ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- disjoint_ball_closedBall_iff
theorem disjoint_ball_closedBall_iff (hδ : 0 < δ) (hε : 0 ≤ ε) :
    Disjoint (ball x δ) (closedBall y ε) ↔ δ + ε ≤ dist x y 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x y : E} {δ ε : ℝ} (hδ : 0 < δ) (hε : 0 ≤ ε)
Goal: Disjoint (Metric.ball x δ) (Metric.closedBall y ε) ↔ δ + ε ≤ dist x y
-- disjoint_closedBall_ball_iff
theorem disjoint_closedBall_ball_iff (hδ : 0 ≤ δ) (hε : 0 < ε) :
    Disjoint (closedBall x δ) (ball y ε) ↔ δ + ε ≤ dist x y 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x y : E} {δ ε : ℝ} (hδ : 0 ≤ δ) (hε : 0 < ε)
Goal: Disjoint (Metric.closedBall x δ) (Metric.ball y ε) ↔ δ + ε ≤ dist x y
-- disjoint_closedBall_closedBall_iff
theorem disjoint_closedBall_closedBall_iff (hδ : 0 ≤ δ) (hε : 0 ≤ ε) :
    Disjoint (closedBall x δ) (closedBall y ε) ↔ δ + ε < dist x y 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x y : E} {δ ε : ℝ} (hδ : 0 ≤ δ) (hε : 0 ≤ ε)
Goal: Disjoint (Metric.closedBall x δ) (Metric.closedBall y ε) ↔ δ + ε < dist x y
-- infEdist_thickening
@[simp]
theorem infEdist_thickening (hδ : 0 < δ) (s : Set E) (x : E) :
    infEdist x (thickening δ s) = infEdist x s - ENNReal.ofReal δ 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ : ℝ} (hδ : 0 < δ) (s : Set E) (x : E)
Goal: EMetric.infEdist x (Metric.thickening δ s) = EMetric.infEdist x s - ENNReal.ofReal δ
-- thickening_thickening
@[simp]
theorem thickening_thickening (hε : 0 < ε) (hδ : 0 < δ) (s : Set E) :
    thickening ε (thickening δ s) = thickening (ε + δ) s 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 < ε) (hδ : 0 < δ) (s : Set E)
Goal: Metric.thickening ε (Metric.thickening δ s) = Metric.thickening (ε + δ) s
-- cthickening_thickening
@[simp]
theorem cthickening_thickening (hε : 0 ≤ ε) (hδ : 0 < δ) (s : Set E) :
    cthickening ε (thickening δ s) = cthickening (ε + δ) s 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 ≤ ε) (hδ : 0 < δ) (s : Set E)
Goal: Metric.cthickening ε (Metric.thickening δ s) = Metric.cthickening (ε + δ) s
-- closure_thickening
@[simp]
theorem closure_thickening (hδ : 0 < δ) (s : Set E) :
    closure (thickening δ s) = cthickening δ s 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ : ℝ} (hδ : 0 < δ) (s : Set E)
Goal: closure (Metric.thickening δ s) = Metric.cthickening δ s
-- infEdist_cthickening
@[simp]
theorem infEdist_cthickening (δ : ℝ) (s : Set E) (x : E) :
    infEdist x (cthickening δ s) = infEdist x s - ENNReal.ofReal δ 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (δ : ℝ) (s : Set E) (x : E)
Goal: EMetric.infEdist x (Metric.cthickening δ s) = EMetric.infEdist x s - ENNReal.ofReal δ
-- thickening_cthickening
@[simp]
theorem thickening_cthickening (hε : 0 < ε) (hδ : 0 ≤ δ) (s : Set E) :
    thickening ε (cthickening δ s) = thickening (ε + δ) s 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 < ε) (hδ : 0 ≤ δ) (s : Set E)
Goal: Metric.thickening ε (Metric.cthickening δ s) = Metric.thickening (ε + δ) s
-- cthickening_cthickening
@[simp]
theorem cthickening_cthickening (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (s : Set E) :
    cthickening ε (cthickening δ s) = cthickening (ε + δ) s 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (s : Set E)
Goal: Metric.cthickening ε (Metric.cthickening δ s) = Metric.cthickening (ε + δ) s
-- thickening_ball
@[simp]
theorem thickening_ball (hε : 0 < ε) (hδ : 0 < δ) (x : E) :
    thickening ε (ball x δ) = ball x (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 < ε) (hδ : 0 < δ) (x : E)
Goal: Metric.thickening ε (Metric.ball x δ) = Metric.ball x (ε + δ)
-- thickening_closedBall
@[simp]
theorem thickening_closedBall (hε : 0 < ε) (hδ : 0 ≤ δ) (x : E) :
    thickening ε (closedBall x δ) = ball x (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 < ε) (hδ : 0 ≤ δ) (x : E)
Goal: Metric.thickening ε (Metric.closedBall x δ) = Metric.ball x (ε + δ)
-- cthickening_ball
@[simp]
theorem cthickening_ball (hε : 0 ≤ ε) (hδ : 0 < δ) (x : E) :
    cthickening ε (ball x δ) = closedBall x (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 ≤ ε) (hδ : 0 < δ) (x : E)
Goal: Metric.cthickening ε (Metric.ball x δ) = Metric.closedBall x (ε + δ)
-- cthickening_closedBall
@[simp]
theorem cthickening_closedBall (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (x : E) :
    cthickening ε (closedBall x δ) = closedBall x (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (x : E)
Goal: Metric.cthickening ε (Metric.closedBall x δ) = Metric.closedBall x (ε + δ)
-- ball_add_ball
theorem ball_add_ball (hε : 0 < ε) (hδ : 0 < δ) (a b : E) :
    ball a ε + ball b δ = ball (a + b) (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 < ε) (hδ : 0 < δ) (a b : E)
Goal: Metric.ball a ε + Metric.ball b δ = Metric.ball (a + b) (ε + δ)
-- ball_sub_ball
theorem ball_sub_ball (hε : 0 < ε) (hδ : 0 < δ) (a b : E) :
    ball a ε - ball b δ = ball (a - b) (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 < ε) (hδ : 0 < δ) (a b : E)
Goal: Metric.ball a ε - Metric.ball b δ = Metric.ball (a - b) (ε + δ)
-- ball_add_closedBall
theorem ball_add_closedBall (hε : 0 < ε) (hδ : 0 ≤ δ) (a b : E) :
    ball a ε + closedBall b δ = ball (a + b) (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 < ε) (hδ : 0 ≤ δ) (a b : E)
Goal: Metric.ball a ε + Metric.closedBall b δ = Metric.ball (a + b) (ε + δ)
-- ball_sub_closedBall
theorem ball_sub_closedBall (hε : 0 < ε) (hδ : 0 ≤ δ) (a b : E) :
    ball a ε - closedBall b δ = ball (a - b) (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 < ε) (hδ : 0 ≤ δ) (a b : E)
Goal: Metric.ball a ε - Metric.closedBall b δ = Metric.ball (a - b) (ε + δ)
-- closedBall_add_ball
theorem closedBall_add_ball (hε : 0 ≤ ε) (hδ : 0 < δ) (a b : E) :
    closedBall a ε + ball b δ = ball (a + b) (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 ≤ ε) (hδ : 0 < δ) (a b : E)
Goal: Metric.closedBall a ε + Metric.ball b δ = Metric.ball (a + b) (ε + δ)
-- closedBall_sub_ball
theorem closedBall_sub_ball (hε : 0 ≤ ε) (hδ : 0 < δ) (a b : E) :
    closedBall a ε - ball b δ = ball (a - b) (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} (hε : 0 ≤ ε) (hδ : 0 < δ) (a b : E)
Goal: Metric.closedBall a ε - Metric.ball b δ = Metric.ball (a - b) (ε + δ)
-- closedBall_add_closedBall
theorem closedBall_add_closedBall [ProperSpace E] (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (a b : E) :
    closedBall a ε + closedBall b δ = closedBall (a + b) (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} [ProperSpace E] (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (a b : E)
Goal: Metric.closedBall a ε + Metric.closedBall b δ = Metric.closedBall (a + b) (ε + δ)
-- closedBall_sub_closedBall
theorem closedBall_sub_closedBall [ProperSpace E] (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (a b : E) :
    closedBall a ε - closedBall b δ = closedBall (a - b) (ε + δ) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ} [ProperSpace E] (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (a b : E)
Goal: Metric.closedBall a ε - Metric.closedBall b δ = Metric.closedBall (a - b) (ε + δ)
-- smul_closedBall
theorem smul_closedBall (c : 𝕜) (x : E) {r : ℝ} (hr : 0 ≤ r) :
    c • closedBall x r = closedBall (c • x) (‖c‖ * r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (c : 𝕜) (x : E) {r : ℝ} (hr : 0 ≤ r)
Goal: c • Metric.closedBall x r = Metric.closedBall (c • x) (‖c‖ * r)
-- smul_closedUnitBall
theorem smul_closedUnitBall (c : 𝕜) : c • closedBall (0 : E) (1 : ℝ) = closedBall (0 : E) ‖c‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (c : 𝕜)
Goal: c • Metric.closedBall 0 1 = Metric.closedBall 0 ‖c‖
-- smul_closedUnitBall_of_nonneg
theorem smul_closedUnitBall_of_nonneg {r : ℝ} (hr : 0 ≤ r) :
    r • closedBall (0 : E) 1 = closedBall (0 : E) r 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {r : ℝ} (hr : 0 ≤ r)
Goal: r • Metric.closedBall 0 1 = Metric.closedBall 0 r
-- NormedSpace.sphere_nonempty
@[simp]
theorem NormedSpace.sphere_nonempty [Nontrivial E] {x : E} {r : ℝ} :
    (sphere x r).Nonempty ↔ 0 ≤ r 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E] {x : E} {r : ℝ}
Goal: (Metric.sphere x r).Nonempty ↔ 0 ≤ r
-- smul_sphere
theorem smul_sphere [Nontrivial E] (c : 𝕜) (x : E) {r : ℝ} (hr : 0 ≤ r) :
    c • sphere x r = sphere (c • x) (‖c‖ * r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace ℝ E] [Nontrivial E] (c : 𝕜) (x : E) {r : ℝ} (hr : 0 ≤ r)
Goal: c • Metric.sphere x r = Metric.sphere (c • x) (‖c‖ * r)
-- affinity_unitBall
theorem affinity_unitBall {r : ℝ} (hr : 0 < r) (x : E) : x +ᵥ r • ball (0 : E) 1 = ball x r 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {r : ℝ} (hr : 0 < r) (x : E)
Goal: x +ᵥ r • Metric.ball 0 1 = Metric.ball x r
-- affinity_unitClosedBall
theorem affinity_unitClosedBall {r : ℝ} (hr : 0 ≤ r) (x : E) :
    x +ᵥ r • closedBall (0 : E) 1 = closedBall x r 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {r : ℝ} (hr : 0 ≤ r) (x : E)
Goal: x +ᵥ r • Metric.closedBall 0 1 = Metric.closedBall x r
-- absorbs_iff_norm
lemma absorbs_iff_norm : Absorbs 𝕜 A B ↔ ∃ r, ∀ c : 𝕜, r ≤ ‖c‖ → B ⊆ c • A 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] {A B : Set E}
Goal: Absorbs 𝕜 A B ↔ ∃ r, ∀ (c : 𝕜), r ≤ ‖c‖ → B ⊆ c • A
-- Absorbs.exists_pos
lemma Absorbs.exists_pos (h : Absorbs 𝕜 A B) : ∃ r > 0, ∀ c : 𝕜, r ≤ ‖c‖ → B ⊆ c • A 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] {A B : Set E} (h : Absorbs 𝕜 A B)
Goal: ∃ r > 0, ∀ (c : 𝕜), r ≤ ‖c‖ → B ⊆ c • A
-- balanced_iff_smul_mem
theorem balanced_iff_smul_mem : Balanced 𝕜 s ↔ ∀ ⦃a : 𝕜⦄, ‖a‖ ≤ 1 → ∀ ⦃x : E⦄, x ∈ s → a • x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] {s : Set E}
Goal: Balanced 𝕜 s ↔ ∀ ⦃a : 𝕜⦄, ‖a‖ ≤ 1 → ∀ ⦃x : E⦄, x ∈ s → a • x ∈ s
-- balanced_iff_closedBall_smul
theorem balanced_iff_closedBall_smul : Balanced 𝕜 s ↔ Metric.closedBall (0 : 𝕜) 1 • s ⊆ s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] {s : Set E}
Goal: Balanced 𝕜 s ↔ Metric.closedBall 0 1 • s ⊆ s
-- balanced_empty
@[simp]
theorem balanced_empty : Balanced 𝕜 (∅ : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E]
Goal: Balanced 𝕜 ∅
-- balanced_univ
@[simp]
theorem balanced_univ : Balanced 𝕜 (univ : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E]
Goal: Balanced 𝕜 Set.univ
-- Balanced.union
theorem Balanced.union (hA : Balanced 𝕜 A) (hB : Balanced 𝕜 B) : Balanced 𝕜 (A ∪ B) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] {A B : Set E} (hA : Balanced 𝕜 A) (hB : Balanced 𝕜 B)
Goal: Balanced 𝕜 (A ∪ B)
-- Balanced.inter
theorem Balanced.inter (hA : Balanced 𝕜 A) (hB : Balanced 𝕜 B) : Balanced 𝕜 (A ∩ B) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] {A B : Set E} (hA : Balanced 𝕜 A) (hB : Balanced 𝕜 B)
Goal: Balanced 𝕜 (A ∩ B)
-- balanced_iUnion
theorem balanced_iUnion {f : ι → Set E} (h : ∀ i, Balanced 𝕜 (f i)) : Balanced 𝕜 (⋃ i, f i) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {ι : Sort u_4} [SeminormedRing 𝕜] [SMul 𝕜 E] {f : ι → Set E} (h : ∀ (i : ι), Balanced 𝕜 (f i))
Goal: Balanced 𝕜 (⋃ i, f i)
-- balanced_iUnion₂
theorem balanced_iUnion₂ {f : ∀ i, κ i → Set E} (h : ∀ i j, Balanced 𝕜 (f i j)) :
    Balanced 𝕜 (⋃ (i) (j), f i j) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {ι : Sort u_4} {κ : ι → Sort u_5} [SeminormedRing 𝕜] [SMul 𝕜 E] {f : (i : ι) → κ i → Set E} (h : ∀ (i : ι) (j : κ i), Balanced 𝕜 (f i j))
Goal: Balanced 𝕜 (⋃ i, ⋃ j, f i j)
-- balanced_iInter
theorem balanced_iInter {f : ι → Set E} (h : ∀ i, Balanced 𝕜 (f i)) : Balanced 𝕜 (⋂ i, f i) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {ι : Sort u_4} [SeminormedRing 𝕜] [SMul 𝕜 E] {f : ι → Set E} (h : ∀ (i : ι), Balanced 𝕜 (f i))
Goal: Balanced 𝕜 (⋂ i, f i)
-- balanced_iInter₂
theorem balanced_iInter₂ {f : ∀ i, κ i → Set E} (h : ∀ i j, Balanced 𝕜 (f i j)) :
    Balanced 𝕜 (⋂ (i) (j), f i j) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {ι : Sort u_4} {κ : ι → Sort u_5} [SeminormedRing 𝕜] [SMul 𝕜 E] {f : (i : ι) → κ i → Set E} (h : ∀ (i : ι) (j : κ i), Balanced 𝕜 (f i j))
Goal: Balanced 𝕜 (⋂ i, ⋂ j, f i j)
-- Balanced.smul
theorem Balanced.smul (a : 𝕝) (hs : Balanced 𝕜 s) : Balanced 𝕜 (a • s) 
Arguments: {𝕜 : Type u_1} {𝕝 : Type u_2} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] {s : Set E} [SMul 𝕝 E] [SMulCommClass 𝕜 𝕝 E] (a : 𝕝) (hs : Balanced 𝕜 s)
Goal: Balanced 𝕜 (a • s)
-- Balanced.neg
theorem Balanced.neg : Balanced 𝕜 s → Balanced 𝕜 (-s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E}
Goal: Balanced 𝕜 s → Balanced 𝕜 (-s)
-- balanced_neg
@[simp]
theorem balanced_neg : Balanced 𝕜 (-s) ↔ Balanced 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E}
Goal: Balanced 𝕜 (-s) ↔ Balanced 𝕜 s
-- Balanced.neg_mem_iff
theorem Balanced.neg_mem_iff [NormOneClass 𝕜] (h : Balanced 𝕜 s) {x : E} : -x ∈ s ↔ x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} [NormOneClass 𝕜] (h : Balanced 𝕜 s) {x : E}
Goal: -x ∈ s ↔ x ∈ s
-- Balanced.neg_eq
theorem Balanced.neg_eq [NormOneClass 𝕜] (h : Balanced 𝕜 s) : -s = s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} [NormOneClass 𝕜] (h : Balanced 𝕜 s)
Goal: -s = s
-- Balanced.add
theorem Balanced.add (hs : Balanced 𝕜 s) (ht : Balanced 𝕜 t) : Balanced 𝕜 (s + t) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (hs : Balanced 𝕜 s) (ht : Balanced 𝕜 t)
Goal: Balanced 𝕜 (s + t)
-- Balanced.sub
theorem Balanced.sub (hs : Balanced 𝕜 s) (ht : Balanced 𝕜 t) : Balanced 𝕜 (s - t) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (hs : Balanced 𝕜 s) (ht : Balanced 𝕜 t)
Goal: Balanced 𝕜 (s - t)
-- balanced_zero
theorem balanced_zero : Balanced 𝕜 (0 : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E]
Goal: Balanced 𝕜 0
-- absorbs_iff_eventually_nhdsWithin_zero
theorem absorbs_iff_eventually_nhdsWithin_zero :
    Absorbs 𝕜 s t ↔ ∀ᶠ c : 𝕜 in 𝓝[≠] 0, MapsTo (c • ·) t s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedDivisionRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E}
Goal: Absorbs 𝕜 s t ↔ ∀ᶠ (c : 𝕜) in 𝓝[≠] 0, Set.MapsTo (fun x => c • x) t s
-- absorbent_iff_eventually_nhdsWithin_zero
theorem absorbent_iff_eventually_nhdsWithin_zero :
    Absorbent 𝕜 s ↔ ∀ x : E, ∀ᶠ c : 𝕜 in 𝓝[≠] 0, c • x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedDivisionRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E}
Goal: Absorbent 𝕜 s ↔ ∀ (x : E), ∀ᶠ (c : 𝕜) in 𝓝[≠] 0, c • x ∈ s
-- absorbs_iff_eventually_nhds_zero
theorem absorbs_iff_eventually_nhds_zero (h₀ : 0 ∈ s) :
    Absorbs 𝕜 s t ↔ ∀ᶠ c : 𝕜 in 𝓝 0, MapsTo (c • ·) t s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedDivisionRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (h₀ : 0 ∈ s)
Goal: Absorbs 𝕜 s t ↔ ∀ᶠ (c : 𝕜) in 𝓝 0, Set.MapsTo (fun x => c • x) t s
-- Absorbs.eventually_nhds_zero
theorem Absorbs.eventually_nhds_zero (h : Absorbs 𝕜 s t) (h₀ : 0 ∈ s) :
    ∀ᶠ c : 𝕜 in 𝓝 0, MapsTo (c • ·) t s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedDivisionRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (h : Absorbs 𝕜 s t) (h₀ : 0 ∈ s)
Goal: ∀ᶠ (c : 𝕜) in 𝓝 0, Set.MapsTo (fun x => c • x) t s
-- Balanced.smul_mono
theorem Balanced.smul_mono (hs : Balanced 𝕝 s) {a : 𝕝} {b : 𝕜} (h : ‖a‖ ≤ ‖b‖) : a • s ⊆ b • s 
Arguments: {𝕜 : Type u_1} {𝕝 : Type u_2} {E : Type u_3} [NormedField 𝕜] [NormedRing 𝕝] [NormedSpace 𝕜 𝕝] [AddCommGroup E] [Module 𝕜 E] [SMulWithZero 𝕝 E] [IsScalarTower 𝕜 𝕝 E] {s : Set E} (hs : Balanced 𝕝 s) {a : 𝕝} {b : 𝕜} (h : ‖a‖ ≤ ‖b‖)
Goal: a • s ⊆ b • s
-- Balanced.smul_mem_mono
theorem Balanced.smul_mem_mono [SMulCommClass 𝕝 𝕜 E] (hs : Balanced 𝕝 s) {a : 𝕜} {b : 𝕝}
    (ha : a • x ∈ s) (hba : ‖b‖ ≤ ‖a‖) : b • x ∈ s 
Arguments: {𝕜 : Type u_1} {𝕝 : Type u_2} {E : Type u_3} [NormedField 𝕜] [NormedRing 𝕝] [NormedSpace 𝕜 𝕝] [AddCommGroup E] [Module 𝕜 E] [SMulWithZero 𝕝 E] [IsScalarTower 𝕜 𝕝 E] {s : Set E} {x : E} [SMulCommClass 𝕝 𝕜 E] (hs : Balanced 𝕝 s) {a : 𝕜} {b : 𝕝} (ha : a • x ∈ s) (hba : ‖b‖ ≤ ‖a‖)
Goal: b • x ∈ s
-- Balanced.subset_smul
theorem Balanced.subset_smul (hA : Balanced 𝕜 A) (ha : 1 ≤ ‖a‖) : A ⊆ a • A 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {A : Set E} {a : 𝕜} (hA : Balanced 𝕜 A) (ha : 1 ≤ ‖a‖)
Goal: A ⊆ a • A
-- Balanced.smul_congr
theorem Balanced.smul_congr (hs : Balanced 𝕜 A) (h : ‖a‖ = ‖b‖) : a • A = b • A 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {A : Set E} {a b : 𝕜} (hs : Balanced 𝕜 A) (h : ‖a‖ = ‖b‖)
Goal: a • A = b • A
-- Balanced.smul_eq
theorem Balanced.smul_eq (hA : Balanced 𝕜 A) (ha : ‖a‖ = 1) : a • A = A 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {A : Set E} {a : 𝕜} (hA : Balanced 𝕜 A) (ha : ‖a‖ = 1)
Goal: a • A = A
-- Balanced.absorbs_self
theorem Balanced.absorbs_self (hA : Balanced 𝕜 A) : Absorbs 𝕜 A A 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {A : Set E} (hA : Balanced 𝕜 A)
Goal: Absorbs 𝕜 A A
-- Balanced.smul_mem_iff
theorem Balanced.smul_mem_iff (hs : Balanced 𝕜 s) (h : ‖a‖ = ‖b‖) : a • x ∈ s ↔ b • x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} {a b : 𝕜} (hs : Balanced 𝕜 s) (h : ‖a‖ = ‖b‖)
Goal: a • x ∈ s ↔ b • x ∈ s
-- absorbent_nhds_zero
theorem absorbent_nhds_zero (hA : A ∈ 𝓝 (0 : E)) : Absorbent 𝕜 A 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {A : Set E} [TopologicalSpace E] [ContinuousSMul 𝕜 E] (hA : A ∈ 𝓝 0)
Goal: Absorbent 𝕜 A
-- Balanced.zero_insert_interior
theorem Balanced.zero_insert_interior (hA : Balanced 𝕜 A) :
    Balanced 𝕜 (insert 0 (interior A)) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {A : Set E} [TopologicalSpace E] [ContinuousSMul 𝕜 E] (hA : Balanced 𝕜 A)
Goal: Balanced 𝕜 (insert 0 (interior A))
-- balanced_zero_union_interior
@[deprecated Balanced.zero_insert_interior (since := "2024-02-03")]
theorem balanced_zero_union_interior (hA : Balanced 𝕜 A) : Balanced 𝕜 ((0 : Set E) ∪ interior A) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {A : Set E} [TopologicalSpace E] [ContinuousSMul 𝕜 E] (hA : Balanced 𝕜 A)
Goal: Balanced 𝕜 (0 ∪ interior A)
-- Balanced.interior
protected theorem Balanced.interior (hA : Balanced 𝕜 A) (h : (0 : E) ∈ interior A) :
    Balanced 𝕜 (interior A) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {A : Set E} [TopologicalSpace E] [ContinuousSMul 𝕜 E] (hA : Balanced 𝕜 A) (h : 0 ∈ interior A)
Goal: Balanced 𝕜 (interior A)
-- Balanced.closure
protected theorem Balanced.closure (hA : Balanced 𝕜 A) : Balanced 𝕜 (closure A) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {A : Set E} [TopologicalSpace E] [ContinuousSMul 𝕜 E] (hA : Balanced 𝕜 A)
Goal: Balanced 𝕜 (closure A)
-- Absorbent.zero_mem'
@[deprecated Absorbent.zero_mem (since := "2024-02-02")]
theorem Absorbent.zero_mem' (hs : Absorbent 𝕜 s) : (0 : E) ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : Absorbent 𝕜 s)
Goal: 0 ∈ s
-- Balanced.convexHull
protected theorem Balanced.convexHull (hs : Balanced 𝕜 s) : Balanced 𝕜 (convexHull ℝ s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} [Module ℝ E] [SMulCommClass ℝ 𝕜 E] (hs : Balanced 𝕜 s)
Goal: Balanced 𝕜 ((convexHull ℝ) s)
-- balanced_iff_neg_mem
theorem balanced_iff_neg_mem (hs : Convex ℝ s) : Balanced ℝ s ↔ ∀ ⦃x⦄, x ∈ s → -x ∈ s 
Arguments: {E : Type u_3} [AddCommGroup E] [Module ℝ E] {s : Set E} (hs : Convex ℝ s)
Goal: Balanced ℝ s ↔ ∀ ⦃x : E⦄, x ∈ s → -x ∈ s
-- Real.fourierCoeff_tsum_comp_add
theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
    (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖)
    (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m 
Arguments: {f : C(ℝ, ℂ)} (hf :
  ∀ (K : TopologicalSpace.Compacts ℝ),
    Summable fun n => ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖) (m : ℤ)
Goal: fourierCoeff ⋯.lift m = 𝓕 ⇑f ↑m
-- Real.tsum_eq_tsum_fourierIntegral
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
    (h_norm :
      ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖)
    (h_sum : Summable fun n : ℤ => 𝓕 f n) (x : ℝ) :
    ∑' n : ℤ, f (x + n) = ∑' n : ℤ, 𝓕 f n * fourier n (x : UnitAddCircle) 
Arguments: {f : C(ℝ, ℂ)} (h_norm :
  ∀ (K : TopologicalSpace.Compacts ℝ),
    Summable fun n => ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖) (h_sum : Summable fun n => 𝓕 ⇑f ↑n) (x : ℝ)
Goal: ∑' (n : ℤ), f (x + ↑n) = ∑' (n : ℤ), 𝓕 ⇑f ↑n * (fourier n) ↑x
-- isBigO_norm_Icc_restrict_atTop
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
    (hf : f =O[atTop] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
    (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atTop] fun x : ℝ => |x| ^ (-b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : ⇑f =O[Filter.atTop] fun x => |x| ^ (-b)) (R S : ℝ)
Goal: (fun x => ‖ContinuousMap.restrict (Set.Icc (x + R) (x + S)) f‖) =O[Filter.atTop] fun x => |x| ^ (-b)
-- isBigO_norm_Icc_restrict_atBot
theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
    (hf : f =O[atBot] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
    (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atBot] fun x : ℝ => |x| ^ (-b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : ⇑f =O[Filter.atBot] fun x => |x| ^ (-b)) (R S : ℝ)
Goal: (fun x => ‖ContinuousMap.restrict (Set.Icc (x + R) (x + S)) f‖) =O[Filter.atBot] fun x => |x| ^ (-b)
-- isBigO_norm_restrict_cocompact
theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
    (hf : f =O[cocompact ℝ] fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
    (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) =O[cocompact ℝ] (|·| ^ (-b)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b) (hf : ⇑f =O[Filter.cocompact ℝ] fun x => |x| ^ (-b)) (K : TopologicalSpace.Compacts ℝ)
Goal: (fun x => ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))‖) =O[Filter.cocompact ℝ] fun x => |x| ^ (-b)
-- Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable
theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ → ℂ} (hc : Continuous f)
    {b : ℝ} (hb : 1 < b) (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
    (hFf : Summable fun n : ℤ => 𝓕 f n) (x : ℝ) :
    ∑' n : ℤ, f (x + n) = ∑' n : ℤ, 𝓕 f n * fourier n (x : UnitAddCircle) 
Arguments: {f : ℝ → ℂ} (hc : Continuous f) {b : ℝ} (hb : 1 < b) (hf : f =O[Filter.cocompact ℝ] fun x => |x| ^ (-b)) (hFf : Summable fun n => 𝓕 f ↑n) (x : ℝ)
Goal: ∑' (n : ℤ), f (x + ↑n) = ∑' (n : ℤ), 𝓕 f ↑n * (fourier n) ↑x