ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DifferentiableOn.dist
theorem DifferentiableOn.dist (hf : DifferentiableOn ℝ f s) (hg : DifferentiableOn ℝ g s)
    (hne : ∀ x ∈ s, f x ≠ g x) : DifferentiableOn ℝ (fun y => dist (f y) (g y)) s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] {f g : G → E} {s : Set G} (hf : DifferentiableOn ℝ f s) (hg : DifferentiableOn ℝ g s) (hne : ∀ x ∈ s, f x ≠ g x)
Goal: DifferentiableOn ℝ (fun y => dist (f y) (g y)) s
-- differentiableWithinAt_euclidean
theorem differentiableWithinAt_euclidean :
    DifferentiableWithinAt 𝕜 f t y ↔ ∀ i, DifferentiableWithinAt 𝕜 (fun x => f x i) t y 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {H : Type u_3} [RCLike 𝕜] [NormedAddCommGroup H] [NormedSpace 𝕜 H] [Fintype ι] {f : H → EuclideanSpace 𝕜 ι} {t : Set H} {y : H}
Goal: DifferentiableWithinAt 𝕜 f t y ↔ ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => f x i) t y
-- differentiableAt_euclidean
theorem differentiableAt_euclidean :
    DifferentiableAt 𝕜 f y ↔ ∀ i, DifferentiableAt 𝕜 (fun x => f x i) y 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {H : Type u_3} [RCLike 𝕜] [NormedAddCommGroup H] [NormedSpace 𝕜 H] [Fintype ι] {f : H → EuclideanSpace 𝕜 ι} {y : H}
Goal: DifferentiableAt 𝕜 f y ↔ ∀ (i : ι), DifferentiableAt 𝕜 (fun x => f x i) y
-- differentiableOn_euclidean
theorem differentiableOn_euclidean :
    DifferentiableOn 𝕜 f t ↔ ∀ i, DifferentiableOn 𝕜 (fun x => f x i) t 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {H : Type u_3} [RCLike 𝕜] [NormedAddCommGroup H] [NormedSpace 𝕜 H] [Fintype ι] {f : H → EuclideanSpace 𝕜 ι} {t : Set H}
Goal: DifferentiableOn 𝕜 f t ↔ ∀ (i : ι), DifferentiableOn 𝕜 (fun x => f x i) t
-- differentiable_euclidean
theorem differentiable_euclidean : Differentiable 𝕜 f ↔ ∀ i, Differentiable 𝕜 fun x => f x i 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {H : Type u_3} [RCLike 𝕜] [NormedAddCommGroup H] [NormedSpace 𝕜 H] [Fintype ι] {f : H → EuclideanSpace 𝕜 ι}
Goal: Differentiable 𝕜 f ↔ ∀ (i : ι), Differentiable 𝕜 fun x => f x i
-- hasStrictFDerivAt_euclidean
theorem hasStrictFDerivAt_euclidean :
    HasStrictFDerivAt f f' y ↔
      ∀ i, HasStrictFDerivAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') y 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {H : Type u_3} [RCLike 𝕜] [NormedAddCommGroup H] [NormedSpace 𝕜 H] [Fintype ι] {f : H → EuclideanSpace 𝕜 ι} {f' : H →L[𝕜] EuclideanSpace 𝕜 ι} {y : H}
Goal: HasStrictFDerivAt f f' y ↔ ∀ (i : ι), HasStrictFDerivAt (fun x => f x i) ((EuclideanSpace.proj i).comp f') y
-- hasFDerivWithinAt_euclidean
theorem hasFDerivWithinAt_euclidean :
    HasFDerivWithinAt f f' t y ↔
      ∀ i, HasFDerivWithinAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') t y 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {H : Type u_3} [RCLike 𝕜] [NormedAddCommGroup H] [NormedSpace 𝕜 H] [Fintype ι] {f : H → EuclideanSpace 𝕜 ι} {f' : H →L[𝕜] EuclideanSpace 𝕜 ι} {t : Set H} {y : H}
Goal: HasFDerivWithinAt f f' t y ↔ ∀ (i : ι), HasFDerivWithinAt (fun x => f x i) ((EuclideanSpace.proj i).comp f') t y
-- contDiffWithinAt_euclidean
theorem contDiffWithinAt_euclidean {n : ℕ∞} :
    ContDiffWithinAt 𝕜 n f t y ↔ ∀ i, ContDiffWithinAt 𝕜 n (fun x => f x i) t y 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {H : Type u_3} [RCLike 𝕜] [NormedAddCommGroup H] [NormedSpace 𝕜 H] [Fintype ι] {f : H → EuclideanSpace 𝕜 ι} {t : Set H} {y : H} {n : ℕ∞}
Goal: ContDiffWithinAt 𝕜 n f t y ↔ ∀ (i : ι), ContDiffWithinAt 𝕜 n (fun x => f x i) t y
-- contDiffAt_euclidean
theorem contDiffAt_euclidean {n : ℕ∞} :
    ContDiffAt 𝕜 n f y ↔ ∀ i, ContDiffAt 𝕜 n (fun x => f x i) y 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {H : Type u_3} [RCLike 𝕜] [NormedAddCommGroup H] [NormedSpace 𝕜 H] [Fintype ι] {f : H → EuclideanSpace 𝕜 ι} {y : H} {n : ℕ∞}
Goal: ContDiffAt 𝕜 n f y ↔ ∀ (i : ι), ContDiffAt 𝕜 n (fun x => f x i) y
-- contDiffOn_euclidean
theorem contDiffOn_euclidean {n : ℕ∞} :
    ContDiffOn 𝕜 n f t ↔ ∀ i, ContDiffOn 𝕜 n (fun x => f x i) t 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {H : Type u_3} [RCLike 𝕜] [NormedAddCommGroup H] [NormedSpace 𝕜 H] [Fintype ι] {f : H → EuclideanSpace 𝕜 ι} {t : Set H} {n : ℕ∞}
Goal: ContDiffOn 𝕜 n f t ↔ ∀ (i : ι), ContDiffOn 𝕜 n (fun x => f x i) t
-- contDiff_euclidean
theorem contDiff_euclidean {n : ℕ∞} : ContDiff 𝕜 n f ↔ ∀ i, ContDiff 𝕜 n fun x => f x i 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {H : Type u_3} [RCLike 𝕜] [NormedAddCommGroup H] [NormedSpace 𝕜 H] [Fintype ι] {f : H → EuclideanSpace 𝕜 ι} {n : ℕ∞}
Goal: ContDiff 𝕜 n f ↔ ∀ (i : ι), ContDiff 𝕜 n fun x => f x i
-- PartialHomeomorph.contDiff_univUnitBall
theorem PartialHomeomorph.contDiff_univUnitBall : ContDiff ℝ n (univUnitBall : E → E) 
Arguments: {n : ℕ∞} {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
Goal: ContDiff ℝ n ↑PartialHomeomorph.univUnitBall
-- PartialHomeomorph.contDiffOn_univUnitBall_symm
theorem PartialHomeomorph.contDiffOn_univUnitBall_symm :
    ContDiffOn ℝ n univUnitBall.symm (ball (0 : E) 1) 
Arguments: {n : ℕ∞} {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
Goal: ContDiffOn ℝ n (↑PartialHomeomorph.univUnitBall.symm) (Metric.ball 0 1)
-- Homeomorph.contDiff_unitBall
theorem Homeomorph.contDiff_unitBall : ContDiff ℝ n fun x : E => (unitBall x : E) 
Arguments: {n : ℕ∞} {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
Goal: ContDiff ℝ n fun x => ↑(Homeomorph.unitBall x)
-- PartialHomeomorph.contDiff_unitBallBall
theorem contDiff_unitBallBall (hr : 0 < r) : ContDiff ℝ n (unitBallBall c r hr) 
Arguments: {n : ℕ∞} {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {c : E} {r : ℝ} (hr : 0 < r)
Goal: ContDiff ℝ n ↑(PartialHomeomorph.unitBallBall c r hr)
-- PartialHomeomorph.contDiff_unitBallBall_symm
theorem contDiff_unitBallBall_symm (hr : 0 < r) : ContDiff ℝ n (unitBallBall c r hr).symm 
Arguments: {n : ℕ∞} {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {c : E} {r : ℝ} (hr : 0 < r)
Goal: ContDiff ℝ n ↑(PartialHomeomorph.unitBallBall c r hr).symm
-- PartialHomeomorph.contDiff_univBall
theorem contDiff_univBall : ContDiff ℝ n (univBall c r) 
Arguments: {n : ℕ∞} {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {c : E} {r : ℝ}
Goal: ContDiff ℝ n ↑(PartialHomeomorph.univBall c r)
-- PartialHomeomorph.contDiffOn_univBall_symm
theorem contDiffOn_univBall_symm :
    ContDiffOn ℝ n (univBall c r).symm (ball c r) 
Arguments: {n : ℕ∞} {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {c : E} {r : ℝ}
Goal: ContDiffOn ℝ n (↑(PartialHomeomorph.univBall c r).symm) (Metric.ball c r)
-- BoxIntegral.Box.lower_lt_upper
structure Box (ι : Type*) where
    (lower upper : ι → ℝ)
    lower_lt_upper : ∀ i, lower i < upper i
Arguments: {ι : Type u_2} (self : BoxIntegral.Box ι) (i : ι)
Goal: self.lower i < self.upper i
-- BoxIntegral.Box.lower_le_upper
theorem lower_le_upper : I.lower ≤ I.upper 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: I.lower ≤ I.upper
-- BoxIntegral.Box.lower_ne_upper
theorem lower_ne_upper (i) : I.lower i ≠ I.upper i 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (i : ι)
Goal: I.lower i ≠ I.upper i
-- BoxIntegral.Box.mem_mk
@[simp]
theorem mem_mk {l u x : ι → ℝ} {H} : x ∈ mk l u H ↔ ∀ i, x i ∈ Ioc (l i) (u i) 
Arguments: {ι : Type u_1} {l u x : ι → ℝ} {H : ∀ (i : ι), l i < u i}
Goal: x ∈ { lower := l, upper := u, lower_lt_upper := H } ↔ ∀ (i : ι), x i ∈ Set.Ioc (l i) (u i)
-- BoxIntegral.Box.mem_coe
@[simp, norm_cast]
theorem mem_coe : x ∈ (I : Set (ι → ℝ)) ↔ x ∈ I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) {x : ι → ℝ}
Goal: x ∈ ↑I ↔ x ∈ I
-- BoxIntegral.Box.mem_def
theorem mem_def : x ∈ I ↔ ∀ i, x i ∈ Ioc (I.lower i) (I.upper i) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) {x : ι → ℝ}
Goal: x ∈ I ↔ ∀ (i : ι), x i ∈ Set.Ioc (I.lower i) (I.upper i)
-- BoxIntegral.Box.mem_univ_Ioc
theorem mem_univ_Ioc {I : Box ι} : (x ∈ pi univ fun i ↦ Ioc (I.lower i) (I.upper i)) ↔ x ∈ I 
Arguments: {ι : Type u_1} {x : ι → ℝ} {I : BoxIntegral.Box ι}
Goal: (x ∈ Set.univ.pi fun i => Set.Ioc (I.lower i) (I.upper i)) ↔ x ∈ I
-- BoxIntegral.Box.coe_eq_pi
theorem coe_eq_pi : (I : Set (ι → ℝ)) = pi univ fun i ↦ Ioc (I.lower i) (I.upper i) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: ↑I = Set.univ.pi fun i => Set.Ioc (I.lower i) (I.upper i)
-- BoxIntegral.Box.upper_mem
@[simp]
theorem upper_mem : I.upper ∈ I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: I.upper ∈ I
-- BoxIntegral.Box.exists_mem
theorem exists_mem : ∃ x, x ∈ I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: ∃ x, x ∈ I
-- BoxIntegral.Box.nonempty_coe
theorem nonempty_coe : Set.Nonempty (I : Set (ι → ℝ)) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: (↑I).Nonempty
-- BoxIntegral.Box.coe_ne_empty
@[simp]
theorem coe_ne_empty : (I : Set (ι → ℝ)) ≠ ∅ 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: ↑I ≠ ∅
-- BoxIntegral.Box.empty_ne_coe
@[simp]
theorem empty_ne_coe : ∅ ≠ (I : Set (ι → ℝ)) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: ∅ ≠ ↑I
-- BoxIntegral.Box.le_def
theorem le_def : I ≤ J ↔ ∀ x ∈ I, x ∈ J 
Arguments: {ι : Type u_1} (I J : BoxIntegral.Box ι)
Goal: I ≤ J ↔ ∀ x ∈ I, x ∈ J
-- BoxIntegral.Box.le_TFAE
theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
    Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] 
Arguments: {ι : Type u_1} (I J : BoxIntegral.Box ι)
Goal: [I ≤ J, ↑I ⊆ ↑J, Set.Icc I.lower I.upper ⊆ Set.Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE
-- BoxIntegral.Box.coe_subset_coe
@[simp, norm_cast]
theorem coe_subset_coe : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι}
Goal: ↑I ⊆ ↑J ↔ I ≤ J
-- BoxIntegral.Box.le_iff_bounds
theorem le_iff_bounds : I ≤ J ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι}
Goal: I ≤ J ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
-- BoxIntegral.Box.injective_coe
theorem injective_coe : Injective ((↑) : Box ι → Set (ι → ℝ)) 
Arguments: {ι : Type u_1}
Goal: Function.Injective BoxIntegral.Box.toSet
-- BoxIntegral.Box.coe_inj
@[simp, norm_cast]
theorem coe_inj : (I : Set (ι → ℝ)) = J ↔ I = J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι}
Goal: ↑I = ↑J ↔ I = J
-- BoxIntegral.Box.ext
@[ext]
theorem ext (H : ∀ x, x ∈ I ↔ x ∈ J) : I = J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (H : ∀ (x : ι → ℝ), x ∈ I ↔ x ∈ J)
Goal: I = J
-- BoxIntegral.Box.ne_of_disjoint_coe
theorem ne_of_disjoint_coe (h : Disjoint (I : Set (ι → ℝ)) J) : I ≠ J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (h : Disjoint ↑I ↑J)
Goal: I ≠ J
-- BoxIntegral.Box.Icc_def
theorem Icc_def : Box.Icc I = Icc I.lower I.upper 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι}
Goal: BoxIntegral.Box.Icc I = Set.Icc I.lower I.upper
-- BoxIntegral.Box.upper_mem_Icc
@[simp]
theorem upper_mem_Icc (I : Box ι) : I.upper ∈ Box.Icc I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: I.upper ∈ BoxIntegral.Box.Icc I
-- BoxIntegral.Box.lower_mem_Icc
@[simp]
theorem lower_mem_Icc (I : Box ι) : I.lower ∈ Box.Icc I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: I.lower ∈ BoxIntegral.Box.Icc I
-- BoxIntegral.Box.isCompact_Icc
protected theorem isCompact_Icc (I : Box ι) : IsCompact (Box.Icc I) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: IsCompact (BoxIntegral.Box.Icc I)
-- BoxIntegral.Box.Icc_eq_pi
theorem Icc_eq_pi : Box.Icc I = pi univ fun i ↦ Icc (I.lower i) (I.upper i) 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι}
Goal: BoxIntegral.Box.Icc I = Set.univ.pi fun i => Set.Icc (I.lower i) (I.upper i)
-- BoxIntegral.Box.le_iff_Icc
theorem le_iff_Icc : I ≤ J ↔ Box.Icc I ⊆ Box.Icc J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι}
Goal: I ≤ J ↔ BoxIntegral.Box.Icc I ⊆ BoxIntegral.Box.Icc J
-- BoxIntegral.Box.antitone_lower
theorem antitone_lower : Antitone fun I : Box ι ↦ I.lower 
Arguments: {ι : Type u_1}
Goal: Antitone fun I => I.lower
-- BoxIntegral.Box.monotone_upper
theorem monotone_upper : Monotone fun I : Box ι ↦ I.upper 
Arguments: {ι : Type u_1}
Goal: Monotone fun I => I.upper
-- BoxIntegral.Box.coe_subset_Icc
theorem coe_subset_Icc : ↑I ⊆ Box.Icc I 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι}
Goal: ↑I ⊆ BoxIntegral.Box.Icc I
-- BoxIntegral.Box.coe_bot
@[simp, norm_cast]
theorem coe_bot : ((⊥ : WithBot (Box ι)) : Set (ι → ℝ)) = ∅ 
Arguments: {ι : Type u_1}
Goal: ↑⊥ = ∅
-- BoxIntegral.Box.coe_coe
@[simp, norm_cast]
theorem coe_coe : ((I : WithBot (Box ι)) : Set (ι → ℝ)) = I 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι}
Goal: ↑↑I = ↑I
-- BoxIntegral.Box.isSome_iff
theorem isSome_iff : ∀ {I : WithBot (Box ι)}, I.isSome ↔ (I : Set (ι → ℝ)).Nonempty
  
Arguments: {ι : Type u_1} {I : WithBot (BoxIntegral.Box ι)}
Goal: Option.isSome I = true ↔ (↑I).Nonempty
-- BoxIntegral.Box.biUnion_coe_eq_coe
theorem biUnion_coe_eq_coe (I : WithBot (Box ι)) :
    ⋃ (J : Box ι) (_ : ↑J = I), (J : Set (ι → ℝ)) = I 
Arguments: {ι : Type u_1} (I : WithBot (BoxIntegral.Box ι))
Goal: ⋃ J, ⋃ (_ : ↑J = I), ↑J = ↑I
-- BoxIntegral.Box.withBotCoe_subset_iff
@[simp, norm_cast]
theorem withBotCoe_subset_iff {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J 
Arguments: {ι : Type u_1} {I J : WithBot (BoxIntegral.Box ι)}
Goal: ↑I ⊆ ↑J ↔ I ≤ J
-- BoxIntegral.Box.withBotCoe_inj
@[simp, norm_cast]
theorem withBotCoe_inj {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) = J ↔ I = J 
Arguments: {ι : Type u_1} {I J : WithBot (BoxIntegral.Box ι)}
Goal: ↑I = ↑J ↔ I = J
-- BoxIntegral.Box.mk'_eq_bot
@[simp]
theorem mk'_eq_bot {l u : ι → ℝ} : mk' l u = ⊥ ↔ ∃ i, u i ≤ l i 
Arguments: {ι : Type u_1} {l u : ι → ℝ}
Goal: BoxIntegral.Box.mk' l u = ⊥ ↔ ∃ i, u i ≤ l i
-- BoxIntegral.Box.mk'_eq_coe
@[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {l u : ι → ℝ}
Goal: BoxIntegral.Box.mk' l u = ↑I ↔ l = I.lower ∧ u = I.upper
-- BoxIntegral.Box.coe_mk'
@[simp]
theorem coe_mk' (l u : ι → ℝ) : (mk' l u : Set (ι → ℝ)) = pi univ fun i ↦ Ioc (l i) (u i) 
Arguments: {ι : Type u_1} (l u : ι → ℝ)
Goal: ↑(BoxIntegral.Box.mk' l u) = Set.univ.pi fun i => Set.Ioc (l i) (u i)
-- BoxIntegral.Box.coe_inf
@[simp]
theorem coe_inf (I J : WithBot (Box ι)) : (↑(I ⊓ J) : Set (ι → ℝ)) = (I : Set _) ∩ J 
Arguments: {ι : Type u_1} (I J : WithBot (BoxIntegral.Box ι))
Goal: ↑(I ⊓ J) = ↑I ∩ ↑J
-- BoxIntegral.Box.disjoint_withBotCoe
@[simp, norm_cast]
theorem disjoint_withBotCoe {I J : WithBot (Box ι)} :
    Disjoint (I : Set (ι → ℝ)) J ↔ Disjoint I J 
Arguments: {ι : Type u_1} {I J : WithBot (BoxIntegral.Box ι)}
Goal: Disjoint ↑I ↑J ↔ Disjoint I J
-- BoxIntegral.Box.disjoint_coe
theorem disjoint_coe : Disjoint (I : WithBot (Box ι)) J ↔ Disjoint (I : Set (ι → ℝ)) J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι}
Goal: Disjoint ↑I ↑J ↔ Disjoint ↑I ↑J
-- BoxIntegral.Box.not_disjoint_coe_iff_nonempty_inter
theorem not_disjoint_coe_iff_nonempty_inter :
    ¬Disjoint (I : WithBot (Box ι)) J ↔ (I ∩ J : Set (ι → ℝ)).Nonempty 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι}
Goal: ¬Disjoint ↑I ↑J ↔ (↑I ∩ ↑J).Nonempty
-- BoxIntegral.Box.face_lower
@[simps (config := { simpRhs := true })]
def face {n} (I : Box (Fin (n + 1))) (i : Fin (n + 1)) : Box (Fin n) 
Arguments: {n : ℕ} (I : BoxIntegral.Box (Fin (n + 1))) (i : Fin (n + 1))
Goal: ∀ (a : Fin n), (I.face i).lower a = I.lower (i.succAbove a)
-- BoxIntegral.Box.face_upper
@[simps (config := { simpRhs := true })]
def face {n} (I : Box (Fin (n + 1))) (i : Fin (n + 1)) : Box (Fin n) 
Arguments: {n : ℕ} (I : BoxIntegral.Box (Fin (n + 1))) (i : Fin (n + 1))
Goal: ∀ (a : Fin n), (I.face i).upper a = I.upper (i.succAbove a)
-- BoxIntegral.Box.face_mk
@[simp]
theorem face_mk {n} (l u : Fin (n + 1) → ℝ) (h : ∀ i, l i < u i) (i : Fin (n + 1)) :
    face ⟨l, u, h⟩ i = ⟨l ∘ Fin.succAbove i, u ∘ Fin.succAbove i, fun _ ↦ h _⟩ 
Arguments: {n : ℕ} (l u : Fin (n + 1) → ℝ) (h : ∀ (i : Fin (n + 1)), l i < u i) (i : Fin (n + 1))
Goal: { lower := l, upper := u, lower_lt_upper := h }.face i =
  { lower := l ∘ i.succAbove, upper := u ∘ i.succAbove, lower_lt_upper := ⋯ }
-- BoxIntegral.Box.face_mono
@[mono]
theorem face_mono {n} {I J : Box (Fin (n + 1))} (h : I ≤ J) (i : Fin (n + 1)) :
    face I i ≤ face J i 
Arguments: {n : ℕ} {I J : BoxIntegral.Box (Fin (n + 1))} (h : I ≤ J) (i : Fin (n + 1))
Goal: I.face i ≤ J.face i
-- BoxIntegral.Box.monotone_face
theorem monotone_face {n} (i : Fin (n + 1)) : Monotone fun I ↦ face I i 
Arguments: {n : ℕ} (i : Fin (n + 1))
Goal: Monotone fun I => I.face i
-- BoxIntegral.Box.mapsTo_insertNth_face_Icc
theorem mapsTo_insertNth_face_Icc {n} (I : Box (Fin (n + 1))) {i : Fin (n + 1)} {x : ℝ}
    (hx : x ∈ Icc (I.lower i) (I.upper i)) :
    MapsTo (i.insertNth x) (Box.Icc (I.face i)) (Box.Icc I) 
Arguments: {n : ℕ} (I : BoxIntegral.Box (Fin (n + 1))) {i : Fin (n + 1)} {x : ℝ} (hx : x ∈ Set.Icc (I.lower i) (I.upper i))
Goal: Set.MapsTo (i.insertNth x) (BoxIntegral.Box.Icc (I.face i)) (BoxIntegral.Box.Icc I)
-- BoxIntegral.Box.mapsTo_insertNth_face
theorem mapsTo_insertNth_face {n} (I : Box (Fin (n + 1))) {i : Fin (n + 1)} {x : ℝ}
    (hx : x ∈ Ioc (I.lower i) (I.upper i)) :
    MapsTo (i.insertNth x) (I.face i : Set (_ → _)) (I : Set (_ → _)) 
Arguments: {n : ℕ} (I : BoxIntegral.Box (Fin (n + 1))) {i : Fin (n + 1)} {x : ℝ} (hx : x ∈ Set.Ioc (I.lower i) (I.upper i))
Goal: Set.MapsTo (i.insertNth x) ↑(I.face i) ↑I
-- BoxIntegral.Box.continuousOn_face_Icc
theorem continuousOn_face_Icc {X} [TopologicalSpace X] {n} {f : (Fin (n + 1) → ℝ) → X}
    {I : Box (Fin (n + 1))} (h : ContinuousOn f (Box.Icc I)) {i : Fin (n + 1)} {x : ℝ}
    (hx : x ∈ Icc (I.lower i) (I.upper i)) :
    ContinuousOn (f ∘ i.insertNth x) (Box.Icc (I.face i)) 
Arguments: {X : Type u_2} [TopologicalSpace X] {n : ℕ} {f : (Fin (n + 1) → ℝ) → X} {I : BoxIntegral.Box (Fin (n + 1))} (h : ContinuousOn f (BoxIntegral.Box.Icc I)) {i : Fin (n + 1)} {x : ℝ} (hx : x ∈ Set.Icc (I.lower i) (I.upper i))
Goal: ContinuousOn (f ∘ i.insertNth x) (BoxIntegral.Box.Icc (I.face i))
-- BoxIntegral.Box.Ioo_subset_coe
theorem Ioo_subset_coe (I : Box ι) : Box.Ioo I ⊆ I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: BoxIntegral.Box.Ioo I ⊆ ↑I
-- BoxIntegral.Box.Ioo_subset_Icc
protected theorem Ioo_subset_Icc (I : Box ι) : Box.Ioo I ⊆ Box.Icc I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: BoxIntegral.Box.Ioo I ⊆ BoxIntegral.Box.Icc I
-- BoxIntegral.Box.iUnion_Ioo_of_tendsto
theorem iUnion_Ioo_of_tendsto [Finite ι] {I : Box ι} {J : ℕ → Box ι} (hJ : Monotone J)
    (hl : Tendsto (lower ∘ J) atTop (𝓝 I.lower)) (hu : Tendsto (upper ∘ J) atTop (𝓝 I.upper)) :
    ⋃ n, Box.Ioo (J n) = Box.Ioo I 
Arguments: {ι : Type u_1} [Finite ι] {I : BoxIntegral.Box ι} {J : ℕ → BoxIntegral.Box ι} (hJ : Monotone J) (hl : Filter.Tendsto (BoxIntegral.Box.lower ∘ J) Filter.atTop (𝓝 I.lower)) (hu : Filter.Tendsto (BoxIntegral.Box.upper ∘ J) Filter.atTop (𝓝 I.upper))
Goal: ⋃ n, BoxIntegral.Box.Ioo (J n) = BoxIntegral.Box.Ioo I
-- BoxIntegral.Box.exists_seq_mono_tendsto
theorem exists_seq_mono_tendsto (I : Box ι) :
    ∃ J : ℕ →o Box ι,
      (∀ n, Box.Icc (J n) ⊆ Box.Ioo I) ∧
        Tendsto (lower ∘ J) atTop (𝓝 I.lower) ∧ Tendsto (upper ∘ J) atTop (𝓝 I.upper) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: ∃ J,
  (∀ (n : ℕ), BoxIntegral.Box.Icc (J n) ⊆ BoxIntegral.Box.Ioo I) ∧
    Filter.Tendsto (BoxIntegral.Box.lower ∘ ⇑J) Filter.atTop (𝓝 I.lower) ∧
      Filter.Tendsto (BoxIntegral.Box.upper ∘ ⇑J) Filter.atTop (𝓝 I.upper)
-- BoxIntegral.Box.distortion_eq_of_sub_eq_div
theorem distortion_eq_of_sub_eq_div {I J : Box ι} {r : ℝ}
    (h : ∀ i, I.upper i - I.lower i = (J.upper i - J.lower i) / r) :
    distortion I = distortion J 
Arguments: {ι : Type u_1} [Fintype ι] {I J : BoxIntegral.Box ι} {r : ℝ} (h : ∀ (i : ι), I.upper i - I.lower i = (J.upper i - J.lower i) / r)
Goal: I.distortion = J.distortion
-- BoxIntegral.Box.nndist_le_distortion_mul
theorem nndist_le_distortion_mul (I : Box ι) (i : ι) :
    nndist I.lower I.upper ≤ I.distortion * nndist (I.lower i) (I.upper i) 
Arguments: {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) (i : ι)
Goal: nndist I.lower I.upper ≤ I.distortion * nndist (I.lower i) (I.upper i)
-- BoxIntegral.Box.dist_le_distortion_mul
theorem dist_le_distortion_mul (I : Box ι) (i : ι) :
    dist I.lower I.upper ≤ I.distortion * (I.upper i - I.lower i) 
Arguments: {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) (i : ι)
Goal: dist I.lower I.upper ≤ ↑I.distortion * (I.upper i - I.lower i)
-- BoxIntegral.Box.diam_Icc_le_of_distortion_le
theorem diam_Icc_le_of_distortion_le (I : Box ι) (i : ι) {c : ℝ≥0} (h : I.distortion ≤ c) :
    diam (Box.Icc I) ≤ c * (I.upper i - I.lower i) 
Arguments: {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) (i : ι) {c : ℝ≥0} (h : I.distortion ≤ c)
Goal: Metric.diam (BoxIntegral.Box.Icc I) ≤ ↑c * (I.upper i - I.lower i)
-- BoxIntegral.Box.mem_splitCenterBox
theorem mem_splitCenterBox {s : Set ι} {y : ι → ℝ} :
    y ∈ I.splitCenterBox s ↔ y ∈ I ∧ ∀ i, (I.lower i + I.upper i) / 2 < y i ↔ i ∈ s 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {s : Set ι} {y : ι → ℝ}
Goal: y ∈ I.splitCenterBox s ↔ y ∈ I ∧ ∀ (i : ι), (I.lower i + I.upper i) / 2 < y i ↔ i ∈ s
-- BoxIntegral.Box.splitCenterBox_le
theorem splitCenterBox_le (I : Box ι) (s : Set ι) : I.splitCenterBox s ≤ I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (s : Set ι)
Goal: I.splitCenterBox s ≤ I
-- BoxIntegral.Box.disjoint_splitCenterBox
theorem disjoint_splitCenterBox (I : Box ι) {s t : Set ι} (h : s ≠ t) :
    Disjoint (I.splitCenterBox s : Set (ι → ℝ)) (I.splitCenterBox t) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) {s t : Set ι} (h : s ≠ t)
Goal: Disjoint ↑(I.splitCenterBox s) ↑(I.splitCenterBox t)
-- BoxIntegral.Box.injective_splitCenterBox
theorem injective_splitCenterBox (I : Box ι) : Injective I.splitCenterBox 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: Function.Injective I.splitCenterBox
-- BoxIntegral.Box.exists_mem_splitCenterBox
@[simp]
theorem exists_mem_splitCenterBox {I : Box ι} {x : ι → ℝ} : (∃ s, x ∈ I.splitCenterBox s) ↔ x ∈ I 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {x : ι → ℝ}
Goal: (∃ s, x ∈ I.splitCenterBox s) ↔ x ∈ I
-- BoxIntegral.Box.splitCenterBoxEmb_apply
@[simps]
def splitCenterBoxEmb (I : Box ι) : Set ι ↪ Box ι 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (s : Set ι)
Goal: I.splitCenterBoxEmb s = I.splitCenterBox s
-- BoxIntegral.Box.iUnion_coe_splitCenterBox
@[simp]
theorem iUnion_coe_splitCenterBox (I : Box ι) : ⋃ s, (I.splitCenterBox s : Set (ι → ℝ)) = I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: ⋃ s, ↑(I.splitCenterBox s) = ↑I
-- BoxIntegral.Box.upper_sub_lower_splitCenterBox
@[simp]
theorem upper_sub_lower_splitCenterBox (I : Box ι) (s : Set ι) (i : ι) :
    (I.splitCenterBox s).upper i - (I.splitCenterBox s).lower i = (I.upper i - I.lower i) / 2 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (s : Set ι) (i : ι)
Goal: (I.splitCenterBox s).upper i - (I.splitCenterBox s).lower i = (I.upper i - I.lower i) / 2
-- BoxIntegral.Box.subbox_induction_on'
@[elab_as_elim]
theorem subbox_induction_on' {p : Box ι → Prop} (I : Box ι)
    (H_ind : ∀ J ≤ I, (∀ s, p (splitCenterBox J s)) → p J)
    (H_nhds : ∀ z ∈ Box.Icc I, ∃ U ∈ 𝓝[Box.Icc I] z, ∀ J ≤ I, ∀ (m : ℕ), z ∈ Box.Icc J →
      Box.Icc J ⊆ U → (∀ i, J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J) :
    p I 
Arguments: {ι : Type u_1} {p : BoxIntegral.Box ι → Prop} (I : BoxIntegral.Box ι) (H_ind : ∀ J ≤ I, (∀ (s : Set ι), p (J.splitCenterBox s)) → p J) (H_nhds :
  ∀ z ∈ BoxIntegral.Box.Icc I,
    ∃ U ∈ 𝓝[BoxIntegral.Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ BoxIntegral.Box.Icc J →
            BoxIntegral.Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J)
Goal: p I
-- unitary_iff_isStarNormal_and_spectrum_subset_circle
lemma unitary_iff_isStarNormal_and_spectrum_subset_circle {u : A} :
    u ∈ unitary A ↔ IsStarNormal u ∧ spectrum ℂ u ⊆ circle 
Arguments: {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] {u : A}
Goal: u ∈ unitary A ↔ IsStarNormal u ∧ spectrum ℂ u ⊆ ↑circle