ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- hasDerivAt_inv
theorem hasDerivAt_inv (x_ne_zero : x ≠ 0) : HasDerivAt (fun y => y⁻¹) (-(x ^ 2)⁻¹) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} (x_ne_zero : x ≠ 0)
Goal: HasDerivAt (fun y => y⁻¹) (-(x ^ 2)⁻¹) x
-- hasDerivWithinAt_inv
theorem hasDerivWithinAt_inv (x_ne_zero : x ≠ 0) (s : Set 𝕜) :
    HasDerivWithinAt (fun x => x⁻¹) (-(x ^ 2)⁻¹) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} (x_ne_zero : x ≠ 0) (s : Set 𝕜)
Goal: HasDerivWithinAt (fun x => x⁻¹) (-(x ^ 2)⁻¹) s x
-- differentiableAt_inv
theorem differentiableAt_inv : DifferentiableAt 𝕜 (fun x => x⁻¹) x ↔ x ≠ 0 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜}
Goal: DifferentiableAt 𝕜 (fun x => x⁻¹) x ↔ x ≠ 0
-- differentiableWithinAt_inv
theorem differentiableWithinAt_inv (x_ne_zero : x ≠ 0) :
    DifferentiableWithinAt 𝕜 (fun x => x⁻¹) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} (x_ne_zero : x ≠ 0)
Goal: DifferentiableWithinAt 𝕜 (fun x => x⁻¹) s x
-- differentiableOn_inv
theorem differentiableOn_inv : DifferentiableOn 𝕜 (fun x : 𝕜 => x⁻¹) { x | x ≠ 0 } 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜]
Goal: DifferentiableOn 𝕜 (fun x => x⁻¹) {x | x ≠ 0}
-- deriv_inv
theorem deriv_inv : deriv (fun x => x⁻¹) x = -(x ^ 2)⁻¹ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜}
Goal: deriv (fun x => x⁻¹) x = -(x ^ 2)⁻¹
-- deriv_inv'
@[simp]
theorem deriv_inv' : (deriv fun x : 𝕜 => x⁻¹) = fun x => -(x ^ 2)⁻¹ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜]
Goal: (deriv fun x => x⁻¹) = fun x => -(x ^ 2)⁻¹
-- derivWithin_inv
theorem derivWithin_inv (x_ne_zero : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => x⁻¹) s x = -(x ^ 2)⁻¹ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} (x_ne_zero : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun x => x⁻¹) s x = -(x ^ 2)⁻¹
-- hasFDerivAt_inv
theorem hasFDerivAt_inv (x_ne_zero : x ≠ 0) :
    HasFDerivAt (fun x => x⁻¹) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) : 𝕜 →L[𝕜] 𝕜) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} (x_ne_zero : x ≠ 0)
Goal: HasFDerivAt (fun x => x⁻¹) (ContinuousLinearMap.smulRight 1 (-(x ^ 2)⁻¹)) x
-- hasFDerivWithinAt_inv
theorem hasFDerivWithinAt_inv (x_ne_zero : x ≠ 0) :
    HasFDerivWithinAt (fun x => x⁻¹) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) : 𝕜 →L[𝕜] 𝕜) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} (x_ne_zero : x ≠ 0)
Goal: HasFDerivWithinAt (fun x => x⁻¹) (ContinuousLinearMap.smulRight 1 (-(x ^ 2)⁻¹)) s x
-- fderiv_inv
theorem fderiv_inv : fderiv 𝕜 (fun x => x⁻¹) x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜}
Goal: fderiv 𝕜 (fun x => x⁻¹) x = ContinuousLinearMap.smulRight 1 (-(x ^ 2)⁻¹)
-- fderivWithin_inv
theorem fderivWithin_inv (x_ne_zero : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun x => x⁻¹) s x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} (x_ne_zero : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (fun x => x⁻¹) s x = ContinuousLinearMap.smulRight 1 (-(x ^ 2)⁻¹)
-- HasDerivWithinAt.inv
theorem HasDerivWithinAt.inv (hc : HasDerivWithinAt c c' s x) (hx : c x ≠ 0) :
    HasDerivWithinAt (fun y => (c y)⁻¹) (-c' / c x ^ 2) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {c : 𝕜 → 𝕜} {c' : 𝕜} (hc : HasDerivWithinAt c c' s x) (hx : c x ≠ 0)
Goal: HasDerivWithinAt (fun y => (c y)⁻¹) (-c' / c x ^ 2) s x
-- HasDerivAt.inv
theorem HasDerivAt.inv (hc : HasDerivAt c c' x) (hx : c x ≠ 0) :
    HasDerivAt (fun y => (c y)⁻¹) (-c' / c x ^ 2) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {c : 𝕜 → 𝕜} {c' : 𝕜} (hc : HasDerivAt c c' x) (hx : c x ≠ 0)
Goal: HasDerivAt (fun y => (c y)⁻¹) (-c' / c x ^ 2) x
-- DifferentiableWithinAt.inv
theorem DifferentiableWithinAt.inv (hf : DifferentiableWithinAt 𝕜 h S z) (hz : h z ≠ 0) :
    DifferentiableWithinAt 𝕜 (fun x => (h x)⁻¹) S z 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {h : E → 𝕜} {z : E} {S : Set E} (hf : DifferentiableWithinAt 𝕜 h S z) (hz : h z ≠ 0)
Goal: DifferentiableWithinAt 𝕜 (fun x => (h x)⁻¹) S z
-- DifferentiableAt.inv
@[simp]
theorem DifferentiableAt.inv (hf : DifferentiableAt 𝕜 h z) (hz : h z ≠ 0) :
    DifferentiableAt 𝕜 (fun x => (h x)⁻¹) z 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {h : E → 𝕜} {z : E} (hf : DifferentiableAt 𝕜 h z) (hz : h z ≠ 0)
Goal: DifferentiableAt 𝕜 (fun x => (h x)⁻¹) z
-- DifferentiableOn.inv
theorem DifferentiableOn.inv (hf : DifferentiableOn 𝕜 h S) (hz : ∀ x ∈ S, h x ≠ 0) :
    DifferentiableOn 𝕜 (fun x => (h x)⁻¹) S 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {h : E → 𝕜} {S : Set E} (hf : DifferentiableOn 𝕜 h S) (hz : ∀ x ∈ S, h x ≠ 0)
Goal: DifferentiableOn 𝕜 (fun x => (h x)⁻¹) S
-- Differentiable.inv
@[simp]
theorem Differentiable.inv (hf : Differentiable 𝕜 h) (hz : ∀ x, h x ≠ 0) :
    Differentiable 𝕜 fun x => (h x)⁻¹ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {h : E → 𝕜} (hf : Differentiable 𝕜 h) (hz : ∀ (x : E), h x ≠ 0)
Goal: Differentiable 𝕜 fun x => (h x)⁻¹
-- derivWithin_inv'
theorem derivWithin_inv' (hc : DifferentiableWithinAt 𝕜 c s x) (hx : c x ≠ 0)
    (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {c : 𝕜 → 𝕜} (hc : DifferentiableWithinAt 𝕜 c s x) (hx : c x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun x => (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2
-- deriv_inv''
@[simp]
theorem deriv_inv'' (hc : DifferentiableAt 𝕜 c x) (hx : c x ≠ 0) :
    deriv (fun x => (c x)⁻¹) x = -deriv c x / c x ^ 2 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {c : 𝕜 → 𝕜} (hc : DifferentiableAt 𝕜 c x) (hx : c x ≠ 0)
Goal: deriv (fun x => (c x)⁻¹) x = -deriv c x / c x ^ 2
-- HasDerivWithinAt.div
theorem HasDerivWithinAt.div (hc : HasDerivWithinAt c c' s x) (hd : HasDerivWithinAt d d' s x)
    (hx : d x ≠ 0) :
    HasDerivWithinAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c d : 𝕜 → 𝕜'} {c' d' : 𝕜'} (hc : HasDerivWithinAt c c' s x) (hd : HasDerivWithinAt d d' s x) (hx : d x ≠ 0)
Goal: HasDerivWithinAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) s x
-- HasStrictDerivAt.div
theorem HasStrictDerivAt.div (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x)
    (hx : d x ≠ 0) : HasStrictDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c d : 𝕜 → 𝕜'} {c' d' : 𝕜'} (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x) (hx : d x ≠ 0)
Goal: HasStrictDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x
-- HasDerivAt.div
theorem HasDerivAt.div (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x) (hx : d x ≠ 0) :
    HasDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c d : 𝕜 → 𝕜'} {c' d' : 𝕜'} (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x) (hx : d x ≠ 0)
Goal: HasDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x
-- DifferentiableWithinAt.div
theorem DifferentiableWithinAt.div (hc : DifferentiableWithinAt 𝕜 c s x)
    (hd : DifferentiableWithinAt 𝕜 d s x) (hx : d x ≠ 0) :
    DifferentiableWithinAt 𝕜 (fun x => c x / d x) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c d : 𝕜 → 𝕜'} (hc : DifferentiableWithinAt 𝕜 c s x) (hd : DifferentiableWithinAt 𝕜 d s x) (hx : d x ≠ 0)
Goal: DifferentiableWithinAt 𝕜 (fun x => c x / d x) s x
-- DifferentiableAt.div
@[simp]
theorem DifferentiableAt.div (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x)
    (hx : d x ≠ 0) : DifferentiableAt 𝕜 (fun x => c x / d x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c d : 𝕜 → 𝕜'} (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x) (hx : d x ≠ 0)
Goal: DifferentiableAt 𝕜 (fun x => c x / d x) x
-- DifferentiableOn.div
theorem DifferentiableOn.div (hc : DifferentiableOn 𝕜 c s) (hd : DifferentiableOn 𝕜 d s)
    (hx : ∀ x ∈ s, d x ≠ 0) : DifferentiableOn 𝕜 (fun x => c x / d x) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c d : 𝕜 → 𝕜'} (hc : DifferentiableOn 𝕜 c s) (hd : DifferentiableOn 𝕜 d s) (hx : ∀ x ∈ s, d x ≠ 0)
Goal: DifferentiableOn 𝕜 (fun x => c x / d x) s
-- Differentiable.div
@[simp]
theorem Differentiable.div (hc : Differentiable 𝕜 c) (hd : Differentiable 𝕜 d) (hx : ∀ x, d x ≠ 0) :
    Differentiable 𝕜 fun x => c x / d x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c d : 𝕜 → 𝕜'} (hc : Differentiable 𝕜 c) (hd : Differentiable 𝕜 d) (hx : ∀ (x : 𝕜), d x ≠ 0)
Goal: Differentiable 𝕜 fun x => c x / d x
-- derivWithin_div
theorem derivWithin_div (hc : DifferentiableWithinAt 𝕜 c s x) (hd : DifferentiableWithinAt 𝕜 d s x)
    (hx : d x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => c x / d x) s x =
      (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c d : 𝕜 → 𝕜'} (hc : DifferentiableWithinAt 𝕜 c s x) (hd : DifferentiableWithinAt 𝕜 d s x) (hx : d x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun x => c x / d x) s x = (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2
-- deriv_div
@[simp]
theorem deriv_div (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x) (hx : d x ≠ 0) :
    deriv (fun x => c x / d x) x = (deriv c x * d x - c x * deriv d x) / d x ^ 2 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c d : 𝕜 → 𝕜'} (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x) (hx : d x ≠ 0)
Goal: deriv (fun x => c x / d x) x = (deriv c x * d x - c x * deriv d x) / d x ^ 2
-- HasDerivAtFilter.star
protected nonrec theorem HasDerivAtFilter.star (h : HasDerivAtFilter f f' x L) :
    HasDerivAtFilter (fun x => star (f x)) (star f') x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} [StarRing 𝕜] [TrivialStar 𝕜] [StarAddMonoid F] [ContinuousStar F] [StarModule 𝕜 F] {f' : F} {x : 𝕜} {L : Filter 𝕜} (h : HasDerivAtFilter f f' x L)
Goal: HasDerivAtFilter (fun x => star (f x)) (star f') x L
-- HasDerivWithinAt.star
protected nonrec theorem HasDerivWithinAt.star (h : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => star (f x)) (star f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} [StarRing 𝕜] [TrivialStar 𝕜] [StarAddMonoid F] [ContinuousStar F] [StarModule 𝕜 F] {f' : F} {s : Set 𝕜} {x : 𝕜} (h : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => star (f x)) (star f') s x
-- HasDerivAt.star
protected nonrec theorem HasDerivAt.star (h : HasDerivAt f f' x) :
    HasDerivAt (fun x => star (f x)) (star f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} [StarRing 𝕜] [TrivialStar 𝕜] [StarAddMonoid F] [ContinuousStar F] [StarModule 𝕜 F] {f' : F} {x : 𝕜} (h : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => star (f x)) (star f') x
-- HasStrictDerivAt.star
protected nonrec theorem HasStrictDerivAt.star (h : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => star (f x)) (star f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} [StarRing 𝕜] [TrivialStar 𝕜] [StarAddMonoid F] [ContinuousStar F] [StarModule 𝕜 F] {f' : F} {x : 𝕜} (h : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => star (f x)) (star f') x
-- derivWithin.star
protected theorem derivWithin.star (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun y => star (f y)) s x = star (derivWithin f s x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} [StarRing 𝕜] [TrivialStar 𝕜] [StarAddMonoid F] [ContinuousStar F] [StarModule 𝕜 F] {s : Set 𝕜} {x : 𝕜} (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun y => star (f y)) s x = star (derivWithin f s x)
-- deriv.star
protected theorem deriv.star : deriv (fun y => star (f y)) x = star (deriv f x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} [StarRing 𝕜] [TrivialStar 𝕜] [StarAddMonoid F] [ContinuousStar F] [StarModule 𝕜 F] {x : 𝕜}
Goal: deriv (fun y => star (f y)) x = star (deriv f x)
-- deriv.star'
@[simp]
protected theorem deriv.star' : (deriv fun y => star (f y)) = fun x => star (deriv f x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} [StarRing 𝕜] [TrivialStar 𝕜] [StarAddMonoid F] [ContinuousStar F] [StarModule 𝕜 F]
Goal: (deriv fun y => star (f y)) = fun x => star (deriv f x)
-- ContDiffBump.rIn_pos
structure ContDiffBump (c : E) where
    (rIn rOut : ℝ)
  rIn_pos : 0 < rIn
  rIn_lt_rOut : rIn < rOut
Arguments: {E : Type u_1} {c : E} (self : ContDiffBump c)
Goal: 0 < self.rIn
-- ContDiffBump.rIn_lt_rOut
structure ContDiffBump (c : E) where
    (rIn rOut : ℝ)
  rIn_pos : 0 < rIn
  rIn_lt_rOut : rIn < rOut
Arguments: {E : Type u_1} {c : E} (self : ContDiffBump c)
Goal: self.rIn < self.rOut
-- ContDiffBumpBase.support
-- Porting note(#5171): linter not yet ported; was @[nolint has_nonempty_instance]
structure ContDiffBumpBase (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] where
    toFun : ℝ → E → ℝ
  mem_Icc : ∀ (R : ℝ) (x : E), toFun R x ∈ Icc (0 : ℝ) 1
  symmetric : ∀ (R : ℝ) (x : E), toFun R (-x) = toFun R x
  smooth : ContDiffOn ℝ ⊤ (uncurry toFun) (Ioi (1 : ℝ) ×ˢ (univ : Set E))
  eq_one : ∀ R : ℝ, 1 < R → ∀ x : E, ‖x‖ ≤ 1 → toFun R x = 1
  support : ∀ R : ℝ, 1 < R → Function.support (toFun R) = Metric.ball (0 : E) R
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] (self : ContDiffBumpBase E) (R : ℝ)
Goal: 1 < R → Function.support (self.toFun R) = Metric.ball 0 R
-- ContDiffBumpBase.smooth
-- Porting note(#5171): linter not yet ported; was @[nolint has_nonempty_instance]
structure ContDiffBumpBase (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] where
    toFun : ℝ → E → ℝ
  mem_Icc : ∀ (R : ℝ) (x : E), toFun R x ∈ Icc (0 : ℝ) 1
  symmetric : ∀ (R : ℝ) (x : E), toFun R (-x) = toFun R x
  smooth : ContDiffOn ℝ ⊤ (uncurry toFun) (Ioi (1 : ℝ) ×ˢ (univ : Set E))
  eq_one : ∀ R : ℝ, 1 < R → ∀ x : E, ‖x‖ ≤ 1 → toFun R x = 1
  support : ∀ R : ℝ, 1 < R → Function.support (toFun R) = Metric.ball (0 : E) R
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] (self : ContDiffBumpBase E)
Goal: ContDiffOn ℝ ⊤ (Function.uncurry self.toFun) (Set.Ioi 1 ×ˢ Set.univ)
-- ContDiffBumpBase.eq_one
-- Porting note(#5171): linter not yet ported; was @[nolint has_nonempty_instance]
structure ContDiffBumpBase (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] where
    toFun : ℝ → E → ℝ
  mem_Icc : ∀ (R : ℝ) (x : E), toFun R x ∈ Icc (0 : ℝ) 1
  symmetric : ∀ (R : ℝ) (x : E), toFun R (-x) = toFun R x
  smooth : ContDiffOn ℝ ⊤ (uncurry toFun) (Ioi (1 : ℝ) ×ˢ (univ : Set E))
  eq_one : ∀ R : ℝ, 1 < R → ∀ x : E, ‖x‖ ≤ 1 → toFun R x = 1
  support : ∀ R : ℝ, 1 < R → Function.support (toFun R) = Metric.ball (0 : E) R
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] (self : ContDiffBumpBase E) (R : ℝ)
Goal: 1 < R → ∀ (x : E), ‖x‖ ≤ 1 → self.toFun R x = 1
-- ContDiffBumpBase.symmetric
-- Porting note(#5171): linter not yet ported; was @[nolint has_nonempty_instance]
structure ContDiffBumpBase (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] where
    toFun : ℝ → E → ℝ
  mem_Icc : ∀ (R : ℝ) (x : E), toFun R x ∈ Icc (0 : ℝ) 1
  symmetric : ∀ (R : ℝ) (x : E), toFun R (-x) = toFun R x
  smooth : ContDiffOn ℝ ⊤ (uncurry toFun) (Ioi (1 : ℝ) ×ˢ (univ : Set E))
  eq_one : ∀ R : ℝ, 1 < R → ∀ x : E, ‖x‖ ≤ 1 → toFun R x = 1
  support : ∀ R : ℝ, 1 < R → Function.support (toFun R) = Metric.ball (0 : E) R
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] (self : ContDiffBumpBase E) (R : ℝ) (x : E)
Goal: self.toFun R (-x) = self.toFun R x
-- ContDiffBumpBase.mem_Icc
-- Porting note(#5171): linter not yet ported; was @[nolint has_nonempty_instance]
structure ContDiffBumpBase (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] where
    toFun : ℝ → E → ℝ
  mem_Icc : ∀ (R : ℝ) (x : E), toFun R x ∈ Icc (0 : ℝ) 1
  symmetric : ∀ (R : ℝ) (x : E), toFun R (-x) = toFun R x
  smooth : ContDiffOn ℝ ⊤ (uncurry toFun) (Ioi (1 : ℝ) ×ˢ (univ : Set E))
  eq_one : ∀ R : ℝ, 1 < R → ∀ x : E, ‖x‖ ≤ 1 → toFun R x = 1
  support : ∀ R : ℝ, 1 < R → Function.support (toFun R) = Metric.ball (0 : E) R
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] (self : ContDiffBumpBase E) (R : ℝ) (x : E)
Goal: self.toFun R x ∈ Set.Icc 0 1
-- HasContDiffBump.out
class HasContDiffBump (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] : Prop where
  out : Nonempty (ContDiffBumpBase E)
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [self : HasContDiffBump E]
Goal: Nonempty (ContDiffBumpBase E)
-- ContDiffBump.rOut_pos
theorem rOut_pos {c : E} (f : ContDiffBump c) : 0 < f.rOut 
Arguments: {E : Type u_1} {c : E} (f : ContDiffBump c)
Goal: 0 < f.rOut
-- ContDiffBump.one_lt_rOut_div_rIn
theorem one_lt_rOut_div_rIn {c : E} (f : ContDiffBump c) : 1 < f.rOut / f.rIn 
Arguments: {E : Type u_1} {c : E} (f : ContDiffBump c)
Goal: 1 < f.rOut / f.rIn
-- ContDiffBump.apply
protected theorem apply (x : E) :
    f x = (someContDiffBumpBase E).toFun (f.rOut / f.rIn) (f.rIn⁻¹ • (x - c)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) (x : E)
Goal: ↑f x = (someContDiffBumpBase E).toFun (f.rOut / f.rIn) (f.rIn⁻¹ • (x - c))
-- ContDiffBump.sub
protected theorem sub (x : E) : f (c - x) = f (c + x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) (x : E)
Goal: ↑f (c - x) = ↑f (c + x)
-- ContDiffBump.neg
protected theorem neg (f : ContDiffBump (0 : E)) (x : E) : f (-x) = f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] (f : ContDiffBump 0) (x : E)
Goal: ↑f (-x) = ↑f x
-- ContDiffBump.one_of_mem_closedBall
theorem one_of_mem_closedBall (hx : x ∈ closedBall c f.rIn) : f x = 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) {x : E} (hx : x ∈ Metric.closedBall c f.rIn)
Goal: ↑f x = 1
-- ContDiffBump.nonneg
theorem nonneg : 0 ≤ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) {x : E}
Goal: 0 ≤ ↑f x
-- ContDiffBump.nonneg'
theorem nonneg' (x : E) : 0 ≤ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) (x : E)
Goal: 0 ≤ ↑f x
-- ContDiffBump.le_one
theorem le_one : f x ≤ 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) {x : E}
Goal: ↑f x ≤ 1
-- ContDiffBump.support_eq
theorem support_eq : Function.support f = Metric.ball c f.rOut 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c)
Goal: Function.support ↑f = Metric.ball c f.rOut
-- ContDiffBump.tsupport_eq
theorem tsupport_eq : tsupport f = closedBall c f.rOut 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c)
Goal: tsupport ↑f = Metric.closedBall c f.rOut
-- ContDiffBump.pos_of_mem_ball
theorem pos_of_mem_ball (hx : x ∈ ball c f.rOut) : 0 < f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) {x : E} (hx : x ∈ Metric.ball c f.rOut)
Goal: 0 < ↑f x
-- ContDiffBump.zero_of_le_dist
theorem zero_of_le_dist (hx : f.rOut ≤ dist x c) : f x = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) {x : E} (hx : f.rOut ≤ dist x c)
Goal: ↑f x = 0
-- ContDiffBump.hasCompactSupport
protected theorem hasCompactSupport [FiniteDimensional ℝ E] : HasCompactSupport f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) [FiniteDimensional ℝ E]
Goal: HasCompactSupport ↑f
-- ContDiffBump.eventuallyEq_one_of_mem_ball
theorem eventuallyEq_one_of_mem_ball (h : x ∈ ball c f.rIn) : f =ᶠ[𝓝 x] 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) {x : E} (h : x ∈ Metric.ball c f.rIn)
Goal: ↑f =ᶠ[𝓝 x] 1
-- ContDiffBump.eventuallyEq_one
theorem eventuallyEq_one : f =ᶠ[𝓝 c] 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c)
Goal: ↑f =ᶠ[𝓝 c] 1
-- ContDiffWithinAt.contDiffBump
protected theorem _root_.ContDiffWithinAt.contDiffBump {c g : X → E} {s : Set X}
    {f : ∀ x, ContDiffBump (c x)} {x : X} (hc : ContDiffWithinAt ℝ n c s x)
    (hr : ContDiffWithinAt ℝ n (fun x => (f x).rIn) s x)
    (hR : ContDiffWithinAt ℝ n (fun x => (f x).rOut) s x)
    (hg : ContDiffWithinAt ℝ n g s x) :
    ContDiffWithinAt ℝ n (fun x => f x (g x)) s x 
Arguments: {E : Type u_1} {X : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup X] [NormedSpace ℝ X] [HasContDiffBump E] {n : ℕ∞} {c g : X → E} {s : Set X} {f : (x : X) → ContDiffBump (c x)} {x : X} (hc : ContDiffWithinAt ℝ n c s x) (hr : ContDiffWithinAt ℝ n (fun x => (f x).rIn) s x) (hR : ContDiffWithinAt ℝ n (fun x => (f x).rOut) s x) (hg : ContDiffWithinAt ℝ n g s x)
Goal: ContDiffWithinAt ℝ n (fun x => ↑(f x) (g x)) s x
-- ContDiffAt.contDiffBump
protected nonrec theorem _root_.ContDiffAt.contDiffBump {c g : X → E} {f : ∀ x, ContDiffBump (c x)}
    {x : X} (hc : ContDiffAt ℝ n c x) (hr : ContDiffAt ℝ n (fun x => (f x).rIn) x)
    (hR : ContDiffAt ℝ n (fun x => (f x).rOut) x) (hg : ContDiffAt ℝ n g x) :
    ContDiffAt ℝ n (fun x => f x (g x)) x 
Arguments: {E : Type u_1} {X : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup X] [NormedSpace ℝ X] [HasContDiffBump E] {n : ℕ∞} {c g : X → E} {f : (x : X) → ContDiffBump (c x)} {x : X} (hc : ContDiffAt ℝ n c x) (hr : ContDiffAt ℝ n (fun x => (f x).rIn) x) (hR : ContDiffAt ℝ n (fun x => (f x).rOut) x) (hg : ContDiffAt ℝ n g x)
Goal: ContDiffAt ℝ n (fun x => ↑(f x) (g x)) x
-- ContDiff.contDiffBump
theorem _root_.ContDiff.contDiffBump {c g : X → E} {f : ∀ x, ContDiffBump (c x)}
    (hc : ContDiff ℝ n c) (hr : ContDiff ℝ n fun x => (f x).rIn)
    (hR : ContDiff ℝ n fun x => (f x).rOut) (hg : ContDiff ℝ n g) :
    ContDiff ℝ n fun x => f x (g x) 
Arguments: {E : Type u_1} {X : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup X] [NormedSpace ℝ X] [HasContDiffBump E] {n : ℕ∞} {c g : X → E} {f : (x : X) → ContDiffBump (c x)} (hc : ContDiff ℝ n c) (hr : ContDiff ℝ n fun x => (f x).rIn) (hR : ContDiff ℝ n fun x => (f x).rOut) (hg : ContDiff ℝ n g)
Goal: ContDiff ℝ n fun x => ↑(f x) (g x)
-- ContDiffBump.contDiff
protected theorem contDiff : ContDiff ℝ n f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) {n : ℕ∞}
Goal: ContDiff ℝ n ↑f