l3lab/ntp-mathlib huggingface dataset:
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 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 < ↑⌈Real.sqrt (log ↑N)⌉₊

Next tactic:
rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      

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:
refine' log_pos _
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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      

Proof state:
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_lt_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]
      exact hN.trans' (by norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      

Proof state:
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_lt' (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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (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
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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    

Proof state:
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:
apply le_sqrt_of_sq_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)
    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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    

Proof state:
case h
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ (50 / 19) ^ 2 ≤ log ↑N

Next tactic:
have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast 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
    · norm_num1
    · apply log_nonneg
      rw [one_le_cast]
      exact hN.trans' (by norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ ↑12 * log 2 ≤ log ↑N

Next tactic:
rw [← log_rpow zero_lt_two, rpow_nat_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]
      exact hN.trans' (by norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ log (2 ^ 12) ≤ log ↑N

Next tactic:
exact log_le_log (by positivity) (mod_cast 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
    · norm_num1
    · apply log_nonneg
      rw [one_le_cast]
      exact hN.trans' (by norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 0 < 2 ^ 12

Next tactic:
positivity
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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    

Proof state:
case h
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
this : ↑12 * log 2 ≤ log ↑N
⊢ (50 / 19) ^ 2 ≤ log ↑N

Next tactic:
refine' le_trans _ this
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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    

Proof state:
case h
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
this : ↑12 * log 2 ≤ log ↑N
⊢ (50 / 19) ^ 2 ≤ ↑12 * log 2

Next tactic:
rw [← div_le_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]
    · norm_num; exact le_sqrt_log hN
    · norm_num1
    · apply log_nonneg
      rw [one_le_cast]
      exact hN.trans' (by norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    rw [← div_le_iff']
    · 

Proof state:
case h
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
this : ↑12 * log 2 ≤ log ↑N
⊢ (50 / 19) ^ 2 / ↑12 ≤ log 2

Next tactic:
exact log_two_gt_d9.le.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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    rw [← div_le_iff']
    · exact log_two_gt_d9.le.trans' (by 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
this : ↑12 * log 2 ≤ log ↑N
⊢ (50 / 19) ^ 2 / ↑12 ≤ 0.6931471803

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
    · apply log_nonneg
      rw [one_le_cast]
      exact hN.trans' (by norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    rw [← div_le_iff']
    · exact log_two_gt_d9.le.trans' (by norm_num1)
    · 

Proof state:
case h
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
this : ↑12 * log 2 ≤ log ↑N
⊢ 0 < ↑12

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
    · apply log_nonneg
      rw [one_le_cast]
      exact hN.trans' (by norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    rw [← div_le_iff']
    · exact log_two_gt_d9.le.trans' (by norm_num1)
    · norm_num1
  · 

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

Next tactic:
rw [cast_pos]
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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    rw [← div_le_iff']
    · exact log_two_gt_d9.le.trans' (by norm_num1)
    · norm_num1
  · rw [cast_pos]
    

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

Next tactic:
exact hN.trans_lt' (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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    rw [← div_le_iff']
    · exact log_two_gt_d9.le.trans' (by norm_num1)
    · norm_num1
  · rw [cast_pos]
    exact hN.trans_lt' (by 

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

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
    · apply log_nonneg
      rw [one_le_cast]
      exact hN.trans' (by norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    rw [← div_le_iff']
    · exact log_two_gt_d9.le.trans' (by norm_num1)
    · norm_num1
  · rw [cast_pos]
    exact hN.trans_lt' (by norm_num1)
  · 

Proof state:
case h
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 0 < 2 / (1 - 2 / rexp 1)

Next tactic:
refine' div_pos zero_lt_two _
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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    rw [← div_le_iff']
    · exact log_two_gt_d9.le.trans' (by norm_num1)
    · norm_num1
  · rw [cast_pos]
    exact hN.trans_lt' (by norm_num1)
  · refine' div_pos zero_lt_two _
    

Proof state:
case h
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 0 < 1 - 2 / rexp 1

Next tactic:
rw [sub_pos, div_lt_one (exp_pos _)]
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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    rw [← div_le_iff']
    · exact log_two_gt_d9.le.trans' (by norm_num1)
    · norm_num1
  · rw [cast_pos]
    exact hN.trans_lt' (by norm_num1)
  · refine' div_pos zero_lt_two _
    rw [sub_pos, div_lt_one (exp_pos _)]
    

Proof state:
case h
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 2 < rexp 1

Next tactic:
exact lt_of_le_of_lt (by norm_num1) exp_one_gt_d9
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 norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    rw [← div_le_iff']
    · exact log_two_gt_d9.le.trans' (by norm_num1)
    · norm_num1
  · rw [cast_pos]
    exact hN.trans_lt' (by norm_num1)
  · refine' div_pos zero_lt_two _
    rw [sub_pos, div_lt_one (exp_pos _)]
    exact lt_of_le_of_lt (by 

Proof state:
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 2 ≤ 2.7182818283

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
    · apply log_nonneg
      rw [one_le_cast]
      exact hN.trans' (by norm_num1)
    · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]
      refine' log_pos _
      rw [one_lt_cast]
      exact hN.trans_lt' (by norm_num1)
    apply le_sqrt_of_sq_le
    have : (12 : ℕ) * log 2 ≤ log N := by
      rw [← log_rpow zero_lt_two, rpow_nat_cast]
      exact log_le_log (by positivity) (mod_cast hN)
    refine' le_trans _ this
    rw [← div_le_iff']
    · exact log_two_gt_d9.le.trans' (by norm_num1)
    · norm_num1
  · rw [cast_pos]
    exact hN.trans_lt' (by norm_num1)
  · refine' div_pos zero_lt_two _
    rw [sub_pos, div_lt_one (exp_pos _)]
    exact lt_of_le_of_lt (by norm_num1) exp_one_gt_d9
  

Proof state:
case h₁
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
hN : 4096 ≤ N
⊢ 0 < ↑N ^ (↑(nValue N))⁻¹

Next tactic:
positivity
theorem roth_lower_bound_explicit (hN : 4096 ≤ N) :
    (N : ℝ) * exp (-4 * sqrt (log N)) < rothNumberNat 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 * rexp (-4 * Real.sqrt (log ↑N)) < ↑(rothNumberNat N)

Next tactic:
let n := nValue N
theorem roth_lower_bound_explicit (hN : 4096 ≤ N) :
    (N : ℝ) * exp (-4 * sqrt (log N)) < rothNumberNat N := by
  let n := nValue N
  

Proof state:
α : Type u_1
β : Type u_2
n✝ d k N : ℕ
x : Fin n✝ → ℕ
hN : 4096 ≤ N
n : ℕ := nValue N
⊢ ↑N * rexp (-4 * Real.sqrt (log ↑N)) < ↑(rothNumberNat N)

Next tactic:
have hn : 0 < (n : ℝ) := cast_pos.2 (nValue_pos <| hN.trans' <| by norm_num1)
theorem roth_lower_bound_explicit (hN : 4096 ≤ N) :
    (N : ℝ) * exp (-4 * sqrt (log N)) < rothNumberNat N := by
  let n := nValue N
  have hn : 0 < (n : ℝ) := cast_pos.2 (nValue_pos <| 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
n : ℕ := nValue N
⊢ 2 ≤ 4096

Next tactic:
norm_num1
theorem roth_lower_bound_explicit (hN : 4096 ≤ N) :
    (N : ℝ) * exp (-4 * sqrt (log N)) < rothNumberNat N := by
  let n := nValue N
  have hn : 0 < (n : ℝ) := cast_pos.2 (nValue_pos <| hN.trans' <| by norm_num1)
  

Proof state:
α : Type u_1
β : Type u_2
n✝ d k N : ℕ
x : Fin n✝ → ℕ
hN : 4096 ≤ N
n : ℕ := nValue N
hn : 0 < ↑n
⊢ ↑N * rexp (-4 * Real.sqrt (log ↑N)) < ↑(rothNumberNat N)

Next tactic:
have hd : 0 < dValue N := dValue_pos (hN.trans' <| by norm_num1)
theorem roth_lower_bound_explicit (hN : 4096 ≤ N) :
    (N : ℝ) * exp (-4 * sqrt (log N)) < rothNumberNat N := by
  let n := nValue N
  have hn : 0 < (n : ℝ) := cast_pos.2 (nValue_pos <| hN.trans' <| by norm_num1)
  have hd : 0 < dValue N := dValue_pos (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
n : ℕ := nValue N
hn : 0 < ↑n
⊢ 8 ≤ 4096

Next tactic:
norm_num1
theorem roth_lower_bound_explicit (hN : 4096 ≤ N) :
    (N : ℝ) * exp (-4 * sqrt (log N)) < rothNumberNat N := by
  let n := nValue N
  have hn : 0 < (n : ℝ) := cast_pos.2 (nValue_pos <| hN.trans' <| by norm_num1)
  have hd : 0 < dValue N := dValue_pos (hN.trans' <| by norm_num1)
  

Proof state:
α : Type u_1
β : Type u_2
n✝ d k N : ℕ
x : Fin n✝ → ℕ
hN : 4096 ≤ N
n : ℕ := nValue N
hn : 0 < ↑n
hd : 0 < dValue N
⊢ ↑N * rexp (-4 * Real.sqrt (log ↑N)) < ↑(rothNumberNat N)

Next tactic:
have hN₀ : 0 < (N : ℝ) := cast_pos.2 (hN.trans' <| by norm_num1)
theorem roth_lower_bound_explicit (hN : 4096 ≤ N) :
    (N : ℝ) * exp (-4 * sqrt (log N)) < rothNumberNat N := by
  let n := nValue N
  have hn : 0 < (n : ℝ) := cast_pos.2 (nValue_pos <| hN.trans' <| by norm_num1)
  have hd : 0 < dValue N := dValue_pos (hN.trans' <| by norm_num1)
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (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
n : ℕ := nValue N
hn : 0 < ↑n
hd : 0 < dValue N
⊢ succ 0 ≤ 4096

Next tactic:
norm_num1
theorem roth_lower_bound_explicit (hN : 4096 ≤ N) :
    (N : ℝ) * exp (-4 * sqrt (log N)) < rothNumberNat N := by
  let n := nValue N
  have hn : 0 < (n : ℝ) := cast_pos.2 (nValue_pos <| hN.trans' <| by norm_num1)
  have hd : 0 < dValue N := dValue_pos (hN.trans' <| by norm_num1)
  have hN₀ : 0 < (N : ℝ) := cast_pos.2 (hN.trans' <| by norm_num1)
  

Proof state:
α : Type u_1
β : Type u_2
n✝ d k N : ℕ
x : Fin n✝ → ℕ
hN : 4096 ≤ N
n : ℕ := nValue N
hn : 0 < ↑n
hd : 0 < dValue N
hN₀ : 0 < ↑N
⊢ ↑N * rexp (-4 * Real.sqrt (log ↑N)) < ↑(rothNumberNat N)

Next tactic:
have hn₂ : 2 < n := three_le_nValue $ hN.trans' $ by norm_num1