-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
theorem exists_primitive_element_lt_of_isComplex {w₀ : InfinitePlace K} (hw₀ : IsComplex w₀)
    {B : ℝ≥0} (hB : minkowskiBound K ↑1 < convexBodyLT'Factor K * B) :
    ∃ a : 𝓞 K, ℚ⟮(a : K)⟯ = ⊤ ∧
      ∀ w : InfinitePlace K, w a < Real.sqrt (1 + B ^ 2)

Proof state:
case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ :
  |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧
    |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(if ↑⟨w₀, hw₀⟩ = w₀ then NNReal.sqrt B else 1) ^ 2
w : InfinitePlace K
h_eq : ¬w = w₀
⊢ 1 ≤ 1 + ↑B ^ 2

Tactic:
norm_num

Result:
no goals

Proof state:
case intro.intro.intro.refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ :
  |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧
    |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(if ↑⟨w₀, hw₀⟩ = w₀ then NNReal.sqrt B else 1) ^ 2
⊢ ℚ⟮↑a⟯ = ⊤

Tactic:
exact is_primitive_element_of_infinitePlace_lt h_nz
  (fun w h_ne ↦ by convert if_neg h_ne ▸ h_le w h_ne) (Or.inr h_le₀.1)

Result:
no goals

Proof state:
case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ :
  |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧
    |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(if ↑⟨w₀, hw₀⟩ = w₀ then NNReal.sqrt B else 1) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ w ↑a < √(1 + ↑B ^ 2)

Tactic:
rw [if_pos rfl] at h_le₀

Result:
case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧ |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ w ↑a < √(1 + ↑B ^ 2)

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
⊢ ∃ a, ℚ⟮↑a⟯ = ⊤ ∧ ∀ (w : InfinitePlace K), w ↑a < √(1 + ↑B ^ 2)

Tactic:
have : minkowskiBound K ↑1 <
    volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩) := by
  rw [convexBodyLT'_volume, ← Finset.prod_erase_mul _ _ (Finset.mem_univ w₀)]
  simp_rw [ite_pow, one_pow]
  rw [Finset.prod_ite_eq']
  simp_rw [Finset.not_mem_erase, ite_false, mult, not_isReal_iff_isComplex.mpr hw₀,
    ite_true, ite_false, one_mul, NNReal.sq_sqrt]
  exact hB

Result:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
⊢ ∃ a, ℚ⟮↑a⟯ = ⊤ ∧ ∀ (w : InfinitePlace K), w ↑a < √(1 + ↑B ^ 2)

Proof state:
case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |(w₀.embedding ↑a).re| < 1 ∧ |(w₀.embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ (w₀.embedding ↑a).re ^ 2 + (w₀.embedding ↑a).im ^ 2 < 1 + ↑B ^ 2

Tactic:
refine add_lt_add ?_ ?_

Result:
case pos.refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |(w₀.embedding ↑a).re| < 1 ∧ |(w₀.embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ (w₀.embedding ↑a).re ^ 2 < 1

case pos.refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |(w₀.embedding ↑a).re| < 1 ∧ |(w₀.embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ (w₀.embedding ↑a).im ^ 2 < ↑B ^ 2

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
⊢ minkowskiBound K 1 <
    ↑(convexBodyLT'Factor K) *
      ↑((∏ x ∈ Finset.univ.erase w₀, (if x = w₀ then NNReal.sqrt B else 1) ^ x.mult) *
          (if w₀ = w₀ then NNReal.sqrt B else 1) ^ w₀.mult)

Tactic:
simp_rw [ite_pow, one_pow]

Result:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
⊢ minkowskiBound K 1 <
    ↑(convexBodyLT'Factor K) *
      ↑((∏ x ∈ Finset.univ.erase w₀, if x = w₀ then NNReal.sqrt B ^ x.mult else 1) *
          if True then NNReal.sqrt B ^ w₀.mult else 1)

Proof state:
case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ :
  |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧
    |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(if ↑⟨w₀, hw₀⟩ = w₀ then NNReal.sqrt B else 1) ^ 2
w : InfinitePlace K
h_eq : ¬w = w₀
⊢ ↑1 ≤ √(1 + ↑B ^ 2)

Tactic:
rw [NNReal.coe_one, Real.le_sqrt' zero_lt_one, one_pow]

Result:
case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ :
  |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧
    |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(if ↑⟨w₀, hw₀⟩ = w₀ then NNReal.sqrt B else 1) ^ 2
w : InfinitePlace K
h_eq : ¬w = w₀
⊢ 1 ≤ 1 + ↑B ^ 2