Theorem: NumberField.abs_discr_ge
File: Mathlib/NumberTheory/NumberField/Discriminant.lean

Proof steps:
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤ ↑|discr K|
state_after: 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_bd :
  ↑|(Algebra.norm ℚ) ↑x| ≤
    (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤ ↑|discr K|
tactic: obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr K
state_before: 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_bd :
  ↑|(Algebra.norm ℚ) ↑x| ≤
    (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤ ↑|discr K|
state_after: 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_bd :
  ↑|(Algebra.norm ℚ) ↑x| ≤
    (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤ ↑|discr K|
tactic: have h_nm : (1 : ℝ) ≤ |Algebra.norm ℚ (x : K)| := by
    rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_intCast, Int.cast_le]
    exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
state_before: 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_bd :
  ↑|(Algebra.norm ℚ) ↑x| ≤
    (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤ ↑|discr K|
state_after: 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 : 1 ≤ (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤ ↑|discr K|
tactic: replace h_bd := le_trans h_nm h_bd
state_before: 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 : 1 ≤ (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤ ↑|discr K|
state_after: 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) * ↑(finrank ℚ K).factorial ^ 2) ≤ ↑|discr K|
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤ ↑|discr K|
tactic: 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
state_before: 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) * ↑(finrank ℚ K).factorial ^ 2) ≤ ↑|discr K|
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤ ↑|discr K|
state_after: 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) * ↑(finrank ℚ K).factorial ^ 2) ≤ ↑|discr K|
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤
    ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(finrank ℚ K).factorial ^ 2)
tactic: refine le_trans ?_ h_bd
state_before: 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) * ↑(finrank ℚ K).factorial ^ 2) ≤ ↑|discr K|
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤
    ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(finrank ℚ K).factorial ^ 2)
state_after: 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) * ↑(finrank ℚ K).factorial ^ 2) ≤ ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑n.factorial ^ 2)
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤
    ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (NrComplexPlaces K * 2) * ↑(finrank ℚ K).factorial ^ 2)
tactic: let a : ℕ → ℝ := fun n => (n : ℝ) ^ (n * 2) / ((4 / π) ^ n * (n.factorial : ℝ) ^ 2)
state_before: 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) * ↑(finrank ℚ K).factorial ^ 2) ≤ ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑n.factorial ^ 2)
⊢ ∀ (n : ℕ), 2 ≤ n → 4 / 9 * (3 * π / 4) ^ n ≤ a n
state_after: 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) * ↑(finrank ℚ K).factorial ^ 2) ≤ ↑|discr K|
a : ℕ → ℝ := fun n => ↑n ^ (n * 2) / ((4 / π) ^ n * ↑n.factorial ^ 2)
n : ℕ
hn : 2 ≤ n
⊢ 4 / 9 * (3 * π / 4) ^ n ≤ a n
tactic: intro n hn