l3lab/ntp-mathlib huggingface dataset:
/-- Any `1`-separated set in the ball of radius `2` has cardinality at most `5 ^ dim`. This is
useful to show that the supremum in the definition of `Besicovitch.multiplicity E` is
well behaved. -/
theorem card_le_of_separated (s : Finset E) (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ 5 ^ finrank ℝ E := by
  /- We consider balls of radius `1/2` around the points in `s`. They are disjoint, and all
    contained in the ball of radius `5/2`. A volume argument gives `s.card * (1/2)^dim ≤ (5/2)^dim`,
    i.e., `s.card ≤ 5^dim`. -/
  borelize E
  let μ : Measure E := Measure.addHaar
  let δ : ℝ := (1 : ℝ) / 2
  let ρ : ℝ := (5 : ℝ) / 2
  have ρpos : 0 < ρ := by norm_num
  set A := ⋃ c ∈ s, ball (c : E) δ with hA
  have D : Set.Pairwise (s : Set E) (Disjoint on fun c => ball (c : E) δ) := by
    rintro c hc d hd hcd
    apply ball_disjoint_ball
    rw [dist_eq_norm]
    convert h c hc d hd hcd
    norm_num
  have A_subset : A ⊆ ball (0 : E) ρ := by
    refine' iUnion₂_subset fun x hx => _
    apply ball_subset_ball'
    calc
      δ + dist x 0 ≤ δ + 2 := by rw [dist_zero_right]; exact add_le_add le_rfl (hs x hx)
      _ = 5 / 2 := by norm_num
  have I :
    (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) ≤
      ENNReal.ofReal (ρ ^ finrank ℝ E) * μ (ball 0 1) :=
    calc
      (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) = μ A := by
        rw [hA, measure_biUnion_finset D fun c _ => measurableSet_ball]
        have I : 0 < δ := by norm_num
        simp only [div_pow, μ.addHaar_ball_of_pos _ I]
        simp only [one_div, one_pow, Finset.sum_const, nsmul_eq_mul, mul_assoc]
      _ ≤ μ (ball (0 : E) ρ) := (measure_mono A_subset)
      _ = ENNReal.ofReal (ρ ^ finrank ℝ E) * μ (ball 0 1) := by
        simp only [μ.addHaar_ball_of_pos _ ρpos]
  have J : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) :=
    (ENNReal.mul_le_mul_right (measure_ball_pos _ _ zero_lt_one).ne' measure_ball_lt_top.ne).1 I
  have K : (s.card : ℝ) ≤ (5 : ℝ) ^ finrank ℝ E := by
    have := ENNReal.toReal_le_of_le_ofReal (pow_nonneg ρpos.le _) J
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
μ : Measure E := Measure.addHaar
δ : ℝ := 1 / 2
ρ : ℝ := 5 / 2
ρpos : 0 < ρ
A : Set E := ⋃ c ∈ s, ball c δ
hA : A = ⋃ c ∈ s, ball c δ
D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ)
A_subset : A ⊆ ball 0 ρ
I :
  ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) ≤
    ENNReal.ofReal (ρ ^ finrank ℝ E) * ↑↑μ (ball 0 1)
J : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E)
this : ENNReal.toReal (↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E)) ≤ ρ ^ finrank ℝ E
⊢ ↑(Finset.card s) ≤ 5 ^ finrank ℝ E

Next tactic:
simp? [ENNReal.toReal_mul] at this says
      simp only [one_div, inv_pow, ENNReal.toReal_mul, ENNReal.toReal_nat, inv_nonneg, ge_iff_le,
        div_pow] at this
/-- Any `1`-separated set in the ball of radius `2` has cardinality at most `5 ^ dim`. This is
useful to show that the supremum in the definition of `Besicovitch.multiplicity E` is
well behaved. -/
theorem card_le_of_separated (s : Finset E) (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ 5 ^ finrank ℝ E := by
  /- We consider balls of radius `1/2` around the points in `s`. They are disjoint, and all
    contained in the ball of radius `5/2`. A volume argument gives `s.card * (1/2)^dim ≤ (5/2)^dim`,
    i.e., `s.card ≤ 5^dim`. -/
  borelize E
  let μ : Measure E := Measure.addHaar
  let δ : ℝ := (1 : ℝ) / 2
  let ρ : ℝ := (5 : ℝ) / 2
  have ρpos : 0 < ρ := by norm_num
  set A := ⋃ c ∈ s, ball (c : E) δ with hA
  have D : Set.Pairwise (s : Set E) (Disjoint on fun c => ball (c : E) δ) := by
    rintro c hc d hd hcd
    apply ball_disjoint_ball
    rw [dist_eq_norm]
    convert h c hc d hd hcd
    norm_num
  have A_subset : A ⊆ ball (0 : E) ρ := by
    refine' iUnion₂_subset fun x hx => _
    apply ball_subset_ball'
    calc
      δ + dist x 0 ≤ δ + 2 := by rw [dist_zero_right]; exact add_le_add le_rfl (hs x hx)
      _ = 5 / 2 := by norm_num
  have I :
    (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) ≤
      ENNReal.ofReal (ρ ^ finrank ℝ E) * μ (ball 0 1) :=
    calc
      (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) = μ A := by
        rw [hA, measure_biUnion_finset D fun c _ => measurableSet_ball]
        have I : 0 < δ := by norm_num
        simp only [div_pow, μ.addHaar_ball_of_pos _ I]
        simp only [one_div, one_pow, Finset.sum_const, nsmul_eq_mul, mul_assoc]
      _ ≤ μ (ball (0 : E) ρ) := (measure_mono A_subset)
      _ = ENNReal.ofReal (ρ ^ finrank ℝ E) * μ (ball 0 1) := by
        simp only [μ.addHaar_ball_of_pos _ ρpos]
  have J : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) :=
    (ENNReal.mul_le_mul_right (measure_ball_pos _ _ zero_lt_one).ne' measure_ball_lt_top.ne).1 I
  have K : (s.card : ℝ) ≤ (5 : ℝ) ^ finrank ℝ E := by
    have := ENNReal.toReal_le_of_le_ofReal (pow_nonneg ρpos.le _) J
    simp? [ENNReal.toReal_mul] at this says
      

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
μ : Measure E := Measure.addHaar
δ : ℝ := 1 / 2
ρ : ℝ := 5 / 2
ρpos : 0 < ρ
A : Set E := ⋃ c ∈ s, ball c δ
hA : A = ⋃ c ∈ s, ball c δ
D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ)
A_subset : A ⊆ ball 0 ρ
I :
  ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) ≤
    ENNReal.ofReal (ρ ^ finrank ℝ E) * ↑↑μ (ball 0 1)
J : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E)
this : ENNReal.toReal (↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E)) ≤ ρ ^ finrank ℝ E
⊢ ↑(Finset.card s) ≤ 5 ^ finrank ℝ E

Next tactic:
simp only [one_div, inv_pow, ENNReal.toReal_mul, ENNReal.toReal_nat, inv_nonneg, ge_iff_le,
        div_pow] at this
/-- Any `1`-separated set in the ball of radius `2` has cardinality at most `5 ^ dim`. This is
useful to show that the supremum in the definition of `Besicovitch.multiplicity E` is
well behaved. -/
theorem card_le_of_separated (s : Finset E) (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ 5 ^ finrank ℝ E := by
  /- We consider balls of radius `1/2` around the points in `s`. They are disjoint, and all
    contained in the ball of radius `5/2`. A volume argument gives `s.card * (1/2)^dim ≤ (5/2)^dim`,
    i.e., `s.card ≤ 5^dim`. -/
  borelize E
  let μ : Measure E := Measure.addHaar
  let δ : ℝ := (1 : ℝ) / 2
  let ρ : ℝ := (5 : ℝ) / 2
  have ρpos : 0 < ρ := by norm_num
  set A := ⋃ c ∈ s, ball (c : E) δ with hA
  have D : Set.Pairwise (s : Set E) (Disjoint on fun c => ball (c : E) δ) := by
    rintro c hc d hd hcd
    apply ball_disjoint_ball
    rw [dist_eq_norm]
    convert h c hc d hd hcd
    norm_num
  have A_subset : A ⊆ ball (0 : E) ρ := by
    refine' iUnion₂_subset fun x hx => _
    apply ball_subset_ball'
    calc
      δ + dist x 0 ≤ δ + 2 := by rw [dist_zero_right]; exact add_le_add le_rfl (hs x hx)
      _ = 5 / 2 := by norm_num
  have I :
    (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) ≤
      ENNReal.ofReal (ρ ^ finrank ℝ E) * μ (ball 0 1) :=
    calc
      (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) = μ A := by
        rw [hA, measure_biUnion_finset D fun c _ => measurableSet_ball]
        have I : 0 < δ := by norm_num
        simp only [div_pow, μ.addHaar_ball_of_pos _ I]
        simp only [one_div, one_pow, Finset.sum_const, nsmul_eq_mul, mul_assoc]
      _ ≤ μ (ball (0 : E) ρ) := (measure_mono A_subset)
      _ = ENNReal.ofReal (ρ ^ finrank ℝ E) * μ (ball 0 1) := by
        simp only [μ.addHaar_ball_of_pos _ ρpos]
  have J : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) :=
    (ENNReal.mul_le_mul_right (measure_ball_pos _ _ zero_lt_one).ne' measure_ball_lt_top.ne).1 I
  have K : (s.card : ℝ) ≤ (5 : ℝ) ^ finrank ℝ E := by
    have := ENNReal.toReal_le_of_le_ofReal (pow_nonneg ρpos.le _) J
    simp? [ENNReal.toReal_mul] at this says
      simp only [one_div, inv_pow, ENNReal.toReal_mul, ENNReal.toReal_nat, inv_nonneg, ge_iff_le,
        div_pow] at this
    simpa [div_eq_mul_inv, zero_le_two] using this
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
μ : Measure E := Measure.addHaar
δ : ℝ := 1 / 2
ρ : ℝ := 5 / 2
ρpos : 0 < ρ
A : Set E := ⋃ c ∈ s, ball c δ
hA : A = ⋃ c ∈ s, ball c δ
D : Set.Pairwise (↑s) (Disjoint on fun c => ball c δ)
A_subset : A ⊆ ball 0 ρ
I :
  ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) * ↑↑μ (ball 0 1) ≤
    ENNReal.ofReal (ρ ^ finrank ℝ E) * ↑↑μ (ball 0 1)
J : ↑(Finset.card s) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E)
K : ↑(Finset.card s) ≤ 5 ^ finrank ℝ E
⊢ Finset.card s ≤ 5 ^ finrank ℝ E

Next tactic:
exact mod_cast K
theorem multiplicity_le : multiplicity E ≤ 5 ^ finrank ℝ E := by
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ multiplicity E ≤ 5 ^ finrank ℝ E

Next tactic:
apply csSup_le
theorem multiplicity_le : multiplicity E ≤ 5 ^ finrank ℝ E := by
  apply csSup_le
  · 

Proof state:
case h₁
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ Set.Nonempty {N | ∃ s, Finset.card s = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖}

Next tactic:
refine' ⟨0, ⟨∅, by simp⟩⟩
theorem multiplicity_le : multiplicity E ≤ 5 ^ finrank ℝ E := by
  apply csSup_le
  · refine' ⟨0, ⟨∅, by 

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ Finset.card ∅ = 0 ∧ (∀ c ∈ ∅, ‖c‖ ≤ 2) ∧ ∀ c ∈ ∅, ∀ d ∈ ∅, c ≠ d → 1 ≤ ‖c - d‖

Next tactic:
simp
theorem multiplicity_le : multiplicity E ≤ 5 ^ finrank ℝ E := by
  apply csSup_le
  · refine' ⟨0, ⟨∅, by simp⟩⟩
  · 

Proof state:
case h₂
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ ∀ b ∈ {N | ∃ s, Finset.card s = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖}, b ≤ 5 ^ finrank ℝ E

Next tactic:
rintro _ ⟨s, ⟨rfl, h⟩⟩
theorem multiplicity_le : multiplicity E ≤ 5 ^ finrank ℝ E := by
  apply csSup_le
  · refine' ⟨0, ⟨∅, by simp⟩⟩
  · rintro _ ⟨s, ⟨rfl, h⟩⟩
    

Proof state:
case h₂.intro.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
h : (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ Finset.card s ≤ 5 ^ finrank ℝ E

Next tactic:
exact Besicovitch.card_le_of_separated s h.1 h.2
theorem card_le_multiplicity {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ multiplicity E := by
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ Finset.card s ≤ multiplicity E

Next tactic:
apply le_csSup
theorem card_le_multiplicity {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ multiplicity E := by
  apply le_csSup
  · 

Proof state:
case h₁
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ BddAbove {N | ∃ s, Finset.card s = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖}

Next tactic:
refine' ⟨5 ^ finrank ℝ E, _⟩
theorem card_le_multiplicity {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ multiplicity E := by
  apply le_csSup
  · refine' ⟨5 ^ finrank ℝ E, _⟩
    

Proof state:
case h₁
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ 5 ^ finrank ℝ E ∈
    upperBounds {N | ∃ s, Finset.card s = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖}

Next tactic:
rintro _ ⟨s, ⟨rfl, h⟩⟩
theorem card_le_multiplicity {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ multiplicity E := by
  apply le_csSup
  · refine' ⟨5 ^ finrank ℝ E, _⟩
    rintro _ ⟨s, ⟨rfl, h⟩⟩
    

Proof state:
case h₁.intro.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s✝ : Finset E
hs : ∀ c ∈ s✝, ‖c‖ ≤ 2
h's : ∀ c ∈ s✝, ∀ d ∈ s✝, c ≠ d → 1 ≤ ‖c - d‖
s : Finset E
h : (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ Finset.card s ≤ 5 ^ finrank ℝ E

Next tactic:
exact Besicovitch.card_le_of_separated s h.1 h.2
theorem card_le_multiplicity {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ multiplicity E := by
  apply le_csSup
  · refine' ⟨5 ^ finrank ℝ E, _⟩
    rintro _ ⟨s, ⟨rfl, h⟩⟩
    exact Besicovitch.card_le_of_separated s h.1 h.2
  · 

Proof state:
case h₂
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ Finset.card s ∈ {N | ∃ s, Finset.card s = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖}

Next tactic:
simp only [mem_setOf_eq, Ne.def]
theorem card_le_multiplicity {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ multiplicity E := by
  apply le_csSup
  · refine' ⟨5 ^ finrank ℝ E, _⟩
    rintro _ ⟨s, ⟨rfl, h⟩⟩
    exact Besicovitch.card_le_of_separated s h.1 h.2
  · simp only [mem_setOf_eq, Ne.def]
    

Proof state:
case h₂
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ ∃ s_1, Finset.card s_1 = Finset.card s ∧ (∀ c ∈ s_1, ‖c‖ ≤ 2) ∧ ∀ c ∈ s_1, ∀ d ∈ s_1, ¬c = d → 1 ≤ ‖c - d‖

Next tactic:
exact ⟨s, rfl, hs, h's⟩
/-- If `δ` is small enough, a `(1-δ)`-separated set in the ball of radius `2` also has cardinality
at most `multiplicity E`. -/
theorem exists_goodδ :
    ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ 2) →
      (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E := by
  classical
  /- This follows from a compactness argument: otherwise, one could extract a converging
    subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
    `N = multiplicity E + 1`. To formalize this, we work with functions `Fin N → E`.
     -/
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ ∃ δ,
    0 < δ ∧
      δ < 1 ∧
        ∀ (s : Finset E),
          (∀ c ∈ s, ‖c‖ ≤ 2) → (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → Finset.card s ≤ multiplicity E

Next tactic:
by_contra! h
/-- If `δ` is small enough, a `(1-δ)`-separated set in the ball of radius `2` also has cardinality
at most `multiplicity E`. -/
theorem exists_goodδ :
    ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ 2) →
      (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E := by
  classical
  /- This follows from a compactness argument: otherwise, one could extract a converging
    subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
    `N = multiplicity E + 1`. To formalize this, we work with functions `Fin N → E`.
     -/
  by_contra! h
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ),
    0 < δ →
      δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < Finset.card s
⊢ False

Next tactic:
set N := multiplicity E + 1 with hN
/-- If `δ` is small enough, a `(1-δ)`-separated set in the ball of radius `2` also has cardinality
at most `multiplicity E`. -/
theorem exists_goodδ :
    ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ 2) →
      (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E := by
  classical
  /- This follows from a compactness argument: otherwise, one could extract a converging
    subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
    `N = multiplicity E + 1`. To formalize this, we work with functions `Fin N → E`.
     -/
  by_contra! h
  set N := multiplicity E + 1 with hN
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ),
    0 < δ →
      δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < Finset.card s
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
⊢ False

Next tactic:
have :
    ∀ δ : ℝ, 0 < δ → ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧
      ∀ i j, i ≠ j → 1 - δ ≤ ‖f i - f j‖ := by
    intro δ hδ
    rcases lt_or_le δ 1 with (hδ' | hδ')
    · rcases h δ hδ hδ' with ⟨s, hs, h's, s_card⟩
      obtain ⟨f, f_inj, hfs⟩ : ∃ f : Fin N → E, Function.Injective f ∧ range f ⊆ ↑s := by
        have : Fintype.card (Fin N) ≤ s.card := by simp only [Fintype.card_fin]; exact s_card
        rcases Function.Embedding.exists_of_card_le_finset this with ⟨f, hf⟩
        exact ⟨f, f.injective, hf⟩
      simp only [range_subset_iff, Finset.mem_coe] at hfs
      refine' ⟨f, fun i => hs _ (hfs i), fun i j hij => h's _ (hfs i) _ (hfs j) (f_inj.ne hij)⟩
    · exact
        ⟨fun _ => 0, fun i => by simp; norm_num, fun i j _ => by
          simpa only [norm_zero, sub_nonpos, sub_self]⟩
/-- If `δ` is small enough, a `(1-δ)`-separated set in the ball of radius `2` also has cardinality
at most `multiplicity E`. -/
theorem exists_goodδ :
    ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ 2) →
      (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E := by
  classical
  /- This follows from a compactness argument: otherwise, one could extract a converging
    subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
    `N = multiplicity E + 1`. To formalize this, we work with functions `Fin N → E`.
     -/
  by_contra! h
  set N := multiplicity E + 1 with hN
  have :
    ∀ δ : ℝ, 0 < δ → ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧
      ∀ i j, i ≠ j → 1 - δ ≤ ‖f i - f j‖ := by
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ),
    0 < δ →
      δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < Finset.card s
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
⊢ ∀ (δ : ℝ), 0 < δ → ∃ f, (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ ∀ (i j : Fin N), i ≠ j → 1 - δ ≤ ‖f i - f j‖

Next tactic:
intro δ hδ
/-- If `δ` is small enough, a `(1-δ)`-separated set in the ball of radius `2` also has cardinality
at most `multiplicity E`. -/
theorem exists_goodδ :
    ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ 2) →
      (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E := by
  classical
  /- This follows from a compactness argument: otherwise, one could extract a converging
    subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
    `N = multiplicity E + 1`. To formalize this, we work with functions `Fin N → E`.
     -/
  by_contra! h
  set N := multiplicity E + 1 with hN
  have :
    ∀ δ : ℝ, 0 < δ → ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧
      ∀ i j, i ≠ j → 1 - δ ≤ ‖f i - f j‖ := by
    intro δ hδ
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ),
    0 < δ →
      δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < Finset.card s
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
δ : ℝ
hδ : 0 < δ
⊢ ∃ f, (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ ∀ (i j : Fin N), i ≠ j → 1 - δ ≤ ‖f i - f j‖

Next tactic:
rcases lt_or_le δ 1 with (hδ' | hδ')
/-- If `δ` is small enough, a `(1-δ)`-separated set in the ball of radius `2` also has cardinality
at most `multiplicity E`. -/
theorem exists_goodδ :
    ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ 2) →
      (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E := by
  classical
  /- This follows from a compactness argument: otherwise, one could extract a converging
    subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
    `N = multiplicity E + 1`. To formalize this, we work with functions `Fin N → E`.
     -/
  by_contra! h
  set N := multiplicity E + 1 with hN
  have :
    ∀ δ : ℝ, 0 < δ → ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧
      ∀ i j, i ≠ j → 1 - δ ≤ ‖f i - f j‖ := by
    intro δ hδ
    rcases lt_or_le δ 1 with (hδ' | hδ')
    · 

Proof state:
case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ),
    0 < δ →
      δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < Finset.card s
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
δ : ℝ
hδ : 0 < δ
hδ' : δ < 1
⊢ ∃ f, (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ ∀ (i j : Fin N), i ≠ j → 1 - δ ≤ ‖f i - f j‖

Next tactic:
rcases h δ hδ hδ' with ⟨s, hs, h's, s_card⟩
/-- If `δ` is small enough, a `(1-δ)`-separated set in the ball of radius `2` also has cardinality
at most `multiplicity E`. -/
theorem exists_goodδ :
    ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ 2) →
      (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E := by
  classical
  /- This follows from a compactness argument: otherwise, one could extract a converging
    subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
    `N = multiplicity E + 1`. To formalize this, we work with functions `Fin N → E`.
     -/
  by_contra! h
  set N := multiplicity E + 1 with hN
  have :
    ∀ δ : ℝ, 0 < δ → ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧
      ∀ i j, i ≠ j → 1 - δ ≤ ‖f i - f j‖ := by
    intro δ hδ
    rcases lt_or_le δ 1 with (hδ' | hδ')
    · rcases h δ hδ hδ' with ⟨s, hs, h's, s_card⟩
      

Proof state:
case inl.intro.intro.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ),
    0 < δ →
      δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < Finset.card s
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
δ : ℝ
hδ : 0 < δ
hδ' : δ < 1
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖
s_card : multiplicity E < Finset.card s
⊢ ∃ f, (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ ∀ (i j : Fin N), i ≠ j → 1 - δ ≤ ‖f i - f j‖

Next tactic:
obtain ⟨f, f_inj, hfs⟩ : ∃ f : Fin N → E, Function.Injective f ∧ range f ⊆ ↑s := by
        have : Fintype.card (Fin N) ≤ s.card := by simp only [Fintype.card_fin]; exact s_card
        rcases Function.Embedding.exists_of_card_le_finset this with ⟨f, hf⟩
        exact ⟨f, f.injective, hf⟩