l3lab/ntp-mathlib huggingface dataset:
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]
  have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖ := 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
hs : ∀ c ∈ s, ‖c‖ ≤ 2
⊢ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖

Next tactic:
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]
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]
  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]
    

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
⊢ ∀ (a a_1 : Fin n), ¬f a = f a_1 → 1 - goodδ E ≤ ‖f a - f a_1‖

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
    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
    

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
i j : Fin n
hij : ¬f i = f j
⊢ 1 - goodδ E ≤ ‖f i - f j‖

Next tactic:
have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
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]
  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 

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
i j : Fin n
hij : ¬f i = f j
h : i = j
⊢ False

Next tactic:
rw [h] at 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
    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; 

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
i j : Fin n
hij : ¬f j = f j
h : i = j
⊢ False

Next tactic:
exact hij rfl
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]
  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
    

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
i j : Fin n
hij : ¬f i = f j
this : i ≠ j
⊢ 1 - goodδ E ≤ ‖f i - f j‖

Next tactic:
exact h' i j 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
    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
  

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
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖
⊢ n ≤ multiplicity E

Next tactic:
have : s.card ≤ multiplicity E := card_le_multiplicity_of_δ hs h's
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]
  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
  

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
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖
this : Finset.card s ≤ multiplicity E
⊢ n ≤ multiplicity E

Next tactic:
rwa [s_card] at this
theorem exists_normalized_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - a.c j‖ := by
  

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
⊢ 1 - δ ≤ ‖c a i - 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_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - 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✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
⊢ ∀ (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_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - 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✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
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 - c a j‖

Next tactic:
have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
theorem exists_normalized_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - 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✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
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_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - 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✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
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 - c a j‖

Next tactic:
have D : 0 ≤ 1 - δ / 4 := by linarith only [hδ2]
theorem exists_normalized_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - 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✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
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_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - 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✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
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 - c a j‖

Next tactic:
have τpos : 0 < τ := _root_.zero_lt_one.trans_le hτ
theorem exists_normalized_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - 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✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
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 - c a j‖

Next tactic:
have I : (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_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - 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 I : (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✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
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 / 4) * (1 + δ / 4) = 1 - δ ^ 2 / 16

Next tactic:
ring
theorem exists_normalized_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - 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 I : (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 

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
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 ^ 2 / 16 ≤ 1

Next tactic:
linarith only [sq_nonneg δ]
theorem exists_normalized_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - 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 I : (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]
  

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
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
I : (1 - δ / 4) * τ ≤ 1
⊢ 1 - δ ≤ ‖c a i - c a j‖

Next tactic:
have J : 1 - δ ≤ 1 - δ / 4 := by linarith only [δnonneg]
theorem exists_normalized_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (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) : 1 - δ ≤ ‖a.c i - 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 I : (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]
  have J : 1 - δ ≤ 1 - δ / 4 := by 

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
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
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
I : (1 - δ / 4) * τ ≤ 1
⊢ 1 - δ ≤ 1 - δ / 4

Next tactic:
linarith only [δnonneg]