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

Goal state before:
pp: V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
A : Tendsto (fun c ↦ ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
c : ℝ
⊢ c ∈ Set.Ioi 0 →
    ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫) • 𝓕 f w =
      ∫ (w : V), 𝓕 (fun w ↦ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫)) w • f w

Tactic:
(hc : 0 < c)

Goal state after:
V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
A : Tendsto (fun c ↦ ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
c : ℝ
hc : 0 < c
⊢ ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫) • 𝓕 f w =
    ∫ (w : V), 𝓕 (fun w ↦ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫)) w • f w
Module: Mathlib.Analysis.Fourier.Inversion

Goal state before:
pp: V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
A : Tendsto (fun c ↦ ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
B :
  Tendsto (fun c ↦ ∫ (w : V), 𝓕 (fun w ↦ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫)) w • f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
c : ℝ
⊢ c ∈ Set.Ioi 0 →
    ∫ (w : V), 𝓕 (fun w ↦ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫)) w • f w =
      ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w

Tactic:
(hc : 0 < c)

Goal state after:
V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
A : Tendsto (fun c ↦ ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
B :
  Tendsto (fun c ↦ ∫ (w : V), 𝓕 (fun w ↦ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫)) w • f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
c : ℝ
hc : 0 < c
⊢ ∫ (w : V), 𝓕 (fun w ↦ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫)) w • f w =
    ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
hs : 0 < s
this : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0
⊢ ENNReal.ofReal 0 < ⊤

Tactic:
exact ENNReal.ofReal_lt_top

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

Goal state before:
pp: case a
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
⊢ -↑n✝ < s + 1

Tactic:
· rw [Nat.cast_add, Nat.cast_one, neg_add] at hs'
        linarith

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

Goal state before:
pp: V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
v : V
h'f : ContinuousAt f v
φ : V → ℝ := fun w ↦ π ^ (↑(finrank ℝ V) / 2) * rexp (-π ^ 2 * ‖w‖ ^ 2)
A : Tendsto (fun c ↦ ∫ (w : V), (c ^ finrank ℝ V * φ (c • (v - w))) • f w) atTop (𝓝 (f v))
B : Tendsto (fun c ↦ ∫ (w : V), ((c ^ (1 / 2)) ^ finrank ℝ V * φ (c ^ (1 / 2) • (v - w))) • f w) atTop (𝓝 (f v))
c : ℝ
⊢ c ∈ Set.Ioi 0 →
    ∫ (w : V), ((c ^ (1 / 2)) ^ finrank ℝ V * φ (c ^ (1 / 2) • (v - w))) • f w =
      ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w

Tactic:
(hc : 0 < c)

Goal state after:
V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
v : V
h'f : ContinuousAt f v
φ : V → ℝ := fun w ↦ π ^ (↑(finrank ℝ V) / 2) * rexp (-π ^ 2 * ‖w‖ ^ 2)
A : Tendsto (fun c ↦ ∫ (w : V), (c ^ finrank ℝ V * φ (c • (v - w))) • f w) atTop (𝓝 (f v))
B : Tendsto (fun c ↦ ∫ (w : V), ((c ^ (1 / 2)) ^ finrank ℝ V * φ (c ^ (1 / 2) • (v - w))) • f w) atTop (𝓝 (f v))
c : ℝ
hc : 0 < c
⊢ ∫ (w : V), ((c ^ (1 / 2)) ^ finrank ℝ V * φ (c ^ (1 / 2) • (v - w))) • f w =
    ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w
Module: Mathlib.Analysis.Distribution.FourierMultiplier

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
F : Type u_4
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : InnerProductSpace ℝ E
inst✝⁶ : NormedSpace ℂ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℂ 𝕜 F
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : CompleteSpace F
g₁ g₂ : E → 𝕜
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
⊢ fourierMultiplierCLM F g₁ ∘SL fourierMultiplierCLM F g₂ = fourierMultiplierCLM F (g₁ * g₂)

Tactic:
ext1 f
  simp [fourierMultiplierCLM_fourierMultiplierCLM_apply hg₁ hg₂]

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

Goal state before:
pp: V : Type u_1
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
b : ℝ
hb : 0 < b
⊢ ∫ (v : V), rexp (-b * ‖v‖ ^ 2) = (π / b) ^ (↑(Module.finrank ℝ V) / 2)

Tactic:
rw [← ofReal_inj]

Goal state after:
V : Type u_1
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
b : ℝ
hb : 0 < b
⊢ ↑(∫ (v : V), rexp (-b * ‖v‖ ^ 2)) = ↑((π / b) ^ (↑(Module.finrank ℝ V) / 2))
Module: Mathlib.Analysis.Distribution.FourierMultiplier

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : NormedSpace ℂ F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
g₁ g₂ : E → ℂ
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
⊢ fourierMultiplierCLM F g₂ ∘SL fourierMultiplierCLM F g₁ = fourierMultiplierCLM F (g₁ * g₂)

Tactic:
ext1 f
  simp [fourierMultiplierCLM_fourierMultiplierCLM_apply hg₁ hg₂]

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

Goal state before:
pp: ι : Type u_1
E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : NormedSpace ℂ F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
g : ι → E → ℂ
s : Finset ι
hg : ∀ i ∈ s, Function.HasTemperateGrowth (g i)
⊢ (fourierMultiplierCLM F fun x ↦ ∑ i ∈ s, g i x) = ∑ i ∈ s, fourierMultiplierCLM F (g i)

Tactic:
ext f u

Goal state after:
ι : Type u_1
E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : NormedSpace ℂ F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
g : ι → E → ℂ
s : Finset ι
hg : ∀ i ∈ s, Function.HasTemperateGrowth (g i)
f : 𝓢'(E, F)
u : 𝓢(E, ℂ)
⊢ ((fourierMultiplierCLM F fun x ↦ ∑ i ∈ s, g i x) f) u = ((∑ i ∈ s, fourierMultiplierCLM F (g i)) f) u
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g : E → ℂ
hg : Function.HasTemperateGrowth g
c : ℂ
⊢ smulLeftCLM F (c • g) = c • smulLeftCLM F g

Tactic:
ext f u

Goal state after:
E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g : E → ℂ
hg : Function.HasTemperateGrowth g
c : ℂ
f : 𝓢'(E, F)
u : 𝓢(E, ℂ)
⊢ ((smulLeftCLM F (c • g)) f) u = ((c • smulLeftCLM F g) f) u
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g₁ g₂ : E → ℂ
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
⊢ smulLeftCLM F (g₁ + g₂) = smulLeftCLM F g₁ + smulLeftCLM F g₂

Tactic:
ext f u

Goal state after:
E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g₁ g₂ : E → ℂ
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
f : 𝓢'(E, F)
u : 𝓢(E, ℂ)
⊢ ((smulLeftCLM F (g₁ + g₂)) f) u = ((smulLeftCLM F g₁ + smulLeftCLM F g₂) f) u
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g₁ g₂ : E → ℂ
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
⊢ smulLeftCLM F (g₁ - g₂) = smulLeftCLM F g₁ - smulLeftCLM F g₂

Tactic:
ext f u

Goal state after:
E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g₁ g₂ : E → ℂ
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
f : 𝓢'(E, F)
u : 𝓢(E, ℂ)
⊢ ((smulLeftCLM F (g₁ - g₂)) f) u = ((smulLeftCLM F g₁ - smulLeftCLM F g₂) f) u
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g : E → ℂ
hg : Function.HasTemperateGrowth g
⊢ smulLeftCLM F (-g) = -smulLeftCLM F g

Tactic:
ext f u

Goal state after:
E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g : E → ℂ
hg : Function.HasTemperateGrowth g
f : 𝓢'(E, F)
u : 𝓢(E, ℂ)
⊢ ((smulLeftCLM F (-g)) f) u = ((-smulLeftCLM F g) f) u
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g : ι → E → ℂ
s : Finset ι
hg : ∀ i ∈ s, Function.HasTemperateGrowth (g i)
⊢ (smulLeftCLM F fun x ↦ ∑ i ∈ s, g i x) = ∑ i ∈ s, smulLeftCLM F (g i)

Tactic:
ext f u

Goal state after:
ι : Type u_1
E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g : ι → E → ℂ
s : Finset ι
hg : ∀ i ∈ s, Function.HasTemperateGrowth (g i)
f : 𝓢'(E, F)
u : 𝓢(E, ℂ)
⊢ ((smulLeftCLM F fun x ↦ ∑ i ∈ s, g i x) f) u = ((∑ i ∈ s, smulLeftCLM F (g i)) f) u
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
⊢ MemSobolev s 2 ((besselPotential E F (-2)) (Δ f))

Tactic:
besselPotential_neg_two_laplacian_eq f

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
⊢ MemSobolev s 2 (-(2 * π) ^ 2 • (fourierMultiplierCLM F fun x ↦ ↑(‖x‖ ^ 2 * (1 + ‖x‖ ^ 2) ^ (-1))) f)
Module: Mathlib.Data.PNat.Prime

Goal state before:
pp: k m n : ℕ+
⊢ (↑m).Coprime ↑k → (↑n).Coprime ↑k → (↑m * ↑n).Coprime ↑k

Tactic:
apply Nat.Coprime.mul_left

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Moebius

Goal state before:
pp: ⊢ μ * ↑ζ = 1

Tactic:
by
  ext n
  induction n using recOnPosPrimePosCoprime with
  | zero => rw [map_zero, map_zero]
  | one => simp
  | prime_pow p n hp hn =>
    rw [coe_mul_zeta_apply, sum_divisors_prime_pow hp, sum_range_succ']
    simp [moebius_apply_prime_pow, hp.ne_one, hn.ne', hp, hn]
  | coprime a b _ha _hb hab ha' hb' =>
    rw [IsMultiplicative.map_mul_of_coprime _ hab, ha', hb',
      IsMultiplicative.map_mul_of_coprime isMultiplicative_one hab]
    exact isMultiplicative_moebius.mul isMultiplicative_zeta.natCast

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

Goal state before:
pp: case mp
M : Type u_1
inst✝ : CommMonoid M
ζ : M
h : IsPrimitiveRoot ζ 1
⊢ ζ = 1

Tactic:
rw [← pow_one ζ, h.pow_eq_one]

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

Goal state before:
pp: case neg
G : Type u_3
inst✝ : DivisionCommMonoid G
k : ℕ
ζ : G
h : IsPrimitiveRoot ζ k
l : ℤ
h0 : ¬0 ≤ l
l' : ℕ
hl' : ↑l' = -l
⊢ ζ ^ l = 1 ↔ ↑k ∣ l

Tactic:
rw [← dvd_neg, ← hl']

Goal state after:
case neg
G : Type u_3
inst✝ : DivisionCommMonoid G
k : ℕ
ζ : G
h : IsPrimitiveRoot ζ k
l : ℤ
h0 : ¬0 ≤ l
l' : ℕ
hl' : ↑l' = -l
⊢ ζ ^ l = 1 ↔ ↑k ∣ ↑l'
Module: Mathlib.RingTheory.RootsOfUnity.Complex

Goal state before:
pp: n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
⊢ ∃ i_1,
    (Complex.exp (↑i / ↑n * (2 * ↑Real.pi) * Complex.I)).arg = ↑i_1 / ↑n * (2 * Real.pi) ∧
      IsCoprime i_1 ↑n ∧ i_1.natAbs < n

Tactic:
Complex.exp_mul_I

Goal state after:
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
⊢ ∃ i_1,
    (Complex.cos (↑i / ↑n * (2 * ↑Real.pi)) + Complex.sin (↑i / ↑n * (2 * ↑Real.pi)) * Complex.I).arg =
        ↑i_1 / ↑n * (2 * Real.pi) ∧
      IsCoprime i_1 ↑n ∧ i_1.natAbs < n
Module: Mathlib.RingTheory.RootsOfUnity.Minpoly

Goal state before:
pp: case inr
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hdiv : ¬p ∣ n
hpos : n > 0
this : IsIntegrallyClosed ℤ := GCDMonoid.toIsIntegrallyClosed
⊢ eval μ (map (algebraMap ℤ K) ((minpoly ℤ (μ ^ p)).comp (X ^ p))) = 0

Tactic:
map_comp,

Goal state after:
case inr
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hdiv : ¬p ∣ n
hpos : n > 0
this : IsIntegrallyClosed ℤ := GCDMonoid.toIsIntegrallyClosed
⊢ eval μ ((map (algebraMap ℤ K) (minpoly ℤ (μ ^ p))).comp (map (algebraMap ℤ K) (X ^ p))) = 0
Module: Mathlib.RingTheory.RootsOfUnity.Minpoly

Goal state before:
pp: case refine_2
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
hn : ¬n = 0
hpos : 0 < n
P : ℤ[X] := ⋯
Q : ℤ[X] := ⋯
hdiff : ¬P = Q
Pmonic : P.Monic
Qmonic : Q.Monic
Pirr : Irreducible P
Qirr : Irreducible Q
PQprim : (P * Q).IsPrimitive
⊢ P ∣ X ^ n - 1

Tactic:
exact minpoly_dvd_x_pow_sub_one h

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

Goal state before:
pp: case neg.inl
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
hn : ¬n = 0
hpos : 0 < n
P : ℤ[X] := minpoly ℤ μ
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hdiff : ¬P = Q
Pmonic : P.Monic
Qmonic : Q.Monic
Pirr : Irreducible P
Qirr : Irreducible Q
PQprim : (P * Q).IsPrimitive
R : (ZMod p)[X]
prod : map (Int.castRingHom (ZMod p)) P ^ 2 * R ∣ X ^ n - 1
hR : map (Int.castRingHom (ZMod p)) (minpoly ℤ (μ ^ p)) = map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) * R
habs : ↑1 < emultiplicity (map (Int.castRingHom (ZMod p)) P) (X ^ n - 1)
hfree : Squarefree (X ^ n - 1)
hle : emultiplicity (map (Int.castRingHom (ZMod p)) P) (X ^ n - 1) ≤ 1
⊢ False

pp: case neg.inr
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
hn : ¬n = 0
hpos : 0 < n
P : ℤ[X] := minpoly ℤ μ
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hdiff : ¬P = Q
Pmonic : P.Monic
Qmonic : Q.Monic
Pirr : Irreducible P
Qirr : Irreducible Q
PQprim : (P * Q).IsPrimitive
R : (ZMod p)[X]
prod : map (Int.castRingHom (ZMod p)) P ^ 2 * R ∣ X ^ n - 1
hR : map (Int.castRingHom (ZMod p)) (minpoly ℤ (μ ^ p)) = map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) * R
habs : ↑1 < emultiplicity (map (Int.castRingHom (ZMod p)) P) (X ^ n - 1)
hfree : Squarefree (X ^ n - 1)
hunit : IsUnit (map (Int.castRingHom (ZMod p)) P)
⊢ False

Tactic:
· rw [Nat.cast_one] at habs; exact hle.not_gt habs

Goal state after:
case neg.inr
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
hn : ¬n = 0
hpos : 0 < n
P : ℤ[X] := minpoly ℤ μ
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hdiff : ¬P = Q
Pmonic : P.Monic
Qmonic : Q.Monic
Pirr : Irreducible P
Qirr : Irreducible Q
PQprim : (P * Q).IsPrimitive
R : (ZMod p)[X]
prod : map (Int.castRingHom (ZMod p)) P ^ 2 * R ∣ X ^ n - 1
hR : map (Int.castRingHom (ZMod p)) (minpoly ℤ (μ ^ p)) = map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) * R
habs : ↑1 < emultiplicity (map (Int.castRingHom (ZMod p)) P) (X ^ n - 1)
hfree : Squarefree (X ^ n - 1)
hunit : IsUnit (map (Int.castRingHom (ZMod p)) P)
⊢ False
Module: Mathlib.Analysis.Fourier.RiemannLebesgueLemma

Goal state before:
pp: E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
⊢ Tendsto (fun w ↦ ∫ (v : V), 𝐞 (-w v) • f v ∂μ) (cocompact (StrongDual ℝ V)) (𝓝 0)

Tactic:
rw [μ.isAddLeftInvariant_eq_smul volume]

Goal state after:
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
⊢ Tendsto (fun w ↦ ∫ (v : V), 𝐞 (-w v) • f v ∂μ.addHaarScalarFactor volume • volume) (cocompact (StrongDual ℝ V)) (𝓝 0)
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst✝⁷ : CommMonoid M
inst✝⁶ : CommMonoid N
inst✝⁵ : DivisionCommMonoid G
k l : ℕ
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst✝² : CommRing R
inst✝¹ : Algebra R S
inst✝ : NeZero n
μ' : ↥(rootsOfUnity n S) := hμ.toRootsOfUnity
ho : orderOf μ' = n
h1 : ∃ m, ↑1 ↑↑μ' = ↑↑μ' ^ m
h : μ' ^ 1 = μ' ^ h1.choose
⊢ ↑h1.choose = 1

Tactic:
convert! ho ▸ (ZMod.natCast_eq_natCast_iff ..).mpr (pow_eq_pow_iff_modEq.mp h).symm

Goal state after:
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst✝⁷ : CommMonoid M
inst✝⁶ : CommMonoid N
inst✝⁵ : DivisionCommMonoid G
k l : ℕ
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst✝² : CommRing R
inst✝¹ : Algebra R S
inst✝ : NeZero n
μ' : ↥(rootsOfUnity n S) := hμ.toRootsOfUnity
ho : orderOf μ' = n
h1 : ∃ m, ↑1 ↑↑μ' = ↑↑μ' ^ m
h : μ' ^ 1 = μ' ^ h1.choose
⊢ 1 = ↑1
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: n : ℕ
h : 0 < n
R : Type u_2
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
⊢ ζ ^ n = 1 ↔ ∃ i ∈ n.divisors, (cyclotomic i R).IsRoot ζ

Tactic:
by
  rw [← mem_nthRoots h, nthRoots, mem_roots <| X_pow_sub_C_ne_zero h _, C_1, ←
      prod_cyclotomic_eq_X_pow_sub_one h, isRoot_prod]

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

Goal state before:
pp: n : ℕ
K : Type u_2
inst✝¹ : Field K
μ : K
inst✝ : NeZero ↑n
hnpos : 0 < n
hμ : (cyclotomic n K).IsRoot μ
hμn : orderOf μ ∣ n
hnμ : orderOf μ ≠ n
ho : 0 < orderOf μ
i : ℕ
hiμ : (cyclotomic i K).IsRoot μ
hio : i ∣ orderOf μ
key : i < n
key' : i ∣ n
⊢ False

Tactic:
← Polynomial.dvd_iff_isRoot

Goal state after:
n : ℕ
K : Type u_2
inst✝¹ : Field K
μ : K
inst✝ : NeZero ↑n
hnpos : 0 < n
hμ : X - C μ ∣ cyclotomic n K
hμn : orderOf μ ∣ n
hnμ : orderOf μ ≠ n
ho : 0 < orderOf μ
i : ℕ
hiμ : X - C μ ∣ cyclotomic i K
hio : i ∣ orderOf μ
key : i < n
key' : i ∣ n
⊢ False
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]
⊢ P = cyclotomic n R ↔ P * ∏ i ∈ n.properDivisors, cyclotomic i R = X ^ n - 1

Tactic:
by
  nontriviality R
  refine ⟨fun hcycl => ?_, fun hP => ?_⟩
  · rw [hcycl, ← prod_cyclotomic_eq_X_pow_sub_one hpos R, ← Nat.cons_self_properDivisors hpos.ne',
      Finset.prod_cons]
  · have prod_monic : (∏ i ∈ Nat.properDivisors n, cyclotomic i R).Monic := by
      apply monic_prod_of_monic
      intro i _
      exact cyclotomic.monic i R
    rw [@cyclotomic_eq_X_pow_sub_one_div R _ _ hpos, (div_modByMonic_unique P 0 prod_monic _).1]
    refine ⟨by rwa [zero_add, mul_comm], ?_⟩
    rw [degree_zero, bot_lt_iff_ne_bot]
    intro h
    exact Monic.ne_zero prod_monic (degree_eq_bot.1 h)

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

Goal state before:
pp: R : Type u_4
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
η : R
⊢ η ∈ nthRootsFinset p 1 ↔ IsPrimitiveRoot η p ∨ η = 1

Tactic:
by
  classical
  simp [nthRootsFinset_eq_of_prime hp, mem_primitiveRoots hp.pos, or_comm]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
j : ℕ
hj : j ∈ n.properDivisors.erase 1
⊢ 1 < j

Tactic:
exact (Ne.le_iff_lt (Finset.mem_erase.1 hj).1.symm).mp
        (Nat.succ_le_of_lt (Nat.pos_of_mem_properDivisors (Finset.mem_erase.1 hj).2))

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

Goal state before:
pp: case inr
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ (expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R

Tactic:
suffices expand ℤ p (cyclotomic n ℤ) = cyclotomic (n * p) ℤ * cyclotomic n ℤ by
    rw [← map_cyclotomic_int, ← map_expand, this, Polynomial.map_mul, map_cyclotomic_int,
      map_cyclotomic]

Goal state after:
case inr
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ (expand ℤ p) (cyclotomic n ℤ) = cyclotomic (n * p) ℤ * cyclotomic n ℤ
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: p n : ℕ
hp : Nat.Prime p
hdiv : p ∣ n
R : Type u_1
inst✝ : CommRing R
⊢ (expand R p) (cyclotomic n R) = cyclotomic (n * p) R

Tactic:
rcases n.eq_zero_or_pos with (rfl | hzero)

Goal state after:
case inl
p : ℕ
hp : Nat.Prime p
R : Type u_1
inst✝ : CommRing R
hdiv : p ∣ 0
⊢ (expand R p) (cyclotomic 0 R) = cyclotomic (0 * p) R

case inr
p n : ℕ
hp : Nat.Prime p
hdiv : p ∣ n
R : Type u_1
inst✝ : CommRing R
hzero : n > 0
⊢ (expand R p) (cyclotomic n R) = cyclotomic (n * p) R