l3lab/ntp-mathlib huggingface dataset:
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := 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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖

Next tactic:
have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := 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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
⊢ ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j

Next tactic:
simpa only [dist_eq_norm] using a.h
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  

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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖

Next tactic:
have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := 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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
⊢ 0 ≤ δ

Next tactic:
linarith only [hτ, hδ1]
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  

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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖

Next tactic:
have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := 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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
⊢ 0 ≤ 1 - δ / 4

Next tactic:
linarith only [hδ2]
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  

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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖

Next tactic:
have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
  

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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖

Next tactic:
have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
  have hcrj : ‖a.c j‖ ≤ a.r j + 1 := 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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
⊢ ‖c a j‖ ≤ r a j + 1

Next tactic:
simpa only [lastc, lastr, dist_zero_right] using a.inter' j
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
  have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j
  

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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
hcrj : ‖c a j‖ ≤ r a j + 1
⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖

Next tactic:
have I : a.r i ≤ 2 := by
    rcases lt_or_le i (last N) with (H | H)
    · apply (a.hlast i H).1.trans
      simpa only [dist_eq_norm, lastc, sub_zero] using hi
    · have : i = last N := top_le_iff.1 H
      rw [this, lastr]
      exact one_le_two
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
  have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j
  have I : a.r i ≤ 2 := 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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
hcrj : ‖c a j‖ ≤ r a j + 1
⊢ r a i ≤ 2

Next tactic:
rcases lt_or_le i (last N) with (H | H)
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
  have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j
  have I : a.r i ≤ 2 := by
    rcases lt_or_le i (last N) with (H | H)
    · 

Proof state:
case 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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
hcrj : ‖c a j‖ ≤ r a j + 1
H : i < last N
⊢ r a i ≤ 2

Next tactic:
apply (a.hlast i H).1.trans
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
  have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j
  have I : a.r i ≤ 2 := by
    rcases lt_or_le i (last N) with (H | H)
    · apply (a.hlast i H).1.trans
      

Proof state:
case 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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
hcrj : ‖c a j‖ ≤ r a j + 1
H : i < last N
⊢ dist (c a i) (c a (last N)) ≤ 2

Next tactic:
simpa only [dist_eq_norm, lastc, sub_zero] using hi
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
  have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j
  have I : a.r i ≤ 2 := by
    rcases lt_or_le i (last N) with (H | H)
    · apply (a.hlast i H).1.trans
      simpa only [dist_eq_norm, lastc, sub_zero] using hi
    · 

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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
hcrj : ‖c a j‖ ≤ r a j + 1
H : last N ≤ i
⊢ r a i ≤ 2

Next tactic:
have : i = last N := top_le_iff.1 H
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
  have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j
  have I : a.r i ≤ 2 := by
    rcases lt_or_le i (last N) with (H | H)
    · apply (a.hlast i H).1.trans
      simpa only [dist_eq_norm, lastc, sub_zero] using hi
    · have : i = last N := top_le_iff.1 H
      

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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
hcrj : ‖c a j‖ ≤ r a j + 1
H : last N ≤ i
this : i = last N
⊢ r a i ≤ 2

Next tactic:
rw [this, lastr]
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
  have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j
  have I : a.r i ≤ 2 := by
    rcases lt_or_le i (last N) with (H | H)
    · apply (a.hlast i H).1.trans
      simpa only [dist_eq_norm, lastc, sub_zero] using hi
    · have : i = last N := top_le_iff.1 H
      rw [this, lastr]
      

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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
hcrj : ‖c a j‖ ≤ r a j + 1
H : last N ≤ i
this : i = last N
⊢ 1 ≤ 2

Next tactic:
exact one_le_two
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
  have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j
  have I : a.r i ≤ 2 := by
    rcases lt_or_le i (last N) with (H | H)
    · apply (a.hlast i H).1.trans
      simpa only [dist_eq_norm, lastc, sub_zero] using hi
    · have : i = last N := top_le_iff.1 H
      rw [this, lastr]
      exact one_le_two
  

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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
hcrj : ‖c a j‖ ≤ r a j + 1
I : r a i ≤ 2
⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖

Next tactic:
have J : (1 - δ / 4) * τ ≤ 1 :=
    calc
      (1 - δ / 4) * τ ≤ (1 - δ / 4) * (1 + δ / 4) := mul_le_mul_of_nonneg_left hδ1 D
      _ = (1 : ℝ) - δ ^ 2 / 16 := by ring
      _ ≤ 1 := by linarith only [sq_nonneg δ]
theorem exists_normalized_aux2 {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) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
  have ah :
      ∀ i j, i ≠ j → a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
        a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
    simpa only [dist_eq_norm] using a.h
  have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
  have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
  have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
  have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right] using a.inter' j
  have I : a.r i ≤ 2 := by
    rcases lt_or_le i (last N) with (H | H)
    · apply (a.hlast i H).1.trans
      simpa only [dist_eq_norm, lastc, sub_zero] using hi
    · have : i = last N := top_le_iff.1 H
      rw [this, lastr]
      exact one_le_two
  have J : (1 - δ / 4) * τ ≤ 1 :=
    calc
      (1 - δ / 4) * τ ≤ (1 - δ / 4) * (1 + δ / 4) := mul_le_mul_of_nonneg_left hδ1 D
      _ = (1 : ℝ) - δ ^ 2 / 16 := 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
i j : Fin (Nat.succ N)
inej : i ≠ j
hi : ‖c a i‖ ≤ 2
hj : 2 < ‖c a j‖
ah :
  ∀ (i j : Fin (Nat.succ N)),
    i ≠ j → r a i ≤ ‖c a i - c a j‖ ∧ r a j ≤ τ * r a i ∨ r a j ≤ ‖c a j - c a i‖ ∧ r a i ≤ τ * r a j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
hcrj : ‖c a j‖ ≤ r a j + 1
I : r a i ≤ 2
⊢ (1 - δ / 4) * (1 + δ / 4) = 1 - δ ^ 2 / 16

Next tactic:
ring