mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: case pos
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDomain R
S : Type u_3
inst✝⁹ : CommRing S
inst✝⁸ : IsDomain S
inst✝⁷ : IsIntegrallyClosed R
inst✝⁶ : IsIntegrallyClosed S
inst✝⁵ : Algebra R S
inst✝⁴ : Module.Finite R S
inst✝³ : IsTorsionFree R S
inst✝² : IsDedekindDomain R
inst✝¹ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝ : p.IsPrime
hp : p = ⊥
⊢ ∃ s, (relNorm R) P = p ^ s

Tactic:
refine ⟨1, ?_⟩
    have : P.LiesOver ⊥ := hp ▸ hPp
    rw [hp, eq_bot_of_liesOver_bot R P, relNorm_bot, bot_pow (one_ne_zero)]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: case neg
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDomain R
S : Type u_3
inst✝⁹ : CommRing S
inst✝⁸ : IsDomain S
inst✝⁷ : IsIntegrallyClosed R
inst✝⁶ : IsIntegrallyClosed S
inst✝⁵ : Algebra R S
inst✝⁴ : Module.Finite R S
inst✝³ : IsTorsionFree R S
inst✝² : IsDedekindDomain R
inst✝¹ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝ : p.IsPrime
hp : ¬p = ⊥
h : (relNorm R) (map (algebraMap R S) p) ≤ (relNorm R) P
⊢ ∃ s, (relNorm R) P = p ^ s

Tactic:
rw [relNorm_algebraMap S, ← dvd_iff_le, dvd_prime_pow (prime_of_isPrime hp inferInstance)] at h

Goal state after:
case neg
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDomain R
S : Type u_3
inst✝⁹ : CommRing S
inst✝⁸ : IsDomain S
inst✝⁷ : IsIntegrallyClosed R
inst✝⁶ : IsIntegrallyClosed S
inst✝⁵ : Algebra R S
inst✝⁴ : Module.Finite R S
inst✝³ : IsTorsionFree R S
inst✝² : IsDedekindDomain R
inst✝¹ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝ : p.IsPrime
hp : ¬p = ⊥
h : ∃ i ≤ finrank R S, Associated ((relNorm R) P) (p ^ i)
⊢ ∃ s, (relNorm R) P = p ^ s
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (R ∙ p)
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
⊢ P.natDegree = B.dim

Tactic:
← deg_K_P,

Goal state after:
R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (R ∙ p)
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
⊢ P.natDegree = (minpoly K B.gen).natDegree
Module: Mathlib.RingTheory.Prime

Goal state before:
pp: case insert
R : Type u_1
inst✝² : CommMonoidWithZero R
inst✝¹ : IsCancelMulZero R
α : Type u_2
inst✝ : DecidableEq α
p : α → R
i : α
s : Finset α
his : i ∉ s
ih :
  ∀ {x y a : R},
    (∀ i ∈ s, Prime (p i)) →
      x * y = a * ∏ i ∈ s, p i →
        ∃ t u b c, t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i
x y a : R
hp : ∀ i_1 ∈ insert i s, Prime (p i_1)
hx : x * y = a * ∏ i ∈ insert i s, p i
⊢ ∃ t u b c, t ∪ u = insert i s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i

Tactic:
prod_insert his,

Goal state after:
case insert
R : Type u_1
inst✝² : CommMonoidWithZero R
inst✝¹ : IsCancelMulZero R
α : Type u_2
inst✝ : DecidableEq α
p : α → R
i : α
s : Finset α
his : i ∉ s
ih :
  ∀ {x y a : R},
    (∀ i ∈ s, Prime (p i)) →
      x * y = a * ∏ i ∈ s, p i →
        ∃ t u b c, t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i
x y a : R
hp : ∀ i_1 ∈ insert i s, Prime (p i_1)
hx : x * y = a * (p i * ∏ x ∈ s, p x)
⊢ ∃ t u b c, t ∪ u = insert i s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (R ∙ p)
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ R[B.gen] ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr :
  (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
i : ℕ
hi : i ∈ (range (Q.natDegree + 1)).erase 0
⊢ Q.coeff i • B.gen ^ (i + n) = p • Q.coeff i • f (i + n)

Tactic:
← (hf _ (aux i hi)).2,

Goal state after:
R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (R ∙ p)
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ R[B.gen] ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr :
  (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
i : ℕ
hi : i ∈ (range (Q.natDegree + 1)).erase 0
⊢ Q.coeff i • ((algebraMap R L) p * f (i + n)) = p • Q.coeff i • f (i + n)
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
this : NumberField K
inst✝ : IsCyclotomicExtension {2 ^ (0 + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (0 + 1))
⊢ 0 < 2 ^ (0 + 1)

Tactic:
simp only [zero_add, pow_one, Nat.ofNat_pos]

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsIntegralClosure B A L
inst✝² : Algebra.IsSeparable K L
I : Submodule B L
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι K L
hb : ∀ (i : ι), IsIntegral A (b i)
a x : L
ha : a ∈ I
hx : x ∈ Iᵛ
hinv : IsUnit (traceMatrix K ⇑b).det
H :
  (traceMatrix K ⇑b *ᵥ (traceMatrix K ⇑b).cramer fun i ↦ (Algebra.trace K L) (x * a * b i)) =
    (traceMatrix K ⇑b).det • fun i ↦ (Algebra.trace K L) (x * a * b i)
⊢ Function.Injective (traceMatrix K ⇑b).mulVec

Tactic:
mulVec_injective_iff_isUnit,

Goal state after:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsIntegralClosure B A L
inst✝² : Algebra.IsSeparable K L
I : Submodule B L
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι K L
hb : ∀ (i : ι), IsIntegral A (b i)
a x : L
ha : a ∈ I
hx : x ∈ Iᵛ
hinv : IsUnit (traceMatrix K ⇑b).det
H :
  (traceMatrix K ⇑b *ᵥ (traceMatrix K ⇑b).cramer fun i ↦ (Algebra.trace K L) (x * a * b i)) =
    (traceMatrix K ⇑b).det • fun i ↦ (Algebra.trace K L) (x * a * b i)
⊢ IsUnit (traceMatrix K ⇑b)
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
⊢ ↑(dual A K 1) = 1ᵛ

Tactic:
rw [← coe_one, coe_dual]
  exact one_ne_zero

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝¹ : Module.Finite K L
P : R[X] := minpoly R B.gen
hei : P.IsEisensteinAt (R ∙ p)
hndiv : ¬p ^ 2 ∣ P.coeff 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
this✝ : Module.IsTorsionFree R L
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQ : (aeval B.gen) Q = p • z
hQzero : ¬Q = 0
j : ℕ
hj : j + 1 ∈ range (Q.natDegree + 1)
H : j + 1 + 1 ≤ P.natDegree
Hj : Q.natDegree + 1 = j + 1 + (Q.natDegree - j)
g : ℕ → R
hg : ∀ k ∈ range (j + 1), Q.coeff k • B.gen ^ k = (algebraMap R L) p * g k • B.gen ^ k
f : ℕ → L
hf :
  ∀ (i : ℕ),
    (Polynomial.map (algebraMap R L) (minpoly R B.gen)).natDegree ≤ i →
      f i ∈ R[B.gen] ∧ (algebraMap R L) p * f i = B.gen ^ i
hf₁ :
  ∀ k ∈ (range (Q.natDegree - j)).erase 0,
    Q.coeff (j + 1 + k) • B.gen ^ (j + 1 + k) * B.gen ^ (P.natDegree - (j + 2)) =
      (algebraMap R L) p * Q.coeff (j + 1 + k) • f (k + P.natDegree - 1)
this : p ^ n.succ ∣ Q.coeff j.succ ^ n.succ * (minpoly R B.gen).coeff 0 ^ (j.succ + (P.natDegree - (j + 2)))
⊢ n = j.succ + (P.natDegree - (j + 2))

Tactic:
rw [Nat.succ_eq_add_one, add_assoc, ← Nat.add_sub_assoc H, add_comm (j + 1),
        Nat.add_sub_add_left, ← Nat.add_sub_assoc, Nat.add_sub_add_left, hP, ←
        (minpoly.monic hBint).natDegree_map (algebraMap R K), ←
        minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, natDegree_minpoly, hn, Nat.sub_one,
        Nat.pred_succ]

Goal state after:
case h
R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝¹ : Module.Finite K L
P : R[X] := minpoly R B.gen
hei : P.IsEisensteinAt (R ∙ p)
hndiv : ¬p ^ 2 ∣ P.coeff 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
this✝ : Module.IsTorsionFree R L
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQ : (aeval B.gen) Q = p • z
hQzero : ¬Q = 0
j : ℕ
hj : j + 1 ∈ range (Q.natDegree + 1)
H : j + 1 + 1 ≤ P.natDegree
Hj : Q.natDegree + 1 = j + 1 + (Q.natDegree - j)
g : ℕ → R
hg : ∀ k ∈ range (j + 1), Q.coeff k • B.gen ^ k = (algebraMap R L) p * g k • B.gen ^ k
f : ℕ → L
hf :
  ∀ (i : ℕ),
    (Polynomial.map (algebraMap R L) (minpoly R B.gen)).natDegree ≤ i →
      f i ∈ R[B.gen] ∧ (algebraMap R L) p * f i = B.gen ^ i
hf₁ :
  ∀ k ∈ (range (Q.natDegree - j)).erase 0,
    Q.coeff (j + 1 + k) • B.gen ^ (j + 1 + k) * B.gen ^ (P.natDegree - (j + 2)) =
      (algebraMap R L) p * Q.coeff (j + 1 + k) • f (k + P.natDegree - 1)
this : p ^ n.succ ∣ Q.coeff j.succ ^ n.succ * (minpoly R B.gen).coeff 0 ^ (j.succ + (P.natDegree - (j + 2)))
⊢ j + 1 ≤ P.natDegree
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
⊢ dual A K 1 * (dual A K 1)⁻¹ * I = I

Tactic:
mul_inv_cancel₀,

Goal state after:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
⊢ 1 * I = I

A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
⊢ dual A K 1 ≠ 0
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.PID

Goal state before:
pp: K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {5} ℚ K
this : Fact (Nat.Prime 5)
⊢ ↑|(-1) ^ ((5 - 1) / 2) * ↑5 ^ (5 - 2)| <
    (2 * (π / 4) ^ nrComplexPlaces K * (↑(Module.finrank ℚ K) ^ Module.finrank ℚ K / ↑(Module.finrank ℚ K)!)) ^ 2

Tactic:
IsCyclotomicExtension.finrank (n := 5) K
    (irreducible_rat (by simp)),

Goal state after:
K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {5} ℚ K
this : Fact (Nat.Prime 5)
⊢ ↑|(-1) ^ ((5 - 1) / 2) * ↑5 ^ (5 - 2)| < (2 * (π / 4) ^ nrComplexPlaces K * (↑(φ 5) ^ φ 5 / ↑(φ 5)!)) ^ 2
Module: Mathlib.NumberTheory.NumberField.Units.DirichletTheorem

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ finrank ℤ (Additive ((𝓞 K)ˣ ⧸ torsion K)) = rank K

Tactic:
← LinearEquiv.finrank_eq (logEmbeddingEquiv K).symm,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ finrank ℤ ↥(unitLattice K) = rank K
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: case inl
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
x : 𝓞 K
hx : S'.a ^ 3 - 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 + 1 = λ ^ 4 * y
⊢ λ ^ 4 ∣ S'.c ^ 3

Tactic:
refine ⟨S'.u⁻¹ * (x + y), ?_⟩
    symm
    calc _ = S'.u⁻¹ * (λ ^ 4 * x + λ ^ 4 * y) := by ring
    _ = S'.u⁻¹ * (S'.a ^ 3 + S'.b ^ 3) := by rw [← hx, ← hy]; ring
    _ = S'.u⁻¹ * (S'.u * S'.c ^ 3) := by rw [S'.H]
    _ = S'.c ^ 3 := by simp

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b p : ℕ
hb : 0 < b
hp : 1 ≤ p
hi_bsquared : 1 ≤ b ^ 2
⊢ b ^ (2 * p) - 1 - (b ^ 2 - 1) = b ^ (2 * p) - (1 + (b ^ 2 - 1))

Tactic:
rw [Nat.sub_sub]

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: case inr
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
x : 𝓞 K
hx : S'.a ^ 3 + 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 - 1 = λ ^ 4 * y
⊢ λ ^ 4 ∣ S'.c ^ 3

Tactic:
refine ⟨S'.u⁻¹ * (x + y), ?_⟩
    symm
    calc _ = S'.u⁻¹ * (λ ^ 4 * x + λ ^ 4 * y) := by ring
    _ = S'.u⁻¹ * (S'.a ^ 3 + S'.b ^ 3) := by rw [← hx, ← hy]; ring
    _ = S'.u⁻¹ * (S'.u * S'.c ^ 3) := by rw [S'.H]
    _ = S'.c ^ 3 := by simp

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: m n : ℕ
cop : m.Coprime n
hm : 1 < m
hn : 1 < n
⊢ FrobeniusNumber (m * n - m - n) {m, n}

Tactic:
simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push Not
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    simp only [Nat.sub_sub, smul_eq_mul] at h
    zify [hmn] at h ⊢
    rw [← sub_eq_zero] at h ⊢
    rw [← h]
    ring
  · intro k hk
    dsimp at hk
    contrapose! hk
    obtain ⟨a, b, h⟩ := exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le m n k
      (by simp [cop.gcd_eq_one]) (by grind [pred_mul, mul_pred, pred_eq_sub_one])
    exact ⟨a, b, succ_inj.mp (congrArg succ h)⟩

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: case refine_1
m n✝¹ : ℕ
s✝ t✝ : Set ℕ
n✝ : ℕ
s : Submodule ℕ ℕ
t : Finset ℕ
n : ℕ
hts : ↑t ⊆ ↑s
hn : ∀ m ≥ n, setGcd ↑s ∣ m → m ∈ Ideal.span ↑t
⊢ ↑(t ∪ {m ∈ Finset.range n | m ∈ s}) ⊆ ↑s

pp: case refine_2
m✝ n✝¹ : ℕ
s✝ t✝ : Set ℕ
n✝ : ℕ
s : Submodule ℕ ℕ
t : Finset ℕ
n : ℕ
hts : ↑t ⊆ ↑s
hn : ∀ m ≥ n, setGcd ↑s ∣ m → m ∈ Ideal.span ↑t
m : ℕ
hm : m ∈ s
⊢ m ∈ Submodule.span ℕ ↑(t ∪ {m ∈ Finset.range n | m ∈ s})

Tactic:
· simpa using ⟨hts, fun _ ↦ And.right⟩

Goal state after:
case refine_2
m✝ n✝¹ : ℕ
s✝ t✝ : Set ℕ
n✝ : ℕ
s : Submodule ℕ ℕ
t : Finset ℕ
n : ℕ
hts : ↑t ⊆ ↑s
hn : ∀ m ≥ n, setGcd ↑s ∣ m → m ∈ Ideal.span ↑t
m : ℕ
hm : m ∈ s
⊢ m ∈ Submodule.span ℕ ↑(t ∪ {m ∈ Finset.range n | m ∈ s})
Module: Mathlib.NumberTheory.Harmonic.Bounds

Goal state before:
pp: n : ℕ
⊢ harmonic n = ∑ i ∈ Finset.Icc 1 n, (↑i)⁻¹

Tactic:
rw [harmonic, Finset.range_eq_Ico, Finset.sum_Ico_add' (fun (i : ℕ) ↦ (i : ℚ)⁻¹) 0 n (c := 1)]
  simp only [Finset.Ico_add_one_right_eq_Icc]

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.Bounds

Goal state before:
pp: n : ℕ
⊢ ↑(harmonic n) ≤ 1 + Real.log ↑n

Tactic:
by_cases hn0 : n = 0

Goal state after:
case pos
n : ℕ
hn0 : n = 0
⊢ ↑(harmonic n) ≤ 1 + Real.log ↑n

case neg
n : ℕ
hn0 : ¬n = 0
⊢ ↑(harmonic n) ≤ 1 + Real.log ↑n
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : λ ^ 2 ∣ λ ^ 4
X : 𝓞 K
hX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X
⊢ ∃ n, λ ^ 2 ∣ ↑S.u₄ - ↑n

Tactic:
rcases lambda_pow_four_dvd_cube_sub_one_or_add_one_of_lambda_not_dvd hζ S.lambda_not_dvd_Y with
    HY | HY <;> rcases lambda_pow_four_dvd_cube_sub_one_or_add_one_of_lambda_not_dvd
      hζ S.lambda_not_dvd_Z with HZ | HZ

Goal state after:
case inl.inl
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : λ ^ 2 ∣ λ ^ 4
X : 𝓞 K
hX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X
HY : λ ^ 4 ∣ S.Y ^ 3 - 1
HZ : λ ^ 4 ∣ S.Z ^ 3 - 1
⊢ ∃ n, λ ^ 2 ∣ ↑S.u₄ - ↑n

case inl.inr
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : λ ^ 2 ∣ λ ^ 4
X : 𝓞 K
hX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X
HY : λ ^ 4 ∣ S.Y ^ 3 - 1
HZ : λ ^ 4 ∣ S.Z ^ 3 + 1
⊢ ∃ n, λ ^ 2 ∣ ↑S.u₄ - ↑n

case inr.inl
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : λ ^ 2 ∣ λ ^ 4
X : 𝓞 K
hX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X
HY : λ ^ 4 ∣ S.Y ^ 3 + 1
HZ : λ ^ 4 ∣ S.Z ^ 3 - 1
⊢ ∃ n, λ ^ 2 ∣ ↑S.u₄ - ↑n

case inr.inr
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : λ ^ 2 ∣ λ ^ 4
X : 𝓞 K
hX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X
HY : λ ^ 4 ∣ S.Y ^ 3 + 1
HZ : λ ^ 4 ∣ S.Z ^ 3 + 1
⊢ ∃ n, λ ^ 2 ∣ ↑S.u₄ - ↑n
Module: Mathlib.NumberTheory.Height.MvPolynomial

Goal state before:
pp: case inr
K : Type u_1
inst✝² : Field K
ι : Type u_2
ι' : Type u_3
inst✝¹ : Fintype ι
inst✝ : Finite ι'
v : AbsoluteValue K ℝ
A : ι' × ι → K
x : ι → K
h✝ : Nonempty ι'
⊢ ⨆ j, v (∑ i, A (j, i) * x i) ≤ (↑(Nat.card ι) * ⨆ ji, v (A ji)) * ⨆ i, v (x i)

Tactic:
refine ciSup_le fun j ↦ ?_

Goal state after:
case inr
K : Type u_1
inst✝² : Field K
ι : Type u_2
ι' : Type u_3
inst✝¹ : Fintype ι
inst✝ : Finite ι'
v : AbsoluteValue K ℝ
A : ι' × ι → K
x : ι → K
h✝ : Nonempty ι'
j : ι'
⊢ v (∑ i, A (j, i) * x i) ≤ (↑(Nat.card ι) * ⨆ ji, v (A ji)) * ⨆ i, v (x i)
Module: Mathlib.NumberTheory.Height.Basic

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : AdmissibleAbsValues K
ι : Type u_2
ι' : Type u_3
inst✝¹ : Finite ι
inst✝ : Finite ι'
x : ι → K
hx : x ≠ 0
i : ι
hi : x i ≠ 0 i
this : Nonempty ι
hx' : Sum.elim x 0 ≠ 0
v : AbsoluteValue K ℝ
⊢ ⨆ j, v (Sum.elim x 0 j) ≤ ⨆ i, v (x i)

Tactic:
refine ciSup_le fun j ↦ ?_

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : AdmissibleAbsValues K
ι : Type u_2
ι' : Type u_3
inst✝¹ : Finite ι
inst✝ : Finite ι'
x : ι → K
hx : x ≠ 0
i : ι
hi : x i ≠ 0 i
this : Nonempty ι
hx' : Sum.elim x 0 ≠ 0
v : AbsoluteValue K ℝ
j : ι ⊕ ι'
⊢ v (Sum.elim x 0 j) ≤ ⨆ i, v (x i)
Module: Mathlib.NumberTheory.Height.MvPolynomial

Goal state before:
pp: case inr.inr
K : Type u_1
inst✝² : Field K
ι : Type u_2
ι' : Type u_3
inst✝¹ : Fintype ι
inst✝ : Finite ι'
v : AbsoluteValue K ℝ
hv : IsNonarchimedean ⇑v
A : ι' × ι → K
x : ι → K
h✝¹ : Nonempty ι
h✝ : Nonempty ι'
⊢ ⨆ j, v (∑ i, A (j, i) * x i) ≤ (⨆ ji, v (A ji)) * ⨆ i, v (x i)

Tactic:
refine ciSup_le fun j ↦ ?_

Goal state after:
case inr.inr
K : Type u_1
inst✝² : Field K
ι : Type u_2
ι' : Type u_3
inst✝¹ : Fintype ι
inst✝ : Finite ι'
v : AbsoluteValue K ℝ
hv : IsNonarchimedean ⇑v
A : ι' × ι → K
x : ι → K
h✝¹ : Nonempty ι
h✝ : Nonempty ι'
j : ι'
⊢ v (∑ i, A (j, i) * x i) ≤ (⨆ ji, v (A ji)) * ⨆ i, v (x i)
Module: Mathlib.NumberTheory.Height.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : AdmissibleAbsValues K
x : K
n : ℕ
⊢ mulHeight₁ (x ^ n) = mulHeight₁ x ^ n

Tactic:
simp only [mulHeight₁_eq_mulHeight, ← mulHeight_pow _ n]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : AdmissibleAbsValues K
x : K
n : ℕ
⊢ mulHeight ![x ^ n, 1] = mulHeight (![x, 1] ^ n)
Module: Mathlib.NumberTheory.Height.MvPolynomial

Goal state before:
pp: K : Type u_4
inst✝¹ : Field K
ι : Type u_5
inst✝ : Finite ι
v : AbsoluteValue K ℝ
p : MvPolynomial ι K
N : ℕ
hp : p.IsHomogeneous N
x : ι → K
⊢ v (∑ d ∈ p.support, coeff d p * ∏ i ∈ d.support, x i ^ d i) ≤ (∑ m ∈ p.support, v (coeff m p)) * (⨆ i, v (x i)) ^ N

Tactic:
Finset.sum_mul

Goal state after:
K : Type u_4
inst✝¹ : Field K
ι : Type u_5
inst✝ : Finite ι
v : AbsoluteValue K ℝ
p : MvPolynomial ι K
N : ℕ
hp : p.IsHomogeneous N
x : ι → K
⊢ v (∑ d ∈ p.support, coeff d p * ∏ i ∈ d.support, x i ^ d i) ≤ ∑ i ∈ p.support, v (coeff i p) * (⨆ i, v (x i)) ^ N