awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
F : (δ : ℝ) → Fin N → E
hF : ∀ (δ : ℝ), 0 < δ → (∀ (i : Fin N), ‖F δ i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖F δ i - F δ j‖
f : Fin N → E
hf : ∀ (i : Fin N), ‖f i‖ ≤ 2
h'f : Pairwise fun (i j : Fin N) ↦ 1 ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
s_card : s.card = N
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this : s.card ≤ multiplicity E
⊢ False
have this₁ : multiplicity E + 1 ≤ multiplicity E
Tactic: rw [s_card, hN] at this
Cleared hypotheses: 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
⊢ 1 <
    1 +
      Exists.choose (p := fun (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)
          ⋯ /
        4
have linarith : 1 +
      Exists.choose (p := fun (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)
          sorry /
        4 ≤
    1 →
  4 / 4 = 1
Tactic: linarith [(exists_goodδ E).choose_spec.1]
-- 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
linarith :
  1 +
        Exists.choose (p := fun (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)
            ⋯ /
          4 ≤
      1 →
    4 / 4 = 1
⊢ 1 <
    1 +
      Exists.choose (p := fun (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)
          ⋯ /
        4
have linarith₁ : 1 +
      Exists.choose (p := fun (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)
          sorry /
        4 ≤
    1 →
  HMul.hMul (α := ℝ) 1 4 = 4
Tactic: linarith [(exists_goodδ E).choose_spec.1]
-- 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
linarith :
  1 +
        Exists.choose (p := fun (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)
            ⋯ /
          4 ≤
      1 →
    4 / 4 = 1
linarith₁ :
  1 +
        Exists.choose (p := fun (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)
            ⋯ /
          4 ≤
      1 →
    HMul.hMul (α := ℝ) 1 4 = 4
⊢ 1 <
    1 +
      Exists.choose (p := fun (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)
          ⋯ /
        4
have linarith₂ : 1 +
      Exists.choose (p := fun (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)
          sorry /
        4 ≤
    1 →
  GT.gt (α := ℝ) 4 0
Tactic: linarith [(exists_goodδ E).choose_spec.1]
-- 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
linarith :
  1 +
        Exists.choose (p := fun (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)
            ⋯ /
          4 ≤
      1 →
    4 / 4 = 1
linarith₁ :
  1 +
        Exists.choose (p := fun (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)
            ⋯ /
          4 ≤
      1 →
    HMul.hMul (α := ℝ) 1 4 = 4
linarith₂ :
  1 +
        Exists.choose (p := fun (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)
            ⋯ /
          4 ≤
      1 →
    GT.gt (α := ℝ) 4 0
⊢ 1 <
    1 +
      Exists.choose (p := fun (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)
          ⋯ /
        4
have linarith₃ : 1 +
      Exists.choose (p := fun (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)
          sorry /
        4 ≤
    1 →
  HMul.hMul (α := ℝ) 4 1 +
          1 *
            Exists.choose (p := fun (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)
              sorry -
        HMul.hMul (α := ℝ) 4 1 +
      (0 -
        Exists.choose (p := fun (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)
          sorry) =
    0
Tactic: linarith [(exists_goodδ E).choose_spec.1]
-- 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
i j : Fin n
hij : f i = f j
⊢ i = j
have by_contra : ∀ (h : ¬i = j), 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
n : ℕ
f : Fin n → E
h_1 : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
i j : Fin n
hij : f i = f j
h : ¬i = j
⊢ False
have this : 1 - goodδ E ≤ ‖f i - f j‖
Tactic: have : 1 - goodδ E ≤ ‖f i - f j‖ := h' 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
n : ℕ
f : Fin n → E
h_1 : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
i j : Fin n
hij : f i = f j
h : ¬i = j
this : 1 - goodδ E ≤ ‖f i - f j‖
⊢ False
have this₁ : 1 - goodδ E ≤ 0
Tactic: simp only [hij, norm_zero, sub_self] at this
Cleared hypotheses: 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
n : ℕ
f : Fin n → E
h_1 : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
i j : Fin n
hij : f i = f j
h : ¬i = j
this : 1 - goodδ E ≤ 0
⊢ False
have linarith : 1 - goodδ E - 0 + (goodδ E - 1) = 0
Tactic: linarith [goodδ_lt_one E]
-- 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
⊢ n ≤ multiplicity E
have finj : Function.Injective f
Tactic: have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' h
    simp only [hij, norm_zero, sub_self] at this
    linarith [goodδ_lt_one 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
⊢ s.card = n
have rw : Finset.card (α := Fin n) Finset.univ = n
Tactic: rw [Finset.card_image_of_injective _ finj]
-- 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
⊢ n ≤ multiplicity E
have s_card : s.card = n
Tactic: have s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n
-- 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
s_card : s.card = n
⊢ n ≤ multiplicity E
have hs : ∀ c ∈ s, ‖c‖ ≤ 2
Tactic: have hs : ∀ c ∈ s, ‖c‖ ≤ 2 := by
    simp only [s, h, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, imp_true_iff, true_and]
-- 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
s_card : s.card = n
hs : ∀ c ∈ s, ‖c‖ ≤ 2
⊢ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖
have simp : ∀ (a a_1 : Fin n), ¬f a = f a_1 → 1 - goodδ E ≤ ‖f a - f a_1‖
Tactic: simp only [s, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne, forall_apply_eq_imp_iff, true_and]
-- 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
n : ℕ
f : Fin n → E
h_1 : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
s_card : s.card = n
hs : ∀ c ∈ s, ‖c‖ ≤ 2
i j : Fin n
hij : ¬f i = f j
h : i = j
⊢ False
have hij₁ : ¬f j = f j
Tactic: rw [h] at hij
Cleared hypotheses: hij, 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
s_card : s.card = n
hs : ∀ c ∈ s, ‖c‖ ≤ 2
i j : Fin n
hij : ¬f i = f j
⊢ 1 - goodδ E ≤ ‖f i - f j‖
have this : i ≠ j
Tactic: have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
-- 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
s_card : s.card = n
hs : ∀ c ∈ s, ‖c‖ ≤ 2
⊢ n ≤ multiplicity E
have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖
Tactic: have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖ := by
    simp only [s, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne, forall_apply_eq_imp_iff, true_and]
    intro i j hij
    have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
    exact h' this
-- 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
s_card : s.card = n
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖
⊢ n ≤ multiplicity E
have this : s.card ≤ multiplicity E
Tactic: have : s.card ≤ multiplicity E := card_le_multiplicity_of_δ hs h's
-- 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
s_card : s.card = n
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖
this : s.card ≤ multiplicity E
⊢ n ≤ multiplicity E
have rwa : ∀ (this : n ≤ multiplicity E), n ≤ multiplicity E
Tactic: rwa [s_card] at this
Cleared hypotheses: 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
⊢ n ≤ multiplicity E
have have : Function.Injective f
Tactic: classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' h
    simp only [hij, norm_zero, sub_self] at this
    linarith [goodδ_lt_one E]
  let s := Finset.image f Finset.univ
  have s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n
  have hs : ∀ c ∈ s, ‖c‖ ≤ 2 := by
    simp only [s, h, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, imp_true_iff, true_and]
  have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖ := by
    simp only [s, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne, forall_apply_eq_imp_iff, true_and]
    intro i j hij
    have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
    exact h' this
  have : s.card ≤ multiplicity E := card_le_multiplicity_of_δ hs h's
  rwa [s_card] at 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
have : Function.Injective f
⊢ n ≤ multiplicity E
have have₁ : ∀ (finj : Function.Injective f), n ≤ multiplicity E
Tactic: classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' h
    simp only [hij, norm_zero, sub_self] at this
    linarith [goodδ_lt_one E]
  let s := Finset.image f Finset.univ
  have s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n
  have hs : ∀ c ∈ s, ‖c‖ ≤ 2 := by
    simp only [s, h, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, imp_true_iff, true_and]
  have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖ := by
    simp only [s, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne, forall_apply_eq_imp_iff, true_and]
    intro i j hij
    have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
    exact h' this
  have : s.card ≤ multiplicity E := card_le_multiplicity_of_δ hs h's
  rwa [s_card] at 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
have : Function.Injective f
have₁ : ∀ (finj : Function.Injective f), n ≤ multiplicity E
have₂ : Function.Injective f
have₃ have₄ have₅ : ∀ (finj : Function.Injective f), n ≤ multiplicity E
⊢ n ≤ multiplicity E
have have₆ : ∀ (finj : Function.Injective f),
  let s := Finset.image f Finset.univ;
  ∀ (s_card : s.card = n), n ≤ multiplicity E
Tactic: classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' h
    simp only [hij, norm_zero, sub_self] at this
    linarith [goodδ_lt_one E]
  let s := Finset.image f Finset.univ
  have s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n
  have hs : ∀ c ∈ s, ‖c‖ ≤ 2 := by
    simp only [s, h, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, imp_true_iff, true_and]
  have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖ := by
    simp only [s, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne, forall_apply_eq_imp_iff, true_and]
    intro i j hij
    have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
    exact h' this
  have : s.card ≤ multiplicity E := card_le_multiplicity_of_δ hs h's
  rwa [s_card] at 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
have : Function.Injective f
have₁ : ∀ (finj : Function.Injective f), n ≤ multiplicity E
have₂ : Function.Injective f
have₃ have₄ have₅ : ∀ (finj : Function.Injective f), n ≤ multiplicity E
have₆ :
  ∀ (finj : Function.Injective f),
    let s := Finset.image f Finset.univ;
    ∀ (s_card : s.card = n), n ≤ multiplicity E
⊢ n ≤ multiplicity E
have have₇ : ∀ (finj : Function.Injective f),
  let s := Finset.image f Finset.univ;
  ∀ (s_card : s.card = n) (hs : ∀ c ∈ s, ‖c‖ ≤ 2), n ≤ multiplicity E
Tactic: classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' h
    simp only [hij, norm_zero, sub_self] at this
    linarith [goodδ_lt_one E]
  let s := Finset.image f Finset.univ
  have s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n
  have hs : ∀ c ∈ s, ‖c‖ ≤ 2 := by
    simp only [s, h, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, imp_true_iff, true_and]
  have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖ := by
    simp only [s, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne, forall_apply_eq_imp_iff, true_and]
    intro i j hij
    have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
    exact h' this
  have : s.card ≤ multiplicity E := card_le_multiplicity_of_δ hs h's
  rwa [s_card] at 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
have : Function.Injective f
have₁ : ∀ (finj : Function.Injective f), n ≤ multiplicity E
have₂ : Function.Injective f
have₃ have₄ have₅ : ∀ (finj : Function.Injective f), n ≤ multiplicity E
have₆ :
  ∀ (finj : Function.Injective f),
    let s := Finset.image f Finset.univ;
    ∀ (s_card : s.card = n), n ≤ multiplicity E
have₇ :
  ∀ (finj : Function.Injective f),
    let s := Finset.image f Finset.univ;
    ∀ (s_card : s.card = n) (hs : ∀ c ∈ s, ‖c‖ ≤ 2), n ≤ multiplicity E
⊢ n ≤ multiplicity E
have have₈ : ∀ (finj : Function.Injective f),
  let s := Finset.image f Finset.univ;
  ∀ (s_card : s.card = n) (hs : ∀ c ∈ s, ‖c‖ ≤ 2) (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖),
    n ≤ multiplicity E
Tactic: classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' h
    simp only [hij, norm_zero, sub_self] at this
    linarith [goodδ_lt_one E]
  let s := Finset.image f Finset.univ
  have s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n
  have hs : ∀ c ∈ s, ‖c‖ ≤ 2 := by
    simp only [s, h, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, imp_true_iff, true_and]
  have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖ := by
    simp only [s, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne, forall_apply_eq_imp_iff, true_and]
    intro i j hij
    have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
    exact h' this
  have : s.card ≤ multiplicity E := card_le_multiplicity_of_δ hs h's
  rwa [s_card] at 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
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : Pairwise fun (i j : Fin n) ↦ 1 - goodδ E ≤ ‖f i - f j‖
have : Function.Injective f
have₁ : ∀ (finj : Function.Injective f), n ≤ multiplicity E
have₂ : Function.Injective f
have₃ have₄ have₅ : ∀ (finj : Function.Injective f), n ≤ multiplicity E
have₆ :
  ∀ (finj : Function.Injective f),
    let s := Finset.image f Finset.univ;
    ∀ (s_card : s.card = n), n ≤ multiplicity E
have₇ :
  ∀ (finj : Function.Injective f),
    let s := Finset.image f Finset.univ;
    ∀ (s_card : s.card = n) (hs : ∀ c ∈ s, ‖c‖ ≤ 2), n ≤ multiplicity E
have₈ :
  ∀ (finj : Function.Injective f),
    let s := Finset.image f Finset.univ;
    ∀ (s_card : s.card = n) (hs : ∀ c ∈ s, ‖c‖ ≤ 2) (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖),
      n ≤ multiplicity E
⊢ n ≤ multiplicity E
have have₉ : ∀ (finj : Function.Injective f),
  let s := Finset.image f Finset.univ;
  ∀ (s_card : s.card = n) (hs : ∀ c ∈ s, ‖c‖ ≤ 2) (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖)
    (this : s.card ≤ multiplicity E), n ≤ multiplicity E
Tactic: classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' h
    simp only [hij, norm_zero, sub_self] at this
    linarith [goodδ_lt_one E]
  let s := Finset.image f Finset.univ
  have s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n
  have hs : ∀ c ∈ s, ‖c‖ ≤ 2 := by
    simp only [s, h, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, imp_true_iff, true_and]
  have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖ := by
    simp only [s, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne, forall_apply_eq_imp_iff, true_and]
    intro i j hij
    have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
    exact h' this
  have : s.card ≤ multiplicity E := card_le_multiplicity_of_δ hs h's
  rwa [s_card] at this