l3lab/ntp-mathlib huggingface dataset:
theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
  rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
    inv_mul_eq_div, sub_nonneg, le_div_iff]
  · have : (nValue N : ℝ) ≤ 2 * sqrt (log N) := by
      apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans
      rw [two_mul, add_le_add_iff_left]
      apply le_sqrt_of_sq_le
      rw [one_pow, le_log_iff_exp_le hN₀]
      exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃)
    apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _
    

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN₃ : 8 ≤ N
hN₀ : 0 < ↑N
this : ↑(nValue N) ≤ 2 * Real.sqrt (log ↑N)
⊢ log 2 * (2 * Real.sqrt (log ↑N)) ≤ log ↑N

Next tactic:
rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]
theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
  rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
    inv_mul_eq_div, sub_nonneg, le_div_iff]
  · have : (nValue N : ℝ) ≤ 2 * sqrt (log N) := by
      apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans
      rw [two_mul, add_le_add_iff_left]
      apply le_sqrt_of_sq_le
      rw [one_pow, le_log_iff_exp_le hN₀]
      exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃)
    apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _
    rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]
    · 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN₃ : 8 ≤ N
hN₀ : 0 < ↑N
this : ↑(nValue N) ≤ 2 * Real.sqrt (log ↑N)
⊢ log 2 * 2 ≤ Real.sqrt (log ↑N)

Next tactic:
apply log_two_mul_two_le_sqrt_log_eight.trans
theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
  rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
    inv_mul_eq_div, sub_nonneg, le_div_iff]
  · have : (nValue N : ℝ) ≤ 2 * sqrt (log N) := by
      apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans
      rw [two_mul, add_le_add_iff_left]
      apply le_sqrt_of_sq_le
      rw [one_pow, le_log_iff_exp_le hN₀]
      exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃)
    apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _
    rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]
    · apply log_two_mul_two_le_sqrt_log_eight.trans
      

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN₃ : 8 ≤ N
hN₀ : 0 < ↑N
this : ↑(nValue N) ≤ 2 * Real.sqrt (log ↑N)
⊢ Real.sqrt (log 8) ≤ Real.sqrt (log ↑N)

Next tactic:
apply Real.sqrt_le_sqrt
theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
  rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
    inv_mul_eq_div, sub_nonneg, le_div_iff]
  · have : (nValue N : ℝ) ≤ 2 * sqrt (log N) := by
      apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans
      rw [two_mul, add_le_add_iff_left]
      apply le_sqrt_of_sq_le
      rw [one_pow, le_log_iff_exp_le hN₀]
      exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃)
    apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _
    rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]
    · apply log_two_mul_two_le_sqrt_log_eight.trans
      apply Real.sqrt_le_sqrt
      

Proof state:
case h
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN₃ : 8 ≤ N
hN₀ : 0 < ↑N
this : ↑(nValue N) ≤ 2 * Real.sqrt (log ↑N)
⊢ log 8 ≤ log ↑N

Next tactic:
exact log_le_log (by norm_num) (mod_cast hN₃)
theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
  rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
    inv_mul_eq_div, sub_nonneg, le_div_iff]
  · have : (nValue N : ℝ) ≤ 2 * sqrt (log N) := by
      apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans
      rw [two_mul, add_le_add_iff_left]
      apply le_sqrt_of_sq_le
      rw [one_pow, le_log_iff_exp_le hN₀]
      exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃)
    apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _
    rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]
    · apply log_two_mul_two_le_sqrt_log_eight.trans
      apply Real.sqrt_le_sqrt
      exact log_le_log (by 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN₃ : 8 ≤ N
hN₀ : 0 < ↑N
this : ↑(nValue N) ≤ 2 * Real.sqrt (log ↑N)
⊢ 0 < 8

Next tactic:
norm_num
theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
  rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
    inv_mul_eq_div, sub_nonneg, le_div_iff]
  · have : (nValue N : ℝ) ≤ 2 * sqrt (log N) := by
      apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans
      rw [two_mul, add_le_add_iff_left]
      apply le_sqrt_of_sq_le
      rw [one_pow, le_log_iff_exp_le hN₀]
      exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃)
    apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _
    rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]
    · apply log_two_mul_two_le_sqrt_log_eight.trans
      apply Real.sqrt_le_sqrt
      exact log_le_log (by norm_num) (mod_cast hN₃)
    

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN₃ : 8 ≤ N
hN₀ : 0 < ↑N
this : ↑(nValue N) ≤ 2 * Real.sqrt (log ↑N)
⊢ 1 < N

Next tactic:
exact hN₃.trans_lt' (by norm_num)
theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
  rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
    inv_mul_eq_div, sub_nonneg, le_div_iff]
  · have : (nValue N : ℝ) ≤ 2 * sqrt (log N) := by
      apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans
      rw [two_mul, add_le_add_iff_left]
      apply le_sqrt_of_sq_le
      rw [one_pow, le_log_iff_exp_le hN₀]
      exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃)
    apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _
    rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]
    · apply log_two_mul_two_le_sqrt_log_eight.trans
      apply Real.sqrt_le_sqrt
      exact log_le_log (by norm_num) (mod_cast hN₃)
    exact hN₃.trans_lt' (by 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN₃ : 8 ≤ N
hN₀ : 0 < ↑N
this : ↑(nValue N) ≤ 2 * Real.sqrt (log ↑N)
⊢ 1 < 8

Next tactic:
norm_num
theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
  rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
    inv_mul_eq_div, sub_nonneg, le_div_iff]
  · have : (nValue N : ℝ) ≤ 2 * sqrt (log N) := by
      apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans
      rw [two_mul, add_le_add_iff_left]
      apply le_sqrt_of_sq_le
      rw [one_pow, le_log_iff_exp_le hN₀]
      exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃)
    apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _
    rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]
    · apply log_two_mul_two_le_sqrt_log_eight.trans
      apply Real.sqrt_le_sqrt
      exact log_le_log (by norm_num) (mod_cast hN₃)
    exact hN₃.trans_lt' (by norm_num)
  · 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN₃ : 8 ≤ N
hN₀ : 0 < ↑N
⊢ 0 < ↑(nValue N)

Next tactic:
exact cast_pos.2 (nValue_pos <| hN₃.trans' <| by norm_num)
theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
  rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
    inv_mul_eq_div, sub_nonneg, le_div_iff]
  · have : (nValue N : ℝ) ≤ 2 * sqrt (log N) := by
      apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans
      rw [two_mul, add_le_add_iff_left]
      apply le_sqrt_of_sq_le
      rw [one_pow, le_log_iff_exp_le hN₀]
      exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃)
    apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _
    rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]
    · apply log_two_mul_two_le_sqrt_log_eight.trans
      apply Real.sqrt_le_sqrt
      exact log_le_log (by norm_num) (mod_cast hN₃)
    exact hN₃.trans_lt' (by norm_num)
  · exact cast_pos.2 (nValue_pos <| hN₃.trans' <| by 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN₃ : 8 ≤ N
hN₀ : 0 < ↑N
⊢ 2 ≤ 8

Next tactic:
norm_num
theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
  rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
    inv_mul_eq_div, sub_nonneg, le_div_iff]
  · have : (nValue N : ℝ) ≤ 2 * sqrt (log N) := by
      apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans
      rw [two_mul, add_le_add_iff_left]
      apply le_sqrt_of_sq_le
      rw [one_pow, le_log_iff_exp_le hN₀]
      exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃)
    apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _
    rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]
    · apply log_two_mul_two_le_sqrt_log_eight.trans
      apply Real.sqrt_le_sqrt
      exact log_le_log (by norm_num) (mod_cast hN₃)
    exact hN₃.trans_lt' (by norm_num)
  · exact cast_pos.2 (nValue_pos <| hN₃.trans' <| by norm_num)
  · 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN₃ : 8 ≤ N
hN₀ : 0 < ↑N
⊢ ↑N ^ (↑(nValue N))⁻¹ ≠ 0

Next tactic:
exact (rpow_pos_of_pos hN₀ _).ne'
theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
  rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
    inv_mul_eq_div, sub_nonneg, le_div_iff]
  · have : (nValue N : ℝ) ≤ 2 * sqrt (log N) := by
      apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans
      rw [two_mul, add_le_add_iff_left]
      apply le_sqrt_of_sq_le
      rw [one_pow, le_log_iff_exp_le hN₀]
      exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃)
    apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _
    rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]
    · apply log_two_mul_two_le_sqrt_log_eight.trans
      apply Real.sqrt_le_sqrt
      exact log_le_log (by norm_num) (mod_cast hN₃)
    exact hN₃.trans_lt' (by norm_num)
  · exact cast_pos.2 (nValue_pos <| hN₃.trans' <| by norm_num)
  · exact (rpow_pos_of_pos hN₀ _).ne'
  · 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN₃ : 8 ≤ N
hN₀ : 0 < ↑N
⊢ 0 < ↑N ^ (↑(nValue N))⁻¹ / 2

Next tactic:
exact div_pos (rpow_pos_of_pos hN₀ _) zero_lt_two
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
⊢ (2 * dValue N - 1) ^ nValue N ≤ N

Next tactic:
have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
  

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
this : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N
⊢ (2 * dValue N - 1) ^ nValue N ≤ N

Next tactic:
apply this.trans
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
  apply this.trans
  

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
this : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N
⊢ (2 * dValue N) ^ nValue N ≤ N

Next tactic:
suffices ((2 * dValue N) ^ nValue N : ℝ) ≤ N from mod_cast this
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
  apply this.trans
  suffices ((2 * dValue N) ^ nValue N : ℝ) ≤ N from mod_cast this
  

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
this : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N
⊢ (2 * ↑(dValue N)) ^ nValue N ≤ ↑N

Next tactic:
suffices i : (2 * dValue N : ℝ) ≤ (N : ℝ) ^ (nValue N : ℝ)⁻¹
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
  apply this.trans
  suffices ((2 * dValue N) ^ nValue N : ℝ) ≤ N from mod_cast this
  suffices i : (2 * dValue N : ℝ) ≤ (N : ℝ) ^ (nValue N : ℝ)⁻¹
  · 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
this : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N
i : 2 * ↑(dValue N) ≤ ↑N ^ (↑(nValue N))⁻¹
⊢ (2 * ↑(dValue N)) ^ nValue N ≤ ↑N

Next tactic:
rw [← rpow_nat_cast]
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
  apply this.trans
  suffices ((2 * dValue N) ^ nValue N : ℝ) ≤ N from mod_cast this
  suffices i : (2 * dValue N : ℝ) ≤ (N : ℝ) ^ (nValue N : ℝ)⁻¹
  · rw [← rpow_nat_cast]
    

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
this : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N
i : 2 * ↑(dValue N) ≤ ↑N ^ (↑(nValue N))⁻¹
⊢ (2 * ↑(dValue N)) ^ ↑(nValue N) ≤ ↑N

Next tactic:
apply (rpow_le_rpow (mul_nonneg zero_le_two (cast_nonneg _)) i (cast_nonneg _)).trans
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
  apply this.trans
  suffices ((2 * dValue N) ^ nValue N : ℝ) ≤ N from mod_cast this
  suffices i : (2 * dValue N : ℝ) ≤ (N : ℝ) ^ (nValue N : ℝ)⁻¹
  · rw [← rpow_nat_cast]
    apply (rpow_le_rpow (mul_nonneg zero_le_two (cast_nonneg _)) i (cast_nonneg _)).trans
    

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
this : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N
i : 2 * ↑(dValue N) ≤ ↑N ^ (↑(nValue N))⁻¹
⊢ (↑N ^ (↑(nValue N))⁻¹) ^ ↑(nValue N) ≤ ↑N

Next tactic:
rw [← rpow_mul (cast_nonneg _), inv_mul_cancel, rpow_one]
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
  apply this.trans
  suffices ((2 * dValue N) ^ nValue N : ℝ) ≤ N from mod_cast this
  suffices i : (2 * dValue N : ℝ) ≤ (N : ℝ) ^ (nValue N : ℝ)⁻¹
  · rw [← rpow_nat_cast]
    apply (rpow_le_rpow (mul_nonneg zero_le_two (cast_nonneg _)) i (cast_nonneg _)).trans
    rw [← rpow_mul (cast_nonneg _), inv_mul_cancel, rpow_one]
    

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
this : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N
i : 2 * ↑(dValue N) ≤ ↑N ^ (↑(nValue N))⁻¹
⊢ ↑(nValue N) ≠ 0

Next tactic:
rw [cast_ne_zero]
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
  apply this.trans
  suffices ((2 * dValue N) ^ nValue N : ℝ) ≤ N from mod_cast this
  suffices i : (2 * dValue N : ℝ) ≤ (N : ℝ) ^ (nValue N : ℝ)⁻¹
  · rw [← rpow_nat_cast]
    apply (rpow_le_rpow (mul_nonneg zero_le_two (cast_nonneg _)) i (cast_nonneg _)).trans
    rw [← rpow_mul (cast_nonneg _), inv_mul_cancel, rpow_one]
    rw [cast_ne_zero]
    

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
this : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N
i : 2 * ↑(dValue N) ≤ ↑N ^ (↑(nValue N))⁻¹
⊢ nValue N ≠ 0

Next tactic:
apply (nValue_pos hN).ne'
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
  apply this.trans
  suffices ((2 * dValue N) ^ nValue N : ℝ) ≤ N from mod_cast this
  suffices i : (2 * dValue N : ℝ) ≤ (N : ℝ) ^ (nValue N : ℝ)⁻¹
  · rw [← rpow_nat_cast]
    apply (rpow_le_rpow (mul_nonneg zero_le_two (cast_nonneg _)) i (cast_nonneg _)).trans
    rw [← rpow_mul (cast_nonneg _), inv_mul_cancel, rpow_one]
    rw [cast_ne_zero]
    apply (nValue_pos hN).ne'
  

Proof state:
case i
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
this : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N
⊢ 2 * ↑(dValue N) ≤ ↑N ^ (↑(nValue N))⁻¹

Next tactic:
rw [← le_div_iff']
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
  apply this.trans
  suffices ((2 * dValue N) ^ nValue N : ℝ) ≤ N from mod_cast this
  suffices i : (2 * dValue N : ℝ) ≤ (N : ℝ) ^ (nValue N : ℝ)⁻¹
  · rw [← rpow_nat_cast]
    apply (rpow_le_rpow (mul_nonneg zero_le_two (cast_nonneg _)) i (cast_nonneg _)).trans
    rw [← rpow_mul (cast_nonneg _), inv_mul_cancel, rpow_one]
    rw [cast_ne_zero]
    apply (nValue_pos hN).ne'
  rw [← le_div_iff']
  · 

Proof state:
case i
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
this : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N
⊢ ↑(dValue N) ≤ ↑N ^ (↑(nValue N))⁻¹ / 2

Next tactic:
exact floor_le (div_nonneg (rpow_nonneg_of_nonneg (cast_nonneg _) _) zero_le_two)
theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by
  have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N :=
    Nat.pow_le_pow_left (Nat.sub_le _ _) _
  apply this.trans
  suffices ((2 * dValue N) ^ nValue N : ℝ) ≤ N from mod_cast this
  suffices i : (2 * dValue N : ℝ) ≤ (N : ℝ) ^ (nValue N : ℝ)⁻¹
  · rw [← rpow_nat_cast]
    apply (rpow_le_rpow (mul_nonneg zero_le_two (cast_nonneg _)) i (cast_nonneg _)).trans
    rw [← rpow_mul (cast_nonneg _), inv_mul_cancel, rpow_one]
    rw [cast_ne_zero]
    apply (nValue_pos hN).ne'
  rw [← le_div_iff']
  · exact floor_le (div_nonneg (rpow_nonneg_of_nonneg (cast_nonneg _) _) zero_le_two)
  

Proof state:
case i
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 2 ≤ N
this : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N
⊢ 0 < 2

Next tactic:
apply zero_lt_two
theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by
  

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ ↑N ^ (↑(nValue N))⁻¹ / rexp 1 < ↑(dValue N)

Next tactic:
apply div_lt_floor _
theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by
  apply div_lt_floor _
  

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 2 / (1 - 2 / rexp 1) ≤ ↑N ^ (↑(nValue N))⁻¹

Next tactic:
rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv]
theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by
  apply div_lt_floor _
  rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv]
  · 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ log (2 / (1 - 2 / rexp 1)) ≤ log ↑N / ↑(nValue N)

Next tactic:
apply le_trans _ (div_le_div_of_le_left _ _ (ceil_lt_mul _).le)
theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by
  apply div_lt_floor _
  rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv]
  · apply le_trans _ (div_le_div_of_le_left _ _ (ceil_lt_mul _).le)
    

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ log (2 / (1 - 2 / rexp 1)) ≤ log ↑N / (1.38 * Real.sqrt (log ↑N))
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 0 ≤ log ↑N
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 0 < ↑⌈Real.sqrt (log ↑N)⌉₊
α : Type u_1 β : Type u_2 n d k N : ℕ x : Fin n → ℕ hN : 4096 ≤ N ⊢ 50 / 19 ≤ Real.sqrt (log ↑N)

Next tactic:
rw [mul_comm, ← div_div, div_sqrt, le_div_iff]
theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by
  apply div_lt_floor _
  rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv]
  · apply le_trans _ (div_le_div_of_le_left _ _ (ceil_lt_mul _).le)
    rw [mul_comm, ← div_div, div_sqrt, le_div_iff]
    · 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ log (2 / (1 - 2 / rexp 1)) * 1.38 ≤ Real.sqrt (log ↑N)

Next tactic:
norm_num
theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by
  apply div_lt_floor _
  rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv]
  · apply le_trans _ (div_le_div_of_le_left _ _ (ceil_lt_mul _).le)
    rw [mul_comm, ← div_div, div_sqrt, le_div_iff]
    · norm_num; 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ log (2 / (1 - 2 / rexp 1)) * (69 / 50) ≤ Real.sqrt (log ↑N)

Next tactic:
exact le_sqrt_log hN
theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by
  apply div_lt_floor _
  rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv]
  · apply le_trans _ (div_le_div_of_le_left _ _ (ceil_lt_mul _).le)
    rw [mul_comm, ← div_div, div_sqrt, le_div_iff]
    · norm_num; exact le_sqrt_log hN
    · 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 0 < 1.38

Next tactic:
norm_num1
theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by
  apply div_lt_floor _
  rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv]
  · apply le_trans _ (div_le_div_of_le_left _ _ (ceil_lt_mul _).le)
    rw [mul_comm, ← div_div, div_sqrt, le_div_iff]
    · norm_num; exact le_sqrt_log hN
    · norm_num1
    · 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 0 ≤ log ↑N

Next tactic:
apply log_nonneg
theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by
  apply div_lt_floor _
  rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv]
  · apply le_trans _ (div_le_div_of_le_left _ _ (ceil_lt_mul _).le)
    rw [mul_comm, ← div_div, div_sqrt, le_div_iff]
    · norm_num; exact le_sqrt_log hN
    · norm_num1
    · apply log_nonneg
      

Proof state:
case hx
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 1 ≤ ↑N

Next tactic:
rw [one_le_cast]
theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by
  apply div_lt_floor _
  rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv]
  · apply le_trans _ (div_le_div_of_le_left _ _ (ceil_lt_mul _).le)
    rw [mul_comm, ← div_div, div_sqrt, le_div_iff]
    · norm_num; exact le_sqrt_log hN
    · norm_num1
    · apply log_nonneg
      rw [one_le_cast]
      

Proof state:
case hx
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 1 ≤ N

Next tactic:
exact hN.trans' (by norm_num1)
theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by
  apply div_lt_floor _
  rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv]
  · apply le_trans _ (div_le_div_of_le_left _ _ (ceil_lt_mul _).le)
    rw [mul_comm, ← div_div, div_sqrt, le_div_iff]
    · norm_num; exact le_sqrt_log hN
    · norm_num1
    · apply log_nonneg
      rw [one_le_cast]
      exact hN.trans' (by 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 1 ≤ 4096

Next tactic:
norm_num1