ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Seminorm.preimage_metric_closedBall
theorem preimage_metric_closedBall {r : ℝ} : p ⁻¹' Metric.closedBall 0 r = { x | p x ≤ r } 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {r : ℝ}
Goal: ⇑p ⁻¹' Metric.closedBall 0 r = {x | p x ≤ r}
-- Seminorm.ball_zero_eq_preimage_ball
theorem ball_zero_eq_preimage_ball {r : ℝ} : p.ball 0 r = p ⁻¹' Metric.ball 0 r 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {r : ℝ}
Goal: p.ball 0 r = ⇑p ⁻¹' Metric.ball 0 r
-- Seminorm.closedBall_zero_eq_preimage_closedBall
theorem closedBall_zero_eq_preimage_closedBall {r : ℝ} :
    p.closedBall 0 r = p ⁻¹' Metric.closedBall 0 r 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {r : ℝ}
Goal: p.closedBall 0 r = ⇑p ⁻¹' Metric.closedBall 0 r
-- Seminorm.ball_bot
@[simp]
theorem ball_bot {r : ℝ} (x : E) (hr : 0 < r) : ball (⊥ : Seminorm 𝕜 E) x r = Set.univ 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {r : ℝ} (x : E) (hr : 0 < r)
Goal: ⊥.ball x r = Set.univ
-- Seminorm.closedBall_bot
@[simp]
theorem closedBall_bot {r : ℝ} (x : E) (hr : 0 < r) :
    closedBall (⊥ : Seminorm 𝕜 E) x r = Set.univ 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {r : ℝ} (x : E) (hr : 0 < r)
Goal: ⊥.closedBall x r = Set.univ
-- Seminorm.balanced_ball_zero
theorem balanced_ball_zero (r : ℝ) : Balanced 𝕜 (ball p 0 r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) (r : ℝ)
Goal: Balanced 𝕜 (p.ball 0 r)
-- Seminorm.balanced_closedBall_zero
theorem balanced_closedBall_zero (r : ℝ) : Balanced 𝕜 (closedBall p 0 r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) (r : ℝ)
Goal: Balanced 𝕜 (p.closedBall 0 r)
-- Seminorm.ball_finset_sup_eq_iInter
theorem ball_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
    (hr : 0 < r) : ball (s.sup p) x r = ⋂ i ∈ s, ball (p i) x r 
Arguments: {𝕜 : Type u_3} {E : Type u_7} {ι : Type u_12} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 < r)
Goal: (s.sup p).ball x r = ⋂ i ∈ s, (p i).ball x r
-- Seminorm.closedBall_finset_sup_eq_iInter
theorem closedBall_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
    (hr : 0 ≤ r) : closedBall (s.sup p) x r = ⋂ i ∈ s, closedBall (p i) x r 
Arguments: {𝕜 : Type u_3} {E : Type u_7} {ι : Type u_12} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 ≤ r)
Goal: (s.sup p).closedBall x r = ⋂ i ∈ s, (p i).closedBall x r
-- Seminorm.ball_finset_sup
theorem ball_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 < r) :
    ball (s.sup p) x r = s.inf fun i => ball (p i) x r 
Arguments: {𝕜 : Type u_3} {E : Type u_7} {ι : Type u_12} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 < r)
Goal: (s.sup p).ball x r = s.inf fun i => (p i).ball x r
-- Seminorm.closedBall_finset_sup
theorem closedBall_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 ≤ r) :
    closedBall (s.sup p) x r = s.inf fun i => closedBall (p i) x r 
Arguments: {𝕜 : Type u_3} {E : Type u_7} {ι : Type u_12} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 ≤ r)
Goal: (s.sup p).closedBall x r = s.inf fun i => (p i).closedBall x r
-- Seminorm.ball_eq_emptyset
@[simp]
theorem ball_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0) : p.ball x r = ∅ 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0)
Goal: p.ball x r = ∅
-- Seminorm.closedBall_eq_emptyset
@[simp]
theorem closedBall_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0) :
    p.closedBall x r = ∅ 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0)
Goal: p.closedBall x r = ∅
-- Seminorm.closedBall_smul_ball
theorem closedBall_smul_ball (p : Seminorm 𝕜 E) {r₁ : ℝ} (hr₁ : r₁ ≠ 0) (r₂ : ℝ) :
    Metric.closedBall (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {r₁ : ℝ} (hr₁ : r₁ ≠ 0) (r₂ : ℝ)
Goal: Metric.closedBall 0 r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂)
-- Seminorm.ball_smul_closedBall
theorem ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
    Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0)
Goal: Metric.ball 0 r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂)
-- Seminorm.ball_smul_ball
theorem ball_smul_ball (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
    Metric.ball (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ)
Goal: Metric.ball 0 r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂)
-- Seminorm.closedBall_smul_closedBall
theorem closedBall_smul_closedBall (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
    Metric.closedBall (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.closedBall 0 (r₁ * r₂) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ)
Goal: Metric.closedBall 0 r₁ • p.closedBall 0 r₂ ⊆ p.closedBall 0 (r₁ * r₂)
-- Seminorm.neg_mem_ball_zero
theorem neg_mem_ball_zero {r : ℝ} {x : E} : -x ∈ ball p 0 r ↔ x ∈ ball p 0 r 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {r : ℝ} {x : E}
Goal: -x ∈ p.ball 0 r ↔ x ∈ p.ball 0 r
-- Seminorm.neg_ball
@[simp]
theorem neg_ball (p : Seminorm 𝕜 E) (r : ℝ) (x : E) : -ball p x r = ball p (-x) r 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) (r : ℝ) (x : E)
Goal: -p.ball x r = p.ball (-x) r
-- Seminorm.closedBall_iSup
theorem closedBall_iSup {ι : Sort*} {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E)
    {r : ℝ} (hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {ι : Sort u_13} {p : ι → Seminorm 𝕜 E} (hp : BddAbove (Set.range p)) (e : E) {r : ℝ} (hr : 0 < r)
Goal: (⨆ i, p i).closedBall e r = ⋂ i, (p i).closedBall e r
-- Seminorm.ball_norm_mul_subset
theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
    p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ}
Goal: p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r
-- Seminorm.smul_ball_zero
theorem smul_ball_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : k ≠ 0) :
    k • p.ball 0 r = p.ball 0 (‖k‖ * r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : k ≠ 0)
Goal: k • p.ball 0 r = p.ball 0 (‖k‖ * r)
-- Seminorm.smul_closedBall_subset
theorem smul_closedBall_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
    k • p.closedBall 0 r ⊆ p.closedBall 0 (‖k‖ * r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ}
Goal: k • p.closedBall 0 r ⊆ p.closedBall 0 (‖k‖ * r)
-- Seminorm.smul_closedBall_zero
theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
    k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖)
Goal: k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r)
-- Seminorm.ball_zero_absorbs_ball_zero
theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
    Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁)
Goal: Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂)
-- Seminorm.absorbent_ball_zero
protected theorem absorbent_ball_zero (hr : 0 < r) : Absorbent 𝕜 (ball p (0 : E) r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {r : ℝ} (hr : 0 < r)
Goal: Absorbent 𝕜 (p.ball 0 r)
-- Seminorm.absorbent_closedBall_zero
protected theorem absorbent_closedBall_zero (hr : 0 < r) : Absorbent 𝕜 (closedBall p (0 : E) r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {r : ℝ} (hr : 0 < r)
Goal: Absorbent 𝕜 (p.closedBall 0 r)
-- Seminorm.absorbent_ball
protected theorem absorbent_ball (hpr : p x < r) : Absorbent 𝕜 (ball p x r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {r : ℝ} {x : E} (hpr : p x < r)
Goal: Absorbent 𝕜 (p.ball x r)
-- Seminorm.absorbent_closedBall
protected theorem absorbent_closedBall (hpr : p x < r) : Absorbent 𝕜 (closedBall p x r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {r : ℝ} {x : E} (hpr : p x < r)
Goal: Absorbent 𝕜 (p.closedBall x r)
-- Seminorm.smul_ball_preimage
@[simp]
theorem smul_ball_preimage (p : Seminorm 𝕜 E) (y : E) (r : ℝ) (a : 𝕜) (ha : a ≠ 0) :
    (a • ·) ⁻¹' p.ball y r = p.ball (a⁻¹ • y) (r / ‖a‖) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) (y : E) (r : ℝ) (a : 𝕜) (ha : a ≠ 0)
Goal: (fun x => a • x) ⁻¹' p.ball y r = p.ball (a⁻¹ • y) (r / ‖a‖)
-- Seminorm.convexOn
protected theorem convexOn : ConvexOn ℝ univ p 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [NormedSpace ℝ 𝕜] [Module 𝕜 E] [SMul ℝ E] [IsScalarTower ℝ 𝕜 E] (p : Seminorm 𝕜 E)
Goal: ConvexOn ℝ Set.univ ⇑p
-- Seminorm.convex_ball
theorem convex_ball : Convex ℝ (ball p x r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [NormedSpace ℝ 𝕜] [Module 𝕜 E] [Module ℝ E] [IsScalarTower ℝ 𝕜 E] (p : Seminorm 𝕜 E) (x : E) (r : ℝ)
Goal: Convex ℝ (p.ball x r)
-- Seminorm.convex_closedBall
theorem convex_closedBall : Convex ℝ (closedBall p x r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [NormedSpace ℝ 𝕜] [Module 𝕜 E] [Module ℝ E] [IsScalarTower ℝ 𝕜 E] (p : Seminorm 𝕜 E) (x : E) (r : ℝ)
Goal: Convex ℝ (p.closedBall x r)
-- Seminorm.coe_restrictScalars
@[simp]
theorem coe_restrictScalars (p : Seminorm 𝕜' E) : (p.restrictScalars 𝕜 : E → ℝ) = p 
Arguments: (𝕜 : Type u_3) {E : Type u_7} {𝕜' : Type u_13} [NormedField 𝕜] [SeminormedRing 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormOneClass 𝕜'] [AddCommGroup E] [Module 𝕜' E] [SMul 𝕜 E] [IsScalarTower 𝕜 𝕜' E] (p : Seminorm 𝕜' E)
Goal: ⇑(Seminorm.restrictScalars 𝕜 p) = ⇑p
-- Seminorm.restrictScalars_ball
@[simp]
theorem restrictScalars_ball (p : Seminorm 𝕜' E) : (p.restrictScalars 𝕜).ball = p.ball 
Arguments: (𝕜 : Type u_3) {E : Type u_7} {𝕜' : Type u_13} [NormedField 𝕜] [SeminormedRing 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormOneClass 𝕜'] [AddCommGroup E] [Module 𝕜' E] [SMul 𝕜 E] [IsScalarTower 𝕜 𝕜' E] (p : Seminorm 𝕜' E)
Goal: (Seminorm.restrictScalars 𝕜 p).ball = p.ball
-- Seminorm.restrictScalars_closedBall
@[simp]
theorem restrictScalars_closedBall (p : Seminorm 𝕜' E) :
    (p.restrictScalars 𝕜).closedBall = p.closedBall 
Arguments: (𝕜 : Type u_3) {E : Type u_7} {𝕜' : Type u_13} [NormedField 𝕜] [SeminormedRing 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormOneClass 𝕜'] [AddCommGroup E] [Module 𝕜' E] [SMul 𝕜 E] [IsScalarTower 𝕜 𝕜' E] (p : Seminorm 𝕜' E)
Goal: (Seminorm.restrictScalars 𝕜 p).closedBall = p.closedBall
-- Seminorm.continuousAt_zero_of_forall'
theorem continuousAt_zero_of_forall' [TopologicalSpace E] {p : Seminorm 𝕝 E}
    (hp : ∀ r > 0, p.closedBall 0 r ∈ (𝓝 0 : Filter E)) :
    ContinuousAt p 0 
Arguments: {𝕝 : Type u_6} {E : Type u_7} [SeminormedRing 𝕝] [AddCommGroup E] [Module 𝕝 E] [TopologicalSpace E] {p : Seminorm 𝕝 E} (hp : ∀ r > 0, p.closedBall 0 r ∈ 𝓝 0)
Goal: ContinuousAt (⇑p) 0
-- Seminorm.continuousAt_zero'
theorem continuousAt_zero' [TopologicalSpace E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E}
    {r : ℝ} (hp : p.closedBall 0 r ∈ (𝓝 0 : Filter E)) : ContinuousAt p 0 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E} {r : ℝ} (hp : p.closedBall 0 r ∈ 𝓝 0)
Goal: ContinuousAt (⇑p) 0
-- Seminorm.continuousAt_zero_of_forall
theorem continuousAt_zero_of_forall [TopologicalSpace E] {p : Seminorm 𝕝 E}
    (hp : ∀ r > 0, p.ball 0 r ∈ (𝓝 0 : Filter E)) :
    ContinuousAt p 0 
Arguments: {𝕝 : Type u_6} {E : Type u_7} [SeminormedRing 𝕝] [AddCommGroup E] [Module 𝕝 E] [TopologicalSpace E] {p : Seminorm 𝕝 E} (hp : ∀ r > 0, p.ball 0 r ∈ 𝓝 0)
Goal: ContinuousAt (⇑p) 0
-- Seminorm.continuousAt_zero
theorem continuousAt_zero [TopologicalSpace E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E} {r : ℝ}
    (hp : p.ball 0 r ∈ (𝓝 0 : Filter E)) : ContinuousAt p 0 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E} {r : ℝ} (hp : p.ball 0 r ∈ 𝓝 0)
Goal: ContinuousAt (⇑p) 0
-- Seminorm.uniformContinuous_of_continuousAt_zero
protected theorem uniformContinuous_of_continuousAt_zero [UniformSpace E] [UniformAddGroup E]
    {p : Seminorm 𝕝 E} (hp : ContinuousAt p 0) : UniformContinuous p 
Arguments: {𝕝 : Type u_6} {E : Type u_7} [SeminormedRing 𝕝] [AddCommGroup E] [Module 𝕝 E] [UniformSpace E] [UniformAddGroup E] {p : Seminorm 𝕝 E} (hp : ContinuousAt (⇑p) 0)
Goal: UniformContinuous ⇑p
-- Seminorm.continuous_of_continuousAt_zero
protected theorem continuous_of_continuousAt_zero [TopologicalSpace E] [TopologicalAddGroup E]
    {p : Seminorm 𝕝 E} (hp : ContinuousAt p 0) : Continuous p 
Arguments: {𝕝 : Type u_6} {E : Type u_7} [SeminormedRing 𝕝] [AddCommGroup E] [Module 𝕝 E] [TopologicalSpace E] [TopologicalAddGroup E] {p : Seminorm 𝕝 E} (hp : ContinuousAt (⇑p) 0)
Goal: Continuous ⇑p
-- Seminorm.uniformContinuous_of_forall
protected theorem uniformContinuous_of_forall [UniformSpace E] [UniformAddGroup E]
    {p : Seminorm 𝕝 E} (hp : ∀ r > 0, p.ball 0 r ∈ (𝓝 0 : Filter E)) :
    UniformContinuous p 
Arguments: {𝕝 : Type u_6} {E : Type u_7} [SeminormedRing 𝕝] [AddCommGroup E] [Module 𝕝 E] [UniformSpace E] [UniformAddGroup E] {p : Seminorm 𝕝 E} (hp : ∀ r > 0, p.ball 0 r ∈ 𝓝 0)
Goal: UniformContinuous ⇑p
-- Seminorm.uniformContinuous
protected theorem uniformContinuous [UniformSpace E] [UniformAddGroup E] [ContinuousConstSMul 𝕜 E]
    {p : Seminorm 𝕜 E} {r : ℝ} (hp : p.ball 0 r ∈ (𝓝 0 : Filter E)) :
    UniformContinuous p 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [UniformSpace E] [UniformAddGroup E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E} {r : ℝ} (hp : p.ball 0 r ∈ 𝓝 0)
Goal: UniformContinuous ⇑p
-- Seminorm.uniformContinuous_of_forall'
protected theorem uniformContinuous_of_forall' [UniformSpace E] [UniformAddGroup E]
    {p : Seminorm 𝕝 E} (hp : ∀ r > 0, p.closedBall 0 r ∈ (𝓝 0 : Filter E)) :
    UniformContinuous p 
Arguments: {𝕝 : Type u_6} {E : Type u_7} [SeminormedRing 𝕝] [AddCommGroup E] [Module 𝕝 E] [UniformSpace E] [UniformAddGroup E] {p : Seminorm 𝕝 E} (hp : ∀ r > 0, p.closedBall 0 r ∈ 𝓝 0)
Goal: UniformContinuous ⇑p
-- Seminorm.uniformContinuous'
protected theorem uniformContinuous' [UniformSpace E] [UniformAddGroup E] [ContinuousConstSMul 𝕜 E]
    {p : Seminorm 𝕜 E} {r : ℝ} (hp : p.closedBall 0 r ∈ (𝓝 0 : Filter E)) :
    UniformContinuous p 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [UniformSpace E] [UniformAddGroup E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E} {r : ℝ} (hp : p.closedBall 0 r ∈ 𝓝 0)
Goal: UniformContinuous ⇑p
-- Seminorm.continuous_of_forall
protected theorem continuous_of_forall [TopologicalSpace E] [TopologicalAddGroup E]
    {p : Seminorm 𝕝 E} (hp : ∀ r > 0, p.ball 0 r ∈ (𝓝 0 : Filter E)) :
    Continuous p 
Arguments: {𝕝 : Type u_6} {E : Type u_7} [SeminormedRing 𝕝] [AddCommGroup E] [Module 𝕝 E] [TopologicalSpace E] [TopologicalAddGroup E] {p : Seminorm 𝕝 E} (hp : ∀ r > 0, p.ball 0 r ∈ 𝓝 0)
Goal: Continuous ⇑p
-- Seminorm.continuous
protected theorem continuous [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E]
    {p : Seminorm 𝕜 E} {r : ℝ} (hp : p.ball 0 r ∈ (𝓝 0 : Filter E)) : Continuous p 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E} {r : ℝ} (hp : p.ball 0 r ∈ 𝓝 0)
Goal: Continuous ⇑p
-- Seminorm.continuous_of_forall'
protected theorem continuous_of_forall' [TopologicalSpace E] [TopologicalAddGroup E]
    {p : Seminorm 𝕝 E} (hp : ∀ r > 0, p.closedBall 0 r ∈ (𝓝 0 : Filter E)) :
    Continuous p 
Arguments: {𝕝 : Type u_6} {E : Type u_7} [SeminormedRing 𝕝] [AddCommGroup E] [Module 𝕝 E] [TopologicalSpace E] [TopologicalAddGroup E] {p : Seminorm 𝕝 E} (hp : ∀ r > 0, p.closedBall 0 r ∈ 𝓝 0)
Goal: Continuous ⇑p
-- Seminorm.continuous'
protected theorem continuous' [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E]
    {p : Seminorm 𝕜 E} {r : ℝ} (hp : p.closedBall 0 r ∈ (𝓝 0 : Filter E)) :
    Continuous p 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E} {r : ℝ} (hp : p.closedBall 0 r ∈ 𝓝 0)
Goal: Continuous ⇑p
-- Seminorm.continuous_of_le
theorem continuous_of_le [TopologicalSpace E] [TopologicalAddGroup E]
    {p q : Seminorm 𝕝 E} (hq : Continuous q) (hpq : p ≤ q) : Continuous p 
Arguments: {𝕝 : Type u_6} {E : Type u_7} [SeminormedRing 𝕝] [AddCommGroup E] [Module 𝕝 E] [TopologicalSpace E] [TopologicalAddGroup E] {p q : Seminorm 𝕝 E} (hq : Continuous ⇑q) (hpq : p ≤ q)
Goal: Continuous ⇑p
-- Seminorm.ball_mem_nhds
lemma ball_mem_nhds [TopologicalSpace E] {p : Seminorm 𝕝 E} (hp : Continuous p) {r : ℝ}
    (hr : 0 < r) : p.ball 0 r ∈ (𝓝 0 : Filter E) 
Arguments: {𝕝 : Type u_6} {E : Type u_7} [SeminormedRing 𝕝] [AddCommGroup E] [Module 𝕝 E] [TopologicalSpace E] {p : Seminorm 𝕝 E} (hp : Continuous ⇑p) {r : ℝ} (hr : 0 < r)
Goal: p.ball 0 r ∈ 𝓝 0
-- Seminorm.uniformSpace_eq_of_hasBasis
lemma uniformSpace_eq_of_hasBasis
    {ι} [UniformSpace E] [UniformAddGroup E] [ContinuousConstSMul 𝕜 E]
    {p' : ι → Prop} {s : ι → Set E} (p : Seminorm 𝕜 E) (hb : (𝓝 0 : Filter E).HasBasis p' s)
    (h₁ : ∃ r, p.closedBall 0 r ∈ 𝓝 0) (h₂ : ∀ i, p' i → ∃ r > 0, p.ball 0 r ⊆ s i) :
    ‹UniformSpace E› = p.toAddGroupSeminorm.toSeminormedAddGroup.toUniformSpace 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {ι : Sort u_13} [UniformSpace E] [UniformAddGroup E] [ContinuousConstSMul 𝕜 E] {p' : ι → Prop} {s : ι → Set E} (p : Seminorm 𝕜 E) (hb : (𝓝 0).HasBasis p' s) (h₁ : ∃ r, p.closedBall 0 r ∈ 𝓝 0) (h₂ : ∀ (i : ι), p' i → ∃ r > 0, p.ball 0 r ⊆ s i)
Goal: inst✝ = PseudoMetricSpace.toUniformSpace
-- Seminorm.uniformity_eq_of_hasBasis
lemma uniformity_eq_of_hasBasis
    {ι} [UniformSpace E] [UniformAddGroup E] [ContinuousConstSMul 𝕜 E]
    {p' : ι → Prop} {s : ι → Set E} (p : Seminorm 𝕜 E) (hb : (𝓝 0 : Filter E).HasBasis p' s)
    (h₁ : ∃ r, p.closedBall 0 r ∈ 𝓝 0) (h₂ : ∀ i, p' i → ∃ r > 0, p.ball 0 r ⊆ s i) :
    𝓤 E = ⨅ r > 0, 𝓟 {x | p (x.1 - x.2) < r} 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {ι : Sort u_13} [UniformSpace E] [UniformAddGroup E] [ContinuousConstSMul 𝕜 E] {p' : ι → Prop} {s : ι → Set E} (p : Seminorm 𝕜 E) (hb : (𝓝 0).HasBasis p' s) (h₁ : ∃ r, p.closedBall 0 r ∈ 𝓝 0) (h₂ : ∀ (i : ι), p' i → ∃ r > 0, p.ball 0 r ⊆ s i)
Goal: 𝓤 E = ⨅ r, ⨅ (_ : r > 0), 𝓟 {x | p (x.1 - x.2) < r}
-- Seminorm.rescale_to_shell_zpow
lemma rescale_to_shell_zpow (p : Seminorm 𝕜 E) {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ}
    (εpos : 0 < ε) {x : E} (hx : p x ≠ 0) : ∃ n : ℤ, c^n ≠ 0 ∧
    p (c^n • x) < ε ∧ (ε / ‖c‖ ≤ p (c^n • x)) ∧ (‖c^n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E} (hx : p x ≠ 0)
Goal: ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x
-- Seminorm.rescale_to_shell
lemma rescale_to_shell (p : Seminorm 𝕜 E) {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E}
    (hx : p x ≠ 0) :
    ∃d : 𝕜, d ≠ 0 ∧ p (d • x) < ε ∧ (ε/‖c‖ ≤ p (d • x)) ∧ (‖d‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : Seminorm 𝕜 E) {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E} (hx : p x ≠ 0)
Goal: ∃ d, d ≠ 0 ∧ p (d • x) < ε ∧ ε / ‖c‖ ≤ p (d • x) ∧ ‖d‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x
-- Seminorm.bound_of_shell
lemma bound_of_shell
    (p q : Seminorm 𝕜 E) {ε C : ℝ} (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < ‖c‖)
    (hf : ∀ x, ε / ‖c‖ ≤ p x → p x < ε → q x ≤ C * p x) {x : E} (hx : p x ≠ 0) :
    q x ≤ C * p x 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p q : Seminorm 𝕜 E) {ε C : ℝ} (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < ‖c‖) (hf : ∀ (x : E), ε / ‖c‖ ≤ p x → p x < ε → q x ≤ C * p x) {x : E} (hx : p x ≠ 0)
Goal: q x ≤ C * p x
-- Seminorm.bound_of_shell_smul
lemma bound_of_shell_smul
    (p q : Seminorm 𝕜 E) {ε : ℝ} {C : ℝ≥0} (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < ‖c‖)
    (hf : ∀ x, ε / ‖c‖ ≤ p x → p x < ε → q x ≤ (C • p) x) {x : E} (hx : p x ≠ 0) :
    q x ≤ (C • p) x 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p q : Seminorm 𝕜 E) {ε : ℝ} {C : ℝ≥0} (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < ‖c‖) (hf : ∀ (x : E), ε / ‖c‖ ≤ p x → p x < ε → q x ≤ (C • p) x) {x : E} (hx : p x ≠ 0)
Goal: q x ≤ (C • p) x
-- Seminorm.bound_of_shell_sup
lemma bound_of_shell_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι)
    (q : Seminorm 𝕜 E) {ε : ℝ} {C : ℝ≥0} (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < ‖c‖)
    (hf : ∀ x, (∀ i ∈ s, p i x < ε) → ∀ j ∈ s, ε / ‖c‖ ≤ p j x → q x ≤ (C • p j) x)
    {x : E} (hx : ∃ j, j ∈ s ∧ p j x ≠ 0) :
    q x ≤ (C • s.sup p) x 
Arguments: {𝕜 : Type u_3} {E : Type u_7} {ι : Type u_12} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : ι → Seminorm 𝕜 E) (s : Finset ι) (q : Seminorm 𝕜 E) {ε : ℝ} {C : ℝ≥0} (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < ‖c‖) (hf : ∀ (x : E), (∀ i ∈ s, (p i) x < ε) → ∀ j ∈ s, ε / ‖c‖ ≤ (p j) x → q x ≤ (C • p j) x) {x : E} (hx : ∃ j ∈ s, (p j) x ≠ 0)
Goal: q x ≤ (C • s.sup p) x
-- Seminorm.bddAbove_of_absorbent
lemma bddAbove_of_absorbent {ι : Sort*} {p : ι → Seminorm 𝕜 E} {s : Set E} (hs : Absorbent 𝕜 s)
    (h : ∀ x ∈ s, BddAbove (range (p · x))) : BddAbove (range p) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {ι : Sort u_13} {p : ι → Seminorm 𝕜 E} {s : Set E} (hs : Absorbent 𝕜 s) (h : ∀ x ∈ s, BddAbove (Set.range fun x_1 => (p x_1) x))
Goal: BddAbove (Set.range p)
-- coe_normSeminorm
@[simp]
theorem coe_normSeminorm : ⇑(normSeminorm 𝕜 E) = norm 
Arguments: (𝕜 : Type u_3) (E : Type u_7) [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: ⇑(normSeminorm 𝕜 E) = norm
-- ball_normSeminorm
@[simp]
theorem ball_normSeminorm : (normSeminorm 𝕜 E).ball = Metric.ball 
Arguments: (𝕜 : Type u_3) (E : Type u_7) [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: (normSeminorm 𝕜 E).ball = Metric.ball
-- absorbent_ball_zero
theorem absorbent_ball_zero (hr : 0 < r) : Absorbent 𝕜 (Metric.ball (0 : E) r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ} (hr : 0 < r)
Goal: Absorbent 𝕜 (Metric.ball 0 r)
-- absorbent_ball
theorem absorbent_ball (hx : ‖x‖ < r) : Absorbent 𝕜 (Metric.ball x r) 
Arguments: {𝕜 : Type u_3} {E : Type u_7} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ} {x : E} (hx : ‖x‖ < r)
Goal: Absorbent 𝕜 (Metric.ball x r)