ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContDiff.div
theorem ContDiff.div [CompleteSpace 𝕜] {f g : E → 𝕜} {n} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g)
    (h0 : ∀ x, g x ≠ 0) : ContDiff 𝕜 n fun x => f x / g x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] {f g : E → 𝕜} {n : ℕ∞} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) (h0 : ∀ (x : E), g x ≠ 0)
Goal: ContDiff 𝕜 n fun x => f x / g x
-- contDiffAt_map_inverse
theorem contDiffAt_map_inverse [CompleteSpace E] (e : E ≃L[𝕜] F) :
    ContDiffAt 𝕜 n inverse (e : E →L[𝕜] F) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} [CompleteSpace E] (e : E ≃L[𝕜] F)
Goal: ContDiffAt 𝕜 n ContinuousLinearMap.inverse ↑e
-- PartialHomeomorph.contDiffAt_symm
theorem PartialHomeomorph.contDiffAt_symm [CompleteSpace E] (f : PartialHomeomorph E F)
    {f₀' : E ≃L[𝕜] F} {a : F} (ha : a ∈ f.target)
    (hf₀' : HasFDerivAt f (f₀' : E →L[𝕜] F) (f.symm a)) (hf : ContDiffAt 𝕜 n f (f.symm a)) :
    ContDiffAt 𝕜 n f.symm a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} [CompleteSpace E] (f : PartialHomeomorph E F) {f₀' : E ≃L[𝕜] F} {a : F} (ha : a ∈ f.target) (hf₀' : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)) (hf : ContDiffAt 𝕜 n (↑f) (↑f.symm a))
Goal: ContDiffAt 𝕜 n (↑f.symm) a
-- Homeomorph.contDiff_symm
theorem Homeomorph.contDiff_symm [CompleteSpace E] (f : E ≃ₜ F) {f₀' : E → E ≃L[𝕜] F}
    (hf₀' : ∀ a, HasFDerivAt f (f₀' a : E →L[𝕜] F) a) (hf : ContDiff 𝕜 n (f : E → F)) :
    ContDiff 𝕜 n (f.symm : F → E) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} [CompleteSpace E] (f : E ≃ₜ F) {f₀' : E → E ≃L[𝕜] F} (hf₀' : ∀ (a : E), HasFDerivAt (⇑f) (↑(f₀' a)) a) (hf : ContDiff 𝕜 n ⇑f)
Goal: ContDiff 𝕜 n ⇑f.symm
-- PartialHomeomorph.contDiffAt_symm_deriv
theorem PartialHomeomorph.contDiffAt_symm_deriv [CompleteSpace 𝕜] (f : PartialHomeomorph 𝕜 𝕜)
    {f₀' a : 𝕜} (h₀ : f₀' ≠ 0) (ha : a ∈ f.target) (hf₀' : HasDerivAt f f₀' (f.symm a))
    (hf : ContDiffAt 𝕜 n f (f.symm a)) : ContDiffAt 𝕜 n f.symm a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : ℕ∞} [CompleteSpace 𝕜] (f : PartialHomeomorph 𝕜 𝕜) {f₀' a : 𝕜} (h₀ : f₀' ≠ 0) (ha : a ∈ f.target) (hf₀' : HasDerivAt (↑f) f₀' (↑f.symm a)) (hf : ContDiffAt 𝕜 n (↑f) (↑f.symm a))
Goal: ContDiffAt 𝕜 n (↑f.symm) a
-- Homeomorph.contDiff_symm_deriv
theorem Homeomorph.contDiff_symm_deriv [CompleteSpace 𝕜] (f : 𝕜 ≃ₜ 𝕜) {f' : 𝕜 → 𝕜}
    (h₀ : ∀ x, f' x ≠ 0) (hf' : ∀ x, HasDerivAt f (f' x) x) (hf : ContDiff 𝕜 n (f : 𝕜 → 𝕜)) :
    ContDiff 𝕜 n (f.symm : 𝕜 → 𝕜) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : ℕ∞} [CompleteSpace 𝕜] (f : 𝕜 ≃ₜ 𝕜) {f' : 𝕜 → 𝕜} (h₀ : ∀ (x : 𝕜), f' x ≠ 0) (hf' : ∀ (x : 𝕜), HasDerivAt (⇑f) (f' x) x) (hf : ContDiff 𝕜 n ⇑f)
Goal: ContDiff 𝕜 n ⇑f.symm
-- PartialHomeomorph.restrContDiff_source
@[simps! apply symm_apply source target]
def restrContDiff (f : PartialHomeomorph E F) (n : ℕ) : PartialHomeomorph E F 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : PartialHomeomorph E F) (n : ℕ)
Goal: (PartialHomeomorph.restrContDiff 𝕜 f n).source =
  f.source ∩ {x | ContDiffAt 𝕜 (↑n) (↑f) x ∧ ContDiffAt 𝕜 (↑n) (↑f.symm) (↑f x)}
-- PartialHomeomorph.restrContDiff_apply
@[simps! apply symm_apply source target]
def restrContDiff (f : PartialHomeomorph E F) (n : ℕ) : PartialHomeomorph E F 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : PartialHomeomorph E F) (n : ℕ)
Goal: ∀ (a : E), ↑(PartialHomeomorph.restrContDiff 𝕜 f n) a = ↑f a
-- PartialHomeomorph.restrContDiff_target
@[simps! apply symm_apply source target]
def restrContDiff (f : PartialHomeomorph E F) (n : ℕ) : PartialHomeomorph E F 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : PartialHomeomorph E F) (n : ℕ)
Goal: (PartialHomeomorph.restrContDiff 𝕜 f n).target =
  f.target ∩ {y | ContDiffAt 𝕜 (↑n) (↑f.symm) y ∧ ContDiffAt 𝕜 (↑n) (↑f) (↑f.symm y)}
-- PartialHomeomorph.restrContDiff_symm_apply
@[simps! apply symm_apply source target]
def restrContDiff (f : PartialHomeomorph E F) (n : ℕ) : PartialHomeomorph E F 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : PartialHomeomorph E F) (n : ℕ)
Goal: ∀ (a : F), ↑(PartialHomeomorph.restrContDiff 𝕜 f n).symm a = ↑f.symm a
-- PartialHomeomorph.contDiffOn_restrContDiff_source
lemma contDiffOn_restrContDiff_source (f : PartialHomeomorph E F) (n : ℕ) :
    ContDiffOn 𝕜 n f (f.restrContDiff 𝕜 n).source 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : PartialHomeomorph E F) (n : ℕ)
Goal: ContDiffOn 𝕜 (↑n) (↑f) (PartialHomeomorph.restrContDiff 𝕜 f n).source
-- PartialHomeomorph.contDiffOn_restrContDiff_target
lemma contDiffOn_restrContDiff_target (f : PartialHomeomorph E F) (n : ℕ) :
    ContDiffOn 𝕜 n f.symm (f.restrContDiff 𝕜 n).target 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : PartialHomeomorph E F) (n : ℕ)
Goal: ContDiffOn 𝕜 (↑n) (↑f.symm) (PartialHomeomorph.restrContDiff 𝕜 f n).target
-- contDiffOn_succ_iff_derivWithin
theorem contDiffOn_succ_iff_derivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s₂) :
    ContDiffOn 𝕜 (n + 1 : ℕ) f₂ s₂ ↔
      DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 n (derivWithin f₂ s₂) s₂ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₂ : 𝕜 → F} {s₂ : Set 𝕜} {n : ℕ} (hs : UniqueDiffOn 𝕜 s₂)
Goal: ContDiffOn 𝕜 (↑(n + 1)) f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (derivWithin f₂ s₂) s₂
-- contDiffOn_succ_iff_deriv_of_isOpen
theorem contDiffOn_succ_iff_deriv_of_isOpen {n : ℕ} (hs : IsOpen s₂) :
    ContDiffOn 𝕜 (n + 1 : ℕ) f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 n (deriv f₂) s₂ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₂ : 𝕜 → F} {s₂ : Set 𝕜} {n : ℕ} (hs : IsOpen s₂)
Goal: ContDiffOn 𝕜 (↑(n + 1)) f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (deriv f₂) s₂
-- contDiffOn_top_iff_derivWithin
theorem contDiffOn_top_iff_derivWithin (hs : UniqueDiffOn 𝕜 s₂) :
    ContDiffOn 𝕜 ∞ f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 ∞ (derivWithin f₂ s₂) s₂ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₂ : 𝕜 → F} {s₂ : Set 𝕜} (hs : UniqueDiffOn 𝕜 s₂)
Goal: ContDiffOn 𝕜 ⊤ f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 ⊤ (derivWithin f₂ s₂) s₂
-- contDiffOn_top_iff_deriv_of_isOpen
theorem contDiffOn_top_iff_deriv_of_isOpen (hs : IsOpen s₂) :
    ContDiffOn 𝕜 ∞ f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 ∞ (deriv f₂) s₂ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₂ : 𝕜 → F} {s₂ : Set 𝕜} (hs : IsOpen s₂)
Goal: ContDiffOn 𝕜 ⊤ f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 ⊤ (deriv f₂) s₂
-- ContDiffOn.derivWithin
protected theorem ContDiffOn.derivWithin (hf : ContDiffOn 𝕜 n f₂ s₂) (hs : UniqueDiffOn 𝕜 s₂)
    (hmn : m + 1 ≤ n) : ContDiffOn 𝕜 m (derivWithin f₂ s₂) s₂ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m n : ℕ∞} {f₂ : 𝕜 → F} {s₂ : Set 𝕜} (hf : ContDiffOn 𝕜 n f₂ s₂) (hs : UniqueDiffOn 𝕜 s₂) (hmn : m + 1 ≤ n)
Goal: ContDiffOn 𝕜 m (derivWithin f₂ s₂) s₂
-- ContDiffOn.deriv_of_isOpen
theorem ContDiffOn.deriv_of_isOpen (hf : ContDiffOn 𝕜 n f₂ s₂) (hs : IsOpen s₂) (hmn : m + 1 ≤ n) :
    ContDiffOn 𝕜 m (deriv f₂) s₂ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m n : ℕ∞} {f₂ : 𝕜 → F} {s₂ : Set 𝕜} (hf : ContDiffOn 𝕜 n f₂ s₂) (hs : IsOpen s₂) (hmn : m + 1 ≤ n)
Goal: ContDiffOn 𝕜 m (deriv f₂) s₂
-- ContDiffOn.continuousOn_derivWithin
theorem ContDiffOn.continuousOn_derivWithin (h : ContDiffOn 𝕜 n f₂ s₂) (hs : UniqueDiffOn 𝕜 s₂)
    (hn : 1 ≤ n) : ContinuousOn (derivWithin f₂ s₂) s₂ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f₂ : 𝕜 → F} {s₂ : Set 𝕜} (h : ContDiffOn 𝕜 n f₂ s₂) (hs : UniqueDiffOn 𝕜 s₂) (hn : 1 ≤ n)
Goal: ContinuousOn (derivWithin f₂ s₂) s₂
-- ContDiffOn.continuousOn_deriv_of_isOpen
theorem ContDiffOn.continuousOn_deriv_of_isOpen (h : ContDiffOn 𝕜 n f₂ s₂) (hs : IsOpen s₂)
    (hn : 1 ≤ n) : ContinuousOn (deriv f₂) s₂ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f₂ : 𝕜 → F} {s₂ : Set 𝕜} (h : ContDiffOn 𝕜 n f₂ s₂) (hs : IsOpen s₂) (hn : 1 ≤ n)
Goal: ContinuousOn (deriv f₂) s₂
-- contDiff_succ_iff_deriv
theorem contDiff_succ_iff_deriv {n : ℕ} :
    ContDiff 𝕜 (n + 1 : ℕ) f₂ ↔ Differentiable 𝕜 f₂ ∧ ContDiff 𝕜 n (deriv f₂) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₂ : 𝕜 → F} {n : ℕ}
Goal: ContDiff 𝕜 (↑(n + 1)) f₂ ↔ Differentiable 𝕜 f₂ ∧ ContDiff 𝕜 (↑n) (deriv f₂)
-- contDiff_one_iff_deriv
theorem contDiff_one_iff_deriv : ContDiff 𝕜 1 f₂ ↔ Differentiable 𝕜 f₂ ∧ Continuous (deriv f₂) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₂ : 𝕜 → F}
Goal: ContDiff 𝕜 1 f₂ ↔ Differentiable 𝕜 f₂ ∧ Continuous (deriv f₂)
-- contDiff_top_iff_deriv
theorem contDiff_top_iff_deriv :
    ContDiff 𝕜 ∞ f₂ ↔ Differentiable 𝕜 f₂ ∧ ContDiff 𝕜 ∞ (deriv f₂) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₂ : 𝕜 → F}
Goal: ContDiff 𝕜 ⊤ f₂ ↔ Differentiable 𝕜 f₂ ∧ ContDiff 𝕜 ⊤ (deriv f₂)
-- ContDiff.continuous_deriv
theorem ContDiff.continuous_deriv (h : ContDiff 𝕜 n f₂) (hn : 1 ≤ n) : Continuous (deriv f₂) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f₂ : 𝕜 → F} (h : ContDiff 𝕜 n f₂) (hn : 1 ≤ n)
Goal: Continuous (deriv f₂)
-- ContDiff.iterate_deriv
theorem ContDiff.iterate_deriv :
    ∀ (n : ℕ) {f₂ : 𝕜 → F}, ContDiff 𝕜 ∞ f₂ → ContDiff 𝕜 ∞ (deriv^[n] f₂)
  
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (n : ℕ) {f₂ : 𝕜 → F}
Goal: ContDiff 𝕜 ⊤ f₂ → ContDiff 𝕜 ⊤ (deriv^[n] f₂)
-- ContDiff.iterate_deriv'
theorem ContDiff.iterate_deriv' (n : ℕ) :
    ∀ (k : ℕ) {f₂ : 𝕜 → F}, ContDiff 𝕜 (n + k : ℕ) f₂ → ContDiff 𝕜 n (deriv^[k] f₂)
  
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (n k : ℕ) {f₂ : 𝕜 → F}
Goal: ContDiff 𝕜 (↑(n + k)) f₂ → ContDiff 𝕜 (↑n) (deriv^[k] f₂)
-- HasFTaylorSeriesUpToOn.restrictScalars
theorem HasFTaylorSeriesUpToOn.restrictScalars (h : HasFTaylorSeriesUpToOn n f p' s) :
    HasFTaylorSeriesUpToOn n f (fun x => (p' x).restrictScalars 𝕜) s 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞} {𝕜' : Type u_3} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {p' : E → FormalMultilinearSeries 𝕜' E F} (h : HasFTaylorSeriesUpToOn n f p' s)
Goal: HasFTaylorSeriesUpToOn n f (fun x => FormalMultilinearSeries.restrictScalars 𝕜 (p' x)) s
-- ContDiffWithinAt.restrict_scalars
theorem ContDiffWithinAt.restrict_scalars (h : ContDiffWithinAt 𝕜' n f s x) :
    ContDiffWithinAt 𝕜 n f s x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} {𝕜' : Type u_3} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] (h : ContDiffWithinAt 𝕜' n f s x)
Goal: ContDiffWithinAt 𝕜 n f s x
-- ContDiffOn.restrict_scalars
theorem ContDiffOn.restrict_scalars (h : ContDiffOn 𝕜' n f s) : ContDiffOn 𝕜 n f s 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞} {𝕜' : Type u_3} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] (h : ContDiffOn 𝕜' n f s)
Goal: ContDiffOn 𝕜 n f s
-- ContDiffAt.restrict_scalars
theorem ContDiffAt.restrict_scalars (h : ContDiffAt 𝕜' n f x) : ContDiffAt 𝕜 n f x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {n : ℕ∞} {𝕜' : Type u_3} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] (h : ContDiffAt 𝕜' n f x)
Goal: ContDiffAt 𝕜 n f x
-- ContDiff.restrict_scalars
theorem ContDiff.restrict_scalars (h : ContDiff 𝕜' n f) : ContDiff 𝕜 n f 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {n : ℕ∞} {𝕜' : Type u_3} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] (h : ContDiff 𝕜' n f)
Goal: ContDiff 𝕜 n f
-- SimpleGraph.Subgraph.Preconnected.coe
protected structure Preconnected (H : G.Subgraph) : Prop where
  protected coe : H.coe.Preconnected
Arguments: {V : Type u} {G : SimpleGraph V} {H : G.Subgraph} (self : H.Preconnected)
Goal: H.coe.Preconnected
-- SimpleGraph.Subgraph.preconnected_iff
protected lemma preconnected_iff {H : G.Subgraph} :
    H.Preconnected ↔ H.coe.Preconnected 
Arguments: {V : Type u} {G : SimpleGraph V} {H : G.Subgraph}
Goal: H.Preconnected ↔ H.coe.Preconnected
-- SimpleGraph.Subgraph.Connected.coe
protected structure Connected (H : G.Subgraph) : Prop where
  protected coe : H.coe.Connected
Arguments: {V : Type u} {G : SimpleGraph V} {H : G.Subgraph} (self : H.Connected)
Goal: H.coe.Connected
-- SimpleGraph.Subgraph.connected_iff'
protected lemma connected_iff' {H : G.Subgraph} :
    H.Connected ↔ H.coe.Connected 
Arguments: {V : Type u} {G : SimpleGraph V} {H : G.Subgraph}
Goal: H.Connected ↔ H.coe.Connected
-- SimpleGraph.Subgraph.connected_iff
protected lemma connected_iff {H : G.Subgraph} :
    H.Connected ↔ H.Preconnected ∧ H.verts.Nonempty 
Arguments: {V : Type u} {G : SimpleGraph V} {H : G.Subgraph}
Goal: H.Connected ↔ H.Preconnected ∧ H.verts.Nonempty
-- SimpleGraph.Subgraph.Connected.preconnected
protected lemma Connected.preconnected {H : G.Subgraph} (h : H.Connected) : H.Preconnected 
Arguments: {V : Type u} {G : SimpleGraph V} {H : G.Subgraph} (h : H.Connected)
Goal: H.Preconnected
-- SimpleGraph.Subgraph.Connected.nonempty
protected lemma Connected.nonempty {H : G.Subgraph} (h : H.Connected) : H.verts.Nonempty 
Arguments: {V : Type u} {G : SimpleGraph V} {H : G.Subgraph} (h : H.Connected)
Goal: H.verts.Nonempty
-- SimpleGraph.Subgraph.singletonSubgraph_connected
theorem singletonSubgraph_connected {v : V} : (G.singletonSubgraph v).Connected 
Arguments: {V : Type u} {G : SimpleGraph V} {v : V}
Goal: (G.singletonSubgraph v).Connected
-- SimpleGraph.Subgraph.subgraphOfAdj_connected
@[simp]
theorem subgraphOfAdj_connected {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).Connected 
Arguments: {V : Type u} {G : SimpleGraph V} {v w : V} (hvw : G.Adj v w)
Goal: (G.subgraphOfAdj hvw).Connected
-- SimpleGraph.Subgraph.top_induce_pair_connected_of_adj
lemma top_induce_pair_connected_of_adj {u v : V} (huv : G.Adj u v) :
    ((⊤ : G.Subgraph).induce {u, v}).Connected 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (huv : G.Adj u v)
Goal: (⊤.induce {u, v}).Connected
-- SimpleGraph.Subgraph.Connected.mono
@[mono]
protected lemma Connected.mono {H H' : G.Subgraph} (hle : H ≤ H') (hv : H.verts = H'.verts)
    (h : H.Connected) : H'.Connected 
Arguments: {V : Type u} {G : SimpleGraph V} {H H' : G.Subgraph} (hle : H ≤ H') (hv : H.verts = H'.verts) (h : H.Connected)
Goal: H'.Connected
-- SimpleGraph.Subgraph.Connected.mono'
protected lemma Connected.mono' {H H' : G.Subgraph}
    (hle : ∀ v w, H.Adj v w → H'.Adj v w) (hv : H.verts = H'.verts)
    (h : H.Connected) : H'.Connected 
Arguments: {V : Type u} {G : SimpleGraph V} {H H' : G.Subgraph} (hle : ∀ (v w : V), H.Adj v w → H'.Adj v w) (hv : H.verts = H'.verts) (h : H.Connected)
Goal: H'.Connected
-- SimpleGraph.Subgraph.Connected.sup
protected lemma Connected.sup {H K : G.Subgraph}
    (hH : H.Connected) (hK : K.Connected) (hn : (H ⊓ K).verts.Nonempty) :
    (H ⊔ K).Connected 
Arguments: {V : Type u} {G : SimpleGraph V} {H K : G.Subgraph} (hH : H.Connected) (hK : K.Connected) (hn : (H ⊓ K).verts.Nonempty)
Goal: (H ⊔ K).Connected
-- SimpleGraph.Walk.toSubgraph_cons_nil_eq_subgraphOfAdj
theorem toSubgraph_cons_nil_eq_subgraphOfAdj (h : G.Adj u v) :
    (cons h nil).toSubgraph = G.subgraphOfAdj h 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (h : G.Adj u v)
Goal: (SimpleGraph.Walk.cons h SimpleGraph.Walk.nil).toSubgraph = G.subgraphOfAdj h
-- SimpleGraph.Walk.mem_verts_toSubgraph
theorem mem_verts_toSubgraph (p : G.Walk u v) : w ∈ p.toSubgraph.verts ↔ w ∈ p.support 
Arguments: {V : Type u} {G : SimpleGraph V} {u v w : V} (p : G.Walk u v)
Goal: w ∈ p.toSubgraph.verts ↔ w ∈ p.support
-- SimpleGraph.Walk.start_mem_verts_toSubgraph
lemma start_mem_verts_toSubgraph (p : G.Walk u v) : u ∈ p.toSubgraph.verts 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (p : G.Walk u v)
Goal: u ∈ p.toSubgraph.verts
-- SimpleGraph.Walk.end_mem_verts_toSubgraph
lemma end_mem_verts_toSubgraph (p : G.Walk u v) : v ∈ p.toSubgraph.verts 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (p : G.Walk u v)
Goal: v ∈ p.toSubgraph.verts
-- SimpleGraph.Walk.verts_toSubgraph
@[simp]
theorem verts_toSubgraph (p : G.Walk u v) : p.toSubgraph.verts = { w | w ∈ p.support } 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (p : G.Walk u v)
Goal: p.toSubgraph.verts = {w | w ∈ p.support}
-- SimpleGraph.Walk.mem_edges_toSubgraph
theorem mem_edges_toSubgraph (p : G.Walk u v) {e : Sym2 V} :
    e ∈ p.toSubgraph.edgeSet ↔ e ∈ p.edges 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (p : G.Walk u v) {e : Sym2 V}
Goal: e ∈ p.toSubgraph.edgeSet ↔ e ∈ p.edges
-- SimpleGraph.Walk.edgeSet_toSubgraph
@[simp]
theorem edgeSet_toSubgraph (p : G.Walk u v) : p.toSubgraph.edgeSet = { e | e ∈ p.edges } 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (p : G.Walk u v)
Goal: p.toSubgraph.edgeSet = {e | e ∈ p.edges}
-- SimpleGraph.Walk.toSubgraph_append
@[simp]
theorem toSubgraph_append (p : G.Walk u v) (q : G.Walk v w) :
    (p.append q).toSubgraph = p.toSubgraph ⊔ q.toSubgraph 
Arguments: {V : Type u} {G : SimpleGraph V} {u v w : V} (p : G.Walk u v) (q : G.Walk v w)
Goal: (p.append q).toSubgraph = p.toSubgraph ⊔ q.toSubgraph
-- SimpleGraph.Walk.toSubgraph_reverse
@[simp]
theorem toSubgraph_reverse (p : G.Walk u v) : p.reverse.toSubgraph = p.toSubgraph 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (p : G.Walk u v)
Goal: p.reverse.toSubgraph = p.toSubgraph
-- SimpleGraph.Walk.toSubgraph_rotate
@[simp]
theorem toSubgraph_rotate [DecidableEq V] (c : G.Walk v v) (h : u ∈ c.support) :
    (c.rotate h).toSubgraph = c.toSubgraph 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} [DecidableEq V] (c : G.Walk v v) (h : u ∈ c.support)
Goal: (c.rotate h).toSubgraph = c.toSubgraph
-- SimpleGraph.Walk.toSubgraph_map
@[simp]
theorem toSubgraph_map (f : G →g G') (p : G.Walk u v) :
    (p.map f).toSubgraph = p.toSubgraph.map f 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} {u v : V} (f : G →g G') (p : G.Walk u v)
Goal: (SimpleGraph.Walk.map f p).toSubgraph = SimpleGraph.Subgraph.map f p.toSubgraph
-- SimpleGraph.Walk.finite_neighborSet_toSubgraph
@[simp]
theorem finite_neighborSet_toSubgraph (p : G.Walk u v) : (p.toSubgraph.neighborSet w).Finite 
Arguments: {V : Type u} {G : SimpleGraph V} {u v w : V} (p : G.Walk u v)
Goal: (p.toSubgraph.neighborSet w).Finite
-- SimpleGraph.Walk.toSubgraph_le_induce_support
lemma toSubgraph_le_induce_support (p : G.Walk u v) :
    p.toSubgraph ≤ (⊤ : G.Subgraph).induce {v | v ∈ p.support} 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (p : G.Walk u v)
Goal: p.toSubgraph ≤ ⊤.induce {v_1 | v_1 ∈ p.support}
-- SimpleGraph.Walk.toSubgraph_adj_getVert
theorem toSubgraph_adj_getVert {u v} (w : G.Walk u v) {i : ℕ} (hi : i < w.length) :
    w.toSubgraph.Adj (w.getVert i) (w.getVert (i + 1)) 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (w : G.Walk u v) {i : ℕ} (hi : i < w.length)
Goal: w.toSubgraph.Adj (w.getVert i) (w.getVert (i + 1))
-- SimpleGraph.Walk.toSubgraph_adj_iff
theorem toSubgraph_adj_iff {u v u' v'} (w : G.Walk u v) :
    w.toSubgraph.Adj u' v' ↔ ∃ i, s(w.getVert i, w.getVert (i + 1)) =
      s(u', v') ∧ i < w.length 
Arguments: {V : Type u} {G : SimpleGraph V} {u v u' v' : V} (w : G.Walk u v)
Goal: w.toSubgraph.Adj u' v' ↔ ∃ i, s(w.getVert i, w.getVert (i + 1)) = s(u', v') ∧ i < w.length
-- SimpleGraph.Walk.toSubgraph_connected
lemma _root_.SimpleGraph.Walk.toSubgraph_connected {u v : V} (p : G.Walk u v) :
    p.toSubgraph.Connected 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (p : G.Walk u v)
Goal: p.toSubgraph.Connected
-- SimpleGraph.Subgraph.induce_union_connected
lemma induce_union_connected {H : G.Subgraph} {s t : Set V}
    (sconn : (H.induce s).Connected) (tconn : (H.induce t).Connected)
    (sintert : (s ⊓ t).Nonempty) :
    (H.induce (s ∪ t)).Connected 
Arguments: {V : Type u} {G : SimpleGraph V} {H : G.Subgraph} {s t : Set V} (sconn : (H.induce s).Connected) (tconn : (H.induce t).Connected) (sintert : (s ⊓ t).Nonempty)
Goal: (H.induce (s ∪ t)).Connected
-- SimpleGraph.Subgraph.Connected.adj_union
lemma Connected.adj_union {H K : G.Subgraph}
    (Hconn : H.Connected) (Kconn : K.Connected) {u v : V} (uH : u ∈ H.verts) (vK : v ∈ K.verts)
    (huv : G.Adj u v) :
    ((⊤ : G.Subgraph).induce {u, v} ⊔ H ⊔ K).Connected 
Arguments: {V : Type u} {G : SimpleGraph V} {H K : G.Subgraph} (Hconn : H.Connected) (Kconn : K.Connected) {u v : V} (uH : u ∈ H.verts) (vK : v ∈ K.verts) (huv : G.Adj u v)
Goal: (⊤.induce {u, v} ⊔ H ⊔ K).Connected
-- SimpleGraph.Subgraph.preconnected_iff_forall_exists_walk_subgraph
lemma preconnected_iff_forall_exists_walk_subgraph (H : G.Subgraph) :
    H.Preconnected ↔ ∀ {u v}, u ∈ H.verts → v ∈ H.verts → ∃ p : G.Walk u v, p.toSubgraph ≤ H 
Arguments: {V : Type u} {G : SimpleGraph V} (H : G.Subgraph)
Goal: H.Preconnected ↔ ∀ {u v : V}, u ∈ H.verts → v ∈ H.verts → ∃ p, p.toSubgraph ≤ H
-- SimpleGraph.Subgraph.connected_iff_forall_exists_walk_subgraph
lemma connected_iff_forall_exists_walk_subgraph (H : G.Subgraph) :
    H.Connected ↔
      H.verts.Nonempty ∧
        ∀ {u v}, u ∈ H.verts → v ∈ H.verts → ∃ p : G.Walk u v, p.toSubgraph ≤ H 
Arguments: {V : Type u} {G : SimpleGraph V} (H : G.Subgraph)
Goal: H.Connected ↔ H.verts.Nonempty ∧ ∀ {u v : V}, u ∈ H.verts → v ∈ H.verts → ∃ p, p.toSubgraph ≤ H
-- SimpleGraph.connected_induce_iff
lemma connected_induce_iff {s : Set V} :
    (G.induce s).Connected ↔ ((⊤ : G.Subgraph).induce s).Connected 
Arguments: {V : Type u} {G : SimpleGraph V} {s : Set V}
Goal: (SimpleGraph.induce s G).Connected ↔ (⊤.induce s).Connected