mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: R : Type u_1
S : Type u_2
G : Type u_3
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
inst✝¹¹ : Group G
inst✝¹⁰ : MulSemiringAction G S
inst✝⁹ : IsGaloisGroup G R S
inst✝⁸ : Finite G
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : Module.Finite R S
inst✝⁴ : Flat R S
p : Ideal R
P : Ideal S
inst✝³ : P.LiesOver p
inst✝² : p.IsPrime
inst✝¹ : P.IsPrime
inst✝ : PerfectField p.ResidueField
H : (p.primesOver S).ncard * Nat.card ↥(inertia G P) * P.inertiaDeg R = Nat.card G
⊢ Nat.card ↥(inertia G P) = p.ramificationIdxIn S

Tactic:
rw [← inertiaDegIn_eq_inertiaDeg p P G] at H

Goal state after:
R : Type u_1
S : Type u_2
G : Type u_3
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
inst✝¹¹ : Group G
inst✝¹⁰ : MulSemiringAction G S
inst✝⁹ : IsGaloisGroup G R S
inst✝⁸ : Finite G
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : Module.Finite R S
inst✝⁴ : Flat R S
p : Ideal R
P : Ideal S
inst✝³ : P.LiesOver p
inst✝² : p.IsPrime
inst✝¹ : P.IsPrime
inst✝ : PerfectField p.ResidueField
H : (p.primesOver S).ncard * Nat.card ↥(inertia G P) * p.inertiaDegIn S = Nat.card G
⊢ Nat.card ↥(inertia G P) = p.ramificationIdxIn S
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: case refine_3
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsReal}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLTFactor ↥K) * ↑B
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < ↑(max B 1)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
⊢ (aeval ↑↑x) (minpoly ℤ ↑x) = 0

Tactic:
exact (aeval_algebraMap_eq_zero_iff A (x : K) _).mpr (minpoly.aeval ℤ (x : K))

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Unramified

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsDomain R
inst✝³ : IsDomain S
inst✝² : Module.IsTorsionFree R S
inst✝¹ : CharZero R
inst✝ : Algebra.IsIntegral R S
this✝ : IsFractionRing S (Localization.AtPrime ⊥)
this : Field (Localization.AtPrime ⊥) := IsFractionRing.toField S
⊢ Function.Injective ⇑((algebraMap S (Localization.AtPrime ⊥)).comp (algebraMap R S))

Tactic:
exact (IsFractionRing.injective S _).comp (FaithfulSMul.algebraMap_injective R S)

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: case refine_1
A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
x✝ : L
h : x✝ ∈ ⇑(IsLocalization.map L f hf) '' ↑(I * J)
⊢ x✝ ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))

Tactic:
rcases h with ⟨x, hx, rfl⟩
    replace hx : x ∈ (I : Submodule A K) * (J : Submodule A K) := coe_mul I J ▸ hx
    rw [Submodule.mul_eq_span_mul_set] at hx
    refine span_induction (fun y hy ↦ ?_) (by simp) (fun y z _ _ hy hz ↦ ?_)
      (fun a y _ hy ↦ ?_) hx
    · rcases Set.mem_mul.mp hy with ⟨i, hi, j, hj, rfl⟩
      exact subset_span <| Set.mem_mul.mpr
        ⟨map_f i, ⟨i, hi, by simp⟩, map_f j, ⟨j, hj, by simp⟩, by simp⟩
    · exact map_add map_f y z ▸ Submodule.add_mem _ hy hz
    · rw [Algebra.smul_def, map_mul, map_eq, ← Algebra.smul_def]
      exact smul_mem _ (f a) hy

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

Goal state before:
pp: case refine_2
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsComplex}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLT'Factor ↥K) * ↑(boundOfDiscBdd N)
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < √(1 + ↑(boundOfDiscBdd N) ^ 2)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ (minpoly ℤ ↑x).coeff i ∈ Set.Icc (-↑C) ↑C

Tactic:
rw [Set.mem_Icc, ← abs_le, ← @Int.cast_le ℝ]
      refine (Eq.trans_le ?_ <| Embeddings.coeff_bdd_of_norm_le
          ((le_iff_le (x : K) _).mp (fun w ↦ le_of_lt (hx₂ w))) i).trans ?_
      · rw [minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, coeff_map, eq_intCast,
          Int.norm_cast_rat, Int.norm_eq_abs, Int.cast_abs]
      · refine le_trans ?_ (Nat.le_ceil _)
        rw [val_eq_coe, NNReal.coe_mul, NNReal.coe_pow, NNReal.coe_max, NNReal.coe_one,
          Real.coe_sqrt, NNReal.coe_add 1, NNReal.coe_one, NNReal.coe_pow]
        gcongr
        · exact le_max_right _ 1
        · exact rank_le_rankOfDiscrBdd hK₂
        · rw [NNReal.coe_natCast, Nat.cast_le]
          exact (Nat.choose_le_choose _ (rank_le_rankOfDiscrBdd hK₂)).trans
            (Nat.choose_le_middle _ _)

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

Goal state before:
pp: case refine_3
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsComplex}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLT'Factor ↥K) * ↑(boundOfDiscBdd N)
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < √(1 + ↑(boundOfDiscBdd N) ^ 2)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
⊢ (aeval ↑↑x) (minpoly ℤ ↑x) = 0

Tactic:
exact (aeval_algebraMap_eq_zero_iff A (x : K) _).mpr (minpoly.aeval ℤ (x : K))

Goal state after:
Module: Mathlib.RingTheory.Ideal.Int

Goal state before:
pp: case inl
S : Type u_2
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
I : Ideal S
h✝ : Finite (S ⧸ I)
this : Fintype (S ⧸ I)
h_main : ∀ {d : ℕ}, ↑d ∈ I ↔ ∀ (x : S ⧸ I), d • x = 0
⊢ AddMonoid.exponent (S ⧸ I) ∣ Fintype.card (S ⧸ I)

Tactic:
exact AddGroup.exponent_dvd_card (G := S ⧸ I)

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹⁸ : CommRing R
inst✝¹⁷ : CommRing S
inst✝¹⁶ : IsDomain R
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Algebra R S
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹³ : CommRing Rₘ
inst✝¹² : CommRing Sₘ
inst✝¹¹ : Algebra R Rₘ
inst✝¹⁰ : IsTorsionFree R S
inst✝⁹ : Algebra.IsSeparable K L
M : Submonoid R
inst✝⁸ : IsLocalization M Rₘ
inst✝⁷ : Algebra Rₘ Sₘ
inst✝⁶ : Algebra S Sₘ
inst✝⁵ : Algebra R Sₘ
inst✝⁴ : IsScalarTower R Rₘ Sₘ
inst✝³ : IsScalarTower R S Sₘ
inst✝² : IsLocalization (algebraMapSubmonoid S M) Sₘ
inst✝¹ : Algebra (FractionRing Rₘ) (FractionRing Sₘ)
inst✝ : IsScalarTower Rₘ (FractionRing Rₘ) (FractionRing Sₘ)
hM : M ≤ R⁰
⊢ Algebra.IsSeparable (FractionRing Rₘ) (FractionRing Sₘ)

Tactic:
let M' := algebraMapSubmonoid S M
  have hM' : algebraMapSubmonoid S M ≤ S⁰ := algebraMapSubmonoid_le_nonZeroDivisors_of_faithfulSMul
    _ hM
  let f₁ : Rₘ →+* K := map _ (T := R⁰) (RingHom.id R) hM
  let f₂ : Sₘ →+* L := map _ (T := S⁰) (RingHom.id S) hM'
  algebraize [f₁, f₂]
  have := localization_isScalarTower_of_submonoid_le Rₘ K _ _ hM
  have := localization_isScalarTower_of_submonoid_le Sₘ L _ _ hM'
  have := isFractionRing_of_isDomain_of_isLocalization M Rₘ K
  have := isFractionRing_of_isDomain_of_isLocalization M' Sₘ L
  have : IsDomain Rₘ := isDomain_of_le_nonZeroDivisors _ hM
  apply Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv Rₘ K).symm.toRingEquiv
    (FractionRing.algEquiv Sₘ L).symm.toRingEquiv
  apply ringHom_ext R⁰
  ext
  simp only [RingHom.coe_comp,
      RingHom.coe_coe, Function.comp_apply, ← algebraMap_apply]
  rw [algebraMap_apply R Rₘ (FractionRing R), AlgEquiv.coe_ringEquiv, AlgEquiv.commutes,
    algebraMap_apply R S L, algebraMap_apply S Sₘ L, AlgEquiv.coe_ringEquiv, AlgEquiv.commutes]
  simp only [← algebraMap_apply]
  rw [algebraMap_apply R Rₘ (FractionRing Rₘ), ← algebraMap_apply Rₘ, ← algebraMap_apply]

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

Goal state before:
pp: case refine_2.zero
p : ℕ
hp : Fact (Nat.Prime p)
i : ℕ
hi : i < ((cyclotomic p ℤ).comp (X + 1)).natDegree
⊢ ((cyclotomic p ℤ).comp (X + 1)).coeff i ∈ ℤ ∙ ↑p

Tactic:
exact (cyclotomic_comp_X_add_one_isEisensteinAt p).mem hi

Goal state after:
Module: Mathlib.RingTheory.Prime

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : CommRing α
inst✝ : LinearOrder α
p : α
hp : Prime p
h : |p| = p
⊢ Prime p

Tactic:
exact hp

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
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))
⊢ p • z * B.gen ^ n = p • (z * B.gen ^ n)

Tactic:
Algebra.smul_mul_assoc

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))
⊢ p • (z * B.gen ^ n) = p • (z * B.gen ^ n)
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
⊢ IsIntegralClosure (CyclotomicRing p ℤ ℚ) ℤ (CyclotomicField p ℚ)

Tactic:
rw [← pow_one p]
  exact cyclotomicRing_isIntegralClosure_of_prime_pow

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

Goal state before:
pp: K : Type u
inst✝¹ : Field K
ζ : K
k : ℕ
inst✝ : NeZero k
hζ : IsPrimitiveRoot ζ k
⊢ IsPrimitiveRoot ((algebraMap (𝓞 K) K) hζ.toInteger) k

Tactic:
by exact hζ

Goal state after:
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:
by simp only [zero_add, pow_one, Nat.ofNat_pos]

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

Goal state before:
pp: k : ℕ
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
this : NumberField K
⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger ^ 2 ^ k - 1)) = (algebraMap ℤ ℚ) ((-2) ^ 2 ^ k)

Tactic:
simp [hζ.norm_pow_sub_one_two (cyclotomic.irreducible_rat (pow_pos (by decide) _))]

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

Goal state before:
pp: p k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
htwo : p ^ (k + 1) ≠ 2
⊢ ¬∃ n, ↑p ∣ hζ.toInteger - ↑n

Tactic:
by
  intro ⟨n, x, h⟩
  -- Let `pB` be the power basis of `𝓞 K` given by powers of `ζ`.
  let pB := hζ.integralPowerBasisOfPrimePow
  have hdim : pB.dim = p ^ k * (↑p - 1) := by
    simp [integralPowerBasisOfPrimePow_dim, pB, Nat.totient_prime_pow hp.1 (Nat.zero_lt_succ k)]
  replace hdim : 1 < pB.dim := by
    rw [Nat.one_lt_iff_ne_zero_and_ne_one, hdim]
    refine ⟨by simp only [ne_eq, mul_eq_zero, NeZero.ne _, Nat.sub_eq_zero_iff_le, false_or,
      not_le, Nat.Prime.one_lt hp.out], ne_of_gt ?_⟩
    by_cases hk : k = 0
    · simp only [hk, zero_add, pow_one, pow_zero, one_mul, Nat.lt_sub_iff_add_lt,
        Nat.reduceAdd] at htwo ⊢
      exact htwo.symm.lt_of_le hp.1.two_le
    · exact one_lt_mul_of_lt_of_le (one_lt_pow₀ hp.1.one_lt hk)
        (have := Nat.Prime.two_le hp.out; by lia)
  rw [sub_eq_iff_eq_add] at h
  -- We are assuming that `ζ = n + p * x` for some integer `n` and `x : 𝓞 K`. Looking at the
  -- coordinates in the base `pB`, we obtain that `1` is a multiple of `p`, contradiction.
  replace h := pB.basis.ext_elem_iff.1 h ⟨1, hdim⟩
  have := pB.basis_eq_pow ⟨1, hdim⟩
  rw [hζ.integralPowerBasisOfPrimePow_gen] at this
  simp only [PowerBasis.coe_basis, pow_one] at this
  rw [← this, show pB.gen = pB.gen ^ (⟨1, hdim⟩ : Fin pB.dim).1 by simp, ← pB.basis_eq_pow,
    pB.basis.repr_self_apply] at h
  simp only [↓reduceIte, map_add, Finsupp.coe_add, Pi.add_apply] at h
  rw [show (p : 𝓞 K) * x = (p : ℤ) • x by simp, ← pB.basis.coord_apply,
    map_smul, ← zsmul_one, ← pB.basis.coord_apply, map_smul,
    show 1 = pB.gen ^ (⟨0, by lia⟩ : Fin pB.dim).1 by simp, ← pB.basis_eq_pow,
    pB.basis.coord_apply, pB.basis.coord_apply, pB.basis.repr_self_apply] at h
  simp only [smul_eq_mul, Fin.mk.injEq, zero_ne_one, ↓reduceIte, mul_zero, add_zero] at h
  exact (Int.prime_iff_natAbs_prime.2 (by simp [hp.1])).not_dvd_one ⟨_, h⟩

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

Goal state before:
pp: 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✝ : 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
k : ℕ
hk : k ∈ (range (Q.natDegree - j)).erase 0
⊢ 1 ≤ k

Tactic:
refine one_le_iff_ne_zero.2 fun h => ?_
        rw [h] at hk
        simp at hk

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

Goal state before:
pp: 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✝ : 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
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)
hQ :
  (Algebra.norm K) ((algebraMap R L) (Q.coeff (j + 1)) * B.gen ^ (j + 1 + (P.natDegree - (j + 2)))) =
    (Algebra.norm K)
      (p • z * B.gen ^ (P.natDegree - (j + 2)) -
        (algebraMap R L) p *
          (∑ i ∈ (range (Q.natDegree - j)).erase 0, Q.coeff (j + 1 + i) • f (i + P.natDegree - 1) +
            ∑ i ∈ range (j + 1), g i • B.gen ^ i * B.gen ^ (P.natDegree - (j + 2))))
k : ℕ
hk : k ∈ (range (Q.natDegree - j)).erase 0
⊢ 1 ≤ k

Tactic:
refine one_le_iff_ne_zero.2 fun h => ?_
        rw [h] at hk
        simp at hk

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

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u
inst✝¹¹ : CommRing A
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra A L
inst✝⁵ : IsScalarTower A K L
inst✝⁴ : IsDomain A
inst✝³ : IsFractionRing A K
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : IsIntegrallyClosed A
x : L
hx : K[x] = ⊤
hAx : IsIntegral A x
hKx : IsIntegral K x
pb : PowerBasis K L
pbgen : pb.gen = x
hnondeg : (traceForm K L).Nondegenerate
hpb :
  ⇑((traceForm K L).dualBasis hnondeg pb.basis) = fun x ↦
    (minpolyDiv K pb.gen).coeff ↑x / (aeval pb.gen) (derivative (minpoly K pb.gen))
this : Subalgebra.toSubmodule A[x] = span A (Set.range ⇑pb.basis)
⊢ span A (Set.range ⇑((traceForm K L).dualBasis hnondeg pb.basis)) =
    span A (((aeval x) (derivative (minpoly K x)))⁻¹ • Set.range (minpolyDiv A x).coeff)

Tactic:
hpb

Goal state after:
A : Type u_1
K : Type u_2
L : Type u
inst✝¹¹ : CommRing A
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra A L
inst✝⁵ : IsScalarTower A K L
inst✝⁴ : IsDomain A
inst✝³ : IsFractionRing A K
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : IsIntegrallyClosed A
x : L
hx : K[x] = ⊤
hAx : IsIntegral A x
hKx : IsIntegral K x
pb : PowerBasis K L
pbgen : pb.gen = x
hnondeg : (traceForm K L).Nondegenerate
hpb :
  ⇑((traceForm K L).dualBasis hnondeg pb.basis) = fun x ↦
    (minpolyDiv K pb.gen).coeff ↑x / (aeval pb.gen) (derivative (minpoly K pb.gen))
this : Subalgebra.toSubmodule A[x] = span A (Set.range ⇑pb.basis)
⊢ span A (Set.range fun x ↦ (minpolyDiv K pb.gen).coeff ↑x / (aeval pb.gen) (derivative (minpoly K pb.gen))) =
    span A (((aeval x) (derivative (minpoly K x)))⁻¹ • Set.range (minpolyDiv A x).coeff)
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
⊢ ↑⋯.unit ^ 2 = -↑⋯.unit - 1

Tactic:
rw [← neg_add', ← add_eq_zero_iff_eq_neg, ← add_assoc]
  ext; simpa using hζ.isRoot_cyclotomic (by decide)

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

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

Tactic:
rw [Monoid.fg_iff_add_fg, ← AddGroup.fg_iff_addMonoid_fg, ← Module.Finite.iff_addGroup_fg]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Module.Finite ℤ (Additive (𝓞 K)ˣ)
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
⊢ x ^ 3 - x ^ 2 * (↑η ^ 2 + ↑η + 1) + x * (↑η ^ 2 + ↑η + ↑η ^ 3) - ↑η ^ 3 =
    x ^ 3 - x ^ 2 * (↑η ^ 2 + ↑η + 1) + x * (↑η ^ 2 + ↑η + 1) - 1

Tactic:
by
    simp [hζ.toInteger_cube_eq_one]

Goal state after:
Module: Mathlib.NumberTheory.LegendreSymbol.Basic

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : p ≠ 2
a : ℤ
⊢ legendreSym p (-1) * legendreSym p a = χ₄ ↑p * legendreSym p a

Tactic:
legendreSym.at_neg_one hp

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : p ≠ 2
a : ℤ
⊢ χ₄ ↑p * legendreSym p a = χ₄ ↑p * legendreSym p a
Module: Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic

Goal state before:
pp: F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
a : F
⊢ ↑(#{x | x ^ 2 = a}.toFinset) = (quadraticChar F) a + 1

Tactic:
by_cases h₀ : a = 0

Goal state after:
case pos
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
a : F
h₀ : a = 0
⊢ ↑(#{x | x ^ 2 = a}.toFinset) = (quadraticChar F) a + 1

case neg
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
a : F
h₀ : ¬a = 0
⊢ ↑(#{x | x ^ 2 = a}.toFinset) = (quadraticChar F) a + 1
Module: Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic

Goal state before:
pp: case pos
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
a : F
h₀ : ¬a = 0
s : Finset F := {x | x ^ 2 = a}.toFinset
h : IsSquare a
⊢ ↑(#s) = 1 + 1

Tactic:
rcases h with ⟨b, h⟩

Goal state after:
case pos
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
a : F
h₀ : ¬a = 0
s : Finset F := {x | x ^ 2 = a}.toFinset
b : F
h : a = b * b
⊢ ↑(#s) = 1 + 1
Module: Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum

Goal state before:
pp: F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
hF : ringChar F ≠ 2
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : p ≠ 2
⊢ IsSquare ↑p ↔ (quadraticChar (ZMod p)) (↑(χ₄ ↑(Fintype.card F)) * ↑(Fintype.card F)) ≠ -1

Tactic:
classical
  rcases eq_or_ne (ringChar F) p with rfl | hFp
  · obtain ⟨q, hq, hq'⟩ := FiniteField.card F (ringChar F)
    simp [hq']
  · rwa [← Iff.not_left quadraticChar_neg_one_iff_not_isSquare, quadraticChar_odd_prime hF hp]

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

Goal state before:
pp: 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
z : 𝓞 K
hz : S'.c = λ * z
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'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1

Tactic:
left
    exact ⟨⟨x, hx⟩, ⟨y, hy⟩⟩

Goal state after:
Module: Mathlib.NumberTheory.LucasPrimality

Goal state before:
pp: p : ℕ
hP : Nat.Prime p
this : Fact (Nat.Prime p)
g : (ZMod p)ˣ
hg : ∀ (x : (ZMod p)ˣ), x ∈ Subgroup.zpowers g
h1 : (Units.coeHom (ZMod p)) g ^ (p - 1) = 1 ∧ ∀ m < p - 1, 0 < m → (Units.coeHom (ZMod p)) g ^ m ≠ 1
h2 : 0 < p - 1
q : ℕ
hqd : q ∣ p - 1
hq : 1 < q
⊢ ↑g ^ ((p - 1) / q) ≠ 1

Tactic:
exact h1.2 _ (Nat.div_lt_self h2 hq) (Nat.div_pos (Nat.le_of_dvd h2 hqd) (zero_lt_one.trans hq))

Goal state after: