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

Goal state before:
pp: case succ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ emultiplicity 2 ((x ^ 2 ^ multiplicity 2 (n + 1)) ^ k - (y ^ 2 ^ multiplicity 2 (n + 1)) ^ k) =
    emultiplicity 2 (x - y) + emultiplicity 2 ↑(2 ^ multiplicity 2 (n + 1) * k)

Tactic:
emultiplicity_pow_sub_pow_of_prime,

Goal state after:
case succ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ emultiplicity 2 (x ^ 2 ^ multiplicity 2 (n + 1) - y ^ 2 ^ multiplicity 2 (n + 1)) =
    emultiplicity 2 (x - y) + emultiplicity 2 ↑(2 ^ multiplicity 2 (n + 1) * k)

case succ.hp
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ Prime 2

case succ.hxy
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ 2 ∣ x ^ 2 ^ multiplicity 2 (n + 1) - y ^ 2 ^ multiplicity 2 (n + 1)

case succ.hx
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ ¬2 ∣ x ^ 2 ^ multiplicity 2 (n + 1)

case succ.hn
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ ¬2 ∣ ↑k
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a : mixedSpace K
ha : a ∈ integerSet K
⊢ ∃! x, (mixedEmbedding K) ↑x = a

Tactic:
obtain ⟨_, ⟨x, rfl⟩⟩ := mem_integerSet.mp ha
  refine Function.Injective.existsUnique_of_mem_range ?_ (Set.mem_range_self x)
  exact (mixedEmbedding_injective K).comp RingOfIntegers.coe_injective

Goal state after:
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
hx : (mixedEmbedding K) ↑x ∈ integerSet K
⊢ ↑(preimageOfMemIntegerSet ⟨(mixedEmbedding K) ↑x, hx⟩) = x

Tactic:
simp_rw [RingOfIntegers.ext_iff, ← (mixedEmbedding_injective K).eq_iff,
    mixedEmbedding_preimageOfMemIntegerSet]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.CMField

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
u : (𝓞 K)ˣ
⊢ (complexConj K).toRingEquiv ↑↑u = ↑↑u ↔ ∃ v, ↑((algebraMap (𝓞 ↥K⁺) (𝓞 K)) ↑v) = ↑↑u

Tactic:
AlgEquiv.coe_ringEquiv,

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
u : (𝓞 K)ˣ
⊢ (complexConj K) ↑↑u = ↑↑u ↔ ∃ v, ↑((algebraMap (𝓞 ↥K⁺) (𝓞 K)) ↑v) = ↑↑u
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
J : ↥(Ideal (𝓞 K))⁰
⊢ (x ∈ fundamentalCone K ∧ ∃ a, Nonempty (∃ x' ∈ ↑J, (algebraMap (𝓞 K) K) x' = a) ∧ (mixedEmbedding K) a = x) ↔
    x ∈ fundamentalCone K ∧ ∃ a ∈ ↑J, (mixedEmbedding K) ↑a = x

Tactic:
nonempty_prop,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
J : ↥(Ideal (𝓞 K))⁰
⊢ (x ∈ fundamentalCone K ∧ ∃ a, (∃ x' ∈ ↑J, (algebraMap (𝓞 K) K) x' = a) ∧ (mixedEmbedding K) a = x) ↔
    x ∈ fundamentalCone K ∧ ∃ a ∈ ↑J, (mixedEmbedding K) ↑a = x
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: case refine_1.refine_4
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
y : mixedSpace K
h₃ : y ∈ {x | mixedEmbedding.norm x ≤ 1}
h₁ : y ∈ logMap ⁻¹' ZSpan.fundamentalDomain (Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))
h₂ : y ∉ {x | mixedEmbedding.norm x = 0}
⊢ normAtAllPlaces y ∈ {x | mixedEmbedding.norm (mixedSpaceOfRealSpace x) ≤ 1}

Tactic:
rwa [Set.mem_ofPred_eq, ← norm_normAtAllPlaces] at h₃

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Completion.Ramification

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
v : InfinitePlace K
w : InfinitePlace L
inst✝ : w.LiesOver v
⊢ v.mult * Module.finrank v.Completion w.Completion = w.mult

Tactic:
by
  have hv : v = w.comap (algebraMap K L) := Subtype.ext ‹w.LiesOver v›.comp_eq.symm
  rcases w.isUnramified_or_isRamified K with h | h
  · rw [h.finrank_eq_one v, hv, h.eq, mul_one]
  · rw [h.finrank_eq_two v, hv, h.isReal.mult_eq_one, h.isComplex.mult_eq_two, one_mul]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
p f n : ℕ
P : K[X]
hK : Fintype.card K = p ^ f
hn : p.Coprime n
hp : Fact (Nat.Prime p)
hP : P ∣ cyclotomic n K
hPirr : Irreducible P
hPmo : P.Monic
this✝¹ : Fact (Irreducible P)
this✝ : Module.Finite K (AdjoinRoot P)
this : Finite (AdjoinRoot P)
hζ : IsPrimitiveRoot (root P) n
pB : PowerBasis K (AdjoinRoot P) := powerBasis ⋯
⊢ Module.finrank K (AdjoinRoot P) = orderOf (unitOfCoprime (p ^ f) ⋯)

Tactic:
← orderOf_frobeniusAlgEquivOfAlgebraic

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
p f n : ℕ
P : K[X]
hK : Fintype.card K = p ^ f
hn : p.Coprime n
hp : Fact (Nat.Prime p)
hP : P ∣ cyclotomic n K
hPirr : Irreducible P
hPmo : P.Monic
this✝¹ : Fact (Irreducible P)
this✝ : Module.Finite K (AdjoinRoot P)
this : Finite (AdjoinRoot P)
hζ : IsPrimitiveRoot (root P) n
pB : PowerBasis K (AdjoinRoot P) := powerBasis ⋯
⊢ orderOf (frobeniusAlgEquivOfAlgebraic K (AdjoinRoot P)) = orderOf (unitOfCoprime (p ^ f) ⋯)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
w : InfinitePlace K
hw : w ≠ w₀
⊢ mixedEmbedding.norm
      (mixedSpaceOfRealSpace (↑expMap ((completeBasis K).equivFun.symm fun i ↦ if i = w₀ then 0 else x i))) =
    1

Tactic:
rw [← expMapBasis_apply,
    norm_expMapBasis, if_pos rfl, Real.exp_zero, one_pow]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
hx₀ : ¬x = 0
⊢ x ∈ compactSet K ↔ x ∈ ↑expMapBasis '' closure (paramSet K) ∪ {0}

Tactic:
· refine ⟨fun hx ↦ Set.mem_union_left _ (compactSet_eq_union_aux₁ hx₀ hx), fun hx ↦ ?_⟩
    simp only [Set.union_singleton, Set.mem_insert_iff, hx₀, false_or] at hx
    exact compactSet_eq_union_aux₂ hx₀ hx

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

Goal state before:
pp: n m p k : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
P : Ideal (𝓞 K)
hP₁ : P.IsPrime
hP₂ : P.LiesOver 𝒑
inst✝ : IsCyclotomicExtension {n} ℚ K
hn : n = p ^ (k + 1) * m
hm : ¬p ∣ m
this : IsGalois ℚ K
⊢ 𝒑.inertiaDegIn (𝓞 K) = orderOf ↑p

Tactic:
inertiaDegIn_eq n K hn hm

Goal state after:
n m p k : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
P : Ideal (𝓞 K)
hP₁ : P.IsPrime
hP₂ : P.LiesOver 𝒑
inst✝ : IsCyclotomicExtension {n} ℚ K
hn : n = p ^ (k + 1) * m
hm : ¬p ∣ m
this : IsGalois ℚ K
⊢ orderOf ↑p = orderOf ↑p
Module: Mathlib.NumberTheory.NumberField.House

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
α : K
⊢ ↑‖(canonicalEmbedding K) α‖₊ = ↑(univ.sup' ⋯ fun φ ↦ ‖φ α‖₊)

Tactic:
nnnorm_eq,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
α : K
⊢ ↑(univ.sup fun φ ↦ ‖φ α‖₊) = ↑(univ.sup' ⋯ fun φ ↦ ‖φ α‖₊)
Module: Mathlib.Topology.Algebra.Nonarchimedean.Completion

Goal state before:
pp: G : Type u_1
inst✝³ : AddGroup G
inst✝² : UniformSpace G
inst✝¹ : IsUniformAddGroup G
inst✝ : NonarchimedeanAddGroup G
U : Set (Completion G)
hU : U ∈ 𝓝 0
C : Set (Completion G)
C_subset_U : id C ⊆ U
hC : C ∈ 𝓝 0
C_closed : IsClosed C
⊢ C ∈ 𝓝 (toCompl 0)

Tactic:
map_zero

Goal state after:
G : Type u_1
inst✝³ : AddGroup G
inst✝² : UniformSpace G
inst✝¹ : IsUniformAddGroup G
inst✝ : NonarchimedeanAddGroup G
U : Set (Completion G)
hU : U ∈ 𝓝 0
C : Set (Completion G)
C_subset_U : id C ⊆ U
hC : C ∈ 𝓝 0
C_closed : IsClosed C
⊢ C ∈ 𝓝 0
Module: Mathlib.NumberTheory.Ostrowski

Goal state before:
pp: f : AbsoluteValue ℚ ℝ
m n : ℕ
hm : 1 < m
hn : 1 < n
notbdd : ¬∀ (n : ℕ), f ↑n ≤ 1
s t : ℝ
hfm : f ↑m = ↑m ^ s
hfn : f ↑n = ↑n ^ t
⊢ t ≤ s

Tactic:
rw [← rpow_le_rpow_left_iff (x := n) (mod_cast hn), ← hfn]
  apply le_trans <| le_pow_log hm hn notbdd
  rw [hfm, ← rpow_mul (Nat.cast_nonneg m), mul_comm, rpow_mul (Nat.cast_nonneg m),
    rpow_logb (mod_cast zero_lt_of_lt hm) (mod_cast hm.ne') (mod_cast zero_lt_of_lt hn)]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.House

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
h0p : 0 < p
hpq : p < q
inst✝² : Fintype β
cardβ : Fintype.card β = q
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
cardα : Fintype.card α = p
h : ℕ := finrank ℚ K
hphqh : p * h < q * h
h0ph : 0 < p * h
⊢ p * Fintype.card (K →+* ℂ) = p * h

Tactic:
Embeddings.card

Goal state after:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
h0p : 0 < p
hpq : p < q
inst✝² : Fintype β
cardβ : Fintype.card β = q
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
cardα : Fintype.card α = p
h : ℕ := finrank ℚ K
hphqh : p * h < q * h
h0ph : 0 < p * h
⊢ p * finrank ℚ K = p * h
Module: Mathlib.NumberTheory.NumberField.House

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
h0p : 0 < p
hpq : p < q
inst✝² : Fintype β
cardβ : Fintype.card β = q
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
cardα : Fintype.card α = p
h : ℕ := finrank ℚ K
hphqh : p * h < q * h
h0ph : 0 < p * h
hfinp : Fintype.card (α × (K →+* ℂ)) = p * h
⊢ q * Fintype.card (K →+* ℂ) = q * h

Tactic:
Embeddings.card

Goal state after:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
h0p : 0 < p
hpq : p < q
inst✝² : Fintype β
cardβ : Fintype.card β = q
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
cardα : Fintype.card α = p
h : ℕ := finrank ℚ K
hphqh : p * h < q * h
h0ph : 0 < p * h
hfinp : Fintype.card (α × (K →+* ℂ)) = p * h
⊢ q * finrank ℚ K = q * h
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case neg
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
γ : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (if x' - y' = 0 then 0 else (↑p ^ padicValRat p (x' - y'))⁻¹) ≤ (↑p ^ (embedding ↑γ * exp (-1)).log.natAbs)⁻¹
H : ¬x' - y' = 0
⊢ -padicValRat p (x' - y') < (embedding ↑γ).log

Tactic:
simp_all [← zpow_natCast, zpow_pos, inv_le_inv₀, zpow_le_zpow_iff_right₀ hp1', abs_le,
          Int.lt_iff_add_one_le]

Goal state after:
Module: Mathlib.NumberTheory.Padics.HeightOneSpectrum

Goal state before:
pp: R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra R ℚ
inst✝² : IsIntegralClosure R ℤ ℚ
inst✝¹ : IsDedekindDomain R
inst✝ : IsFractionRing R ℚ
v : HeightOneSpectrum R
y : ℚ_[↑(primesEquiv v)]
hy : y ∈ ↑(subring ↑(primesEquiv v))
x : ↥(HeightOneSpectrum.adicCompletionIntegers ℚ v)
hx : (adicCompletionIntegers.padicIntEquiv v) x = ⟨y, hy⟩
⊢ ↑((adicCompletionIntegers.padicIntEquiv v) x) = y

Tactic:
hx

Goal state after:
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Algebra R ℚ
inst✝² : IsIntegralClosure R ℤ ℚ
inst✝¹ : IsDedekindDomain R
inst✝ : IsFractionRing R ℚ
v : HeightOneSpectrum R
y : ℚ_[↑(primesEquiv v)]
hy : y ∈ ↑(subring ↑(primesEquiv v))
x : ↥(HeightOneSpectrum.adicCompletionIntegers ℚ v)
hx : (adicCompletionIntegers.padicIntEquiv v) x = ⟨y, hy⟩
⊢ ↑⟨y, hy⟩ = y
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
x : ℤ
hx : ξ = ↑x
⊢ √↑d ^ 2 = ↑(x * x)

Tactic:
Int.cast_mul,

Goal state after:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
x : ℤ
hx : ξ = ↑x
⊢ √↑d ^ 2 = ↑x * ↑x
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
a b : Solution₁ d
ha : IsFundamental a
hb : IsFundamental b
hx : a.x = b.x
⊢ a.x ^ 2 - 1 = b.x ^ 2 - 1

Tactic:
hx

Goal state after:
d : ℤ
a b : Solution₁ d
ha : IsFundamental a
hb : IsFundamental b
hx : a.x = b.x
⊢ b.x ^ 2 - 1 = b.x ^ 2 - 1
Module: Mathlib.NumberTheory.RamificationInertia.Basic

Goal state before:
pp: R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
p : Ideal R
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : Algebra R K
L : Type u_2
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra S L
inst✝⁹ : IsFractionRing S L
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : Module.Finite R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : IsTorsionFree R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map f p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
exists_sum : ∀ (x : S ⧸ M), ∃ a', (∀ (i : Fin n), a' i ∈ p) ∧ ∑ i, a' i • a i = x
A' : Fin n → Fin n → R
hA'p : ∀ (i i_1 : Fin n), A' i i_1 ∈ p
hA' : ∀ (i : Fin n), ∑ i_1, A' i i_1 • a i_1 = a i
A : Matrix (Fin n) (Fin n) R := Matrix.of A' - 1
B : Matrix (Fin n) (Fin n) R := A.adjugate
A_smul : ∀ (i : Fin n), ∑ j, A i j • a j = 0
d_smul : ∀ (i : Fin n), A.det • a i = 0
x : S
⊢ x ∈ Submodule.restrictScalars R (S ∙ f A.det) → x ∈ M

Tactic:
hx

Goal state after:
R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
p : Ideal R
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : Algebra R K
L : Type u_2
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra S L
inst✝⁹ : IsFractionRing S L
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : Module.Finite R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : IsTorsionFree R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map f p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
exists_sum : ∀ (x : S ⧸ M), ∃ a', (∀ (i : Fin n), a' i ∈ p) ∧ ∑ i, a' i • a i = x
A' : Fin n → Fin n → R
hA'p : ∀ (i i_1 : Fin n), A' i i_1 ∈ p
hA' : ∀ (i : Fin n), ∑ i_1, A' i i_1 • a i_1 = a i
A : Matrix (Fin n) (Fin n) R := Matrix.of A' - 1
B : Matrix (Fin n) (Fin n) R := A.adjugate
A_smul : ∀ (i : Fin n), ∑ j, A i j • a j = 0
d_smul : ∀ (i : Fin n), A.det • a i = 0
x : S
hx : x ∈ Submodule.restrictScalars R (S ∙ f A.det)
⊢ x ∈ M
Module: Mathlib.NumberTheory.RamificationInertia.HilbertTheory

Goal state before:
pp: case refine_3
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁹ : Field K
inst✝³⁸ : Field L
inst✝³⁷ : Algebra K L
inst✝³⁶ : CommRing A
inst✝³⁵ : CommRing B
inst✝³⁴ : Algebra A B
p : Ideal A
P : Ideal B
inst✝³³ : P.LiesOver p
inst✝³² : Algebra A K
inst✝³¹ : IsFractionRing A K
inst✝³⁰ : Algebra A L
inst✝²⁹ : IsScalarTower A K L
inst✝²⁸ : Algebra B L
inst✝²⁷ : IsScalarTower A B L
inst✝²⁶ : IsFractionRing B L
inst✝²⁵ : MulSemiringAction Gal(L/K) B
inst✝²⁴ : SMulDistribClass Gal(L/K) B L
D : Type u_5
𝓞D : Type u_6
inst✝²³ : Field D
inst✝²² : Algebra D L
inst✝²¹ : IsDecompositionField K L P D
inst✝²⁰ : CommRing 𝓞D
inst✝¹⁹ : Algebra 𝓞D D
inst✝¹⁸ : IsFractionRing 𝓞D D
inst✝¹⁷ : Algebra 𝓞D B
inst✝¹⁶ : Algebra 𝓞D L
inst✝¹⁵ : IsScalarTower 𝓞D D L
inst✝¹⁴ : IsScalarTower 𝓞D B L
𝓟D : Ideal 𝓞D
hD : P.LiesOver 𝓟D
inst✝¹³ : IsGalois K L
inst✝¹² : IsDedekindDomain A
inst✝¹¹ : IsDedekindDomain B
inst✝¹⁰ : Module.Finite A B
inst✝⁹ : Module.IsTorsionFree A B
inst✝⁸ : Algebra A 𝓞D
inst✝⁷ : Module.Finite A 𝓞D
inst✝⁶ : IsScalarTower A 𝓞D B
inst✝⁵ : IsDedekindDomain 𝓞D
inst✝⁴ : 𝓟D.LiesOver p
inst✝³ : FiniteDimensional K L
inst✝² : Ring.HasFiniteQuotients A
inst✝¹ : 𝓟D.IsMaximal
inst✝ : P.IsMaximal
hp : p ≠ ⊥
left✝⁴ : Module.Finite 𝓞D B
left✝³ : Module.IsTorsionFree 𝓞D B
left✝² : Module.IsTorsionFree A 𝓞D
left✝¹ : IsGaloisGroup Gal(L/K) A B
left✝ : IsGaloisGroup (↥(stabilizer Gal(L/K) P)) 𝓞D B
h𝓟 : 𝓟D ≠ ⊥
⊢ 𝓟D.ramificationIdxIn B ≤ p.ramificationIdxIn B

Tactic:
rw [ramificationIdxIn_eq_ramificationIdx p P Gal(L/K),
      ramificationIdxIn_eq_ramificationIdx _ P (stabilizer Gal(L/K) P)]
    exact 𝓟D.ramificationIdx_above_le P

Goal state after:
Module: Mathlib.NumberTheory.Padics.Hensel

Goal state before:
pp: p : ℕ
inst✝² : Fact (Nat.Prime p)
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2
hnsol : (Polynomial.aeval a) F ≠ 0
⊢ ‖soln - a‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖

Tactic:
rw [soln_dist_to_a, div_lt_iff₀ (deriv_norm_pos _), ← sq] <;> assumption

Goal state after:
Module: Mathlib.NumberTheory.Padics.Hensel

Goal state before:

Tactic:
‖z - soln‖ = ‖z - a + (a - soln)‖ := by rw [sub_add_sub_cancel]
      _ ≤ max ‖z - a‖ ‖a - soln‖ := PadicInt.nonarchimedean _ _
      _ < ‖F.derivative.aeval a‖ :=
        max_lt hnlt ((norm_sub_rev soln a ▸ (soln_dist_to_a_lt_deriv hnorm)) hnsol)

Goal state after:
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: s : BoundingSieve
d : ℕ
hd : d ∣ s.prodPrimes
⊢ 0 < s.nu d

Tactic:
calc
    0 < ∏ p ∈ d.primeFactors, s.nu p := by
      apply prod_pos
      intro p hpd
      have hp_prime : p.Prime := prime_of_mem_primeFactors hpd
      have hp_dvd : p ∣ s.prodPrimes := (dvd_of_mem_primeFactors hpd).trans hd
      exact s.nu_pos_of_prime p hp_prime hp_dvd
    _ = s.nu d := prod_primeFactors_nu hd

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

Goal state before:
pp: case refine_2
R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
p : Ideal R
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : Algebra R K
L : Type u_2
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra S L
inst✝⁹ : IsFractionRing S L
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : Module.Finite R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : IsTorsionFree R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map f p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
exists_sum : ∀ (x : S ⧸ M), ∃ a', (∀ (i : Fin n), a' i ∈ p) ∧ ∑ i, a' i • a i = x
A' : Fin n → Fin n → R
hA'p : ∀ (i i_1 : Fin n), A' i i_1 ∈ p
hA' : ∀ (i : Fin n), ∑ i_1, A' i i_1 • a i_1 = a i
A : Matrix (Fin n) (Fin n) R := Matrix.of A' - 1
B : Matrix (Fin n) (Fin n) R := A.adjugate
A_smul : ∀ (i : Fin n), ∑ j, A i j • a j = 0
d_smul : ∀ (i : Fin n), A.det • a i = 0
span_d : Submodule.restrictScalars R (S ∙ f A.det) ≤ M
x : L
⊢ x ∈ Submodule.restrictScalars K (span {(algebraMap R L) A.det}) → x ∈ Submodule.span K (⇑(algebraMap S L) '' b)

Tactic:
hx

Goal state after:
case refine_2
R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
p : Ideal R
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : Algebra R K
L : Type u_2
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra S L
inst✝⁹ : IsFractionRing S L
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : Module.Finite R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : IsTorsionFree R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map f p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
exists_sum : ∀ (x : S ⧸ M), ∃ a', (∀ (i : Fin n), a' i ∈ p) ∧ ∑ i, a' i • a i = x
A' : Fin n → Fin n → R
hA'p : ∀ (i i_1 : Fin n), A' i i_1 ∈ p
hA' : ∀ (i : Fin n), ∑ i_1, A' i i_1 • a i_1 = a i
A : Matrix (Fin n) (Fin n) R := Matrix.of A' - 1
B : Matrix (Fin n) (Fin n) R := A.adjugate
A_smul : ∀ (i : Fin n), ∑ j, A i j • a j = 0
d_smul : ∀ (i : Fin n), A.det • a i = 0
span_d : Submodule.restrictScalars R (S ∙ f A.det) ≤ M
x : L
hx : x ∈ Submodule.restrictScalars K (span {(algebraMap R L) A.det})
⊢ x ∈ Submodule.span K (⇑(algebraMap S L) '' b)
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: s : BoundingSieve
d : ℕ
hd : d ∣ s.prodPrimes
⊢ (selbergTerms d * ∑ l ∈ s.prodPrimes.divisors, if l ∣ d then (selbergTerms l)⁻¹ else 0) = selbergTerms d * (s.nu d)⁻¹

Tactic:
rw [← nu_inv_eq_sum_divisors_inv_selbergTerms hd]

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

Goal state before:
pp: case refine_2.refine_2
R : Type u
inst✝¹⁴ : CommRing R
S : Type v
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
p : Ideal R
K : Type u_1
inst✝¹¹ : Field K
inst✝¹⁰ : Algebra R K
L : Type u_2
inst✝⁹ : Field L
inst✝⁸ : Algebra S L
inst✝⁷ : IsFractionRing S L
hRK : IsFractionRing R K
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : Algebra K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsScalarTower R S L
hp : p.IsMaximal
inst✝ : Module.Finite R S
ι : Type v := Free.ChooseBasisIndex (R ⧸ p) (S ⧸ map f p)
b : Basis ι (R ⧸ p) (S ⧸ map f p) := Free.chooseBasis (R ⧸ p) (S ⧸ map f p)
b' : ι → S := fun i ↦ ⋯.choose
b_eq_b' : ⇑b = ⇑(↑R (Submodule.mkQ (map f p))) ∘ b'
b'' : ι → L := ⇑(algebraMap S L) ∘ b'
b''_li : LinearIndependent K b''
b''_sp : Submodule.span K (Set.range b'') = ⊤
⊢ finrank (R ⧸ p) (S ⧸ map f p) = finrank K L

Tactic:
let c : Basis ι K L := Basis.mk b''_li b''_sp.ge
      rw [finrank_eq_card_basis b, finrank_eq_card_basis c]

Goal state after:
Module: Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
hpi : Prime ↑p
hp1 : p % 2 = 1
hp3 : p % 4 ≠ 3
k : ℕ
k_lt_p : k < p
hk : -1 = ↑k * ↑k
hpk : p ∣ k ^ 2 + 1
hkmul : ↑k ^ 2 + 1 = { re := ↑k, im := 1 } * { re := ↑k, im := -1 }
hk₀ : k ≠ 0
hkltp : 1 + k * k < p * p
x✝ : ↑p ∣ { re := ↑k, im := -1 }
x : ℤ[i]
hx : { re := ↑k, im := -1 } = ↑p * x
⊢ (norm (↑p * x)).natAbs = { re := ↑k, im := -1 }.norm.natAbs

Tactic:
hx

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
hpi : Prime ↑p
hp1 : p % 2 = 1
hp3 : p % 4 ≠ 3
k : ℕ
k_lt_p : k < p
hk : -1 = ↑k * ↑k
hpk : p ∣ k ^ 2 + 1
hkmul : ↑k ^ 2 + 1 = { re := ↑k, im := 1 } * { re := ↑k, im := -1 }
hk₀ : k ≠ 0
hkltp : 1 + k * k < p * p
x✝ : ↑p ∣ { re := ↑k, im := -1 }
x : ℤ[i]
hx : { re := ↑k, im := -1 } = ↑p * x
⊢ (norm (↑p * x)).natAbs = (norm (↑p * x)).natAbs