l3lab/ntp-mathlib huggingface dataset:
theorem abs_discr_ge (h : 1 < finrank ℚ K) :
    (4 / 9 : ℝ) * (3 * π / 4) ^ finrank ℚ K ≤ |discr K| := by
  -- We use `exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr` to get a nonzero
  -- algebraic integer `x` of small norm and the fact that `1 ≤ |Norm x|` to get a lower bound
  -- on `sqrt |discr K|`.
  obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr K
  have h_nm : (1:ℝ) ≤ |(Algebra.norm ℚ) (x:K)| := by
    rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_coe_int, Int.cast_le]
    exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
  replace h_bd := le_trans h_nm h_bd
  rw [← inv_mul_le_iff (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
    (by positivity), ← Int.cast_abs, div_pow, mul_pow, ← pow_mul, ← pow_mul] at h_bd
  refine le_trans ?_ h_bd
  -- The sequence `a n` is a lower bound for `|discr K|`. We prove below by induction an uniform
  -- lower bound for this sequence from which we deduce the result.
  let a : ℕ → ℝ := fun n => (n:ℝ) ^ (n * 2) / ((4 / π) ^ n * (n.factorial:ℝ) ^ 2)
  suffices ∀ n, 2 ≤ n → (4 / 9 : ℝ) * (3 * π / 4) ^ n ≤ a n by
    refine le_trans (this (finrank ℚ K) h) ?_
    refine div_le_div_of_le_left (by positivity) (by positivity) ?_
    refine mul_le_mul_of_nonneg_right (pow_le_pow_right ?_ ?_) (by positivity)
    · rw [_root_.le_div_iff Real.pi_pos, one_mul]
      exact Real.pi_le_four
    · rw [← card_add_two_mul_card_eq_rank, mul_comm]
      exact Nat.le_add_left _ _
  

Proof state:
case intro.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
x : ↥(𝓞 K)
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd :
  ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(Nat.factorial (finrank ℚ K)) ^ 2) ≤
    ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑(Nat.factorial n) ^ 2)
⊢ ∀ (n : ℕ), 2 ≤ n → 4 / 9 * (3 * π / 4) ^ n ≤ a n

Next tactic:
intro n hn
theorem abs_discr_ge (h : 1 < finrank ℚ K) :
    (4 / 9 : ℝ) * (3 * π / 4) ^ finrank ℚ K ≤ |discr K| := by
  -- We use `exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr` to get a nonzero
  -- algebraic integer `x` of small norm and the fact that `1 ≤ |Norm x|` to get a lower bound
  -- on `sqrt |discr K|`.
  obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr K
  have h_nm : (1:ℝ) ≤ |(Algebra.norm ℚ) (x:K)| := by
    rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_coe_int, Int.cast_le]
    exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
  replace h_bd := le_trans h_nm h_bd
  rw [← inv_mul_le_iff (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
    (by positivity), ← Int.cast_abs, div_pow, mul_pow, ← pow_mul, ← pow_mul] at h_bd
  refine le_trans ?_ h_bd
  -- The sequence `a n` is a lower bound for `|discr K|`. We prove below by induction an uniform
  -- lower bound for this sequence from which we deduce the result.
  let a : ℕ → ℝ := fun n => (n:ℝ) ^ (n * 2) / ((4 / π) ^ n * (n.factorial:ℝ) ^ 2)
  suffices ∀ n, 2 ≤ n → (4 / 9 : ℝ) * (3 * π / 4) ^ n ≤ a n by
    refine le_trans (this (finrank ℚ K) h) ?_
    refine div_le_div_of_le_left (by positivity) (by positivity) ?_
    refine mul_le_mul_of_nonneg_right (pow_le_pow_right ?_ ?_) (by positivity)
    · rw [_root_.le_div_iff Real.pi_pos, one_mul]
      exact Real.pi_le_four
    · rw [← card_add_two_mul_card_eq_rank, mul_comm]
      exact Nat.le_add_left _ _
  intro n hn
  

Proof state:
case intro.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
x : ↥(𝓞 K)
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd :
  ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(Nat.factorial (finrank ℚ K)) ^ 2) ≤
    ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑(Nat.factorial n) ^ 2)
n : ℕ
hn : 2 ≤ n
⊢ 4 / 9 * (3 * π / 4) ^ n ≤ a n

Next tactic:
induction n, hn using Nat.le_induction with
  | base => exact le_of_eq <| by norm_num [Nat.factorial_two]; field_simp; ring
  | succ m _ h_m =>
      suffices (3:ℝ) ≤ (1 + 1 / m : ℝ) ^ (2 * m) by
        convert_to _ ≤ (a m) * (1 + 1 / m : ℝ) ^ (2 * m) / (4 / π)
        · simp_rw [add_mul, one_mul, pow_succ, Nat.factorial_succ]
          field_simp; ring
        · rw [_root_.le_div_iff (by positivity), pow_succ]
          convert (mul_le_mul h_m this (by positivity) (by positivity)) using 1
          field_simp; ring
      refine le_trans (le_of_eq (by field_simp; norm_num)) (one_add_mul_le_pow ?_ (2 * m))
      exact le_trans (by norm_num : (-2:ℝ) ≤ 0) (by positivity)
theorem abs_discr_ge (h : 1 < finrank ℚ K) :
    (4 / 9 : ℝ) * (3 * π / 4) ^ finrank ℚ K ≤ |discr K| := by
  -- We use `exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr` to get a nonzero
  -- algebraic integer `x` of small norm and the fact that `1 ≤ |Norm x|` to get a lower bound
  -- on `sqrt |discr K|`.
  obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr K
  have h_nm : (1:ℝ) ≤ |(Algebra.norm ℚ) (x:K)| := by
    rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_coe_int, Int.cast_le]
    exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
  replace h_bd := le_trans h_nm h_bd
  rw [← inv_mul_le_iff (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
    (by positivity), ← Int.cast_abs, div_pow, mul_pow, ← pow_mul, ← pow_mul] at h_bd
  refine le_trans ?_ h_bd
  -- The sequence `a n` is a lower bound for `|discr K|`. We prove below by induction an uniform
  -- lower bound for this sequence from which we deduce the result.
  let a : ℕ → ℝ := fun n => (n:ℝ) ^ (n * 2) / ((4 / π) ^ n * (n.factorial:ℝ) ^ 2)
  suffices ∀ n, 2 ≤ n → (4 / 9 : ℝ) * (3 * π / 4) ^ n ≤ a n by
    refine le_trans (this (finrank ℚ K) h) ?_
    refine div_le_div_of_le_left (by positivity) (by positivity) ?_
    refine mul_le_mul_of_nonneg_right (pow_le_pow_right ?_ ?_) (by positivity)
    · rw [_root_.le_div_iff Real.pi_pos, one_mul]
      exact Real.pi_le_four
    · rw [← card_add_two_mul_card_eq_rank, mul_comm]
      exact Nat.le_add_left _ _
  intro n hn
  induction n, hn using Nat.le_induction with
  

Proof state:
case intro.intro.base
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
x : ↥(𝓞 K)
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd :
  ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(Nat.factorial (finrank ℚ K)) ^ 2) ≤
    ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑(Nat.factorial n) ^ 2)
n : ℕ
⊢ 4 / 9 * (3 * π / 4) ^ 2 ≤ a 2

Next tactic:
| base => exact le_of_eq <| by norm_num [Nat.factorial_two]; field_simp; ring
theorem abs_discr_ge (h : 1 < finrank ℚ K) :
    (4 / 9 : ℝ) * (3 * π / 4) ^ finrank ℚ K ≤ |discr K| := by
  -- We use `exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr` to get a nonzero
  -- algebraic integer `x` of small norm and the fact that `1 ≤ |Norm x|` to get a lower bound
  -- on `sqrt |discr K|`.
  obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr K
  have h_nm : (1:ℝ) ≤ |(Algebra.norm ℚ) (x:K)| := by
    rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_coe_int, Int.cast_le]
    exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
  replace h_bd := le_trans h_nm h_bd
  rw [← inv_mul_le_iff (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
    (by positivity), ← Int.cast_abs, div_pow, mul_pow, ← pow_mul, ← pow_mul] at h_bd
  refine le_trans ?_ h_bd
  -- The sequence `a n` is a lower bound for `|discr K|`. We prove below by induction an uniform
  -- lower bound for this sequence from which we deduce the result.
  let a : ℕ → ℝ := fun n => (n:ℝ) ^ (n * 2) / ((4 / π) ^ n * (n.factorial:ℝ) ^ 2)
  suffices ∀ n, 2 ≤ n → (4 / 9 : ℝ) * (3 * π / 4) ^ n ≤ a n by
    refine le_trans (this (finrank ℚ K) h) ?_
    refine div_le_div_of_le_left (by positivity) (by positivity) ?_
    refine mul_le_mul_of_nonneg_right (pow_le_pow_right ?_ ?_) (by positivity)
    · rw [_root_.le_div_iff Real.pi_pos, one_mul]
      exact Real.pi_le_four
    · rw [← card_add_two_mul_card_eq_rank, mul_comm]
      exact Nat.le_add_left _ _
  intro n hn
  induction n, hn using Nat.le_induction with
  | base => 

Proof state:
case intro.intro.base
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
x : ↥(𝓞 K)
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd :
  ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(Nat.factorial (finrank ℚ K)) ^ 2) ≤
    ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑(Nat.factorial n) ^ 2)
n : ℕ
⊢ 4 / 9 * (3 * π / 4) ^ 2 ≤ a 2

Next tactic:
exact le_of_eq <| by norm_num [Nat.factorial_two]; field_simp; ring
theorem abs_discr_ge (h : 1 < finrank ℚ K) :
    (4 / 9 : ℝ) * (3 * π / 4) ^ finrank ℚ K ≤ |discr K| := by
  -- We use `exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr` to get a nonzero
  -- algebraic integer `x` of small norm and the fact that `1 ≤ |Norm x|` to get a lower bound
  -- on `sqrt |discr K|`.
  obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr K
  have h_nm : (1:ℝ) ≤ |(Algebra.norm ℚ) (x:K)| := by
    rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_coe_int, Int.cast_le]
    exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
  replace h_bd := le_trans h_nm h_bd
  rw [← inv_mul_le_iff (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
    (by positivity), ← Int.cast_abs, div_pow, mul_pow, ← pow_mul, ← pow_mul] at h_bd
  refine le_trans ?_ h_bd
  -- The sequence `a n` is a lower bound for `|discr K|`. We prove below by induction an uniform
  -- lower bound for this sequence from which we deduce the result.
  let a : ℕ → ℝ := fun n => (n:ℝ) ^ (n * 2) / ((4 / π) ^ n * (n.factorial:ℝ) ^ 2)
  suffices ∀ n, 2 ≤ n → (4 / 9 : ℝ) * (3 * π / 4) ^ n ≤ a n by
    refine le_trans (this (finrank ℚ K) h) ?_
    refine div_le_div_of_le_left (by positivity) (by positivity) ?_
    refine mul_le_mul_of_nonneg_right (pow_le_pow_right ?_ ?_) (by positivity)
    · rw [_root_.le_div_iff Real.pi_pos, one_mul]
      exact Real.pi_le_four
    · rw [← card_add_two_mul_card_eq_rank, mul_comm]
      exact Nat.le_add_left _ _
  intro n hn
  induction n, hn using Nat.le_induction with
  | base => exact le_of_eq <| by 

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
x : ↥(𝓞 K)
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd :
  ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(Nat.factorial (finrank ℚ K)) ^ 2) ≤
    ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑(Nat.factorial n) ^ 2)
n : ℕ
⊢ 4 / 9 * (3 * π / 4) ^ 2 = a 2

Next tactic:
norm_num [Nat.factorial_two]
theorem abs_discr_ge (h : 1 < finrank ℚ K) :
    (4 / 9 : ℝ) * (3 * π / 4) ^ finrank ℚ K ≤ |discr K| := by
  -- We use `exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr` to get a nonzero
  -- algebraic integer `x` of small norm and the fact that `1 ≤ |Norm x|` to get a lower bound
  -- on `sqrt |discr K|`.
  obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr K
  have h_nm : (1:ℝ) ≤ |(Algebra.norm ℚ) (x:K)| := by
    rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_coe_int, Int.cast_le]
    exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
  replace h_bd := le_trans h_nm h_bd
  rw [← inv_mul_le_iff (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
    (by positivity), ← Int.cast_abs, div_pow, mul_pow, ← pow_mul, ← pow_mul] at h_bd
  refine le_trans ?_ h_bd
  -- The sequence `a n` is a lower bound for `|discr K|`. We prove below by induction an uniform
  -- lower bound for this sequence from which we deduce the result.
  let a : ℕ → ℝ := fun n => (n:ℝ) ^ (n * 2) / ((4 / π) ^ n * (n.factorial:ℝ) ^ 2)
  suffices ∀ n, 2 ≤ n → (4 / 9 : ℝ) * (3 * π / 4) ^ n ≤ a n by
    refine le_trans (this (finrank ℚ K) h) ?_
    refine div_le_div_of_le_left (by positivity) (by positivity) ?_
    refine mul_le_mul_of_nonneg_right (pow_le_pow_right ?_ ?_) (by positivity)
    · rw [_root_.le_div_iff Real.pi_pos, one_mul]
      exact Real.pi_le_four
    · rw [← card_add_two_mul_card_eq_rank, mul_comm]
      exact Nat.le_add_left _ _
  intro n hn
  induction n, hn using Nat.le_induction with
  | base => exact le_of_eq <| by norm_num [Nat.factorial_two]; 

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
x : ↥(𝓞 K)
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd :
  ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(Nat.factorial (finrank ℚ K)) ^ 2) ≤
    ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑(Nat.factorial n) ^ 2)
n : ℕ
⊢ 4 / 9 * ((3 * π) ^ 2 / 16) = 16 / (16 / π ^ 2 * 4)

Next tactic:
field_simp
theorem abs_discr_ge (h : 1 < finrank ℚ K) :
    (4 / 9 : ℝ) * (3 * π / 4) ^ finrank ℚ K ≤ |discr K| := by
  -- We use `exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr` to get a nonzero
  -- algebraic integer `x` of small norm and the fact that `1 ≤ |Norm x|` to get a lower bound
  -- on `sqrt |discr K|`.
  obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr K
  have h_nm : (1:ℝ) ≤ |(Algebra.norm ℚ) (x:K)| := by
    rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_coe_int, Int.cast_le]
    exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
  replace h_bd := le_trans h_nm h_bd
  rw [← inv_mul_le_iff (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
    (by positivity), ← Int.cast_abs, div_pow, mul_pow, ← pow_mul, ← pow_mul] at h_bd
  refine le_trans ?_ h_bd
  -- The sequence `a n` is a lower bound for `|discr K|`. We prove below by induction an uniform
  -- lower bound for this sequence from which we deduce the result.
  let a : ℕ → ℝ := fun n => (n:ℝ) ^ (n * 2) / ((4 / π) ^ n * (n.factorial:ℝ) ^ 2)
  suffices ∀ n, 2 ≤ n → (4 / 9 : ℝ) * (3 * π / 4) ^ n ≤ a n by
    refine le_trans (this (finrank ℚ K) h) ?_
    refine div_le_div_of_le_left (by positivity) (by positivity) ?_
    refine mul_le_mul_of_nonneg_right (pow_le_pow_right ?_ ?_) (by positivity)
    · rw [_root_.le_div_iff Real.pi_pos, one_mul]
      exact Real.pi_le_four
    · rw [← card_add_two_mul_card_eq_rank, mul_comm]
      exact Nat.le_add_left _ _
  intro n hn
  induction n, hn using Nat.le_induction with
  | base => exact le_of_eq <| by norm_num [Nat.factorial_two]; field_simp; 

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
x : ↥(𝓞 K)
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd :
  ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(Nat.factorial (finrank ℚ K)) ^ 2) ≤
    ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑(Nat.factorial n) ^ 2)
n : ℕ
⊢ 4 * (3 * π) ^ 2 * (16 * 4) = 16 * π ^ 2 * (9 * 16)

Next tactic:
ring
theorem abs_discr_ge (h : 1 < finrank ℚ K) :
    (4 / 9 : ℝ) * (3 * π / 4) ^ finrank ℚ K ≤ |discr K| := by
  -- We use `exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr` to get a nonzero
  -- algebraic integer `x` of small norm and the fact that `1 ≤ |Norm x|` to get a lower bound
  -- on `sqrt |discr K|`.
  obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr K
  have h_nm : (1:ℝ) ≤ |(Algebra.norm ℚ) (x:K)| := by
    rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_coe_int, Int.cast_le]
    exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
  replace h_bd := le_trans h_nm h_bd
  rw [← inv_mul_le_iff (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
    (by positivity), ← Int.cast_abs, div_pow, mul_pow, ← pow_mul, ← pow_mul] at h_bd
  refine le_trans ?_ h_bd
  -- The sequence `a n` is a lower bound for `|discr K|`. We prove below by induction an uniform
  -- lower bound for this sequence from which we deduce the result.
  let a : ℕ → ℝ := fun n => (n:ℝ) ^ (n * 2) / ((4 / π) ^ n * (n.factorial:ℝ) ^ 2)
  suffices ∀ n, 2 ≤ n → (4 / 9 : ℝ) * (3 * π / 4) ^ n ≤ a n by
    refine le_trans (this (finrank ℚ K) h) ?_
    refine div_le_div_of_le_left (by positivity) (by positivity) ?_
    refine mul_le_mul_of_nonneg_right (pow_le_pow_right ?_ ?_) (by positivity)
    · rw [_root_.le_div_iff Real.pi_pos, one_mul]
      exact Real.pi_le_four
    · rw [← card_add_two_mul_card_eq_rank, mul_comm]
      exact Nat.le_add_left _ _
  intro n hn
  induction n, hn using Nat.le_induction with
  | base => exact le_of_eq <| by norm_num [Nat.factorial_two]; field_simp; ring
  

Proof state:
case intro.intro.succ
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
x : ↥(𝓞 K)
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd :
  ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(Nat.factorial (finrank ℚ K)) ^ 2) ≤
    ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑(Nat.factorial n) ^ 2)
n m : ℕ
hn✝ : 2 ≤ m
h_m : 4 / 9 * (3 * π / 4) ^ m ≤ a m
⊢ 4 / 9 * (3 * π / 4) ^ (m + 1) ≤ a (m + 1)

Next tactic:
| succ m _ h_m =>
      suffices (3:ℝ) ≤ (1 + 1 / m : ℝ) ^ (2 * m) by
        convert_to _ ≤ (a m) * (1 + 1 / m : ℝ) ^ (2 * m) / (4 / π)
        · simp_rw [add_mul, one_mul, pow_succ, Nat.factorial_succ]
          field_simp; ring
        · rw [_root_.le_div_iff (by positivity), pow_succ]
          convert (mul_le_mul h_m this (by positivity) (by positivity)) using 1
          field_simp; ring
      refine le_trans (le_of_eq (by field_simp; norm_num)) (one_add_mul_le_pow ?_ (2 * m))
      exact le_trans (by norm_num : (-2:ℝ) ≤ 0) (by positivity)
theorem abs_discr_ge (h : 1 < finrank ℚ K) :
    (4 / 9 : ℝ) * (3 * π / 4) ^ finrank ℚ K ≤ |discr K| := by
  -- We use `exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr` to get a nonzero
  -- algebraic integer `x` of small norm and the fact that `1 ≤ |Norm x|` to get a lower bound
  -- on `sqrt |discr K|`.
  obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr K
  have h_nm : (1:ℝ) ≤ |(Algebra.norm ℚ) (x:K)| := by
    rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_coe_int, Int.cast_le]
    exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
  replace h_bd := le_trans h_nm h_bd
  rw [← inv_mul_le_iff (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
    (by positivity), ← Int.cast_abs, div_pow, mul_pow, ← pow_mul, ← pow_mul] at h_bd
  refine le_trans ?_ h_bd
  -- The sequence `a n` is a lower bound for `|discr K|`. We prove below by induction an uniform
  -- lower bound for this sequence from which we deduce the result.
  let a : ℕ → ℝ := fun n => (n:ℝ) ^ (n * 2) / ((4 / π) ^ n * (n.factorial:ℝ) ^ 2)
  suffices ∀ n, 2 ≤ n → (4 / 9 : ℝ) * (3 * π / 4) ^ n ≤ a n by
    refine le_trans (this (finrank ℚ K) h) ?_
    refine div_le_div_of_le_left (by positivity) (by positivity) ?_
    refine mul_le_mul_of_nonneg_right (pow_le_pow_right ?_ ?_) (by positivity)
    · rw [_root_.le_div_iff Real.pi_pos, one_mul]
      exact Real.pi_le_four
    · rw [← card_add_two_mul_card_eq_rank, mul_comm]
      exact Nat.le_add_left _ _
  intro n hn
  induction n, hn using Nat.le_induction with
  | base => exact le_of_eq <| by norm_num [Nat.factorial_two]; field_simp; ring
  | succ m _ h_m =>
      

Proof state:
case intro.intro.succ
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
x : ↥(𝓞 K)
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd :
  ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(Nat.factorial (finrank ℚ K)) ^ 2) ≤
    ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑(Nat.factorial n) ^ 2)
n m : ℕ
hn✝ : 2 ≤ m
h_m : 4 / 9 * (3 * π / 4) ^ m ≤ a m
⊢ 4 / 9 * (3 * π / 4) ^ (m + 1) ≤ a (m + 1)

Next tactic:
suffices (3:ℝ) ≤ (1 + 1 / m : ℝ) ^ (2 * m) by
        convert_to _ ≤ (a m) * (1 + 1 / m : ℝ) ^ (2 * m) / (4 / π)
        · simp_rw [add_mul, one_mul, pow_succ, Nat.factorial_succ]
          field_simp; ring
        · rw [_root_.le_div_iff (by positivity), pow_succ]
          convert (mul_le_mul h_m this (by positivity) (by positivity)) using 1
          field_simp; ring
theorem abs_discr_ge (h : 1 < finrank ℚ K) :
    (4 / 9 : ℝ) * (3 * π / 4) ^ finrank ℚ K ≤ |discr K| := by
  -- We use `exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr` to get a nonzero
  -- algebraic integer `x` of small norm and the fact that `1 ≤ |Norm x|` to get a lower bound
  -- on `sqrt |discr K|`.
  obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr K
  have h_nm : (1:ℝ) ≤ |(Algebra.norm ℚ) (x:K)| := by
    rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_coe_int, Int.cast_le]
    exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
  replace h_bd := le_trans h_nm h_bd
  rw [← inv_mul_le_iff (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
    (by positivity), ← Int.cast_abs, div_pow, mul_pow, ← pow_mul, ← pow_mul] at h_bd
  refine le_trans ?_ h_bd
  -- The sequence `a n` is a lower bound for `|discr K|`. We prove below by induction an uniform
  -- lower bound for this sequence from which we deduce the result.
  let a : ℕ → ℝ := fun n => (n:ℝ) ^ (n * 2) / ((4 / π) ^ n * (n.factorial:ℝ) ^ 2)
  suffices ∀ n, 2 ≤ n → (4 / 9 : ℝ) * (3 * π / 4) ^ n ≤ a n by
    refine le_trans (this (finrank ℚ K) h) ?_
    refine div_le_div_of_le_left (by positivity) (by positivity) ?_
    refine mul_le_mul_of_nonneg_right (pow_le_pow_right ?_ ?_) (by positivity)
    · rw [_root_.le_div_iff Real.pi_pos, one_mul]
      exact Real.pi_le_four
    · rw [← card_add_two_mul_card_eq_rank, mul_comm]
      exact Nat.le_add_left _ _
  intro n hn
  induction n, hn using Nat.le_induction with
  | base => exact le_of_eq <| by norm_num [Nat.factorial_two]; field_simp; ring
  | succ m _ h_m =>
      suffices (3:ℝ) ≤ (1 + 1 / m : ℝ) ^ (2 * m) by
        

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
x : ↥(𝓞 K)
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd :
  ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(Nat.factorial (finrank ℚ K)) ^ 2) ≤
    ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑(Nat.factorial n) ^ 2)
n m : ℕ
hn✝ : 2 ≤ m
h_m : 4 / 9 * (3 * π / 4) ^ m ≤ a m
this : 3 ≤ (1 + 1 / ↑m) ^ (2 * m)
⊢ 4 / 9 * (3 * π / 4) ^ (m + 1) ≤ a (m + 1)

Next tactic:
convert_to _ ≤ (a m) * (1 + 1 / m : ℝ) ^ (2 * m) / (4 / π)