mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case e'_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
⊢ ∏ w, w x ^ w.mult = ∏ j, ∏ i with mk i = j, ‖i x‖

Tactic:
]

Goal state after:
case e'_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
⊢ ∏ w, w x ^ w.mult = ∏ j, ∏ i with mk i = j, ‖i x‖
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
w : InfinitePlace K
φ : K →+* ℂ
hφ : φ ∈ {φ | mk φ = w}
⊢ ‖φ x‖ = ‖φ x‖

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ x = (-m) ^ 2 - (-n) ^ 2 ∧
    y = 2 * -m * -n ∧
      z = (-m) ^ 2 + (-n) ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

pp: case h.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_even : x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
⊢ x = (-m) ^ 2 - (-n) ^ 2 ∧
    y = 2 * -m * -n ∧
      z = (-m) ^ 2 + (-n) ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Tactic:
· rw [neg_sq m]
      rw [neg_sq n]
      apply And.intro h_odd.1
      constructor
      · rw [h_odd.2]
        ring
      rcases ht2 with h_pos | h_neg
      · apply And.intro h_pos
        constructor
        · delta Int.gcd
          rw [Int.natAbs_neg, Int.natAbs_neg]
          exact ht3
        · rw [Int.neg_emod_two, Int.neg_emod_two]
          apply And.intro ht4
          lia
      · exfalso
        revert h_pos
        rw [h_neg]
        exact imp_false.mpr (not_lt.mpr (neg_nonpos.mpr (by positivity)))

Goal state after:
case h.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_even : x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
⊢ x = (-m) ^ 2 - (-n) ^ 2 ∧
    y = 2 * -m * -n ∧
      z = (-m) ^ 2 + (-n) ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inr
y z : ℤ
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h : PythagoreanTriple (2 * m * n) y z
h_coprime : (2 * m * n).gcd y = 1
h_parity : 0 = 1
⊢ False

Tactic:
]

Goal state after:
case h.inr
y z : ℤ
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h : PythagoreanTriple (2 * m * n) y z
h_coprime : (2 * m * n).gcd y = 1
h_parity : 0 = 1
⊢ False
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case e'_3
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
⊢ ‖↑((Algebra.norm ℚ) x)‖ = ‖↑((Algebra.norm ℚ) x)‖

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: case e'_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
⊢ IsUnit ((RingOfIntegers.norm ℚ) x) ↔ IsUnit ((RingOfIntegers.norm ℚ) x)

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : (𝓞 K)ˣ
n : ℕ
⊢ (algebraMap (𝓞 K) K) ↑(x ^ n) = (algebraMap (𝓞 K) K) ↑(x ^ n)

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
this : w ↑a < 1
⊢ ∏ w, w ↑a ^ w.mult < ∏ _x ∈ ?m.47, 1

pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
this : w ↑a < 1
⊢ Type ?u.40

pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
this : w ↑a < 1
⊢ Finset ?m.45

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
this : w ↑a < 1
⊢ ∏ w, w ↑a ^ w.mult < ∏ _x ∈ ?m.47, 1

K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
this : w ↑a < 1
⊢ Type ?u.40

K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
this : w ↑a < 1
⊢ Finset ?m.45
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: z k m n : ℤ
right✝ : m.gcd n = 1
h : PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) z
⊢ k * (m ^ 2 - n ^ 2) * (k * (m ^ 2 - n ^ 2)) + k * (2 * m * n) * (k * (2 * m * n)) = (k * (m ^ 2 + n ^ 2)) ^ 2

Tactic:
]

Goal state after:
z k m n : ℤ
right✝ : m.gcd n = 1
h : PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) z
⊢ k * (m ^ 2 - n ^ 2) * (k * (m ^ 2 - n ^ 2)) + k * (2 * m * n) * (k * (2 * m * n)) = (k * (m ^ 2 + n ^ 2)) ^ 2
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
this : w ↑a < 1
z : InfinitePlace K
x✝ : z ∈ Finset.univ
⊢ (if z.IsReal then 1 else 2) ≠ 0

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
this : w ↑a < 1
z : InfinitePlace K
x✝ : z ∈ Finset.univ
⊢ (if z.IsReal then 1 else 2) ≠ 0
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
⊢ IsOfFinOrder x ↔ ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
⊢ IsOfFinOrder x ↔ ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case inr.hi.hab
k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
⊢ IsCoprime a b

Tactic:
]

Goal state after:
case inr.hi.hab
k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
⊢ IsCoprime a b
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
h : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1
n : ℕ
hn : 0 < n
hx : (algebraMap (𝓞 K) K) ↑x ^ n = 1
⊢ ↑↑(x ^ n) = ↑↑1

Tactic:
NumberField.RingOfIntegers.coe_eq_algebraMap,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
h : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1
n : ℕ
hn : 0 < n
hx : (algebraMap (𝓞 K) K) ↑x ^ n = 1
⊢ (algebraMap (𝓞 K) K) ↑(x ^ n) = ↑↑1
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
h : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1
n : ℕ
hn : 0 < n
hx : (algebraMap (𝓞 K) K) ↑x ^ n = 1
⊢ 1 = ↑↑1

Tactic:
NumberField.RingOfIntegers.coe_eq_algebraMap,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
h : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1
n : ℕ
hn : 0 < n
hx : (algebraMap (𝓞 K) K) ↑x ^ n = 1
⊢ 1 = (algebraMap (𝓞 K) K) ↑1
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
h : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1
n : ℕ
hn : 0 < n
hx : (algebraMap (𝓞 K) K) ↑x ^ n = 1
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: z k m n : ℤ
right✝ : m.gcd n = 1
h : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z
⊢ k * (2 * m * n) * (k * (2 * m * n)) + k * (m ^ 2 - n ^ 2) * (k * (m ^ 2 - n ^ 2)) = (k * (m ^ 2 + n ^ 2)) ^ 2

Tactic:
]

Goal state after:
z k m n : ℤ
right✝ : m.gcd n = 1
h : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z
⊢ k * (2 * m * n) * (k * (2 * m * n)) + k * (m ^ 2 - n ^ 2) * (k * (m ^ 2 - n ^ 2)) = (k * (m ^ 2 + n ^ 2)) ^ 2
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
h : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1
n : ℕ
hn : 0 < n
hx : (algebraMap (𝓞 K) K) ↑x ^ n = 1
⊢ x ^ n = 1

Tactic:
by ext; rw [NumberField.RingOfIntegers.coe_eq_algebraMap, coe_pow, hx,
    NumberField.RingOfIntegers.coe_eq_algebraMap, coe_one]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
this : w ↑a < 1
z : InfinitePlace K
x✝ : z ∈ Finset.univ
hz : z = w
⊢ w ↑a < 1

Tactic:
]

Goal state after:
case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
this : w ↑a < 1
z : InfinitePlace K
x✝ : z ∈ Finset.univ
hz : z = w
⊢ w ↑a < 1
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a : K
x✝ : a ∈ (fun x ↦ (algebraMap (𝓞 K) K) ↑x) '' ↑(torsion K)
u : (𝓞 K)ˣ
h_tors : u ∈ ↑(torsion K)
h_ua : (fun x ↦ (algebraMap (𝓞 K) K) ↑x) u = a
⊢ IsIntegral ℤ ((fun x ↦ (algebraMap (𝓞 K) K) ↑x) u)

Tactic:
]

Goal state after:
case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a : K
x✝ : a ∈ (fun x ↦ (algebraMap (𝓞 K) K) ↑x) '' ↑(torsion K)
u : (𝓞 K)ˣ
h_tors : u ∈ ↑(torsion K)
h_ua : (fun x ↦ (algebraMap (𝓞 K) K) ↑x) u = a
⊢ IsIntegral ℤ ((fun x ↦ (algebraMap (𝓞 K) K) ↑x) u)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
⊢ 1 ≤ |(Algebra.norm ℤ) a|

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
a : 𝓞 K
ha : a ≠ 0
h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1
⊢ 1 ≤ |(Algebra.norm ℤ) a|
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
⊢ ∀ (ψ : K →ₐ[ℚ] ℂ), w.embedding.toRatAlgHom ↑x = ψ ↑x → w.embedding.toRatAlgHom = ψ

pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
⊢ ∀ (x : K), (Polynomial.map (algebraMap ℚ ℂ) (minpoly ℚ x)).Splits

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
⊢ ∀ (ψ : K →ₐ[ℚ] ℂ), w.embedding.toRatAlgHom ↑x = ψ ↑x → w.embedding.toRatAlgHom = ψ

K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
⊢ ∀ (x : K), (Polynomial.map (algebraMap ℚ ℂ) (minpoly ℚ x)).Splits
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a : K
x✝ : a ∈ (fun x ↦ (algebraMap (𝓞 K) K) ↑x) '' ↑(torsion K)
u : (𝓞 K)ˣ
h_tors : u ∈ ↑(torsion K)
h_ua : (fun x ↦ (algebraMap (𝓞 K) K) ↑x) u = a
φ : K →+* ℂ
⊢ ‖φ ((fun x ↦ (algebraMap (𝓞 K) K) ↑x) u)‖ ≤ 1

Tactic:
]

Goal state after:
case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a : K
x✝ : a ∈ (fun x ↦ (algebraMap (𝓞 K) K) ↑x) '' ↑(torsion K)
u : (𝓞 K)ˣ
h_tors : u ∈ ↑(torsion K)
h_ua : (fun x ↦ (algebraMap (𝓞 K) K) ↑x) u = a
φ : K →+* ℂ
⊢ ‖φ ((fun x ↦ (algebraMap (𝓞 K) K) ↑x) u)‖ ≤ 1
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding ↑x = ψ ↑x
h : 1 ≤ ‖ψ ↑x‖
⊢ w = mk ψ.toRingHom

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding ↑x = ψ ↑x
h : 1 ≤ ‖ψ ↑x‖
⊢ w = mk ψ.toRingHom
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝ : Field K
k : ℕ+
hc : (↑k).Coprime (torsionOrder K)
ζ : (𝓞 K)ˣ
⊢ ζ ^ ↑k = 1 ↔ ζ = 1

Tactic:
]

Goal state after:
K : Type u_1
inst✝ : Field K
k : ℕ+
hc : (↑k).Coprime (torsionOrder K)
ζ : (𝓞 K)ˣ
⊢ ζ ^ ↑k = 1 ↔ ζ = 1
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
x✝ : ch ≠ 1
hch2 : 2 ≤ ch
⊢ ca.natDegree + 1 ≤ ca.natDegree * ch

Tactic:
]

Goal state after:
k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
x✝ : ch ≠ 1
hch2 : 2 ≤ ch
⊢ ca.natDegree + 1 ≤ ca.natDegree * ch
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝ : Field K
k : ℕ+
hc : (↑k).Coprime (torsionOrder K)
ζ : (𝓞 K)ˣ
h : ζ = 1
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝ : Field K
k : ℕ+
hc : (↑k).Coprime (torsionOrder K)
ζ : (𝓞 K)ˣ
h : ζ ^ ↑k = 1
⊢ ∃ n, 0 < n ∧ ζ ^ n = 1

Tactic:
]

Goal state after:
K : Type u_1
inst✝ : Field K
k : ℕ+
hc : (↑k).Coprime (torsionOrder K)
ζ : (𝓞 K)ˣ
h : ζ ^ ↑k = 1
⊢ ∃ n, 0 < n ∧ ζ ^ n = 1
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
⊢ w.embedding.toRatAlgHom = ψ

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
⊢ w.embedding.toRatAlgHom = ψ
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: case refine_2
K : Type u_1
inst✝ : Field K
k : ℕ+
hc : (↑k).Coprime (torsionOrder K)
ζ : (𝓞 K)ˣ
h : ζ ^ ↑k = 1
hζ : ζ ∈ torsion K
⊢ orderOf ⟨ζ, hζ⟩ ∣ torsionOrder K

Tactic:
]

Goal state after:
case refine_2
K : Type u_1
inst✝ : Field K
k : ℕ+
hc : (↑k).Coprime (torsionOrder K)
ζ : (𝓞 K)ˣ
h : ζ ^ ↑k = 1
hζ : ζ ∈ torsion K
⊢ orderOf ⟨ζ, hζ⟩ ∣ torsionOrder K
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : (𝓞 K)ˣ
⊢ ζ ^ torsionOrder K = 1 ↔ ζ ∈ CommGroup.torsion (𝓞 K)ˣ

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : (𝓞 K)ˣ
⊢ ζ ^ torsionOrder K = 1 ↔ ζ ∈ CommGroup.torsion (𝓞 K)ˣ
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom
hw : w.IsReal
⊢ w.embedding.toRatAlgHom = ψ

Tactic:
]

Goal state after:
case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom
hw : w.IsReal
⊢ w.embedding.toRatAlgHom = ψ
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : (𝓞 K)ˣ
h : ζ ^ torsionOrder K = 1
⊢ ∃ n, 0 < n ∧ ζ ^ n = 1

Tactic:
]

Goal state after:
case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : (𝓞 K)ˣ
h : ζ ^ torsionOrder K = 1
⊢ ∃ n, 0 < n ∧ ζ ^ n = 1
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
x✝ : ch ≠ 1
hch2 : 2 ≤ ch
⊢ ca.natDegree + 1 ≤ ca.natDegree * 2

Tactic:
]

Goal state after:
k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
x✝ : ch ≠ 1
hch2 : 2 ≤ ch
⊢ ca.natDegree + 1 ≤ ca.natDegree * 2
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: case hle
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
φ : K →+* ℂ
⊢ Subgroup.map (Units.complexEmbedding φ) (rootsOfUnity (torsionOrder K) (𝓞 K)) ≤ rootsOfUnity (torsionOrder K) ℂ

Tactic:
]

Goal state after:
case hle
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
φ : K →+* ℂ
⊢ Subgroup.map (Units.complexEmbedding φ) (rootsOfUnity (torsionOrder K) (𝓞 K)) ≤ rootsOfUnity (torsionOrder K) ℂ
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
⊢ (starRingEnd ℂ) (w.embedding ↑x) = w.embedding ↑x

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
⊢ (starRingEnd ℂ) (w.embedding ↑x) = w.embedding ↑x
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: case hcard
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
φ : K →+* ℂ
e : ↥(Subgroup.map (Units.complexEmbedding φ) (torsion K)) ≃ ↥(torsion K) :=
  ((torsion K).equivMapOfInjective (Units.complexEmbedding φ) ⋯).symm.toEquiv
⊢ Nat.card ↥(torsion K) ≤ Nat.card ↥(torsion K)

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
this : orderOf ⟨-1, ⋯⟩ = 2
⊢ orderOf ⟨-1, ⋯⟩ ∣ torsionOrder K

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
this : orderOf ⟨-1, ⋯⟩ = 2
⊢ orderOf ⟨-1, ⋯⟩ ∣ torsionOrder K
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ 2 = 2

Tactic:
]

Goal state after: