l3lab/ntp-mathlib huggingface dataset:
theorem continuous_of_le [TopologicalSpace E] [TopologicalAddGroup E]
    {p q : Seminorm 𝕝 E} (hq : Continuous q) (hpq : p ≤ q) : Continuous p := by
  refine Seminorm.continuous_of_forall (fun r hr ↦ Filter.mem_of_superset
    (IsOpen.mem_nhds ?_ <| q.mem_ball_self hr) (ball_antitone hpq))
  rw [ball_zero_eq]
  

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedRing 𝕝
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕝 E
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalAddGroup E
p q : Seminorm 𝕝 E
hq : Continuous ⇑q
hpq : p ≤ q
r : ℝ
hr : r > 0
⊢ IsOpen {y | q y < r}

Next tactic:
exact isOpen_lt hq continuous_const
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) :=
  have this : Tendsto p (𝓝 0) (𝓝 0) := map_zero p ▸ hp.tendsto 0
  by 

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : SeminormedRing 𝕝
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕝 E
inst✝ : TopologicalSpace E
p : Seminorm 𝕝 E
hp : Continuous ⇑p
r : ℝ
hr : 0 < r
this : Tendsto (⇑p) (𝓝 0) (𝓝 0)
⊢ ball p 0 r ∈ 𝓝 0

Next tactic:
simpa only [p.ball_zero_eq] using this (Iio_mem_nhds hr)
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 := by
  

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedRing 𝕝
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕝 E
ι : Sort u_13
inst✝² : UniformSpace E
inst✝¹ : UniformAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p' : ι → Prop
s : ι → Set E
p : Seminorm 𝕜 E
hb : HasBasis (𝓝 0) p' s
h₁ : ∃ r, closedBall p 0 r ∈ 𝓝 0
h₂ : ∀ (i : ι), p' i → ∃ r > 0, ball p 0 r ⊆ s i
⊢ inst✝² = PseudoMetricSpace.toUniformSpace

Next tactic:
refine UniformAddGroup.ext ‹_› p.toAddGroupSeminorm.toSeminormedAddCommGroup.to_uniformAddGroup ?_
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 := by
  refine UniformAddGroup.ext ‹_› p.toAddGroupSeminorm.toSeminormedAddCommGroup.to_uniformAddGroup ?_
  

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedRing 𝕝
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕝 E
ι : Sort u_13
inst✝² : UniformSpace E
inst✝¹ : UniformAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p' : ι → Prop
s : ι → Set E
p : Seminorm 𝕜 E
hb : HasBasis (𝓝 0) p' s
h₁ : ∃ r, closedBall p 0 r ∈ 𝓝 0
h₂ : ∀ (i : ι), p' i → ∃ r > 0, ball p 0 r ⊆ s i
⊢ 𝓝 0 = 𝓝 0

Next tactic:
apply le_antisymm
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 := by
  refine UniformAddGroup.ext ‹_› p.toAddGroupSeminorm.toSeminormedAddCommGroup.to_uniformAddGroup ?_
  apply le_antisymm
  · 

Proof state:
case a
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedRing 𝕝
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕝 E
ι : Sort u_13
inst✝² : UniformSpace E
inst✝¹ : UniformAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p' : ι → Prop
s : ι → Set E
p : Seminorm 𝕜 E
hb : HasBasis (𝓝 0) p' s
h₁ : ∃ r, closedBall p 0 r ∈ 𝓝 0
h₂ : ∀ (i : ι), p' i → ∃ r > 0, ball p 0 r ⊆ s i
⊢ 𝓝 0 ≤ 𝓝 0

Next tactic:
rw [← @comap_norm_nhds_zero E p.toAddGroupSeminorm.toSeminormedAddGroup, ← tendsto_iff_comap]
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 := by
  refine UniformAddGroup.ext ‹_› p.toAddGroupSeminorm.toSeminormedAddCommGroup.to_uniformAddGroup ?_
  apply le_antisymm
  · rw [← @comap_norm_nhds_zero E p.toAddGroupSeminorm.toSeminormedAddGroup, ← tendsto_iff_comap]
    

Proof state:
case a
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedRing 𝕝
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕝 E
ι : Sort u_13
inst✝² : UniformSpace E
inst✝¹ : UniformAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p' : ι → Prop
s : ι → Set E
p : Seminorm 𝕜 E
hb : HasBasis (𝓝 0) p' s
h₁ : ∃ r, closedBall p 0 r ∈ 𝓝 0
h₂ : ∀ (i : ι), p' i → ∃ r > 0, ball p 0 r ⊆ s i
⊢ Tendsto norm (𝓝 0) (𝓝 0)

Next tactic:
suffices Continuous p from this.tendsto' 0 _ (map_zero p)
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 := by
  refine UniformAddGroup.ext ‹_› p.toAddGroupSeminorm.toSeminormedAddCommGroup.to_uniformAddGroup ?_
  apply le_antisymm
  · rw [← @comap_norm_nhds_zero E p.toAddGroupSeminorm.toSeminormedAddGroup, ← tendsto_iff_comap]
    suffices Continuous p from this.tendsto' 0 _ (map_zero p)
    

Proof state:
case a
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedRing 𝕝
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕝 E
ι : Sort u_13
inst✝² : UniformSpace E
inst✝¹ : UniformAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p' : ι → Prop
s : ι → Set E
p : Seminorm 𝕜 E
hb : HasBasis (𝓝 0) p' s
h₁ : ∃ r, closedBall p 0 r ∈ 𝓝 0
h₂ : ∀ (i : ι), p' i → ∃ r > 0, ball p 0 r ⊆ s i
⊢ Continuous ⇑p

Next tactic:
rcases h₁ with ⟨r, hr⟩
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 := by
  refine UniformAddGroup.ext ‹_› p.toAddGroupSeminorm.toSeminormedAddCommGroup.to_uniformAddGroup ?_
  apply le_antisymm
  · rw [← @comap_norm_nhds_zero E p.toAddGroupSeminorm.toSeminormedAddGroup, ← tendsto_iff_comap]
    suffices Continuous p from this.tendsto' 0 _ (map_zero p)
    rcases h₁ with ⟨r, hr⟩
    

Proof state:
case a.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedRing 𝕝
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕝 E
ι : Sort u_13
inst✝² : UniformSpace E
inst✝¹ : UniformAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p' : ι → Prop
s : ι → Set E
p : Seminorm 𝕜 E
hb : HasBasis (𝓝 0) p' s
h₂ : ∀ (i : ι), p' i → ∃ r > 0, ball p 0 r ⊆ s i
r : ℝ
hr : closedBall p 0 r ∈ 𝓝 0
⊢ Continuous ⇑p

Next tactic:
exact p.continuous' hr
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 := by
  refine UniformAddGroup.ext ‹_› p.toAddGroupSeminorm.toSeminormedAddCommGroup.to_uniformAddGroup ?_
  apply le_antisymm
  · rw [← @comap_norm_nhds_zero E p.toAddGroupSeminorm.toSeminormedAddGroup, ← tendsto_iff_comap]
    suffices Continuous p from this.tendsto' 0 _ (map_zero p)
    rcases h₁ with ⟨r, hr⟩
    exact p.continuous' hr
  · 

Proof state:
case a
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedRing 𝕝
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕝 E
ι : Sort u_13
inst✝² : UniformSpace E
inst✝¹ : UniformAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p' : ι → Prop
s : ι → Set E
p : Seminorm 𝕜 E
hb : HasBasis (𝓝 0) p' s
h₁ : ∃ r, closedBall p 0 r ∈ 𝓝 0
h₂ : ∀ (i : ι), p' i → ∃ r > 0, ball p 0 r ⊆ s i
⊢ 𝓝 0 ≤ 𝓝 0

Next tactic:
rw [(@NormedAddCommGroup.nhds_zero_basis_norm_lt E
      p.toAddGroupSeminorm.toSeminormedAddGroup).le_basis_iff hb]
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 := by
  refine UniformAddGroup.ext ‹_› p.toAddGroupSeminorm.toSeminormedAddCommGroup.to_uniformAddGroup ?_
  apply le_antisymm
  · rw [← @comap_norm_nhds_zero E p.toAddGroupSeminorm.toSeminormedAddGroup, ← tendsto_iff_comap]
    suffices Continuous p from this.tendsto' 0 _ (map_zero p)
    rcases h₁ with ⟨r, hr⟩
    exact p.continuous' hr
  · rw [(@NormedAddCommGroup.nhds_zero_basis_norm_lt E
      p.toAddGroupSeminorm.toSeminormedAddGroup).le_basis_iff hb]
    

Proof state:
case a
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedRing 𝕝
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕝 E
ι : Sort u_13
inst✝² : UniformSpace E
inst✝¹ : UniformAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p' : ι → Prop
s : ι → Set E
p : Seminorm 𝕜 E
hb : HasBasis (𝓝 0) p' s
h₁ : ∃ r, closedBall p 0 r ∈ 𝓝 0
h₂ : ∀ (i : ι), p' i → ∃ r > 0, ball p 0 r ⊆ s i
⊢ ∀ (i' : ι), p' i' → ∃ i, 0 < i ∧ {y | ‖y‖ < i} ⊆ s i'

Next tactic:
simpa only [subset_def, mem_ball_zero] using h₂
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} := by
  

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedRing 𝕝
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕝 E
ι : Sort u_13
inst✝² : UniformSpace E
inst✝¹ : UniformAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p' : ι → Prop
s : ι → Set E
p : Seminorm 𝕜 E
hb : HasBasis (𝓝 0) p' s
h₁ : ∃ r, closedBall p 0 r ∈ 𝓝 0
h₂ : ∀ (i : ι), p' i → ∃ r > 0, ball p 0 r ⊆ s i
⊢ 𝓤 E = ⨅ r, ⨅ (_ : r > 0), 𝓟 {x | p (x.1 - x.2) < r}

Next tactic:
rw [uniformSpace_eq_of_hasBasis p hb h₁ h₂]
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} := by
  rw [uniformSpace_eq_of_hasBasis p hb h₁ h₂]; 

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedRing 𝕝
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕝 E
ι : Sort u_13
inst✝² : UniformSpace E
inst✝¹ : UniformAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p' : ι → Prop
s : ι → Set E
p : Seminorm 𝕜 E
hb : HasBasis (𝓝 0) p' s
h₁ : ∃ r, closedBall p 0 r ∈ 𝓝 0
h₂ : ∀ (i : ι), p' i → ∃ r > 0, ball p 0 r ⊆ s i
⊢ 𝓤 E = ⨅ r, ⨅ (_ : r > 0), 𝓟 {x | p (x.1 - x.2) < r}

Next tactic:
rfl
/-- Let `p` be a seminorm on a vector space over a `NormedField`.
If there is a scalar `c` with `‖c‖>1`, then any `x` such that `p x ≠ 0` can be
moved by scalar multiplication to any `p`-shell of width `‖c‖`. Also recap information on the
value of `p` on the rescaling element that shows up in applications. -/
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) := by
  

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
ε : ℝ
εpos : 0 < ε
x : E
hx : p x ≠ 0
⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x

Next tactic:
have xεpos : 0 < (p x)/ε := by positivity
/-- Let `p` be a seminorm on a vector space over a `NormedField`.
If there is a scalar `c` with `‖c‖>1`, then any `x` such that `p x ≠ 0` can be
moved by scalar multiplication to any `p`-shell of width `‖c‖`. Also recap information on the
value of `p` on the rescaling element that shows up in applications. -/
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) := by
  have xεpos : 0 < (p x)/ε := by 

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
ε : ℝ
εpos : 0 < ε
x : E
hx : p x ≠ 0
⊢ 0 < p x / ε

Next tactic:
positivity
/-- Let `p` be a seminorm on a vector space over a `NormedField`.
If there is a scalar `c` with `‖c‖>1`, then any `x` such that `p x ≠ 0` can be
moved by scalar multiplication to any `p`-shell of width `‖c‖`. Also recap information on the
value of `p` on the rescaling element that shows up in applications. -/
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) := by
  have xεpos : 0 < (p x)/ε := by positivity
  

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
ε : ℝ
εpos : 0 < ε
x : E
hx : p x ≠ 0
xεpos : 0 < p x / ε
⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x

Next tactic:
rcases exists_mem_Ico_zpow xεpos hc with ⟨n, hn⟩
/-- Let `p` be a seminorm on a vector space over a `NormedField`.
If there is a scalar `c` with `‖c‖>1`, then any `x` such that `p x ≠ 0` can be
moved by scalar multiplication to any `p`-shell of width `‖c‖`. Also recap information on the
value of `p` on the rescaling element that shows up in applications. -/
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) := by
  have xεpos : 0 < (p x)/ε := by positivity
  rcases exists_mem_Ico_zpow xεpos hc with ⟨n, hn⟩
  

Proof state:
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
ε : ℝ
εpos : 0 < ε
x : E
hx : p x ≠ 0
xεpos : 0 < p x / ε
n : ℤ
hn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))
⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x

Next tactic:
have cpos : 0 < ‖c‖ := by positivity
/-- Let `p` be a seminorm on a vector space over a `NormedField`.
If there is a scalar `c` with `‖c‖>1`, then any `x` such that `p x ≠ 0` can be
moved by scalar multiplication to any `p`-shell of width `‖c‖`. Also recap information on the
value of `p` on the rescaling element that shows up in applications. -/
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) := by
  have xεpos : 0 < (p x)/ε := by positivity
  rcases exists_mem_Ico_zpow xεpos hc with ⟨n, hn⟩
  have cpos : 0 < ‖c‖ := by 

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
ε : ℝ
εpos : 0 < ε
x : E
hx : p x ≠ 0
xεpos : 0 < p x / ε
n : ℤ
hn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))
⊢ 0 < ‖c‖

Next tactic:
positivity
/-- Let `p` be a seminorm on a vector space over a `NormedField`.
If there is a scalar `c` with `‖c‖>1`, then any `x` such that `p x ≠ 0` can be
moved by scalar multiplication to any `p`-shell of width `‖c‖`. Also recap information on the
value of `p` on the rescaling element that shows up in applications. -/
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) := by
  have xεpos : 0 < (p x)/ε := by positivity
  rcases exists_mem_Ico_zpow xεpos hc with ⟨n, hn⟩
  have cpos : 0 < ‖c‖ := by positivity
  

Proof state:
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
ε : ℝ
εpos : 0 < ε
x : E
hx : p x ≠ 0
xεpos : 0 < p x / ε
n : ℤ
hn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))
cpos : 0 < ‖c‖
⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x

Next tactic:
have cnpos : 0 < ‖c^(n+1)‖ := by rw [norm_zpow]; exact xεpos.trans hn.2
/-- Let `p` be a seminorm on a vector space over a `NormedField`.
If there is a scalar `c` with `‖c‖>1`, then any `x` such that `p x ≠ 0` can be
moved by scalar multiplication to any `p`-shell of width `‖c‖`. Also recap information on the
value of `p` on the rescaling element that shows up in applications. -/
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) := by
  have xεpos : 0 < (p x)/ε := by positivity
  rcases exists_mem_Ico_zpow xεpos hc with ⟨n, hn⟩
  have cpos : 0 < ‖c‖ := by positivity
  have cnpos : 0 < ‖c^(n+1)‖ := by 

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
ε : ℝ
εpos : 0 < ε
x : E
hx : p x ≠ 0
xεpos : 0 < p x / ε
n : ℤ
hn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))
cpos : 0 < ‖c‖
⊢ 0 < ‖c ^ (n + 1)‖

Next tactic:
rw [norm_zpow]
/-- Let `p` be a seminorm on a vector space over a `NormedField`.
If there is a scalar `c` with `‖c‖>1`, then any `x` such that `p x ≠ 0` can be
moved by scalar multiplication to any `p`-shell of width `‖c‖`. Also recap information on the
value of `p` on the rescaling element that shows up in applications. -/
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) := by
  have xεpos : 0 < (p x)/ε := by positivity
  rcases exists_mem_Ico_zpow xεpos hc with ⟨n, hn⟩
  have cpos : 0 < ‖c‖ := by positivity
  have cnpos : 0 < ‖c^(n+1)‖ := by rw [norm_zpow]; 

Proof state:
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
ε : ℝ
εpos : 0 < ε
x : E
hx : p x ≠ 0
xεpos : 0 < p x / ε
n : ℤ
hn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))
cpos : 0 < ‖c‖
⊢ 0 < ‖c‖ ^ (n + 1)

Next tactic:
exact xεpos.trans hn.2
/-- Let `p` be a seminorm on a vector space over a `NormedField`.
If there is a scalar `c` with `‖c‖>1`, then any `x` such that `p x ≠ 0` can be
moved by scalar multiplication to any `p`-shell of width `‖c‖`. Also recap information on the
value of `p` on the rescaling element that shows up in applications. -/
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) := by
  have xεpos : 0 < (p x)/ε := by positivity
  rcases exists_mem_Ico_zpow xεpos hc with ⟨n, hn⟩
  have cpos : 0 < ‖c‖ := by positivity
  have cnpos : 0 < ‖c^(n+1)‖ := by rw [norm_zpow]; exact xεpos.trans hn.2
  

Proof state:
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
ε : ℝ
εpos : 0 < ε
x : E
hx : p x ≠ 0
xεpos : 0 < p x / ε
n : ℤ
hn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))
cpos : 0 < ‖c‖
cnpos : 0 < ‖c ^ (n + 1)‖
⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x

Next tactic:
refine ⟨-(n+1), ?_, ?_, ?_, ?_⟩