eq
theorem hasConstantSpeedOnWith_iff_variationOnFromTo_eq :
    HasConstantSpeedOnWith f s l ↔ LocallyBoundedVariationOn f s ∧
      ∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s), variationOnFromTo f s x y = l * (y - x) 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {s : Set ℝ} {l : ℝ≥0}
Goal: HasConstantSpeedOnWith f s l ↔
  LocallyBoundedVariationOn f s ∧ ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → variationOnFromTo f s x y = ↑l * (y - x)
-- HasConstantSpeedOnWith.union
theorem HasConstantSpeedOnWith.union {t : Set ℝ} (hfs : HasConstantSpeedOnWith f s l)
    (hft : HasConstantSpeedOnWith f t l) {x : ℝ} (hs : IsGreatest s x) (ht : IsLeast t x) :
    HasConstantSpeedOnWith f (s ∪ t) l 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {s : Set ℝ} {l : ℝ≥0} {t : Set ℝ} (hfs : HasConstantSpeedOnWith f s l) (hft : HasConstantSpeedOnWith f t l) {x : ℝ} (hs : IsGreatest s x) (ht : IsLeast t x)
Goal: HasConstantSpeedOnWith f (s ∪ t) l
-- HasConstantSpeedOnWith.Icc_Icc
theorem HasConstantSpeedOnWith.Icc_Icc {x y z : ℝ} (hfs : HasConstantSpeedOnWith f (Icc x y) l)
    (hft : HasConstantSpeedOnWith f (Icc y z) l) : HasConstantSpeedOnWith f (Icc x z) l 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {l : ℝ≥0} {x y z : ℝ} (hfs : HasConstantSpeedOnWith f (Set.Icc x y) l) (hft : HasConstantSpeedOnWith f (Set.Icc y z) l)
Goal: HasConstantSpeedOnWith f (Set.Icc x z) l
-- hasConstantSpeedOnWith_zero_iff
theorem hasConstantSpeedOnWith_zero_iff :
    HasConstantSpeedOnWith f s 0 ↔ ∀ᵉ (x ∈ s) (y ∈ s), edist (f x) (f y) = 0 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {s : Set ℝ}
Goal: HasConstantSpeedOnWith f s 0 ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0
-- HasConstantSpeedOnWith.ratio
theorem HasConstantSpeedOnWith.ratio {l' : ℝ≥0} (hl' : l' ≠ 0) {φ : ℝ → ℝ} (φm : MonotoneOn φ s)
    (hfφ : HasConstantSpeedOnWith (f ∘ φ) s l) (hf : HasConstantSpeedOnWith f (φ '' s) l') ⦃x : ℝ⦄
    (xs : x ∈ s) : EqOn φ (fun y => l / l' * (y - x) + φ x) s 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {s : Set ℝ} {l l' : ℝ≥0} (hl' : l' ≠ 0) {φ : ℝ → ℝ} (φm : MonotoneOn φ s) (hfφ : HasConstantSpeedOnWith (f ∘ φ) s l) (hf : HasConstantSpeedOnWith f (φ '' s) l') ⦃x : ℝ⦄ (xs : x ∈ s)
Goal: Set.EqOn φ (fun y => ↑l / ↑l' * (y - x) + φ x) s
-- HasUnitSpeedOn.union
theorem HasUnitSpeedOn.union {t : Set ℝ} {x : ℝ} (hfs : HasUnitSpeedOn f s)
    (hft : HasUnitSpeedOn f t) (hs : IsGreatest s x) (ht : IsLeast t x) :
    HasUnitSpeedOn f (s ∪ t) 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {s t : Set ℝ} {x : ℝ} (hfs : HasUnitSpeedOn f s) (hft : HasUnitSpeedOn f t) (hs : IsGreatest s x) (ht : IsLeast t x)
Goal: HasUnitSpeedOn f (s ∪ t)
-- HasUnitSpeedOn.Icc_Icc
theorem HasUnitSpeedOn.Icc_Icc {x y z : ℝ} (hfs : HasUnitSpeedOn f (Icc x y))
    (hft : HasUnitSpeedOn f (Icc y z)) : HasUnitSpeedOn f (Icc x z) 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {x y z : ℝ} (hfs : HasUnitSpeedOn f (Set.Icc x y)) (hft : HasUnitSpeedOn f (Set.Icc y z))
Goal: HasUnitSpeedOn f (Set.Icc x z)
-- unique_unit_speed
theorem unique_unit_speed {φ : ℝ → ℝ} (φm : MonotoneOn φ s) (hfφ : HasUnitSpeedOn (f ∘ φ) s)
    (hf : HasUnitSpeedOn f (φ '' s)) ⦃x : ℝ⦄ (xs : x ∈ s) : EqOn φ (fun y => y - x + φ x) s 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {s : Set ℝ} {φ : ℝ → ℝ} (φm : MonotoneOn φ s) (hfφ : HasUnitSpeedOn (f ∘ φ) s) (hf : HasUnitSpeedOn f (φ '' s)) ⦃x : ℝ⦄ (xs : x ∈ s)
Goal: Set.EqOn φ (fun y => y - x + φ x) s
-- unique_unit_speed_on_Icc_zero
theorem unique_unit_speed_on_Icc_zero {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) {φ : ℝ → ℝ}
    (φm : MonotoneOn φ <| Icc 0 s) (φst : φ '' Icc 0 s = Icc 0 t)
    (hfφ : HasUnitSpeedOn (f ∘ φ) (Icc 0 s)) (hf : HasUnitSpeedOn f (Icc 0 t)) :
    EqOn φ id (Icc 0 s) 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) {φ : ℝ → ℝ} (φm : MonotoneOn φ (Set.Icc 0 s)) (φst : φ '' Set.Icc 0 s = Set.Icc 0 t) (hfφ : HasUnitSpeedOn (f ∘ φ) (Set.Icc 0 s)) (hf : HasUnitSpeedOn f (Set.Icc 0 t))
Goal: Set.EqOn φ id (Set.Icc 0 s)
-- edist_naturalParameterization_eq_zero
theorem edist_naturalParameterization_eq_zero {f : α → E} {s : Set α}
    (hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s) {b : α} (bs : b ∈ s) :
    edist (naturalParameterization f s a (variationOnFromTo f s a b)) (f b) = 0 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s) {b : α} (bs : b ∈ s)
Goal: edist (naturalParameterization f s a (variationOnFromTo f s a b)) (f b) = 0
-- has_unit_speed_naturalParameterization
theorem has_unit_speed_naturalParameterization (f : α → E) {s : Set α}
    (hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s) :
    HasUnitSpeedOn (naturalParameterization f s a) (variationOnFromTo f s a '' s) 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s : Set α} (hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s)
Goal: HasUnitSpeedOn (naturalParameterization f s a) (variationOnFromTo f s a '' s)
-- HasDerivAt.comp_const_add
lemma HasDerivAt.comp_const_add {𝕜 : Type*} [NontriviallyNormedField 𝕜] (a x : 𝕜) {𝕜' : Type*}
    [NormedAddCommGroup 𝕜'] [NormedSpace 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h' : 𝕜'}
    (hh : HasDerivAt h h' (a + x)) :
    HasDerivAt (fun x ↦ h (a + x)) h' x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (a x : 𝕜) {𝕜' : Type u_2} [NormedAddCommGroup 𝕜'] [NormedSpace 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h' : 𝕜'} (hh : HasDerivAt h h' (a + x))
Goal: HasDerivAt (fun x => h (a + x)) h' x
-- HasDerivAt.comp_add_const
lemma HasDerivAt.comp_add_const {𝕜 : Type*} [NontriviallyNormedField 𝕜] (x a : 𝕜) {𝕜' : Type*}
    [NormedAddCommGroup 𝕜'] [NormedSpace 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h' : 𝕜'}
    (hh : HasDerivAt h h' (x + a)) :
    HasDerivAt (fun x ↦ h (x + a)) h' x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (x a : 𝕜) {𝕜' : Type u_2} [NormedAddCommGroup 𝕜'] [NormedSpace 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h' : 𝕜'} (hh : HasDerivAt h h' (x + a))
Goal: HasDerivAt (fun x => h (x + a)) h' x
-- deriv_comp_neg
lemma deriv_comp_neg {𝕜 : Type*} [NontriviallyNormedField 𝕜] {F : Type*} [NormedAddCommGroup F]
    [NormedSpace 𝕜 F] (f : 𝕜 → F) (a : 𝕜) : deriv (fun x ↦ f (-x)) a = -deriv f (-a) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : 𝕜 → F) (a : 𝕜)
Goal: deriv (fun x => f (-x)) a = -deriv f (-a)
-- deriv_comp_const_add
lemma deriv_comp_const_add {𝕜 : Type*} [NontriviallyNormedField 𝕜] (a x : 𝕜) {𝕜' : Type*}
    [NormedAddCommGroup 𝕜'] [NormedSpace 𝕜 𝕜'] {h : 𝕜 → 𝕜'}
    (hh : DifferentiableAt 𝕜 h (a + x)) :
    deriv (fun x ↦ h (a + x)) x = deriv h (a + x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (a x : 𝕜) {𝕜' : Type u_2} [NormedAddCommGroup 𝕜'] [NormedSpace 𝕜 𝕜'] {h : 𝕜 → 𝕜'} (hh : DifferentiableAt 𝕜 h (a + x))
Goal: deriv (fun x => h (a + x)) x = deriv h (a + x)
-- deriv_comp_add_const
lemma deriv_comp_add_const {𝕜 : Type*} [NontriviallyNormedField 𝕜] (a x : 𝕜) {𝕜' : Type*}
    [NormedAddCommGroup 𝕜'] [NormedSpace 𝕜 𝕜'] {h : 𝕜 → 𝕜'}
    (hh : DifferentiableAt 𝕜 h (x + a)) :
    deriv (fun x ↦ h (x + a)) x = deriv h (x + a) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (a x : 𝕜) {𝕜' : Type u_2} [NormedAddCommGroup 𝕜'] [NormedSpace 𝕜 𝕜'] {h : 𝕜 → 𝕜'} (hh : DifferentiableAt 𝕜 h (x + a))
Goal: deriv (fun x => h (x + a)) x = deriv h (x + a)
-- AffineMap.hasStrictDerivAt
theorem hasStrictDerivAt : HasStrictDerivAt f (f.linear 1) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 →ᵃ[𝕜] E) {x : 𝕜}
Goal: HasStrictDerivAt (⇑f) (f.linear 1) x
-- AffineMap.hasDerivAtFilter
theorem hasDerivAtFilter : HasDerivAtFilter f (f.linear 1) x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 →ᵃ[𝕜] E) {L : Filter 𝕜} {x : 𝕜}
Goal: HasDerivAtFilter (⇑f) (f.linear 1) x L
-- AffineMap.hasDerivWithinAt
theorem hasDerivWithinAt : HasDerivWithinAt f (f.linear 1) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 →ᵃ[𝕜] E) {s : Set 𝕜} {x : 𝕜}
Goal: HasDerivWithinAt (⇑f) (f.linear 1) s x
-- AffineMap.hasDerivAt
theorem hasDerivAt : HasDerivAt f (f.linear 1) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 →ᵃ[𝕜] E) {x : 𝕜}
Goal: HasDerivAt (⇑f) (f.linear 1) x
-- AffineMap.derivWithin
protected theorem derivWithin (hs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin f s x = f.linear 1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 →ᵃ[𝕜] E) {s : Set 𝕜} {x : 𝕜} (hs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (⇑f) s x = f.linear 1
-- AffineMap.deriv
@[simp] protected theorem deriv : deriv f x = f.linear 1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 →ᵃ[𝕜] E) {x : 𝕜}
Goal: deriv (⇑f) x = f.linear 1
-- AffineMap.differentiableAt
protected theorem differentiableAt : DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 →ᵃ[𝕜] E) {x : 𝕜}
Goal: DifferentiableAt 𝕜 (⇑f) x
-- AffineMap.differentiable
protected theorem differentiable : Differentiable 𝕜 f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 →ᵃ[𝕜] E)
Goal: Differentiable 𝕜 ⇑f
-- AffineMap.differentiableWithinAt
protected theorem differentiableWithinAt : DifferentiableWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 →ᵃ[𝕜] E) {s : Set 𝕜} {x : 𝕜}
Goal: DifferentiableWithinAt 𝕜 (⇑f) s x
-- AffineMap.differentiableOn
protected theorem differentiableOn : DifferentiableOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 →ᵃ[𝕜] E) {s : Set 𝕜}
Goal: DifferentiableOn 𝕜 (⇑f) s
-- AffineMap.hasStrictDerivAt_lineMap
theorem hasStrictDerivAt_lineMap : HasStrictDerivAt (lineMap a b) (b - a) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {a b : E} {x : 𝕜}
Goal: HasStrictDerivAt (⇑(AffineMap.lineMap a b)) (b - a) x
-- AffineMap.hasDerivAt_lineMap
theorem hasDerivAt_lineMap :  HasDerivAt (lineMap a b) (b - a) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {a b : E} {x : 𝕜}
Goal: HasDerivAt (⇑(AffineMap.lineMap a b)) (b - a) x
-- AffineMap.hasDerivWithinAt_lineMap
theorem hasDerivWithinAt_lineMap : HasDerivWithinAt (lineMap a b) (b - a) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {a b : E} {s : Set 𝕜} {x : 𝕜}
Goal: HasDerivWithinAt (⇑(AffineMap.lineMap a b)) (b - a) s x
-- MeasureTheory.convolution_integrand_bound_right_of_le_of_subset
theorem convolution_integrand_bound_right_of_le_of_subset {C : ℝ} (hC : ∀ i, ‖g i‖ ≤ C) {x t : G}
    {s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u) :
    ‖L (f t) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * C) t 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [AddGroup G] [TopologicalSpace G] {C : ℝ} (hC : ∀ (i : G), ‖g i‖ ≤ C) {x t : G} {s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u)
Goal: ‖(L (f t)) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * C) t
-- HasCompactSupport.convolution_integrand_bound_right_of_subset
theorem _root_.HasCompactSupport.convolution_integrand_bound_right_of_subset
    (hcg : HasCompactSupport g) (hg : Continuous g)
    {x t : G} {s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u) :
    ‖L (f t) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖) t 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [AddGroup G] [TopologicalSpace G] (hcg : HasCompactSupport g) (hg : Continuous g) {x t : G} {s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u)
Goal: ‖(L (f t)) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖) t
-- HasCompactSupport.convolution_integrand_bound_right
theorem _root_.HasCompactSupport.convolution_integrand_bound_right (hcg : HasCompactSupport g)
    (hg : Continuous g) {x t : G} {s : Set G} (hx : x ∈ s) :
    ‖L (f t) (g (x - t))‖ ≤ (-tsupport g + s).indicator (fun t => ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖) t 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [AddGroup G] [TopologicalSpace G] (hcg : HasCompactSupport g) (hg : Continuous g) {x t : G} {s : Set G} (hx : x ∈ s)
Goal: ‖(L (f t)) (g (x - t))‖ ≤ (-tsupport g + s).indicator (fun t => ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖) t