l3lab/ntp-mathlib huggingface dataset:
theorem exists_normalized {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) (lastc : a.c (last N) = 0)
    (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) :
    ∃ c' : Fin N.succ → E, (∀ n, ‖c' n‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 - δ ≤ ‖c' i - c' j‖ := by
  let c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
  have norm_c'_le : ∀ i, ‖c' i‖ ≤ 2 := by
    intro i
    simp only
    split_ifs with h; · exact h
    by_cases hi : ‖a.c i‖ = 0 <;> field_simp [norm_smul, hi]; norm_num
  refine' ⟨c', fun n => norm_c'_le n, fun i j inej => _⟩
  -- up to exchanging `i` and `j`, one can assume `‖c i‖ ≤ ‖c j‖`.
  wlog hij : ‖a.c i‖ ≤ ‖a.c j‖ generalizing i j
  · rw [norm_sub_rev]; exact this j i inej.symm (le_of_not_le hij)
  rcases le_or_lt ‖a.c j‖ 2 with (Hj | Hj)
  -- case `‖c j‖ ≤ 2` (and therefore also `‖c i‖ ≤ 2`)
  · simp_rw [Hj, hij.trans Hj, if_true]
    exact exists_normalized_aux1 a lastr hτ δ hδ1 hδ2 i j inej
  -- case `2 < ‖c j‖`
  · have H'j : ‖a.c j‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hj
    

Proof state:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : c a (last N) = 0
lastr : r a (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i
norm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2
i j : Fin (Nat.succ N)
inej : i ≠ j
hij : ‖c a i‖ ≤ ‖c a j‖
Hj : 2 < ‖c a j‖
H'j : ‖c a j‖ ≤ 2 ↔ False
⊢ 1 - δ ≤ ‖c' i - c' j‖

Next tactic:
rcases le_or_lt ‖a.c i‖ 2 with (Hi | Hi)
theorem exists_normalized {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) (lastc : a.c (last N) = 0)
    (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) :
    ∃ c' : Fin N.succ → E, (∀ n, ‖c' n‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 - δ ≤ ‖c' i - c' j‖ := by
  let c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
  have norm_c'_le : ∀ i, ‖c' i‖ ≤ 2 := by
    intro i
    simp only
    split_ifs with h; · exact h
    by_cases hi : ‖a.c i‖ = 0 <;> field_simp [norm_smul, hi]; norm_num
  refine' ⟨c', fun n => norm_c'_le n, fun i j inej => _⟩
  -- up to exchanging `i` and `j`, one can assume `‖c i‖ ≤ ‖c j‖`.
  wlog hij : ‖a.c i‖ ≤ ‖a.c j‖ generalizing i j
  · rw [norm_sub_rev]; exact this j i inej.symm (le_of_not_le hij)
  rcases le_or_lt ‖a.c j‖ 2 with (Hj | Hj)
  -- case `‖c j‖ ≤ 2` (and therefore also `‖c i‖ ≤ 2`)
  · simp_rw [Hj, hij.trans Hj, if_true]
    exact exists_normalized_aux1 a lastr hτ δ hδ1 hδ2 i j inej
  -- case `2 < ‖c j‖`
  · have H'j : ‖a.c j‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hj
    rcases le_or_lt ‖a.c i‖ 2 with (Hi | Hi)
    · -- case `‖c i‖ ≤ 2`
      

Proof state:
case inr.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : c a (last N) = 0
lastr : r a (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i
norm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2
i j : Fin (Nat.succ N)
inej : i ≠ j
hij : ‖c a i‖ ≤ ‖c a j‖
Hj : 2 < ‖c a j‖
H'j : ‖c a j‖ ≤ 2 ↔ False
Hi : ‖c a i‖ ≤ 2
⊢ 1 - δ ≤ ‖c' i - c' j‖

Next tactic:
simp_rw [Hi, if_true, H'j, if_false]
theorem exists_normalized {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) (lastc : a.c (last N) = 0)
    (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) :
    ∃ c' : Fin N.succ → E, (∀ n, ‖c' n‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 - δ ≤ ‖c' i - c' j‖ := by
  let c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
  have norm_c'_le : ∀ i, ‖c' i‖ ≤ 2 := by
    intro i
    simp only
    split_ifs with h; · exact h
    by_cases hi : ‖a.c i‖ = 0 <;> field_simp [norm_smul, hi]; norm_num
  refine' ⟨c', fun n => norm_c'_le n, fun i j inej => _⟩
  -- up to exchanging `i` and `j`, one can assume `‖c i‖ ≤ ‖c j‖`.
  wlog hij : ‖a.c i‖ ≤ ‖a.c j‖ generalizing i j
  · rw [norm_sub_rev]; exact this j i inej.symm (le_of_not_le hij)
  rcases le_or_lt ‖a.c j‖ 2 with (Hj | Hj)
  -- case `‖c j‖ ≤ 2` (and therefore also `‖c i‖ ≤ 2`)
  · simp_rw [Hj, hij.trans Hj, if_true]
    exact exists_normalized_aux1 a lastr hτ δ hδ1 hδ2 i j inej
  -- case `2 < ‖c j‖`
  · have H'j : ‖a.c j‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hj
    rcases le_or_lt ‖a.c i‖ 2 with (Hi | Hi)
    · -- case `‖c i‖ ≤ 2`
      simp_rw [Hi, if_true, H'j, if_false]
      

Proof state:
case inr.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : c a (last N) = 0
lastr : r a (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i
norm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2
i j : Fin (Nat.succ N)
inej : i ≠ j
hij : ‖c a i‖ ≤ ‖c a j‖
Hj : 2 < ‖c a j‖
H'j : ‖c a j‖ ≤ 2 ↔ False
Hi : ‖c a i‖ ≤ 2
⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖

Next tactic:
exact exists_normalized_aux2 a lastc lastr hτ δ hδ1 hδ2 i j inej Hi Hj
theorem exists_normalized {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) (lastc : a.c (last N) = 0)
    (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) :
    ∃ c' : Fin N.succ → E, (∀ n, ‖c' n‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 - δ ≤ ‖c' i - c' j‖ := by
  let c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
  have norm_c'_le : ∀ i, ‖c' i‖ ≤ 2 := by
    intro i
    simp only
    split_ifs with h; · exact h
    by_cases hi : ‖a.c i‖ = 0 <;> field_simp [norm_smul, hi]; norm_num
  refine' ⟨c', fun n => norm_c'_le n, fun i j inej => _⟩
  -- up to exchanging `i` and `j`, one can assume `‖c i‖ ≤ ‖c j‖`.
  wlog hij : ‖a.c i‖ ≤ ‖a.c j‖ generalizing i j
  · rw [norm_sub_rev]; exact this j i inej.symm (le_of_not_le hij)
  rcases le_or_lt ‖a.c j‖ 2 with (Hj | Hj)
  -- case `‖c j‖ ≤ 2` (and therefore also `‖c i‖ ≤ 2`)
  · simp_rw [Hj, hij.trans Hj, if_true]
    exact exists_normalized_aux1 a lastr hτ δ hδ1 hδ2 i j inej
  -- case `2 < ‖c j‖`
  · have H'j : ‖a.c j‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hj
    rcases le_or_lt ‖a.c i‖ 2 with (Hi | Hi)
    · -- case `‖c i‖ ≤ 2`
      simp_rw [Hi, if_true, H'j, if_false]
      exact exists_normalized_aux2 a lastc lastr hτ δ hδ1 hδ2 i j inej Hi Hj
    · -- case `2 < ‖c i‖`
      

Proof state:
case inr.inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : c a (last N) = 0
lastr : r a (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i
norm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2
i j : Fin (Nat.succ N)
inej : i ≠ j
hij : ‖c a i‖ ≤ ‖c a j‖
Hj : 2 < ‖c a j‖
H'j : ‖c a j‖ ≤ 2 ↔ False
Hi : 2 < ‖c a i‖
⊢ 1 - δ ≤ ‖c' i - c' j‖

Next tactic:
have H'i : ‖a.c i‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hi
theorem exists_normalized {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) (lastc : a.c (last N) = 0)
    (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) :
    ∃ c' : Fin N.succ → E, (∀ n, ‖c' n‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 - δ ≤ ‖c' i - c' j‖ := by
  let c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
  have norm_c'_le : ∀ i, ‖c' i‖ ≤ 2 := by
    intro i
    simp only
    split_ifs with h; · exact h
    by_cases hi : ‖a.c i‖ = 0 <;> field_simp [norm_smul, hi]; norm_num
  refine' ⟨c', fun n => norm_c'_le n, fun i j inej => _⟩
  -- up to exchanging `i` and `j`, one can assume `‖c i‖ ≤ ‖c j‖`.
  wlog hij : ‖a.c i‖ ≤ ‖a.c j‖ generalizing i j
  · rw [norm_sub_rev]; exact this j i inej.symm (le_of_not_le hij)
  rcases le_or_lt ‖a.c j‖ 2 with (Hj | Hj)
  -- case `‖c j‖ ≤ 2` (and therefore also `‖c i‖ ≤ 2`)
  · simp_rw [Hj, hij.trans Hj, if_true]
    exact exists_normalized_aux1 a lastr hτ δ hδ1 hδ2 i j inej
  -- case `2 < ‖c j‖`
  · have H'j : ‖a.c j‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hj
    rcases le_or_lt ‖a.c i‖ 2 with (Hi | Hi)
    · -- case `‖c i‖ ≤ 2`
      simp_rw [Hi, if_true, H'j, if_false]
      exact exists_normalized_aux2 a lastc lastr hτ δ hδ1 hδ2 i j inej Hi Hj
    · -- case `2 < ‖c i‖`
      have H'i : ‖a.c i‖ ≤ 2 ↔ False := by 

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : c a (last N) = 0
lastr : r a (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i
norm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2
i j : Fin (Nat.succ N)
inej : i ≠ j
hij : ‖c a i‖ ≤ ‖c a j‖
Hj : 2 < ‖c a j‖
H'j : ‖c a j‖ ≤ 2 ↔ False
Hi : 2 < ‖c a i‖
⊢ ‖c a i‖ ≤ 2 ↔ False

Next tactic:
simpa only [not_le, iff_false_iff] using Hi
theorem exists_normalized {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) (lastc : a.c (last N) = 0)
    (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) :
    ∃ c' : Fin N.succ → E, (∀ n, ‖c' n‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 - δ ≤ ‖c' i - c' j‖ := by
  let c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
  have norm_c'_le : ∀ i, ‖c' i‖ ≤ 2 := by
    intro i
    simp only
    split_ifs with h; · exact h
    by_cases hi : ‖a.c i‖ = 0 <;> field_simp [norm_smul, hi]; norm_num
  refine' ⟨c', fun n => norm_c'_le n, fun i j inej => _⟩
  -- up to exchanging `i` and `j`, one can assume `‖c i‖ ≤ ‖c j‖`.
  wlog hij : ‖a.c i‖ ≤ ‖a.c j‖ generalizing i j
  · rw [norm_sub_rev]; exact this j i inej.symm (le_of_not_le hij)
  rcases le_or_lt ‖a.c j‖ 2 with (Hj | Hj)
  -- case `‖c j‖ ≤ 2` (and therefore also `‖c i‖ ≤ 2`)
  · simp_rw [Hj, hij.trans Hj, if_true]
    exact exists_normalized_aux1 a lastr hτ δ hδ1 hδ2 i j inej
  -- case `2 < ‖c j‖`
  · have H'j : ‖a.c j‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hj
    rcases le_or_lt ‖a.c i‖ 2 with (Hi | Hi)
    · -- case `‖c i‖ ≤ 2`
      simp_rw [Hi, if_true, H'j, if_false]
      exact exists_normalized_aux2 a lastc lastr hτ δ hδ1 hδ2 i j inej Hi Hj
    · -- case `2 < ‖c i‖`
      have H'i : ‖a.c i‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hi
      

Proof state:
case inr.inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : c a (last N) = 0
lastr : r a (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i
norm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2
i j : Fin (Nat.succ N)
inej : i ≠ j
hij : ‖c a i‖ ≤ ‖c a j‖
Hj : 2 < ‖c a j‖
H'j : ‖c a j‖ ≤ 2 ↔ False
Hi : 2 < ‖c a i‖
H'i : ‖c a i‖ ≤ 2 ↔ False
⊢ 1 - δ ≤ ‖c' i - c' j‖

Next tactic:
simp_rw [H'i, if_false, H'j, if_false]
theorem exists_normalized {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) (lastc : a.c (last N) = 0)
    (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) :
    ∃ c' : Fin N.succ → E, (∀ n, ‖c' n‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 - δ ≤ ‖c' i - c' j‖ := by
  let c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
  have norm_c'_le : ∀ i, ‖c' i‖ ≤ 2 := by
    intro i
    simp only
    split_ifs with h; · exact h
    by_cases hi : ‖a.c i‖ = 0 <;> field_simp [norm_smul, hi]; norm_num
  refine' ⟨c', fun n => norm_c'_le n, fun i j inej => _⟩
  -- up to exchanging `i` and `j`, one can assume `‖c i‖ ≤ ‖c j‖`.
  wlog hij : ‖a.c i‖ ≤ ‖a.c j‖ generalizing i j
  · rw [norm_sub_rev]; exact this j i inej.symm (le_of_not_le hij)
  rcases le_or_lt ‖a.c j‖ 2 with (Hj | Hj)
  -- case `‖c j‖ ≤ 2` (and therefore also `‖c i‖ ≤ 2`)
  · simp_rw [Hj, hij.trans Hj, if_true]
    exact exists_normalized_aux1 a lastr hτ δ hδ1 hδ2 i j inej
  -- case `2 < ‖c j‖`
  · have H'j : ‖a.c j‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hj
    rcases le_or_lt ‖a.c i‖ 2 with (Hi | Hi)
    · -- case `‖c i‖ ≤ 2`
      simp_rw [Hi, if_true, H'j, if_false]
      exact exists_normalized_aux2 a lastc lastr hτ δ hδ1 hδ2 i j inej Hi Hj
    · -- case `2 < ‖c i‖`
      have H'i : ‖a.c i‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hi
      simp_rw [H'i, if_false, H'j, if_false]
      

Proof state:
case inr.inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : c a (last N) = 0
lastr : r a (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i
norm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2
i j : Fin (Nat.succ N)
inej : i ≠ j
hij : ‖c a i‖ ≤ ‖c a j‖
Hj : 2 < ‖c a j‖
H'j : ‖c a j‖ ≤ 2 ↔ False
Hi : 2 < ‖c a i‖
H'i : ‖c a i‖ ≤ 2 ↔ False
⊢ 1 - δ ≤ ‖(2 / ‖c a i‖) • c a i - (2 / ‖c a j‖) • c a j‖

Next tactic:
exact exists_normalized_aux3 a lastc lastr hτ δ hδ1 i j inej Hi hij
/-- In a normed vector space `E`, there can be no satellite configuration with `multiplicity E + 1`
points and the parameter `goodτ E`. This will ensure that in the inductive construction to get
the Besicovitch covering families, there will never be more than `multiplicity E` nonempty
families. -/
theorem isEmpty_satelliteConfig_multiplicity :
    IsEmpty (SatelliteConfig E (multiplicity E) (goodτ E)) :=
  ⟨by
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ SatelliteConfig E (multiplicity E) (goodτ E) → False

Next tactic:
intro a
/-- In a normed vector space `E`, there can be no satellite configuration with `multiplicity E + 1`
points and the parameter `goodτ E`. This will ensure that in the inductive construction to get
the Besicovitch covering families, there will never be more than `multiplicity E` nonempty
families. -/
theorem isEmpty_satelliteConfig_multiplicity :
    IsEmpty (SatelliteConfig E (multiplicity E) (goodτ E)) :=
  ⟨by
    intro a
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
a : SatelliteConfig E (multiplicity E) (goodτ E)
⊢ False

Next tactic:
let b := a.centerAndRescale
/-- In a normed vector space `E`, there can be no satellite configuration with `multiplicity E + 1`
points and the parameter `goodτ E`. This will ensure that in the inductive construction to get
the Besicovitch covering families, there will never be more than `multiplicity E` nonempty
families. -/
theorem isEmpty_satelliteConfig_multiplicity :
    IsEmpty (SatelliteConfig E (multiplicity E) (goodτ E)) :=
  ⟨by
    intro a
    let b := a.centerAndRescale
    

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
a : SatelliteConfig E (multiplicity E) (goodτ E)
b : SatelliteConfig E (multiplicity E) (goodτ E) := SatelliteConfig.centerAndRescale a
⊢ False

Next tactic:
rcases b.exists_normalized a.centerAndRescale_center a.centerAndRescale_radius
        (one_lt_goodτ E).le (goodδ E) le_rfl (goodδ_lt_one E).le with
      ⟨c', c'_le_two, hc'⟩
/-- In a normed vector space `E`, there can be no satellite configuration with `multiplicity E + 1`
points and the parameter `goodτ E`. This will ensure that in the inductive construction to get
the Besicovitch covering families, there will never be more than `multiplicity E` nonempty
families. -/
theorem isEmpty_satelliteConfig_multiplicity :
    IsEmpty (SatelliteConfig E (multiplicity E) (goodτ E)) :=
  ⟨by
    intro a
    let b := a.centerAndRescale
    rcases b.exists_normalized a.centerAndRescale_center a.centerAndRescale_radius
        (one_lt_goodτ E).le (goodδ E) le_rfl (goodδ_lt_one E).le with
      ⟨c', c'_le_two, hc'⟩
    

Proof state:
case intro.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
a : SatelliteConfig E (multiplicity E) (goodτ E)
b : SatelliteConfig E (multiplicity E) (goodτ E) := SatelliteConfig.centerAndRescale a
c' : Fin (Nat.succ (multiplicity E)) → E
c'_le_two : ∀ (n : Fin (Nat.succ (multiplicity E))), ‖c' n‖ ≤ 2
hc' : ∀ (i j : Fin (Nat.succ (multiplicity E))), i ≠ j → 1 - goodδ E ≤ ‖c' i - c' j‖
⊢ False

Next tactic:
exact
      lt_irrefl _ ((Nat.lt_succ_self _).trans_le (le_multiplicity_of_δ_of_fin c' c'_le_two hc'))
theorem toConvexCone_injective : Injective ((↑) : PointedCone 𝕜 E → ConvexCone 𝕜 E) :=
  fun _ _ => by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
x✝¹ x✝ : PointedCone 𝕜 E
⊢ ↑x✝¹ = ↑x✝ → x✝¹ = x✝

Next tactic:
simp [toConvexCone]
@[simp]
theorem toConvexCone_pointed (S : PointedCone 𝕜 E) : (S : ConvexCone 𝕜 E).Pointed := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
S : PointedCone 𝕜 E
⊢ ConvexCone.Pointed ↑S

Next tactic:
simp [toConvexCone, ConvexCone.Pointed]
/-- The `PointedCone` constructed from a pointed `ConvexCone`. -/
def _root_.ConvexCone.toPointedCone {S : ConvexCone 𝕜 E} (hS : S.Pointed) : PointedCone 𝕜 E where
  carrier := S
  add_mem' hx hy := S.add_mem hx hy
  zero_mem' := hS
  smul_mem' := fun ⟨c, hc⟩ x hx => by
    

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
S : ConvexCone 𝕜 E
hS : ConvexCone.Pointed S
x✝ : { c // 0 ≤ c }
x : E
hx :
  x ∈
    { toAddSubsemigroup := { carrier := ↑S, add_mem' := (_ : ∀ {a b : E}, a ∈ ↑S → b ∈ ↑S → a + b ∈ S) },
          zero_mem' := hS }.toAddSubsemigroup.carrier
c : 𝕜
hc : 0 ≤ c
⊢ { val := c, property := hc } • x ∈
    { toAddSubsemigroup := { carrier := ↑S, add_mem' := (_ : ∀ {a b : E}, a ∈ ↑S → b ∈ ↑S → a + b ∈ S) },
          zero_mem' := hS }.toAddSubsemigroup.carrier

Next tactic:
simp_rw [SetLike.mem_coe]
/-- The `PointedCone` constructed from a pointed `ConvexCone`. -/
def _root_.ConvexCone.toPointedCone {S : ConvexCone 𝕜 E} (hS : S.Pointed) : PointedCone 𝕜 E where
  carrier := S
  add_mem' hx hy := S.add_mem hx hy
  zero_mem' := hS
  smul_mem' := fun ⟨c, hc⟩ x hx => by
    simp_rw [SetLike.mem_coe]
    

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
S : ConvexCone 𝕜 E
hS : ConvexCone.Pointed S
x✝ : { c // 0 ≤ c }
x : E
hx :
  x ∈
    { toAddSubsemigroup := { carrier := ↑S, add_mem' := (_ : ∀ {a b : E}, a ∈ ↑S → b ∈ ↑S → a + b ∈ S) },
          zero_mem' := hS }.toAddSubsemigroup.carrier
c : 𝕜
hc : 0 ≤ c
⊢ { val := c, property := hc } • x ∈ S

Next tactic:
cases' eq_or_lt_of_le hc with hzero hpos
/-- The `PointedCone` constructed from a pointed `ConvexCone`. -/
def _root_.ConvexCone.toPointedCone {S : ConvexCone 𝕜 E} (hS : S.Pointed) : PointedCone 𝕜 E where
  carrier := S
  add_mem' hx hy := S.add_mem hx hy
  zero_mem' := hS
  smul_mem' := fun ⟨c, hc⟩ x hx => by
    simp_rw [SetLike.mem_coe]
    cases' eq_or_lt_of_le hc with hzero hpos
    · 

Proof state:
case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
S : ConvexCone 𝕜 E
hS : ConvexCone.Pointed S
x✝ : { c // 0 ≤ c }
x : E
hx :
  x ∈
    { toAddSubsemigroup := { carrier := ↑S, add_mem' := (_ : ∀ {a b : E}, a ∈ ↑S → b ∈ ↑S → a + b ∈ S) },
          zero_mem' := hS }.toAddSubsemigroup.carrier
c : 𝕜
hc : 0 ≤ c
hzero : 0 = c
⊢ { val := c, property := hc } • x ∈ S

Next tactic:
unfold ConvexCone.Pointed at hS
/-- The `PointedCone` constructed from a pointed `ConvexCone`. -/
def _root_.ConvexCone.toPointedCone {S : ConvexCone 𝕜 E} (hS : S.Pointed) : PointedCone 𝕜 E where
  carrier := S
  add_mem' hx hy := S.add_mem hx hy
  zero_mem' := hS
  smul_mem' := fun ⟨c, hc⟩ x hx => by
    simp_rw [SetLike.mem_coe]
    cases' eq_or_lt_of_le hc with hzero hpos
    · unfold ConvexCone.Pointed at hS
      

Proof state:
case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
S : ConvexCone 𝕜 E
hS : 0 ∈ S
x✝ : { c // 0 ≤ c }
x : E
hx :
  x ∈
    { toAddSubsemigroup := { carrier := ↑S, add_mem' := (_ : ∀ {a b : E}, a ∈ ↑S → b ∈ ↑S → a + b ∈ S) },
          zero_mem' := hS }.toAddSubsemigroup.carrier
c : 𝕜
hc : 0 ≤ c
hzero : 0 = c
⊢ { val := c, property := hc } • x ∈ S

Next tactic:
convert hS
/-- The `PointedCone` constructed from a pointed `ConvexCone`. -/
def _root_.ConvexCone.toPointedCone {S : ConvexCone 𝕜 E} (hS : S.Pointed) : PointedCone 𝕜 E where
  carrier := S
  add_mem' hx hy := S.add_mem hx hy
  zero_mem' := hS
  smul_mem' := fun ⟨c, hc⟩ x hx => by
    simp_rw [SetLike.mem_coe]
    cases' eq_or_lt_of_le hc with hzero hpos
    · unfold ConvexCone.Pointed at hS
      convert hS
      

Proof state:
case h.e'_4
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
S : ConvexCone 𝕜 E
hS : 0 ∈ S
x✝ : { c // 0 ≤ c }
x : E
hx :
  x ∈
    { toAddSubsemigroup := { carrier := ↑S, add_mem' := (_ : ∀ {a b : E}, a ∈ ↑S → b ∈ ↑S → a + b ∈ S) },
          zero_mem' := hS }.toAddSubsemigroup.carrier
c : 𝕜
hc : 0 ≤ c
hzero : 0 = c
⊢ { val := c, property := hc } • x = 0

Next tactic:
simp [← hzero]
/-- The `PointedCone` constructed from a pointed `ConvexCone`. -/
def _root_.ConvexCone.toPointedCone {S : ConvexCone 𝕜 E} (hS : S.Pointed) : PointedCone 𝕜 E where
  carrier := S
  add_mem' hx hy := S.add_mem hx hy
  zero_mem' := hS
  smul_mem' := fun ⟨c, hc⟩ x hx => by
    simp_rw [SetLike.mem_coe]
    cases' eq_or_lt_of_le hc with hzero hpos
    · unfold ConvexCone.Pointed at hS
      convert hS
      simp [← hzero]
    · 

Proof state:
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
S : ConvexCone 𝕜 E
hS : ConvexCone.Pointed S
x✝ : { c // 0 ≤ c }
x : E
hx :
  x ∈
    { toAddSubsemigroup := { carrier := ↑S, add_mem' := (_ : ∀ {a b : E}, a ∈ ↑S → b ∈ ↑S → a + b ∈ S) },
          zero_mem' := hS }.toAddSubsemigroup.carrier
c : 𝕜
hc : 0 ≤ c
hpos : 0 < c
⊢ { val := c, property := hc } • x ∈ S

Next tactic:
apply ConvexCone.smul_mem
/-- The `PointedCone` constructed from a pointed `ConvexCone`. -/
def _root_.ConvexCone.toPointedCone {S : ConvexCone 𝕜 E} (hS : S.Pointed) : PointedCone 𝕜 E where
  carrier := S
  add_mem' hx hy := S.add_mem hx hy
  zero_mem' := hS
  smul_mem' := fun ⟨c, hc⟩ x hx => by
    simp_rw [SetLike.mem_coe]
    cases' eq_or_lt_of_le hc with hzero hpos
    · unfold ConvexCone.Pointed at hS
      convert hS
      simp [← hzero]
    · apply ConvexCone.smul_mem
      

Proof state:
case inr.hc
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
S : ConvexCone 𝕜 E
hS : ConvexCone.Pointed S
x✝ : { c // 0 ≤ c }
x : E
hx :
  x ∈
    { toAddSubsemigroup := { carrier := ↑S, add_mem' := (_ : ∀ {a b : E}, a ∈ ↑S → b ∈ ↑S → a + b ∈ S) },
          zero_mem' := hS }.toAddSubsemigroup.carrier
c : 𝕜
hc : 0 ≤ c
hpos : 0 < c
⊢ 0 < ↑(RingHom.toMonoidWithZeroHom Nonneg.coeRingHom) { val := c, property := hc }
case inr.hx
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
S : ConvexCone 𝕜 E
hS : ConvexCone.Pointed S
x✝ : { c // 0 ≤ c }
x : E
hx :
  x ∈
    { toAddSubsemigroup := { carrier := ↑S, add_mem' := (_ : ∀ {a b : E}, a ∈ ↑S → b ∈ ↑S → a + b ∈ S) },
          zero_mem' := hS }.toAddSubsemigroup.carrier
c : 𝕜
hc : 0 ≤ c
hpos : 0 < c
⊢ x ∈ S

Next tactic:
convert hpos