ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- dist_right_lineMap
@[simp]
theorem dist_right_lineMap (p₁ p₂ : P) (c : 𝕜) : dist p₂ (lineMap p₁ p₂ c) = ‖1 - c‖ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: dist p₂ ((AffineMap.lineMap p₁ p₂) c) = ‖1 - c‖ * dist p₁ p₂
-- nndist_right_lineMap
@[simp]
theorem nndist_right_lineMap (p₁ p₂ : P) (c : 𝕜) :
    nndist p₂ (lineMap p₁ p₂ c) = ‖1 - c‖₊ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: nndist p₂ ((AffineMap.lineMap p₁ p₂) c) = ‖1 - c‖₊ * nndist p₁ p₂
-- dist_homothety_self
@[simp]
theorem dist_homothety_self (p₁ p₂ : P) (c : 𝕜) :
    dist (homothety p₁ c p₂) p₂ = ‖1 - c‖ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: dist ((AffineMap.homothety p₁ c) p₂) p₂ = ‖1 - c‖ * dist p₁ p₂
-- nndist_homothety_self
@[simp]
theorem nndist_homothety_self (p₁ p₂ : P) (c : 𝕜) :
    nndist (homothety p₁ c p₂) p₂ = ‖1 - c‖₊ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: nndist ((AffineMap.homothety p₁ c) p₂) p₂ = ‖1 - c‖₊ * nndist p₁ p₂
-- dist_self_homothety
@[simp]
theorem dist_self_homothety (p₁ p₂ : P) (c : 𝕜) :
    dist p₂ (homothety p₁ c p₂) = ‖1 - c‖ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: dist p₂ ((AffineMap.homothety p₁ c) p₂) = ‖1 - c‖ * dist p₁ p₂
-- nndist_self_homothety
@[simp]
theorem nndist_self_homothety (p₁ p₂ : P) (c : 𝕜) :
    nndist p₂ (homothety p₁ c p₂) = ‖1 - c‖₊ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜)
Goal: nndist p₂ ((AffineMap.homothety p₁ c) p₂) = ‖1 - c‖₊ * nndist p₁ p₂
-- dist_left_midpoint
@[simp]
theorem dist_left_midpoint (p₁ p₂ : P) : dist p₁ (midpoint 𝕜 p₁ p₂) = ‖(2 : 𝕜)‖⁻¹ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P)
Goal: dist p₁ (midpoint 𝕜 p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂
-- nndist_left_midpoint
@[simp]
theorem nndist_left_midpoint (p₁ p₂ : P) :
    nndist p₁ (midpoint 𝕜 p₁ p₂) = ‖(2 : 𝕜)‖₊⁻¹ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P)
Goal: nndist p₁ (midpoint 𝕜 p₁ p₂) = ‖2‖₊⁻¹ * nndist p₁ p₂
-- dist_midpoint_left
@[simp]
theorem dist_midpoint_left (p₁ p₂ : P) : dist (midpoint 𝕜 p₁ p₂) p₁ = ‖(2 : 𝕜)‖⁻¹ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P)
Goal: dist (midpoint 𝕜 p₁ p₂) p₁ = ‖2‖⁻¹ * dist p₁ p₂
-- nndist_midpoint_left
@[simp]
theorem nndist_midpoint_left (p₁ p₂ : P) :
    nndist (midpoint 𝕜 p₁ p₂) p₁ = ‖(2 : 𝕜)‖₊⁻¹ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P)
Goal: nndist (midpoint 𝕜 p₁ p₂) p₁ = ‖2‖₊⁻¹ * nndist p₁ p₂
-- dist_midpoint_right
@[simp]
theorem dist_midpoint_right (p₁ p₂ : P) :
    dist (midpoint 𝕜 p₁ p₂) p₂ = ‖(2 : 𝕜)‖⁻¹ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P)
Goal: dist (midpoint 𝕜 p₁ p₂) p₂ = ‖2‖⁻¹ * dist p₁ p₂
-- nndist_midpoint_right
@[simp]
theorem nndist_midpoint_right (p₁ p₂ : P) :
    nndist (midpoint 𝕜 p₁ p₂) p₂ = ‖(2 : 𝕜)‖₊⁻¹ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P)
Goal: nndist (midpoint 𝕜 p₁ p₂) p₂ = ‖2‖₊⁻¹ * nndist p₁ p₂
-- dist_right_midpoint
@[simp]
theorem dist_right_midpoint (p₁ p₂ : P) :
    dist p₂ (midpoint 𝕜 p₁ p₂) = ‖(2 : 𝕜)‖⁻¹ * dist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P)
Goal: dist p₂ (midpoint 𝕜 p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂
-- nndist_right_midpoint
@[simp]
theorem nndist_right_midpoint (p₁ p₂ : P) :
    nndist p₂ (midpoint 𝕜 p₁ p₂) = ‖(2 : 𝕜)‖₊⁻¹ * nndist p₁ p₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P)
Goal: nndist p₂ (midpoint 𝕜 p₁ p₂) = ‖2‖₊⁻¹ * nndist p₁ p₂
-- dist_midpoint_midpoint_le'
theorem dist_midpoint_midpoint_le' (p₁ p₂ p₃ p₄ : P) :
    dist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / ‖(2 : 𝕜)‖ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ p₃ p₄ : P)
Goal: dist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / ‖2‖
-- nndist_midpoint_midpoint_le'
theorem nndist_midpoint_midpoint_le' (p₁ p₂ p₃ p₄ : P) :
    nndist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / ‖(2 : 𝕜)‖₊ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ p₃ p₄ : P)
Goal: nndist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / ‖2‖₊
-- dist_pointReflection_left
@[simp] theorem dist_pointReflection_left (p q : P) :
    dist (Equiv.pointReflection p q) p = dist p q 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p q : P)
Goal: dist ((Equiv.pointReflection p) q) p = dist p q
-- dist_left_pointReflection
@[simp] theorem dist_left_pointReflection (p q : P) :
    dist p (Equiv.pointReflection p q) = dist p q 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p q : P)
Goal: dist p ((Equiv.pointReflection p) q) = dist p q
-- dist_pointReflection_right
theorem dist_pointReflection_right (p q : P) :
    dist (Equiv.pointReflection p q) q = ‖(2 : 𝕜)‖ * dist p q 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (𝕜 : Type u_6) [NormedField 𝕜] [NormedSpace 𝕜 V] (p q : P)
Goal: dist ((Equiv.pointReflection p) q) q = ‖2‖ * dist p q
-- dist_right_pointReflection
theorem dist_right_pointReflection (p q : P) :
    dist q (Equiv.pointReflection p q) = ‖(2 : 𝕜)‖ * dist p q 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (𝕜 : Type u_6) [NormedField 𝕜] [NormedSpace 𝕜 V] (p q : P)
Goal: dist q ((Equiv.pointReflection p) q) = ‖2‖ * dist p q
-- antilipschitzWith_lineMap
theorem antilipschitzWith_lineMap {p₁ p₂ : Q} (h : p₁ ≠ p₂) :
    AntilipschitzWith (nndist p₁ p₂)⁻¹ (lineMap p₁ p₂ : 𝕜 → Q) 
Arguments: {W : Type u_4} {Q : Type u_5} [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 W] {p₁ p₂ : Q} (h : p₁ ≠ p₂)
Goal: AntilipschitzWith (nndist p₁ p₂)⁻¹ ⇑(AffineMap.lineMap p₁ p₂)
-- eventually_homothety_mem_of_mem_interior
theorem eventually_homothety_mem_of_mem_interior (x : Q) {s : Set Q} {y : Q} (hy : y ∈ interior s) :
    ∀ᶠ δ in 𝓝 (1 : 𝕜), homothety x δ y ∈ s 
Arguments: {W : Type u_4} {Q : Type u_5} [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] (𝕜 : Type u_6) [NormedField 𝕜] [NormedSpace 𝕜 W] (x : Q) {s : Set Q} {y : Q} (hy : y ∈ interior s)
Goal: ∀ᶠ (δ : 𝕜) in nhds 1, (AffineMap.homothety x δ) y ∈ s
-- eventually_homothety_image_subset_of_finite_subset_interior
theorem eventually_homothety_image_subset_of_finite_subset_interior (x : Q) {s : Set Q} {t : Set Q}
    (ht : t.Finite) (h : t ⊆ interior s) : ∀ᶠ δ in 𝓝 (1 : 𝕜), homothety x δ '' t ⊆ s 
Arguments: {W : Type u_4} {Q : Type u_5} [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] (𝕜 : Type u_6) [NormedField 𝕜] [NormedSpace 𝕜 W] (x : Q) {s t : Set Q} (ht : t.Finite) (h : t ⊆ interior s)
Goal: ∀ᶠ (δ : 𝕜) in nhds 1, ⇑(AffineMap.homothety x δ) '' t ⊆ s
-- dist_midpoint_midpoint_le
theorem dist_midpoint_midpoint_le (p₁ p₂ p₃ p₄ : V) :
    dist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / 2 
Arguments: {V : Type u_2} [SeminormedAddCommGroup V] [NormedSpace ℝ V] (p₁ p₂ p₃ p₄ : V)
Goal: dist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / 2
-- nndist_midpoint_midpoint_le
theorem nndist_midpoint_midpoint_le (p₁ p₂ p₃ p₄ : V) :
    nndist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / 2 
Arguments: {V : Type u_2} [SeminormedAddCommGroup V] [NormedSpace ℝ V] (p₁ p₂ p₃ p₄ : V)
Goal: nndist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / 2
-- DilationEquiv.smulTorsor_apply
@[simps]
def DilationEquiv.smulTorsor (c : P) {k : 𝕜} (hk : k ≠ 0) : E ≃ᵈ P 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] {P : Type u_3} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) {k : 𝕜} (hk : k ≠ 0)
Goal: ∀ (x : E), (DilationEquiv.smulTorsor c hk) x = k • x +ᵥ c
-- DilationEquiv.smulTorsor_symm_apply
@[simps]
def DilationEquiv.smulTorsor (c : P) {k : 𝕜} (hk : k ≠ 0) : E ≃ᵈ P 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] {P : Type u_3} [PseudoMetricSpace P] [NormedAddTorsor E P] (c : P) {k : 𝕜} (hk : k ≠ 0)
Goal: ∀ (x : P), (DilationEquiv.smulTorsor c hk).symm x = (k⁻¹ • fun x => x -ᵥ c) x
-- DilationEquiv.smulTorsor_ratio
@[simp]
lemma DilationEquiv.smulTorsor_ratio {c : P} {k : 𝕜} (hk : k ≠ 0) {x y : E}
    (h : dist x y ≠ 0) : ratio (smulTorsor c hk) = ‖k‖₊ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] {P : Type u_3} [PseudoMetricSpace P] [NormedAddTorsor E P] {c : P} {k : 𝕜} (hk : k ≠ 0) {x y : E} (h : dist x y ≠ 0)
Goal: Dilation.ratio (DilationEquiv.smulTorsor c hk) = ‖k‖₊
-- DilationEquiv.smulTorsor_preimage_ball
@[simp]
lemma DilationEquiv.smulTorsor_preimage_ball {c : P} {k : 𝕜} (hk : k ≠ 0) :
    smulTorsor c hk ⁻¹' (Metric.ball c ‖k‖) = Metric.ball (0 : E) 1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] {P : Type u_3} [PseudoMetricSpace P] [NormedAddTorsor E P] {c : P} {k : 𝕜} (hk : k ≠ 0)
Goal: ⇑(DilationEquiv.smulTorsor c hk) ⁻¹' Metric.ball c ‖k‖ = Metric.ball 0 1
-- Set.Countable.isPathConnected_compl_of_one_lt_rank
theorem Set.Countable.isPathConnected_compl_of_one_lt_rank
    (h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable) :
    IsPathConnected sᶜ 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] (h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable)
Goal: IsPathConnected sᶜ
-- Set.Countable.isConnected_compl_of_one_lt_rank
theorem Set.Countable.isConnected_compl_of_one_lt_rank (h : 1 < Module.rank ℝ E) {s : Set E}
    (hs : s.Countable) : IsConnected sᶜ 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] (h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable)
Goal: IsConnected sᶜ
-- isPathConnected_compl_singleton_of_one_lt_rank
theorem isPathConnected_compl_singleton_of_one_lt_rank (h : 1 < Module.rank ℝ E) (x : E) :
    IsPathConnected {x}ᶜ 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] (h : 1 < Module.rank ℝ E) (x : E)
Goal: IsPathConnected {x}ᶜ
-- isConnected_compl_singleton_of_one_lt_rank
theorem isConnected_compl_singleton_of_one_lt_rank (h : 1 < Module.rank ℝ E) (x : E) :
    IsConnected {x}ᶜ 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] (h : 1 < Module.rank ℝ E) (x : E)
Goal: IsConnected {x}ᶜ
-- isPathConnected_sphere
theorem isPathConnected_sphere (h : 1 < Module.rank ℝ E) (x : E) {r : ℝ} (hr : 0 ≤ r) :
    IsPathConnected (sphere x r) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (h : 1 < Module.rank ℝ E) (x : E) {r : ℝ} (hr : 0 ≤ r)
Goal: IsPathConnected (Metric.sphere x r)
-- isConnected_sphere
theorem isConnected_sphere (h : 1 < Module.rank ℝ E) (x : E) {r : ℝ} (hr : 0 ≤ r) :
    IsConnected (sphere x r) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (h : 1 < Module.rank ℝ E) (x : E) {r : ℝ} (hr : 0 ≤ r)
Goal: IsConnected (Metric.sphere x r)
-- isPreconnected_sphere
theorem isPreconnected_sphere (h : 1 < Module.rank ℝ E) (x : E) (r : ℝ) :
    IsPreconnected (sphere x r) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (h : 1 < Module.rank ℝ E) (x : E) (r : ℝ)
Goal: IsPreconnected (Metric.sphere x r)
-- isPathConnected_compl_of_one_lt_codim
theorem isPathConnected_compl_of_one_lt_codim {E : Submodule ℝ F}
    (hcodim : 1 < Module.rank ℝ (F ⧸ E)) : IsPathConnected (Eᶜ : Set F) 
Arguments: {F : Type u_1} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul ℝ F] {E : Submodule ℝ F} (hcodim : 1 < Module.rank ℝ (F ⧸ E))
Goal: IsPathConnected (↑E)ᶜ
-- isConnected_compl_of_one_lt_codim
theorem isConnected_compl_of_one_lt_codim {E : Submodule ℝ F} (hcodim : 1 < Module.rank ℝ (F ⧸ E)) :
    IsConnected (Eᶜ : Set F) 
Arguments: {F : Type u_1} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul ℝ F] {E : Submodule ℝ F} (hcodim : 1 < Module.rank ℝ (F ⧸ E))
Goal: IsConnected (↑E)ᶜ
-- Submodule.connectedComponentIn_eq_self_of_one_lt_codim
theorem Submodule.connectedComponentIn_eq_self_of_one_lt_codim (E : Submodule ℝ F)
    (hcodim : 1 < Module.rank ℝ (F ⧸ E)) {x : F} (hx : x ∉ E) :
    connectedComponentIn ((E : Set F)ᶜ) x = (E : Set F)ᶜ 
Arguments: {F : Type u_1} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul ℝ F] (E : Submodule ℝ F) (hcodim : 1 < Module.rank ℝ (F ⧸ E)) {x : F} (hx : x ∉ E)
Goal: connectedComponentIn (↑E)ᶜ x = (↑E)ᶜ
-- IsMaxFilter.norm_add_sameRay
theorem IsMaxFilter.norm_add_sameRay (h : IsMaxFilter (norm ∘ f) l c) (hy : SameRay ℝ (f c) y) :
    IsMaxFilter (fun x => ‖f x + y‖) l c 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {f : α → E} {l : Filter α} {c : α} {y : E} (h : IsMaxFilter (norm ∘ f) l c) (hy : SameRay ℝ (f c) y)
Goal: IsMaxFilter (fun x => ‖f x + y‖) l c
-- IsMaxFilter.norm_add_self
theorem IsMaxFilter.norm_add_self (h : IsMaxFilter (norm ∘ f) l c) :
    IsMaxFilter (fun x => ‖f x + f c‖) l c 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {f : α → E} {l : Filter α} {c : α} (h : IsMaxFilter (norm ∘ f) l c)
Goal: IsMaxFilter (fun x => ‖f x + f c‖) l c
-- IsMaxOn.norm_add_sameRay
theorem IsMaxOn.norm_add_sameRay (h : IsMaxOn (norm ∘ f) s c) (hy : SameRay ℝ (f c) y) :
    IsMaxOn (fun x => ‖f x + y‖) s c 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {f : α → E} {s : Set α} {c : α} {y : E} (h : IsMaxOn (norm ∘ f) s c) (hy : SameRay ℝ (f c) y)
Goal: IsMaxOn (fun x => ‖f x + y‖) s c
-- IsMaxOn.norm_add_self
theorem IsMaxOn.norm_add_self (h : IsMaxOn (norm ∘ f) s c) : IsMaxOn (fun x => ‖f x + f c‖) s c 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {f : α → E} {s : Set α} {c : α} (h : IsMaxOn (norm ∘ f) s c)
Goal: IsMaxOn (fun x => ‖f x + f c‖) s c
-- IsLocalMaxOn.norm_add_sameRay
theorem IsLocalMaxOn.norm_add_sameRay (h : IsLocalMaxOn (norm ∘ f) s c) (hy : SameRay ℝ (f c) y) :
    IsLocalMaxOn (fun x => ‖f x + y‖) s c 
Arguments: {X : Type u_2} {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace X] {f : X → E} {s : Set X} {c : X} {y : E} (h : IsLocalMaxOn (norm ∘ f) s c) (hy : SameRay ℝ (f c) y)
Goal: IsLocalMaxOn (fun x => ‖f x + y‖) s c
-- IsLocalMaxOn.norm_add_self
theorem IsLocalMaxOn.norm_add_self (h : IsLocalMaxOn (norm ∘ f) s c) :
    IsLocalMaxOn (fun x => ‖f x + f c‖) s c 
Arguments: {X : Type u_2} {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace X] {f : X → E} {s : Set X} {c : X} (h : IsLocalMaxOn (norm ∘ f) s c)
Goal: IsLocalMaxOn (fun x => ‖f x + f c‖) s c
-- IsLocalMax.norm_add_sameRay
theorem IsLocalMax.norm_add_sameRay (h : IsLocalMax (norm ∘ f) c) (hy : SameRay ℝ (f c) y) :
    IsLocalMax (fun x => ‖f x + y‖) c 
Arguments: {X : Type u_2} {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace X] {f : X → E} {c : X} {y : E} (h : IsLocalMax (norm ∘ f) c) (hy : SameRay ℝ (f c) y)
Goal: IsLocalMax (fun x => ‖f x + y‖) c
-- IsLocalMax.norm_add_self
theorem IsLocalMax.norm_add_self (h : IsLocalMax (norm ∘ f) c) :
    IsLocalMax (fun x => ‖f x + f c‖) c 
Arguments: {X : Type u_2} {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace X] {f : X → E} {c : X} (h : IsLocalMax (norm ∘ f) c)
Goal: IsLocalMax (fun x => ‖f x + f c‖) c
-- RCLike.norm_coe_norm
theorem RCLike.norm_coe_norm {z : E} : ‖(‖z‖ : 𝕜)‖ = ‖z‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] {z : E}
Goal: ‖↑‖z‖‖ = ‖z‖
-- norm_smul_inv_norm
@[simp]
theorem norm_smul_inv_norm {x : E} (hx : x ≠ 0) : ‖(‖x‖⁻¹ : 𝕜) • x‖ = 1 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} (hx : x ≠ 0)
Goal: ‖(↑‖x‖)⁻¹ • x‖ = 1
-- norm_smul_inv_norm'
theorem norm_smul_inv_norm' {r : ℝ} (r_nonneg : 0 ≤ r) {x : E} (hx : x ≠ 0) :
    ‖((r : 𝕜) * (‖x‖ : 𝕜)⁻¹) • x‖ = r 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ} (r_nonneg : 0 ≤ r) {x : E} (hx : x ≠ 0)
Goal: ‖(↑r * (↑‖x‖)⁻¹) • x‖ = r
-- LinearMap.bound_of_sphere_bound
theorem LinearMap.bound_of_sphere_bound {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] 𝕜)
    (h : ∀ z ∈ sphere (0 : E) r, ‖f z‖ ≤ c) (z : E) : ‖f z‖ ≤ c / r * ‖z‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] 𝕜) (h : ∀ z ∈ Metric.sphere 0 r, ‖f z‖ ≤ c) (z : E)
Goal: ‖f z‖ ≤ c / r * ‖z‖
-- LinearMap.bound_of_ball_bound'
theorem LinearMap.bound_of_ball_bound' {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] 𝕜)
    (h : ∀ z ∈ closedBall (0 : E) r, ‖f z‖ ≤ c) (z : E) : ‖f z‖ ≤ c / r * ‖z‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] 𝕜) (h : ∀ z ∈ Metric.closedBall 0 r, ‖f z‖ ≤ c) (z : E)
Goal: ‖f z‖ ≤ c / r * ‖z‖
-- ContinuousLinearMap.opNorm_bound_of_ball_bound
theorem ContinuousLinearMap.opNorm_bound_of_ball_bound {r : ℝ} (r_pos : 0 < r) (c : ℝ)
    (f : E →L[𝕜] 𝕜) (h : ∀ z ∈ closedBall (0 : E) r, ‖f z‖ ≤ c) : ‖f‖ ≤ c / r 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →L[𝕜] 𝕜) (h : ∀ z ∈ Metric.closedBall 0 r, ‖f z‖ ≤ c)
Goal: ‖f‖ ≤ c / r
-- NormedSpace.sphere_nonempty_rclike
theorem NormedSpace.sphere_nonempty_rclike [Nontrivial E] {r : ℝ} (hr : 0 ≤ r) :
    Nonempty (sphere (0 : E) r) 
Arguments: (𝕜 : Type u_1) [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [Nontrivial E] {r : ℝ} (hr : 0 ≤ r)
Goal: Nonempty ↑(Metric.sphere 0 r)
-- ENorm.eq_zero'
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] where
  toFun : V → ℝ≥0∞
  eq_zero' : ∀ x, toFun x = 0 → x = 0
  map_add_le' : ∀ x y : V, toFun (x + y) ≤ toFun x + toFun y
  map_smul_le' : ∀ (c : 𝕜) (x : V), toFun (c • x) ≤ ‖c‖₊ * toFun x
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (self : ENorm 𝕜 V) (x : V)
Goal: ↑self x = 0 → x = 0
-- ENorm.map_smul_le'
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] where
  toFun : V → ℝ≥0∞
  eq_zero' : ∀ x, toFun x = 0 → x = 0
  map_add_le' : ∀ x y : V, toFun (x + y) ≤ toFun x + toFun y
  map_smul_le' : ∀ (c : 𝕜) (x : V), toFun (c • x) ≤ ‖c‖₊ * toFun x
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (self : ENorm 𝕜 V) (c : 𝕜) (x : V)
Goal: ↑self (c • x) ≤ ↑‖c‖₊ * ↑self x
-- ENorm.map_add_le'
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] where
  toFun : V → ℝ≥0∞
  eq_zero' : ∀ x, toFun x = 0 → x = 0
  map_add_le' : ∀ x y : V, toFun (x + y) ≤ toFun x + toFun y
  map_smul_le' : ∀ (c : 𝕜) (x : V), toFun (c • x) ≤ ‖c‖₊ * toFun x
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (self : ENorm 𝕜 V) (x y : V)
Goal: ↑self (x + y) ≤ ↑self x + ↑self y
-- ENorm.coeFn_injective
theorem coeFn_injective : Function.Injective ((↑) : ENorm 𝕜 V → V → ℝ≥0∞) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V]
Goal: Function.Injective ENorm.toFun
-- ENorm.ext
@[ext]
theorem ext {e₁ e₂ : ENorm 𝕜 V} (h : ∀ x, e₁ x = e₂ x) : e₁ = e₂ 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] {e₁ e₂ : ENorm 𝕜 V} (h : ∀ (x : V), ↑e₁ x = ↑e₂ x)
Goal: e₁ = e₂
-- ENorm.ext_iff
theorem ext_iff {e₁ e₂ : ENorm 𝕜 V} : e₁ = e₂ ↔ ∀ x, e₁ x = e₂ x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] {e₁ e₂ : ENorm 𝕜 V}
Goal: e₁ = e₂ ↔ ∀ (x : V), ↑e₁ x = ↑e₂ x
-- ENorm.coe_inj
@[simp, norm_cast]
theorem coe_inj {e₁ e₂ : ENorm 𝕜 V} : (e₁ : V → ℝ≥0∞) = e₂ ↔ e₁ = e₂ 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] {e₁ e₂ : ENorm 𝕜 V}
Goal: ↑e₁ = ↑e₂ ↔ e₁ = e₂
-- ENorm.map_smul
@[simp]
theorem map_smul (c : 𝕜) (x : V) : e (c • x) = ‖c‖₊ * e x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e : ENorm 𝕜 V) (c : 𝕜) (x : V)
Goal: ↑e (c • x) = ↑‖c‖₊ * ↑e x
-- ENorm.map_zero
@[simp]
theorem map_zero : e 0 = 0 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e : ENorm 𝕜 V)
Goal: ↑e 0 = 0
-- ENorm.eq_zero_iff
@[simp]
theorem eq_zero_iff {x : V} : e x = 0 ↔ x = 0 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e : ENorm 𝕜 V) {x : V}
Goal: ↑e x = 0 ↔ x = 0
-- ENorm.map_neg
@[simp]
theorem map_neg (x : V) : e (-x) = e x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e : ENorm 𝕜 V) (x : V)
Goal: ↑e (-x) = ↑e x
-- ENorm.map_sub_rev
theorem map_sub_rev (x y : V) : e (x - y) = e (y - x) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e : ENorm 𝕜 V) (x y : V)
Goal: ↑e (x - y) = ↑e (y - x)
-- ENorm.map_add_le
theorem map_add_le (x y : V) : e (x + y) ≤ e x + e y 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e : ENorm 𝕜 V) (x y : V)
Goal: ↑e (x + y) ≤ ↑e x + ↑e y
-- ENorm.map_sub_le
theorem map_sub_le (x y : V) : e (x - y) ≤ e x + e y 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e : ENorm 𝕜 V) (x y : V)
Goal: ↑e (x - y) ≤ ↑e x + ↑e y
-- ENorm.top_map
theorem top_map {x : V} (hx : x ≠ 0) : (⊤ : ENorm 𝕜 V) x = ⊤ 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] {x : V} (hx : x ≠ 0)
Goal: ↑⊤ x = ⊤