mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: case succ
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) μ
n : ℕ
ih :
  ↑n ≤ N →
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ n f) = fun w ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w n
hn : ↑(n + 1) ≤ N
w : W
m : Fin (n + 1) → V
⊢ (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ (n + 1) f) w) m =
    (fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w (n + 1)) m

Tactic:
have J : Integrable (fderiv ℝ (iteratedFDeriv ℝ n f)) μ := by
      specialize h'f (n + 1) hn
      rwa [iteratedFDeriv_succ_eq_comp_left, Function.comp_def,
          LinearIsometryEquiv.integrable_comp_iff (𝕜 := ℝ) (φ := fderiv ℝ (iteratedFDeriv ℝ n f))]
        at h'f

Goal state after:
case succ
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) μ
n : ℕ
ih :
  ↑n ≤ N →
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ n f) = fun w ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w n
hn : ↑(n + 1) ≤ N
w : W
m : Fin (n + 1) → V
J : Integrable (fderiv ℝ (iteratedFDeriv ℝ n f)) μ
⊢ (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ (n + 1) f) w) m =
    (fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w (n + 1)) m
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace ℝ G
inst✝³ : NormedSpace 𝕜 G
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace ℝ H
inst✝ : NormedSpace 𝕜 H
L : F →L[𝕜] G
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ‖L‖ * (‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖) ≤ ‖L‖ * (SchwartzMap.seminorm 𝕜 k n) f

Tactic:
grw [le_seminorm 𝕜 k n f x]

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
⊢ ∀ (f : 𝓢(E, F)), ‖f.toBoundedContinuousFunction‖ ≤ 1 * ({0}.sup (schwartzSeminormFamily 𝕜 E F)) f

Tactic:
simpa [BoundedContinuousFunction.norm_le (apply_nonneg _ _)] using norm_le_seminorm 𝕜

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) volume
⊢ ∫ (x : ℝ), f |x| = (∫ (x : ℝ) in Iic 0, f |x|) + ∫ (x : ℝ) in Ioi 0, f |x|

Tactic:
by
        rw [← setIntegral_union (Iic_disjoint_Ioi le_rfl) measurableSet_Ioi int_Iic hf,
          Iic_union_Ioi, restrict_univ]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℝ
ha : a < 0
c : ℝ
⊢ ∫ (x : ℝ) in Ioi c, (Complex.exp ↑(a * x)).re = -(Complex.exp ↑(a * c)).re / a

Tactic:
← RCLike.re_to_complex,

Goal state after:
a : ℝ
ha : a < 0
c : ℝ
⊢ ∫ (x : ℝ) in Ioi c, RCLike.re (Complex.exp ↑(a * x)) = -RCLike.re (Complex.exp ↑(a * c)) / a
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: s : ℂ
t : ℝ
ht : 0 < t
hs : s.re < 0
⊢ s.re - Complex.re 1 < -1

Tactic:
Complex.one_re,

Goal state after:
s : ℂ
t : ℝ
ht : 0 < t
hs : s.re < 0
⊢ s.re - 1 < -1
Module: Mathlib.MeasureTheory.Integral.Asymptotics

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : NormedAddCommGroup E
f : α → E
g : α → F
a : α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F
inst✝² : LinearOrder α
inst✝¹ : CompactIccSpace α
inst✝ : atBot.IsMeasurablyGenerated
hf : LocallyIntegrableOn f (Iic a) μ
ho : f =O[atBot] g
hg : IntegrableAtFilter g atBot μ
⊢ IntegrableOn f (Iic a) μ

Tactic:
by
  refine integrableOn_Iic_iff_integrableAtFilter_atBot.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩
  exact ⟨Iic a, Iic_mem_atBot a, hf.aestronglyMeasurable⟩

Goal state after:
Module: Mathlib.Analysis.Fourier.FourierTransform

Goal state before:
pp: V : Type u_1
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
f : V → E
⊢ 𝓕⁻ f = 𝓕 fun x ↦ f (-x)

Tactic:
ext y
  rw [fourierInv_eq_fourier_neg]
  change 𝓕 f (LinearIsometryEquiv.neg ℝ y) = 𝓕 (f ∘ LinearIsometryEquiv.neg ℝ) y
  exact (fourier_comp_linearIsometry _ _ _).symm

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Asymptotics

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
f : α → E
g : α → F
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : SecondCountableTopology α
inst✝⁸ : MeasurableSpace α
μ : Measure α
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : AddCommGroup α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsOrderedAddMonoid α
inst✝³ : CompactIccSpace α
inst✝² : atTop.IsMeasurablyGenerated
inst✝¹ : MeasurableNeg α
inst✝ : μ.IsNegInvariant
hf : LocallyIntegrable f μ
hsymm : norm ∘ f =ᵐ[μ] norm ∘ f ∘ Neg.neg
ho : f =O[atTop] g
hg : IntegrableAtFilter g atTop μ
h_int : IntegrableOn f (Ici 0) μ
h_map_neg : Measure.map Neg.neg (μ.restrict (Ici 0)) = μ.restrict (Iic 0)
⊢ AEStronglyMeasurable f (Measure.map Neg.neg (μ.restrict (Ici 0)))

Tactic:
exact h_map_neg ▸ hf.aestronglyMeasurable.restrict

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
x : ℝ
hx : x ∈ Ioi 0
⊢ ↑(rexp (-x)) * cexp (↑(Real.log x) * ((starRingEnd ℂ) s - 1)) =
    ↑(rexp (-x)) * cexp (↑(Real.log x) * ((starRingEnd ℂ) s - (starRingEnd ℂ) 1))

Tactic:
map_one

Goal state after:
s : ℂ
x : ℝ
hx : x ∈ Ioi 0
⊢ ↑(rexp (-x)) * cexp (↑(Real.log x) * ((starRingEnd ℂ) s - 1)) =
    ↑(rexp (-x)) * cexp (↑(Real.log x) * ((starRingEnd ℂ) s - 1))
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: x y : ℝ
hxy : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
⊢ (√((x, y).1 ^ 2 + (x, y).2 ^ 2), (Complex.equivRealProd.symm (x, y)).arg) ∈ Ioi 0 ×ˢ Ioo (-π) π

Tactic:
simp only [prodMk_mem_set_prod_eq, mem_Ioi, sqrt_pos, mem_Ioo, Complex.neg_pi_lt_arg,
      true_and, Complex.arg_lt_pi_iff]

Goal state after:
x y : ℝ
hxy : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
⊢ 0 < x ^ 2 + y ^ 2 ∧ (0 ≤ (Complex.equivRealProd.symm (x, y)).re ∨ (Complex.equivRealProd.symm (x, y)).im ≠ 0)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
⊢ IntervalIntegrable (fun x ↦ -(↑(rexp (-x)) * ↑x ^ s)) volume 0 X

Tactic:
convert! (Gamma_integrand_intervalIntegrable (s + 1) _ hX).neg

Goal state after:
s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
x✝ : ℝ
⊢ -(↑(rexp (-x✝)) * ↑x✝ ^ s) = (-fun x ↦ ↑(rexp (-x)) * ↑x ^ (s + 1 - 1)) x✝

s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
⊢ 0 < (s + 1).re
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
⊢ (s + 1).partialGamma X = s * s.partialGamma X - ↑(rexp (-X)) * ↑X ^ s

Tactic:
rw [partialGamma, partialGamma, add_sub_cancel_right]
  have F_der_I : ∀ x : ℝ, x ∈ Ioo 0 X → HasDerivAt (fun x => (-x).exp * x ^ s : ℝ → ℂ)
      (-((-x).exp * x ^ s) + (-x).exp * (s * x ^ (s - 1))) x := by
    intro x hx
    have d1 : HasDerivAt (fun y : ℝ => (-y).exp) (-(-x).exp) x := by
      simpa using! (hasDerivAt_neg x).exp
    have d2 : HasDerivAt (fun y : ℝ => (y : ℂ) ^ s) (s * x ^ (s - 1)) x := by
      have t := @HasDerivAt.cpow_const _ _ _ s (hasDerivAt_id ↑x) ?_
      · simpa only [mul_one] using! t.comp_ofReal
      · exact ofReal_mem_slitPlane.2 hx.1
    simpa only [ofReal_neg, neg_mul] using! d1.ofReal_comp.mul d2
  have cont := (continuous_ofReal.comp continuous_neg.rexp).mul (continuous_ofReal_cpow_const hs)
  have der_ible :=
    (Gamma_integrand_deriv_integrable_A hs hX).add (Gamma_integrand_deriv_integrable_B hs hX)
  have int_eval := integral_eq_sub_of_hasDerivAt_of_le hX cont.continuousOn F_der_I der_ible
  -- We are basically done here but manipulating the output into the right form is fiddly.
  apply_fun fun x : ℂ => -x at int_eval
  rw [intervalIntegral.integral_add (Gamma_integrand_deriv_integrable_A hs hX)
      (Gamma_integrand_deriv_integrable_B hs hX),
    intervalIntegral.integral_neg, neg_add, neg_neg] at int_eval
  rw [eq_sub_of_add_eq int_eval, sub_neg_eq_add, neg_sub, add_comm, add_sub]
  have hn : s ≠ 0 := by contrapose! hs; rw [hs, zero_re]
  simp only [Pi.mul_apply, Function.comp_apply, ofReal_zero, zero_cpow hn, mul_zero, add_zero,
    ← intervalIntegral.integral_const_mul]
  congr with x
  ring

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: case hbc
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ 2 * b.im * y ≤ 2 * |b.im| * |c|

Tactic:
(conv_lhs => rw [mul_assoc])

Goal state after:
case hbc
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ 2 * (b.im * y) ≤ 2 * |b.im| * |c|
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: case hbc
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ y ^ 2 ≤ c ^ 2

Tactic:
· rwa [sq_le_sq]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case succ
s : ℂ
n : ℕ
h1 : -s.re < ↑n
k : ℕ
hk : GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
⊢ GammaAux (⌊1 - s.re⌋₊ + (k + 1)) s = Gamma s

Tactic:
rw [← hk, ← add_assoc]
      refine (GammaAux_recurrence2 s (⌊1 - s.re⌋₊ + k) ?_).symm
      rw [Nat.cast_add]
      have i0 := Nat.sub_one_lt_floor (1 - s.re)
      simp only [sub_sub_cancel_left] at i0
      refine lt_add_of_lt_of_nonneg i0 ?_
      rw [← Nat.cast_zero, Nat.cast_le]; exact Nat.zero_le k

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ (starRingEnd ℂ) s + 1 = (starRingEnd ℂ) s + (starRingEnd ℂ) 1

Tactic:
map_one

Goal state after:
case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ (starRingEnd ℂ) s + 1 = (starRingEnd ℂ) s + 1
Module: Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
hL : Continuous fun p ↦ (L p.1) p.2
⊢ char he hL 0 = 1

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
hL : Continuous fun p ↦ (L p.1) p.2
w : W
⊢ char he hL (-w) = star (char he hL w)

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
hL : Continuous fun p ↦ (L p.1) p.2
w : W
⊢ (charAlgHom he hL).toRingHom (AddMonoidAlgebra.single w 1) = char he hL w

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircleMulti

Goal state before:
pp: case e'_2
d : Type u_1
inst✝ : Fintype d
a : d → ℝ
this :
  Measure.map (fun x i ↦ ↑(x i)) (Measure.map Subtype.val (Measure.comap Subtype.val volume)) =
    Measure.map (fun x i ↦ ↑(↑x i)) (Measure.comap Subtype.val volume)
⊢ volume = Measure.pi fun i ↦ volume

Tactic:
simp [volume, AddCircle.haarAddCircle]

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℂ F
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
f : 𝓢(E, F)
u : 𝓢(E, ℂ)
⊢ (Δ ((toTemperedDistributionCLM E F volume) f)) u = ((toTemperedDistributionCLM E F volume) (Δ f)) u

Tactic:
simp [SchwartzMap.integral_smul_laplacian_right_eq_left]

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
F₁ : Type u_5
F₂ : Type u_6
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : InnerProductSpace ℝ E
inst✝⁷ : FiniteDimensional ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
f : 𝓢'(E, F)
⊢ 𝓕⁻ (𝓕 f) = f

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
F₁ : Type u_5
F₂ : Type u_6
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : InnerProductSpace ℝ E
inst✝⁷ : FiniteDimensional ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
f : 𝓢'(E, F)
⊢ 𝓕 (𝓕⁻ f) = f

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Data.PNat.Prime

Goal state before:
pp: n : ℕ+
⊢ gcd 1 n = 1

Tactic:
rw [gcd_eq_left_iff_dvd]
  apply one_dvd

Goal state after:
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : 0 < k
i : ℕ
hi : IsPrimitiveRoot (ζ ^ i) k
a : ℕ
ha : i = i.gcd k * a
b : ℕ
hb : k = i.gcd k * b
this : b = k
⊢ i.Coprime k

Tactic:
by
    rwa [this, eq_comm, Nat.mul_eq_right h0.ne', ← Nat.coprime_iff_gcd_eq_one] at hb

Goal state after:
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: M : Type u_1
N : Type u_2
F : Type u_6
inst✝³ : CommMonoid M
inst✝² : CommMonoid N
k : ℕ
ζ : M
f : F
inst✝¹ : FunLike F M N
inst✝ : MonoidHomClass F M N
h : IsPrimitiveRoot ζ k
hf : Injective ⇑f
⊢ f 1 = 1

Tactic:
map_one

Goal state after:
M : Type u_1
N : Type u_2
F : Type u_6
inst✝³ : CommMonoid M
inst✝² : CommMonoid N
k : ℕ
ζ : M
f : F
inst✝¹ : FunLike F M N
inst✝ : MonoidHomClass F M N
h : IsPrimitiveRoot ζ k
hf : Injective ⇑f
⊢ 1 = 1
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: M : Type u_1
N : Type u_2
F : Type u_6
inst✝³ : CommMonoid M
inst✝² : CommMonoid N
k : ℕ
ζ : M
f : F
inst✝¹ : FunLike F M N
inst✝ : MonoidHomClass F M N
h : IsPrimitiveRoot (f ζ) k
hf : Injective ⇑f
l : ℕ
hl : f ζ ^ l = f 1
⊢ orderOf (f ζ) ∣ l

Tactic:
map_one

Goal state after:
M : Type u_1
N : Type u_2
F : Type u_6
inst✝³ : CommMonoid M
inst✝² : CommMonoid N
k : ℕ
ζ : M
f : F
inst✝¹ : FunLike F M N
inst✝ : MonoidHomClass F M N
h : IsPrimitiveRoot (f ζ) k
hf : Injective ⇑f
l : ℕ
hl : f ζ ^ l = 1
⊢ orderOf (f ζ) ∣ l
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : CommRing K
inst✝ : IsDomain K
ζ : K
n : ℕ
hpos : 0 < n
h : IsPrimitiveRoot ζ n
prod_monic : (∏ i ∈ n.properDivisors, cyclotomic' i K).Monic
⊢ ⊥ < (∏ i ∈ n.properDivisors, cyclotomic' i K).degree

Tactic:
bot_lt_iff_ne_bot

Goal state after:
K : Type u_2
inst✝¹ : CommRing K
inst✝ : IsDomain K
ζ : K
n : ℕ
hpos : 0 < n
h : IsPrimitiveRoot ζ n
prod_monic : (∏ i ∈ n.properDivisors, cyclotomic' i K).Monic
⊢ (∏ i ∈ n.properDivisors, cyclotomic' i K).degree ≠ ⊥
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: R : Type u_4
ζ : R
inst✝² : CommRing R
inst✝¹ : IsDomain R
p : ℕ := ringChar R
n : ℕ
inst✝ : NeZero (p * n)
hζ : IsPrimitiveRoot ζ (p * n)
h : p ≠ 0 ∧ n ≠ 0
this : NeZero p
hp : Fact (Nat.Prime p)
⊢ 1 = (frobenius R p) 1

Tactic:
map_one

Goal state after:
R : Type u_4
ζ : R
inst✝² : CommRing R
inst✝¹ : IsDomain R
p : ℕ := ringChar R
n : ℕ
inst✝ : NeZero (p * n)
hζ : IsPrimitiveRoot ζ (p * n)
h : p ≠ 0 ∧ n ≠ 0
this : NeZero p
hp : Fact (Nat.Prime p)
⊢ 1 = 1
Module: Mathlib.RingTheory.RootsOfUnity.Complex

Goal state before:
pp: case e'_2.e'_5
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
h₂ : ¬i * 2 ≤ n
⊢ ↑n ≠ 0

Tactic:
exact mod_cast hn

Goal state after:
Module: Mathlib.RingTheory.RootsOfUnity.Complex

Goal state before:
pp: case e'_2.e'_6
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
h₂ : ¬i * 2 ≤ n
⊢ ↑n ≠ 0

Tactic:
exact mod_cast hn

Goal state after:
Module: Mathlib.RingTheory.RootsOfUnity.Minpoly

Goal state before:
pp: case h.right
n : ℕ
K : Type u_1
inst✝ : CommRing K
μ : K
h : IsPrimitiveRoot μ n
hpos : 0 < n
⊢ eval₂ (algebraMap ℤ K) μ (X ^ n - 1) = 0

Tactic:
simp only [((IsPrimitiveRoot.iff_def μ n).mp h).left, eval₂_one, eval₂_X_pow, eval₂_sub,
      sub_self]

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

Goal state before:
pp: n : ℕ
R : Type u_1
inst✝ : Ring R
⊢ (map (Int.castRingHom R) (cyclotomic n ℤ)).Monic

Tactic:
exact (int_cyclotomic_spec n).2.2.map _

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
a✝ : Nontrivial R
⊢ cyclotomic n R = (X ^ n - 1) /ₘ ∏ i ∈ n.properDivisors, cyclotomic i R

Tactic:
rw [← prod_cyclotomic_eq_X_pow_sub_one hpos, ← Nat.cons_self_properDivisors hpos.ne',
    Finset.prod_cons]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
a✝ : Nontrivial R
⊢ cyclotomic n R = (cyclotomic n R * ∏ x ∈ n.properDivisors, cyclotomic x R) /ₘ ∏ i ∈ n.properDivisors, cyclotomic i R
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: case right
R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
a✝ : Nontrivial R
prod_monic : (∏ i ∈ n.properDivisors, cyclotomic i R).Monic
⊢ ⊥ < (∏ i ∈ n.properDivisors, cyclotomic i R).degree

Tactic:
bot_lt_iff_ne_bot

Goal state after:
case right
R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
a✝ : Nontrivial R
prod_monic : (∏ i ∈ n.properDivisors, cyclotomic i R).Monic
⊢ (∏ i ∈ n.properDivisors, cyclotomic i R).degree ≠ ⊥
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
P : R[X]
a✝ : Nontrivial R
hP : P * ∏ i ∈ n.properDivisors, cyclotomic i R = X ^ n - 1
prod_monic : (∏ i ∈ n.properDivisors, cyclotomic i R).Monic
⊢ degree 0 < (∏ i ∈ n.properDivisors, cyclotomic i R).degree

Tactic:
degree_zero,

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
P : R[X]
a✝ : Nontrivial R
hP : P * ∏ i ∈ n.properDivisors, cyclotomic i R = X ^ n - 1
prod_monic : (∏ i ∈ n.properDivisors, cyclotomic i R).Monic
⊢ ⊥ < (∏ i ∈ n.properDivisors, cyclotomic i R).degree
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
P : R[X]
a✝ : Nontrivial R
hP : P * ∏ i ∈ n.properDivisors, cyclotomic i R = X ^ n - 1
prod_monic : (∏ i ∈ n.properDivisors, cyclotomic i R).Monic
⊢ ⊥ < (∏ i ∈ n.properDivisors, cyclotomic i R).degree

Tactic:
bot_lt_iff_ne_bot

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
P : R[X]
a✝ : Nontrivial R
hP : P * ∏ i ∈ n.properDivisors, cyclotomic i R = X ^ n - 1
prod_monic : (∏ i ∈ n.properDivisors, cyclotomic i R).Monic
⊢ (∏ i ∈ n.properDivisors, cyclotomic i R).degree ≠ ⊥
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m
hmzero : NeZero m
hprim : n = orderOf (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n))
⊢ n = m

Tactic:
← IsPrimitiveRoot.eq_orderOf hroot

Goal state after:
case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m
hmzero : NeZero m
hprim : n = m
⊢ n = m
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: S T : Set ℕ
A : Type u
B : Type v
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
C : Type w
inst✝³ : CommRing C
inst✝² : Algebra A C
inst✝¹ : Algebra B C
inst✝ : IsScalarTower A B C
hS : IsCyclotomicExtension S A B
hT : IsCyclotomicExtension T B C
h : Function.Injective ⇑(algebraMap B C)
x : C
b : B
f : B →ₐ[A] C := IsScalarTower.toAlgHom A B C
hb : (algebraMap B C) b ∈ adjoin A (⇑f '' {b | ∃ a ∈ S, a ≠ 0 ∧ b ^ a = 1})
y : C
b₁ : B
h₁ : f b₁ = y
n : ℕ
hn : n ∈ S ∧ n ≠ 0 ∧ b₁ ^ n = 1
⊢ f 1 = 1

Tactic:
map_one

Goal state after:
S T : Set ℕ
A : Type u
B : Type v
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
C : Type w
inst✝³ : CommRing C
inst✝² : Algebra A C
inst✝¹ : Algebra B C
inst✝ : IsScalarTower A B C
hS : IsCyclotomicExtension S A B
hT : IsCyclotomicExtension T B C
h : Function.Injective ⇑(algebraMap B C)
x : C
b : B
f : B →ₐ[A] C := IsScalarTower.toAlgHom A B C
hb : (algebraMap B C) b ∈ adjoin A (⇑f '' {b | ∃ a ∈ S, a ≠ 0 ∧ b ^ a = 1})
y : C
b₁ : B
h₁ : f b₁ = y
n : ℕ
hn : n ∈ S ∧ n ≠ 0 ∧ b₁ ^ n = 1
⊢ 1 = 1
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case refine_1.inr
S : Set ℕ
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
H : IsCyclotomicExtension S A B
s : ℕ
hs' : s ≠ 0
inst✝ : NeZero s
x : ℕ
hm : s * x ∈ S
hm' : s * x ≠ 0
ζ : B
hζ : IsPrimitiveRoot ζ (s * x)
h_xnz : x ≠ 0
⊢ IsPrimitiveRoot (ζ ^ x) s

Tactic:
have := hζ.pow_of_dvd h_xnz (dvd_mul_left x s)

Goal state after:
case refine_1.inr
S : Set ℕ
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
H : IsCyclotomicExtension S A B
s : ℕ
hs' : s ≠ 0
inst✝ : NeZero s
x : ℕ
hm : s * x ∈ S
hm' : s * x ≠ 0
ζ : B
hζ : IsPrimitiveRoot ζ (s * x)
h_xnz : x ≠ 0
this : IsPrimitiveRoot (ζ ^ x) (s * x / x)
⊢ IsPrimitiveRoot (ζ ^ x) s