awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
⊢ A ⊆ ball 0 ρ
have refine : ∀ (x : E) (hx : x ∈ s), ball x δ ⊆ ball 0 ρ
Tactic: refine iUnion₂_subset fun x hx => ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
x : E
hx : x ∈ s
⊢ ball x δ ⊆ ball 0 ρ
have h₁ : δ + dist x 0 ≤ ρ
Tactic: apply ball_subset_ball'
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
x : E
hx : x ∈ s
⊢ δ + dist x 0 ≤ δ + 2
have rw : δ + ‖x‖ ≤ δ + 2
Tactic: rw [dist_zero_right]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
x : E
hx : x ∈ s
⊢ δ + dist x 0 ≤ ρ
have δ₁ : δ + dist x 0 ≤ δ + 2
Tactic: 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
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
x : E
hx : x ∈ s
δ₁ : δ + dist x 0 ≤ δ + 2
⊢ δ + dist x 0 ≤ ρ
have δ₂ : δ + 2 = 5 / 2
Tactic: 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
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
⊢ s.card ≤ HPow.hPow (α := ℕ) 5 (finrank ℝ E)
have A_subset : A ⊆ ball 0 ρ
Tactic: 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
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
A_subset : A ⊆ ball 0 ρ
⊢ HMul.hMul (β := ℝ≥0∞) ((↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E))
      ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞) =
    (μ : (a : Set E) → ℝ≥0∞) A
have rw : HMul.hMul (β := ℝ≥0∞) ((↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E))
    ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞) =
  ∑ p ∈ s, (μ : (a : Set E) → ℝ≥0∞) (ball p δ)
Tactic: rw [hA, measure_biUnion_finset D fun c _ => measurableSet_ball]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
A_subset : A ⊆ ball 0 ρ
⊢ HMul.hMul (β := ℝ≥0∞) ((↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E))
      ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞) =
    ∑ p ∈ s, (μ : (a : Set E) → ℝ≥0∞) (ball p δ)
have I : 0 < δ
Tactic: have I : 0 < δ := by norm_num
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
A_subset : A ⊆ ball 0 ρ
I : 0 < δ
⊢ HMul.hMul (β := ℝ≥0∞) ((↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E))
      ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞) =
    ∑ p ∈ s, (μ : (a : Set E) → ℝ≥0∞) (ball p δ)
have simp : HMul.hMul (β := ℝ≥0∞) ((↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E))
    ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞) =
  ∑ x ∈ s, HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal (δ ^ finrank ℝ E)) ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞)
Tactic: simp only [μ.addHaar_ball_of_pos _ I]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
A_subset : A ⊆ ball 0 ρ
⊢ s.card ≤ HPow.hPow (α := ℕ) 5 (finrank ℝ E)
have I : HMul.hMul (β := ℝ≥0∞) ((↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E))
    ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞) ≤
  HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal (ρ ^ finrank ℝ E)) ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞)
Tactic: 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 [μ.addHaar_ball_of_pos _ I]
        simp only [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]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
A_subset : A ⊆ ball 0 ρ
I :
  HMul.hMul (β := ℝ≥0∞) ((↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E))
      ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞) ≤
    HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal (ρ ^ finrank ℝ E)) ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞)
⊢ s.card ≤ HPow.hPow (α := ℕ) 5 (finrank ℝ E)
have J : (↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E)
Tactic: have J : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) :=
    (ENNReal.mul_le_mul_iff_left (measure_ball_pos _ _ zero_lt_one).ne' measure_ball_lt_top.ne).1 I
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
A_subset : A ⊆ ball 0 ρ
I :
  HMul.hMul (β := ℝ≥0∞) ((↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E))
      ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞) ≤
    HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal (ρ ^ finrank ℝ E)) ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞)
J : (↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E)
⊢ (↑s.card : ℝ) ≤ HPow.hPow (α := ℝ) 5 (finrank ℝ E)
have this : ((↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E)).toReal ≤ ρ ^ finrank ℝ E
Tactic: have := ENNReal.toReal_le_of_le_ofReal (pow_nonneg ρpos.le _) J
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : MeasurableSpace E := borel E
this_1 : 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 : (↑s : Set E).Pairwise (Function.onFun (β := Set E) Disjoint fun (c : E) ↦ ball c δ)
A_subset : A ⊆ ball 0 ρ
I :
  HMul.hMul (β := ℝ≥0∞) ((↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E))
      ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞) ≤
    HMul.hMul (β := ℝ≥0∞) (ENNReal.ofReal (ρ ^ finrank ℝ E)) ((μ : (a : Set E) → ℝ≥0∞) (ball 0 1) : ℝ≥0∞)
J : (↑s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E)
⊢ s.card ≤ HPow.hPow (α := ℕ) 5 (finrank ℝ E)
have K : (↑s.card : ℝ) ≤ HPow.hPow (α := ℝ) 5 (finrank ℝ E)
Tactic: have K : (s.card : ℝ) ≤ (5 : ℝ) ^ finrank ℝ E := by
    have := ENNReal.toReal_le_of_le_ofReal (pow_nonneg ρpos.le _) J
    simpa [ρ, δ, div_eq_mul_inv, mul_pow] using this
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
⊢ multiplicity E ≤ HPow.hPow (α := ℕ) 5 (finrank ℝ E)
have h₁ : Set.Nonempty (α := ℕ)
  {N : ℕ | ∃ (s : Finset E), s.card = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖}
Tactic: apply csSup_le
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
h₁ :
  Set.Nonempty (α := ℕ)
    {N : ℕ | ∃ (s : Finset E), s.card = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖}
⊢ multiplicity E ≤ HPow.hPow (α := ℕ) 5 (finrank ℝ E)
have h₂ : ∀ (b : ℕ),
  Membership.mem (γ := Set ℕ)
      {N : ℕ | ∃ (s : Finset E), s.card = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖} b →
    b ≤ HPow.hPow (α := ℕ) 5 (finrank ℝ E)
Tactic: apply csSup_le
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ s.card ≤ multiplicity E
have h₁ : BddAbove (α := ℕ) {N : ℕ | ∃ (s : Finset E), s.card = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖}
Tactic: apply le_csSup
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
h₁ :
  BddAbove (α := ℕ) {N : ℕ | ∃ (s : Finset E), s.card = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖}
⊢ s.card ≤ multiplicity E
have h₂ : Membership.mem (γ := Set ℕ)
  {N : ℕ | ∃ (s : Finset E), s.card = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖} s.card
Tactic: apply le_csSup
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ BddAbove (α := ℕ) {N : ℕ | ∃ (s : Finset E), s.card = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖}
have h₁ : Membership.mem (γ := Set ℕ)
  (upperBounds {N : ℕ | ∃ (s : Finset E), s.card = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖})
  (HPow.hPow (α := ℕ) 5 (finrank ℝ E))
Tactic: refine ⟨5 ^ finrank ℝ E, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
⊢ ∃ (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
have by_contra! : ∀
  (h :
    ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
      ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card),
  False
Tactic: by_contra! h
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
    ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
⊢ False
have hN : let N := multiplicity E + 1;
N = multiplicity E + 1
Tactic: set N := multiplicity E + 1 with hN
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
    ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
δ : ℝ
hδ : 0 < δ
⊢ ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖
have inl : ∀ (hδ' : δ < 1), ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖
Tactic: rcases lt_or_ge δ 1 with (hδ' | hδ')
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
    ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
δ : ℝ
hδ : 0 < δ
inl : ∀ (hδ' : δ < 1), ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖
⊢ ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖
have inr : ∀ (hδ' : 1 ≤ δ), ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖
Tactic: rcases lt_or_ge δ 1 with (hδ' | hδ')
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
    ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
δ : ℝ
hδ : 0 < δ
hδ' : δ < 1
⊢ ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖
have inl : ∀ (s : Finset E) (hs : ∀ c ∈ s, ‖c‖ ≤ 2) (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖)
  (s_card : multiplicity E < s.card),
  ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖
Tactic: rcases h δ hδ hδ' with ⟨s, hs, h's, s_card⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
    ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
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 < s.card
⊢ Fintype.card (Fin N) ≤ s.card
have simp : N ≤ s.card
Tactic: simp only [Fintype.card_fin]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
    ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
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 < s.card
⊢ ∃ (f : Fin N → E), Function.Injective f ∧ range f ⊆ (↑s : Set E)
have this : Fintype.card (Fin N) ≤ s.card
Tactic: have : Fintype.card (Fin N) ≤ s.card := by simp only [Fintype.card_fin]; exact s_card
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
    ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
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 < s.card
this : Fintype.card (Fin N) ≤ s.card
⊢ ∃ (f : Fin N → E), Function.Injective f ∧ range f ⊆ (↑s : Set E)
have rcases : ∀ (f : Fin N ↪ E) (hf : range (ι := Fin N) (⇑f : (a : Fin N) → E) ⊆ (↑s : Set E)),
  ∃ (f : Fin N → E), Function.Injective f ∧ range f ⊆ (↑s : Set E)
Tactic: rcases Function.Embedding.exists_of_card_le_finset this with ⟨f, hf⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
    ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
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 < s.card
⊢ ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖
have inl : ∀ (f : Fin N → E) (f_inj : Function.Injective f) (hfs : range f ⊆ (↑s : Set E)),
  ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖
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⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
    ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
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 < s.card
f : Fin N → E
f_inj : Function.Injective f
hfs : range f ⊆ (↑s : Set E)
⊢ ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖
have hfs₁ : ∀ (y : Fin N), f y ∈ s
Tactic: simp only [range_subset_iff, Finset.mem_coe] at hfs
Cleared hypotheses: hfs
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
    ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
⊢ False
have this : ∀ (δ : ℝ), 0 < δ → ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖
Tactic: have :
    ∀ δ : ℝ, 0 < δ → ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧
      Pairwise fun i j => 1 - δ ≤ ‖f i - f j‖ := by
    intro δ hδ
    rcases lt_or_ge 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
      exact ⟨f, fun i => hs _ (hfs i), fun i j hij => h's _ (hfs i) _ (hfs j) (f_inj.ne hij)⟩
    · exact
        ⟨fun _ => 0, by simp, fun i j _ => by
          simpa only [norm_zero, sub_nonpos, sub_self]⟩