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

Goal state before:
pp: case h
x : ℍ
hxnorm : ‖↑x‖ = 1
hxre : |x.re| ≤ 1 / 2
⊢ ∃ᶠ (x_1 : ℝ≥0) in 𝓝[>] 0, { coe := ↑x + ↑↑x_1 * Complex.I, coe_im_pos := ⋯ } ∈ closure 𝒟ᵒ

pp: case hf
x : ℍ
hxnorm : ‖↑x‖ = 1
hxre : |x.re| ≤ 1 / 2
⊢ Filter.Tendsto (fun t ↦ { coe := ↑x + ↑↑t * Complex.I, coe_im_pos := ⋯ }) (𝓝[>] 0) (𝓝 x)

Tactic:
· apply Filter.Eventually.frequently
    filter_upwards [self_mem_nhdsWithin] with a (ha : 0 < a)
    refine mem_closure_of_one_lt_norm ?_ (by simpa using! hxre)
    suffices 1 < ‖(x : ℂ)‖ ^ 2 + a ^ 2 + 2 * a * x.im by
      rw [← one_lt_normSq_iff]
      convert! this
      simp [← normSq_eq_norm_sq, normSq_apply]
      ring
    rw [hxnorm, one_pow, add_assoc, lt_add_iff_pos_right]
    positivity

Goal state after:
case hf
x : ℍ
hxnorm : ‖↑x‖ = 1
hxre : |x.re| ≤ 1 / 2
⊢ Filter.Tendsto (fun t ↦ { coe := ↑x + ↑↑t * Complex.I, coe_im_pos := ⋯ }) (𝓝[>] 0) (𝓝 x)
Module: Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable

Goal state before:
pp: ⊢ (fun τ ↦ jacobiTheta τ - 1) =O[comap im atTop] fun τ ↦ rexp (-π * τ.im)

Tactic:
IsBigO,

Goal state after:
⊢ ∃ c, IsBigOWith c (comap im atTop) (fun τ ↦ jacobiTheta τ - 1) fun τ ↦ rexp (-π * τ.im)
Module: Mathlib.NumberTheory.ModularForms.LFunction

Goal state before:
pp: case e'_6.e'_6
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
r : ℝ
hpos : 0 < s.re
hs : r + 1 < s.re
hΛ : Λ hk f s = mellin (fun t ↦ (weakFEPair hk f).f t - (weakFEPair hk f).f₀) s
hcoeff : (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ r
hh : 0 < h Γ
hΓ : h Γ ∈ Γ.strictPeriods
t : ℝ
ht : 0 < t
this :
  HasSum
    (fun m ↦ (PowerSeries.coeff m) (qExpansion (h Γ) ⇑f) * Function.Periodic.qParam (h Γ) ↑(↑ofComplex (I * ↑t)) ^ m)
    (f (↑ofComplex (I * ↑t)))
⊢ (weakFEPair hk f).f₀ =
    (PowerSeries.coeff 0) (qExpansion (h Γ) ⇑f) * Function.Periodic.qParam (h Γ) ↑(↑ofComplex (I * ↑t)) ^ 0

Tactic:
simp only [weakFEPair]

Goal state after:
case e'_6.e'_6
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
r : ℝ
hpos : 0 < s.re
hs : r + 1 < s.re
hΛ : Λ hk f s = mellin (fun t ↦ (weakFEPair hk f).f t - (weakFEPair hk f).f₀) s
hcoeff : (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ r
hh : 0 < h Γ
hΓ : h Γ ∈ Γ.strictPeriods
t : ℝ
ht : 0 < t
this :
  HasSum
    (fun m ↦ (PowerSeries.coeff m) (qExpansion (h Γ) ⇑f) * Function.Periodic.qParam (h Γ) ↑(↑ofComplex (I * ↑t)) ^ m)
    (f (↑ofComplex (I * ↑t)))
⊢ valueAtInfty ⇑f =
    (PowerSeries.coeff 0) (qExpansion (h Γ) ⇑f) * Function.Periodic.qParam (h Γ) ↑(↑ofComplex (I * ↑t)) ^ 0
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p x : R
n y : ℕ
hy : y ∈ range n
⊢ y + 2 ≤ n + 1

Tactic:
linarith [Finset.mem_range.mp hy]

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.LFunction

Goal state before:
pp: case e'_5
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
hs : 0 < s.re
hΛ : HasSum (fun n ↦ ↑π ^ (-s) * Gamma s * (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f) / ↑(2 * ↑n / h Γ) ^ s) (Λ hk f s)
⊢ (fun i ↦ (PowerSeries.coeff i) (qExpansion (h Γ) ⇑f) / ↑i ^ s) = fun i ↦
    ↑π ^ (-s) * Gamma s * (PowerSeries.coeff i) (qExpansion (h Γ) ⇑f) / ↑(2 * ↑i / h Γ) ^ s *
      (2 / s.Gammaℂ * ↑(h Γ) ^ (-s))

Tactic:
ext n
    generalize (PowerSeries.coeff n) (qExpansion (h Γ) f) = p
    rw [Gammaℂ, ← div_div, ← div_div, div_self two_ne_zero, one_div, cpow_neg (2 * _), inv_inv,
      ← ofReal_ofNat, mul_cpow_ofReal_nonneg two_pos.le Real.pi_pos.le]
    simp only [ofReal_div, ofReal_mul, ofReal_ofNat, ofReal_natCast]
    have : (2 * n / h Γ : ℂ) ^ s = 2 ^ s * n ^ s / h Γ ^ s := by
      rw [← ofReal_ofNat, ← ofReal_natCast, ← ofReal_mul,
        div_cpow_ofReal_nonneg (by grind) Γ.strictWidthInfty_nonneg, ofReal_mul,
        mul_cpow_ofReal_nonneg zero_le_two n.cast_nonneg]
    rw [this, cpow_neg, cpow_neg]
    have := Gamma_ne_zero_of_re_pos hs
    have := cpow_ne_zero_iff (y := s).mpr (.inl <| ofReal_ne_zero.mpr Γ.strictWidthInfty_pos.ne')
    field_simp

Goal state after:
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝ : (i : ι) → TopologicalSpace (R i)
hAopen : ∀ (i : ι), IsOpen[inst✝ i] (A i)
p : ι → Prop
⊢ IsOpen[topologicalSpace R A cofinite] {f | ∀ (i : ι), p i → ↑f i ∈ A i}

Tactic:
by
  simp_rw +instances [topologicalSpace_eq_iSup cofinite, isOpen_iSup_iff, isOpen_coinduced]
  exact fun S hS ↦ isOpen_forall_imp_mem_of_principal hAopen hS

Goal state after:
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
𝓕✝ 𝓖 : Filter ι
S : ι → Type u_3
inst✝⁴ : (i : ι) → SetLike (S i) (R i)
B : (i : ι) → S i
T : Set ι
𝓕 : Filter ι
inst✝³ : (i : ι) → TopologicalSpace (R i)
inst✝² : (i : ι) → Inv (R i)
inst✝¹ : ∀ (i : ι), InvMemClass (S i) (R i)
inst✝ : ∀ (i : ι), ContinuousInv (R i)
⊢ Continuous[topologicalSpace R (fun i ↦ ↑(B i)) 𝓕, topologicalSpace R (fun i ↦ ↑(B i)) 𝓕] fun a ↦ a⁻¹

Tactic:
by
    rw [continuous_dom]
    intro T hT
    have : ContinuousInv (Πʳ i, [R i, B i]_[𝓟 T]) :=
      isEmbedding_coe_of_principal.continuousInv fun _ ↦ rfl
    exact (continuous_inclusion hT).comp continuous_inv

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : IsMaxRank u
⊢ (span ℤ (Set.range ⇑(basisOfIsMaxRank hu))).toAddSubgroup =
    AddSubgroup.map (logEmbedding K) (AddSubgroup.closure (⇑Additive.toMul ⁻¹' Set.range u))

Tactic:
AddMonoidHom.map_closure,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : IsMaxRank u
⊢ (span ℤ (Set.range ⇑(basisOfIsMaxRank hu))).toAddSubgroup =
    AddSubgroup.closure (⇑(logEmbedding K) '' ⇑Additive.toMul ⁻¹' Set.range u)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realMixedSpace K
this : ↑(FDerivPolarCoordRealSymm K x) = LinearMap.id.prodMap ↑(fderivPiPolarCoordSymm x.2)
⊢ LinearMap.det LinearMap.id * LinearMap.det ↑(fderivPiPolarCoordSymm x.2) = ∏ w, (x.2 w).1

Tactic:
LinearMap.det_id,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realMixedSpace K
this : ↑(FDerivPolarCoordRealSymm K x) = LinearMap.id.prodMap ↑(fderivPiPolarCoordSymm x.2)
⊢ 1 * LinearMap.det ↑(fderivPiPolarCoordSymm x.2) = ∏ w, (x.2 w).1
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : mixedSpace K → E
⊢ ∫ (x : realMixedSpace K) in (mixedEmbedding.polarCoord K).target,
      (∏ w, (x.2 w).1) • f (↑(mixedEmbedding.polarCoord K).symm x) =
    ∫ (x : realMixedSpace K) in (polarCoordReal K).target,
      (∏ w, (x.2 w).1) • f ((mixedSpaceToRealMixedSpace K).symm (↑(polarCoordReal K).symm x))

Tactic:
polarCoord_target_eq_polarCoordReal_target,

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : mixedSpace K → E
⊢ ∫ (x : realMixedSpace K) in (polarCoordReal K).target, (∏ w, (x.2 w).1) • f (↑(mixedEmbedding.polarCoord K).symm x) =
    ∫ (x : realMixedSpace K) in (polarCoordReal K).target,
      (∏ w, (x.2 w).1) • f ((mixedSpaceToRealMixedSpace K).symm (↑(polarCoordReal K).symm x))
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : mixedSpace K → ℝ≥0∞
⊢ ∫⁻ (x : realMixedSpace K) in (mixedEmbedding.polarCoord K).target,
      (∏ w, ENNReal.ofReal (x.2 w).1) * f (↑(mixedEmbedding.polarCoord K).symm x) =
    ∫⁻ (x : realMixedSpace K) in (polarCoordReal K).target,
      (∏ w, ENNReal.ofReal (x.2 w).1) * f ((mixedSpaceToRealMixedSpace K).symm (↑(polarCoordReal K).symm x))

Tactic:
polarCoord_target_eq_polarCoordReal_target,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : mixedSpace K → ℝ≥0∞
⊢ ∫⁻ (x : realMixedSpace K) in (polarCoordReal K).target,
      (∏ w, ENNReal.ofReal (x.2 w).1) * f (↑(mixedEmbedding.polarCoord K).symm x) =
    ∫⁻ (x : realMixedSpace K) in (polarCoordReal K).target,
      (∏ w, ENNReal.ofReal (x.2 w).1) * f ((mixedSpaceToRealMixedSpace K).symm (↑(polarCoordReal K).symm x))
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
φ : K →+* ℂ
⊢ ComplexEmbedding.IsConj φ (complexConj K)

Tactic:
obtain ⟨σ, hσ⟩ := exists_isConj _ φ

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
φ : K →+* ℂ
σ : Gal(K/↥K⁺)
hσ : ComplexEmbedding.IsConj φ σ
⊢ ComplexEmbedding.IsConj φ (complexConj K)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
hx : x ≠ 0
⊢ ∑ w, ↑expMap.symm (normAtAllPlaces ((mixedEmbedding K) x)) w = Real.log ↑|(Algebra.norm ℚ) x|

Tactic:
by
  simp_rw [← prod_eq_abs_norm, Real.log_prod (fun _ _ ↦ pow_ne_zero _ ((map_ne_zero _).mpr hx)),
    Real.log_pow, expMap_symm_apply, normAtAllPlaces_mixedEmbedding]

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✝ : NumberField K
⊢ regulator K / regulator ↥K⁺ = 2 ^ rank K * (regulator K / (2 ^ rank K * regulator ↥K⁺))

Tactic:
← mul_div_assoc,

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : NumberField K
⊢ regulator K / regulator ↥K⁺ = 2 ^ rank K * regulator K / (2 ^ rank K * regulator ↥K⁺)
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.inertiaDeg w = Module.finrank v.Completion w.Completion

Tactic:
rw [inertiaDeg_of_liesOver, Ideal.inertiaDeg_eq_of_isMaximal ⊥]

Goal state after:
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
⊢ Module.finrank (v.Completion ⧸ ⊥) (w.Completion ⧸ ⊥) = Module.finrank v.Completion w.Completion
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h₁ : LinearIndependent ℝ fun w ↦ completeFamily K ↑w
h : completeFamily K w₀ ∈ Submodule.span ℝ (Set.range fun w ↦ completeFamily K ↑w)
⊢ False

Tactic:
have := sum_eq_zero_of_mem_span_completeFamily h

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h₁ : LinearIndependent ℝ fun w ↦ completeFamily K ↑w
h : completeFamily K w₀ ∈ Submodule.span ℝ (Set.range fun w ↦ completeFamily K ↑w)
this : ∑ w, completeFamily K w₀ w = 0
⊢ False
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ (completeBasis K) w₀ = fun w ↦ ↑w.mult

Tactic:
rw [completeBasis, coe_basisOfLinearIndependentOfCardEqFinrank, completeFamily, dif_pos rfl]

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 ⋯
⊢ P.natDegree = orderOf (unitOfCoprime (p ^ f) ⋯)

Tactic:
← powerBasis_dim hPirr.ne_zero,

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 ⋯
⊢ (powerBasis ⋯).dim = orderOf (unitOfCoprime (p ^ f) ⋯)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization

Goal state before:
pp: p n : ℕ
hp : Fact (Nat.Prime p)
P : (ZMod p)[X]
hpn : ¬p ∣ n
hP : P ∣ cyclotomic n (ZMod p)
hPdeg : P.natDegree = orderOf (unitOfCoprime p ⋯)
⊢ p.Coprime n

Tactic:
by simpa using hp.1.coprime_iff_not_dvd.mpr hpn

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ HasFDerivAt (↑expMapBasis) (fderiv_expMapBasis K x) x

Tactic:
change HasFDerivAt (expMap ∘ (completeBasis K).equivFunL.symm) (fderiv_expMapBasis K x) x

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ HasFDerivAt (↑expMap ∘ ⇑(completeBasis K).equivFunL.symm) (fderiv_expMapBasis K x) x
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Galois

Goal state before:
pp: n : ℕ
inst✝⁷ : NeZero n
K : Type u_1
inst✝⁶ : Field K
inst✝⁵ : NumberField K
hK : IsCyclotomicExtension {n} ℚ K
m : ℕ
inst✝⁴ : NeZero m
F : Type u_2
inst✝³ : Field F
inst✝² : NumberField F
hF : IsCyclotomicExtension {m} ℚ F
inst✝¹ : Algebra F K
inst✝ : IsGalois ℚ F
h : m ∣ n
σ : Gal(K/ℚ)
hζ : IsPrimitiveRoot (zeta m ℚ F) m
ζ : F := zeta m ℚ F
this : ζ ^ (↑((galEquivZMod m F) (σ.restrictNormal F))).val = ζ ^ (↑((galEquivZMod n K) σ)).val
⊢ (galEquivZMod m F) (σ.restrictNormal F) = (ZMod.unitsMap h) ((galEquivZMod n K) σ)

Tactic:
rw [(hζ.isOfFinOrder (NeZero.ne _)).pow_inj_mod, ← hζ.eq_orderOf,
      ← ZMod.natCast_eq_natCast_iff', ZMod.natCast_val, ZMod.natCast_val, ZMod.cast_id] at this

Goal state after:
n : ℕ
inst✝⁷ : NeZero n
K : Type u_1
inst✝⁶ : Field K
inst✝⁵ : NumberField K
hK : IsCyclotomicExtension {n} ℚ K
m : ℕ
inst✝⁴ : NeZero m
F : Type u_2
inst✝³ : Field F
inst✝² : NumberField F
hF : IsCyclotomicExtension {m} ℚ F
inst✝¹ : Algebra F K
inst✝ : IsGalois ℚ F
h : m ∣ n
σ : Gal(K/ℚ)
hζ : IsPrimitiveRoot (zeta m ℚ F) m
ζ : F := zeta m ℚ F
this : ↑((galEquivZMod m F) (σ.restrictNormal F)) = (↑((galEquivZMod n K) σ)).cast
⊢ (galEquivZMod m F) (σ.restrictNormal F) = (ZMod.unitsMap h) ((galEquivZMod n K) σ)
Module: Mathlib.RingTheory.Unramified.Dedekind

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Module.Finite A B
inst✝² : IsDedekindDomain A
inst✝¹ : IsDomain B
inst✝ : Algebra.FormallyUnramified A B
q : Ideal B
hq : q ≠ ⊥
hqp : q.IsPrime
⊢ IsDiscreteValuationRing (Localization.AtPrime q)

Tactic:
let q' := IsLocalRing.maximalIdeal (Localization.AtPrime q)

Goal state after:
A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Module.Finite A B
inst✝² : IsDedekindDomain A
inst✝¹ : IsDomain B
inst✝ : Algebra.FormallyUnramified A B
q : Ideal B
hq : q ≠ ⊥
hqp : q.IsPrime
q' : Ideal (Localization.AtPrime q) := IsLocalRing.maximalIdeal (Localization.AtPrime q)
⊢ IsDiscreteValuationRing (Localization.AtPrime q)
Module: Mathlib.NumberTheory.SiegelsLemma

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
A : Matrix α β ℤ
i : α
⊢ N i ≤ 0

Tactic:
by
    apply Finset.sum_nonpos
    intro j _
    simp only [mul_neg, Left.neg_nonpos_iff]
    positivity

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
s : ℝ
hs : 0 ≤ s
⊢ ∑ c, ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = c }) / s = ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s }) / s

Tactic:
have : Fintype {I : (Ideal (𝓞 K))⁰ // absNorm (I : Ideal (𝓞 K)) ≤ s} := by
      simp_rw [← Nat.le_floor_iff hs]
      refine @Fintype.ofFinite _ (finite_setOfPred_absNorm_le₀ ⌊s⌋₊)

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
s : ℝ
hs : 0 ≤ s
this : Fintype { I // ↑(absNorm ↑I) ≤ s }
⊢ ∑ c, ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = c }) / s = ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s }) / s
Module: Mathlib.NumberTheory.SiegelsLemma

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
A : Matrix α β ℤ
hn : m < n
hm : 0 < m
x : β → ℤ
hxT : x ∈ T
y : β → ℤ
hyT : y ∈ T
hneq : x ≠ y
hfeq : A *ᵥ x = A *ᵥ y
n_mul_norm_A_pow_e_nonneg : 0 ≤ (↑n * max 1 ‖A‖) ^ e
i : β
j : Unit
⊢ ‖replicateCol Unit (x - y) i j‖ ≤ (↑n * max 1 ‖A‖) ^ e

Tactic:
simp only [replicateCol_apply, Pi.sub_apply]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
A : Matrix α β ℤ
hn : m < n
hm : 0 < m
x : β → ℤ
hxT : x ∈ T
y : β → ℤ
hyT : y ∈ T
hneq : x ≠ y
hfeq : A *ᵥ x = A *ᵥ y
n_mul_norm_A_pow_e_nonneg : 0 ≤ (↑n * max 1 ‖A‖) ^ e
i : β
j : Unit
⊢ ‖x i - y i‖ ≤ (↑n * max 1 ‖A‖) ^ e
Module: Mathlib.NumberTheory.NumberField.House

Goal state before:
pp: case calc_6
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)
inst✝² : Fintype β
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
Apos : 0 ≤ A
this : 0 ≤ c K
kr : α × (K →+* ℂ)
lu : β × (K →+* ℂ)
⊢ c K * A * supOfBasis K ≤ c₂ K * A

Tactic:
· rw [mul_right_comm, c₂]
      gcongr
      exacts [supOfBasis_nonneg _, le_max_right ..]

Goal state after:
Module: Mathlib.NumberTheory.Padics.Measure.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
R : Type u_3
E : Type u_4
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : CommRing R
inst✝³ : TopologicalSpace R
inst✝² : IsTopologicalRing R
inst✝¹ : Module R E
inst✝ : ContinuousSMul R E
μ : AbstractMeasure X R R
ν : AbstractMeasure Y R R
f : C(X × Y, R)
y : Y
⊢ Continuous fun x ↦ f (x, y)

Tactic:
by continuity

Goal state after:
Module: Mathlib.NumberTheory.Padics.Measure.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
R : Type u_3
E : Type u_4
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : CommRing R
inst✝³ : TopologicalSpace R
inst✝² : IsTopologicalRing R
inst✝¹ : Module R E
inst✝ : ContinuousSMul R E
μ : AbstractMeasure X R R
ν : AbstractMeasure Y R R
f : C(X × Y, R)
x : X
⊢ Continuous fun y ↦ f (x, y)

Tactic:
by continuity

Goal state after:
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
⊢ (d * (a.y * b.y)) ^ 2 < (a.x * b.x) ^ 2

Tactic:
mul_pow a.x,

Goal state after:
d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
⊢ (d * (a.y * b.y)) ^ 2 < a.x ^ 2 * b.x ^ 2
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
hM : ∀ (m : ℤ), {q | q.num ^ 2 - d * ↑q.den ^ 2 = m}.Finite
⊢ {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}.Finite

Tactic:
refine (congr_arg _ (ext fun x => ?_)).mp (Finite.biUnion (finite_Ioo (-M) M) fun m _ => hM m)

Goal state after:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
hM : ∀ (m : ℤ), {q | q.num ^ 2 - d * ↑q.den ^ 2 = m}.Finite
x : ℚ
⊢ x ∈ ⋃ i ∈ Ioo (-M) M, {q | q.num ^ 2 - d * ↑q.den ^ 2 = i} ↔ x ∈ {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
⊢ ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1

Tactic:
have hax := a.prop

Goal state after:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
hax : a.x ^ 2 - d * a.y ^ 2 = 1
⊢ ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₂ : 0 < a.y
ax : ℕ
hax : ↑ax ^ 2 - d * a.y ^ 2 = 1
ha₁ : 1 < ax
⊢ ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1

Tactic:
exact ⟨ax, ha₁, a.y, ha₂, hax⟩

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

Goal state before:
pp: K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : CommRing B
P : Ideal B
E : Type u_6
inst✝⁶ : Field E
inst✝⁵ : Algebra E L
inst✝⁴ : MulSemiringAction Gal(L/K) B
E' : Type u_9
inst✝³ : Field E'
inst✝² : Algebra E' L
inst✝¹ : IsInertiaField K L P E
inst✝ : IsInertiaField K L P E'
x : E
⊢ (algebraMap E' L) ((ringEquiv K L P E E') x) = (algebraMap E L) x

Tactic:
simp [IsInertiaField.ringEquiv, IsGaloisGroup.ringEquiv]

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

Goal state before:
pp: K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : CommRing B
P : Ideal B
E : Type u_6
inst✝⁶ : Field E
inst✝⁵ : Algebra E L
inst✝⁴ : MulSemiringAction Gal(L/K) B
E' : Type u_9
inst✝³ : Field E'
inst✝² : Algebra E' L
inst✝¹ : IsInertiaField K L P E
inst✝ : IsInertiaField K L P E'
x : E'
⊢ (algebraMap E L) ((ringEquiv K L P E E').symm x) = (algebraMap E' L) x

Tactic:
simp [IsInertiaField.ringEquiv, IsGaloisGroup.ringEquiv]

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

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁸ : Field K
inst✝¹⁷ : Field L
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
p : Ideal A
P : Ideal B
inst✝¹² : P.LiesOver p
inst✝¹¹ : FiniteDimensional K L
inst✝¹⁰ : MulSemiringAction Gal(L/K) B
inst✝⁹ : IsGaloisGroup Gal(L/K) A B
inst✝⁸ : IsDedekindDomain A
inst✝⁷ : IsDedekindDomain B
inst✝⁶ : Module.Finite A B
inst✝⁵ : Module.IsTorsionFree A B
inst✝⁴ : Ring.HasFiniteQuotients A
inst✝³ : P.IsMaximal
E : Type u_6
inst✝² : Field E
inst✝¹ : Algebra E L
inst✝ : IsInertiaField K L P E
hp : p ≠ ⊥
this✝ : p.IsMaximal
this : Finite (A ⧸ p)
⊢ Module.finrank E L = p.ramificationIdxIn B

Tactic:
← IsGaloisGroup.card_eq_finrank (inertia Gal(L/K) P) E L,

Goal state after:
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁸ : Field K
inst✝¹⁷ : Field L
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
p : Ideal A
P : Ideal B
inst✝¹² : P.LiesOver p
inst✝¹¹ : FiniteDimensional K L
inst✝¹⁰ : MulSemiringAction Gal(L/K) B
inst✝⁹ : IsGaloisGroup Gal(L/K) A B
inst✝⁸ : IsDedekindDomain A
inst✝⁷ : IsDedekindDomain B
inst✝⁶ : Module.Finite A B
inst✝⁵ : Module.IsTorsionFree A B
inst✝⁴ : Ring.HasFiniteQuotients A
inst✝³ : P.IsMaximal
E : Type u_6
inst✝² : Field E
inst✝¹ : Algebra E L
inst✝ : IsInertiaField K L P E
hp : p ≠ ⊥
this✝ : p.IsMaximal
this : Finite (A ⧸ p)
⊢ Nat.card ↥(inertia Gal(L/K) P) = p.ramificationIdxIn B
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: case h.inr
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ a.y → ↑m = a.x → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : ↑x = a.x
hax : 0 < ↑x
hy : 0 < a.y
hx₁ : 1 < a.x
hxx₁ : 0 < (a * a₁⁻¹).x
⊢ ∃ n, a = a₁ ^ n

Tactic:
have hxx₂ := h.mul_inv_x_lt_x hx₁ hy

Goal state after:
case h.inr
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ a.y → ↑m = a.x → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : ↑x = a.x
hax : 0 < ↑x
hy : 0 < a.y
hx₁ : 1 < a.x
hxx₁ : 0 < (a * a₁⁻¹).x
hxx₂ : (a * a₁⁻¹).x < a.x
⊢ ∃ n, a = a₁ ^ n