l3lab/ntp-mathlib huggingface dataset:
/-- 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⟩
      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]⟩
  -- For `δ > 0`, `F δ` is a function from `fin N` to the ball of radius `2` for which two points
  -- in the image are separated by `1 - δ`.
  choose! F hF using this
  -- Choose a converging subsequence when `δ → 0`.
  have : ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 ≤ ‖f i - f j‖ := by
    obtain ⟨u, _, zero_lt_u, hu⟩ :
      ∃ u : ℕ → ℝ,
        (∀ m n : ℕ, m < n → u n < u m) ∧ (∀ n : ℕ, 0 < u n) ∧ Filter.Tendsto u Filter.atTop (𝓝 0) :=
      exists_seq_strictAnti_tendsto (0 : ℝ)
    

Proof state:
case 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
F : ℝ → Fin N → E
hF : ∀ (δ : ℝ), 0 < δ → (∀ (i : Fin N), ‖F δ i‖ ≤ 2) ∧ ∀ (i j : Fin N), i ≠ j → 1 - δ ≤ ‖F δ i - F δ j‖
u : ℕ → ℝ
left✝ : ∀ (m n : ℕ), m < n → u n < u m
zero_lt_u : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
⊢ ∃ f, (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ ∀ (i j : Fin N), i ≠ j → 1 ≤ ‖f i - f j‖

Next tactic:
have A : ∀ n, F (u n) ∈ closedBall (0 : Fin N → E) 2 := by
      intro n
      simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right,
        (hF (u n) (zero_lt_u n)).left, forall_const]
/-- 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⟩
      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]⟩
  -- For `δ > 0`, `F δ` is a function from `fin N` to the ball of radius `2` for which two points
  -- in the image are separated by `1 - δ`.
  choose! F hF using this
  -- Choose a converging subsequence when `δ → 0`.
  have : ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 ≤ ‖f i - f j‖ := by
    obtain ⟨u, _, zero_lt_u, hu⟩ :
      ∃ u : ℕ → ℝ,
        (∀ m n : ℕ, m < n → u n < u m) ∧ (∀ n : ℕ, 0 < u n) ∧ Filter.Tendsto u Filter.atTop (𝓝 0) :=
      exists_seq_strictAnti_tendsto (0 : ℝ)
    have A : ∀ n, F (u n) ∈ closedBall (0 : Fin N → E) 2 := 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
F : ℝ → Fin N → E
hF : ∀ (δ : ℝ), 0 < δ → (∀ (i : Fin N), ‖F δ i‖ ≤ 2) ∧ ∀ (i j : Fin N), i ≠ j → 1 - δ ≤ ‖F δ i - F δ j‖
u : ℕ → ℝ
left✝ : ∀ (m n : ℕ), m < n → u n < u m
zero_lt_u : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
⊢ ∀ (n : ℕ), F (u n) ∈ closedBall 0 2

Next tactic:
intro n
/-- 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⟩
      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]⟩
  -- For `δ > 0`, `F δ` is a function from `fin N` to the ball of radius `2` for which two points
  -- in the image are separated by `1 - δ`.
  choose! F hF using this
  -- Choose a converging subsequence when `δ → 0`.
  have : ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 ≤ ‖f i - f j‖ := by
    obtain ⟨u, _, zero_lt_u, hu⟩ :
      ∃ u : ℕ → ℝ,
        (∀ m n : ℕ, m < n → u n < u m) ∧ (∀ n : ℕ, 0 < u n) ∧ Filter.Tendsto u Filter.atTop (𝓝 0) :=
      exists_seq_strictAnti_tendsto (0 : ℝ)
    have A : ∀ n, F (u n) ∈ closedBall (0 : Fin N → E) 2 := by
      intro n
      

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
F : ℝ → Fin N → E
hF : ∀ (δ : ℝ), 0 < δ → (∀ (i : Fin N), ‖F δ i‖ ≤ 2) ∧ ∀ (i j : Fin N), i ≠ j → 1 - δ ≤ ‖F δ i - F δ j‖
u : ℕ → ℝ
left✝ : ∀ (m n : ℕ), m < n → u n < u m
zero_lt_u : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
n : ℕ
⊢ F (u n) ∈ closedBall 0 2

Next tactic:
simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right,
        (hF (u n) (zero_lt_u n)).left, forall_const]
/-- 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⟩
      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]⟩
  -- For `δ > 0`, `F δ` is a function from `fin N` to the ball of radius `2` for which two points
  -- in the image are separated by `1 - δ`.
  choose! F hF using this
  -- Choose a converging subsequence when `δ → 0`.
  have : ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 ≤ ‖f i - f j‖ := by
    obtain ⟨u, _, zero_lt_u, hu⟩ :
      ∃ u : ℕ → ℝ,
        (∀ m n : ℕ, m < n → u n < u m) ∧ (∀ n : ℕ, 0 < u n) ∧ Filter.Tendsto u Filter.atTop (𝓝 0) :=
      exists_seq_strictAnti_tendsto (0 : ℝ)
    have A : ∀ n, F (u n) ∈ closedBall (0 : Fin N → E) 2 := by
      intro n
      simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right,
        (hF (u n) (zero_lt_u n)).left, forall_const]
    

Proof state:
case 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
F : ℝ → Fin N → E
hF : ∀ (δ : ℝ), 0 < δ → (∀ (i : Fin N), ‖F δ i‖ ≤ 2) ∧ ∀ (i j : Fin N), i ≠ j → 1 - δ ≤ ‖F δ i - F δ j‖
u : ℕ → ℝ
left✝ : ∀ (m n : ℕ), m < n → u n < u m
zero_lt_u : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
A : ∀ (n : ℕ), F (u n) ∈ closedBall 0 2
⊢ ∃ f, (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ ∀ (i j : Fin N), i ≠ j → 1 ≤ ‖f i - f j‖

Next tactic:
obtain ⟨f, fmem, φ, φ_mono, hf⟩ :
      ∃ f ∈ closedBall (0 : Fin N → E) 2,
        ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto ((F ∘ u) ∘ φ) atTop (𝓝 f) :=
      IsCompact.tendsto_subseq (isCompact_closedBall _ _) A
theorem one_lt_goodτ : 1 < goodτ E := by
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ 1 < goodτ E

Next tactic:
dsimp [goodτ, goodδ]
theorem one_lt_goodτ : 1 < goodτ E := by
  dsimp [goodτ, goodδ]; 

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ 1 <
    1 +
      Exists.choose
          (_ :
            ∃ 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) /
        4

Next tactic:
linarith [(exists_goodδ E).choose_spec.1]
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
⊢ n ≤ multiplicity E

Next tactic:
classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j 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 [h, forall_apply_eq_imp_iff, forall_const, 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 [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]
    intro i j hij
    have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
    exact h' i j this
  have : s.card ≤ multiplicity E := card_le_multiplicity_of_δ hs h's
  rwa [s_card] at this
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
⊢ n ≤ multiplicity E

Next tactic:
have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j h
    simp only [hij, norm_zero, sub_self] at this
    linarith [goodδ_lt_one E]
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
⊢ Function.Injective f

Next tactic:
intro i j hij
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    intro i j hij
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
i j : Fin n
hij : f i = f j
⊢ i = j

Next tactic:
by_contra h
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h✝ : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
i j : Fin n
hij : f i = f j
h : ¬i = j
⊢ False

Next tactic:
have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j h
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j h
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h✝ : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 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

Next tactic:
simp only [hij, norm_zero, sub_self] at this
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j h
    simp only [hij, norm_zero, sub_self] at this
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h✝ : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 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

Next tactic:
linarith [goodδ_lt_one E]
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j h
    simp only [hij, norm_zero, sub_self] at this
    linarith [goodδ_lt_one E]
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
⊢ n ≤ multiplicity E

Next tactic:
let s := Finset.image f Finset.univ
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j h
    simp only [hij, norm_zero, sub_self] at this
    linarith [goodδ_lt_one E]
  let s := Finset.image f Finset.univ
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
⊢ n ≤ multiplicity E

Next tactic:
have s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j 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 

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
⊢ Finset.card s = n

Next tactic:
rw [Finset.card_image_of_injective _ finj]
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j 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]; 

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
⊢ Finset.card Finset.univ = n

Next tactic:
exact Finset.card_fin n
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j 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
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
s_card : Finset.card s = n
⊢ n ≤ multiplicity E

Next tactic:
have hs : ∀ c ∈ s, ‖c‖ ≤ 2 := by
    simp only [h, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, imp_true_iff, true_and]
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j 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
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
s_card : Finset.card s = n
⊢ ∀ c ∈ s, ‖c‖ ≤ 2

Next tactic:
simp only [h, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, imp_true_iff, true_and]
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : ∀ i j, i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by
  classical
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 - goodδ E ≤ ‖f i - f j‖ := h' i j 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 [h, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, imp_true_iff, true_and]
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ
f : Fin n → E
h : ∀ (i : Fin n), ‖f i‖ ≤ 2
h' : ∀ (i j : Fin n), i ≠ j → 1 - goodδ E ≤ ‖f i - f j‖
finj : Function.Injective f
s : Finset E := Finset.image f Finset.univ
s_card : Finset.card s = n
hs : ∀ c ∈ s, ‖c‖ ≤ 2
⊢ n ≤ multiplicity E

Next tactic:
have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖ := by
    simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]
    intro i j hij
    have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
    exact h' i j this