mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
P : ℕ → X →L[𝕜] X
hrank : ∀ (n : ℕ), Module.finrank 𝕜 ↥(↑(P n)).range = n
hcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x
n : ℕ
U : Submodule 𝕜 X := (↑(succSub P n)).range
V : Submodule 𝕜 X := (↑(P n)).range
W : Submodule 𝕜 X := (↑(P (n + 1))).range
hV : V ≤ W
hUW : U ≤ W
hW : W = U ⊔ V
x : X
x✝ : x ∈ U ⊓ V
y : X
hy : ↑(succSub P n) y = x
z : X
hz : ↑(P n) z = x
⊢ (P n) (↑(succSub P n) y) = 0

Tactic:
ContinuousLinearMap.coe_coe

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
P : ℕ → X →L[𝕜] X
hrank : ∀ (n : ℕ), Module.finrank 𝕜 ↥(↑(P n)).range = n
hcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x
n : ℕ
U : Submodule 𝕜 X := (↑(succSub P n)).range
V : Submodule 𝕜 X := (↑(P n)).range
W : Submodule 𝕜 X := (↑(P (n + 1))).range
hV : V ≤ W
hUW : U ≤ W
hW : W = U ⊔ V
x : X
x✝ : x ∈ U ⊓ V
y : X
hy : ↑(succSub P n) y = x
z : X
hz : ↑(P n) z = x
⊢ (P n) ((succSub P n) y) = 0
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : AddCircle T → E
⊢ ∫ (t : AddCircle T), f t ∂haarAddCircle = T⁻¹ • ∫ (t : AddCircle T), f t

Tactic:
volume_eq_smul_haarAddCircle,

Goal state after:
T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : AddCircle T → E
⊢ ∫ (t : AddCircle T), f t ∂haarAddCircle = T⁻¹ • ∫ (t : AddCircle T), f t ∂ENNReal.ofReal T • haarAddCircle
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: 𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
K : Submodule 𝕜 E
hK : CompleteSpace ↥K
this : ∀ (b : Bool), CompleteSpace ↥(bif b then K else Kᗮ)
⊢ ⊤ ≤ (⨆ i, bif i then K else Kᗮ).topologicalClosure

Tactic:
refine le_trans ?_ (Submodule.le_topologicalClosure _)

Goal state after:
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
K : Submodule 𝕜 E
hK : CompleteSpace ↥K
this : ∀ (b : Bool), CompleteSpace ↥(bif b then K else Kᗮ)
⊢ ⊤ ≤ ⨆ i, bif i then K else Kᗮ
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
n : ℤ
a : ℝ
this : ∀ (x : ℝ), (fourier (-n)) ↑x • f ↑x = (fun z ↦ (fourier (-n)) z • f z) ↑x
⊢ ∫ (t : AddCircle T), (fourier (-n)) t • f t ∂haarAddCircle = (1 / T) • ∫ (b : AddCircle T), (fourier (-n)) b • f b

Tactic:
volume_eq_smul_haarAddCircle,

Goal state after:
T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
n : ℤ
a : ℝ
this : ∀ (x : ℝ), (fourier (-n)) ↑x • f ↑x = (fun z ↦ (fourier (-n)) z • f z) ↑x
⊢ ∫ (t : AddCircle T), (fourier (-n)) t • f t ∂haarAddCircle =
    (1 / T) • ∫ (b : AddCircle T), (fourier (-n)) b • f b ∂ENNReal.ofReal T • haarAddCircle
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ 1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f x * (fourier (-n)) ↑x =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))

Tactic:
rw [integral_mul_deriv_eq_deriv_mul_of_hasDeriv_right hf
    (fun x _ ↦ has_antideriv_at_fourier_neg hT hn x |>.continuousAt |>.continuousWithinAt) hff'
    (fun x _ ↦ has_antideriv_at_fourier_neg hT hn x |>.hasDerivWithinAt) hf'
    (((map_continuous (fourier (-n))).comp (AddCircle.continuous_mk' _)).intervalIntegrable _ _)]

Goal state after:
a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, f' x * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (-⇑f) x‖ = ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖

Tactic:
by
  rw [iteratedFDeriv_neg_apply, norm_neg]

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

Goal state before:
pp: case h
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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f g : 𝓢(E, F)
k n : ℕ
x : E
⊢ ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f + ⇑g) x‖ ≤
    @SchwartzMap.seminormAux E F inst✝³ inst✝² inst✝¹ inst✝ k n f +
      @SchwartzMap.seminormAux E F inst✝³ inst✝² inst✝¹ inst✝ k n g

Tactic:
decay_add_le_aux k n f g x,

Goal state after:
case h
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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f g : 𝓢(E, F)
k n : ℕ
x : E
⊢ ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ + ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑g) x‖ ≤
    @SchwartzMap.seminormAux E F inst✝³ inst✝² inst✝¹ inst✝ k n f +
      @SchwartzMap.seminormAux E F inst✝³ inst✝² inst✝¹ inst✝ k n g
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✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace ℝ E
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace ℝ F
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : NormedField 𝕜'
inst✝⁸ : NormedAddCommGroup D
inst✝⁷ : NormedSpace ℝ D
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace 𝕜' G
inst✝¹ : SMulCommClass ℝ 𝕜' G
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
A : 𝓢(D, E) → G
hadd : ∀ (f g : 𝓢(D, E)), A (f + g) = A f + A g
hsmul : ∀ (a : 𝕜) (f : 𝓢(D, E)), A (a • f) = σ a • A f
hbound : ∃ s C, 0 ≤ C ∧ ∀ (f : 𝓢(D, E)), ‖A f‖ ≤ C * (s.sup (schwartzSeminormFamily 𝕜 D E)) f
f : 𝓢(D, E) →ₛₗ[σ] G := ⋯
⊢ ∃ s C,
    (normSeminorm 𝕜' G).comp { toFun := fun x ↦ A x, map_add' := hadd, map_smul' := hsmul } ≤
      C • s.sup (schwartzSeminormFamily 𝕜 D E)

Tactic:
rcases hbound with ⟨s, C, hC, h⟩

Goal state after:
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✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace ℝ E
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace ℝ F
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : NormedField 𝕜'
inst✝⁸ : NormedAddCommGroup D
inst✝⁷ : NormedSpace ℝ D
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace 𝕜' G
inst✝¹ : SMulCommClass ℝ 𝕜' G
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
A : 𝓢(D, E) → G
hadd : ∀ (f g : 𝓢(D, E)), A (f + g) = A f + A g
hsmul : ∀ (a : 𝕜) (f : 𝓢(D, E)), A (a • f) = σ a • A f
f : 𝓢(D, E) →ₛₗ[σ] G := { toFun := fun x ↦ A x, map_add' := hadd, map_smul' := hsmul }
s : Finset (ℕ × ℕ)
C : ℝ
hC : 0 ≤ C
h : ∀ (f : 𝓢(D, E)), ‖A f‖ ≤ C * (s.sup (schwartzSeminormFamily 𝕜 D E)) f
⊢ ∃ s C,
    (normSeminorm 𝕜' G).comp { toFun := fun x ↦ A x, map_add' := hadd, map_smul' := hsmul } ≤
      C • s.sup (schwartzSeminormFamily 𝕜 D E)
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) in Iic 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x

Tactic:
rw [← neg_zero, ← integral_comp_neg_Iic, neg_zero]

Goal state after:
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) in Iic 0, f |x| = ∫ (x : ℝ) in Iic 0, f (-x)
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: s t : ℝ
ht : 0 < t
h : -1 ≤ s
H : IntegrableOn (fun x ↦ x ^ s) (Ioi t) volume
H' : IntegrableOn (fun x ↦ x ^ s) (Ioi (max 1 t)) volume
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ioi (max 1 t)), ‖a⁻¹‖ ≤ a ^ s

Tactic:
filter_upwards [ae_restrict_mem measurableSet_Ioi] with x hx

Goal state after:
s t : ℝ
ht : 0 < t
h : -1 ≤ s
H : IntegrableOn (fun x ↦ x ^ s) (Ioi t) volume
H' : IntegrableOn (fun x ↦ x ^ s) (Ioi (max 1 t)) volume
x : ℝ
hx : x ∈ Ioi (max 1 t)
⊢ ‖x⁻¹‖ ≤ x ^ s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
X : ℝ
hs : 0 < s.re
hX : 0 ≤ X
⊢ IntervalIntegrable (fun x ↦ ↑(rexp (-x)) * ↑x ^ (s - 1)) volume 0 X

Tactic:
rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hX]
  exact IntegrableOn.mono_set (GammaIntegral_convergent hs) Ioc_subset_Ioi_self

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

Goal state before:
pp: case right.inl
x y : ℝ
hxy : (x, y) ∈ {q | 0 < q.1}
⊢ 0 ≤ (Complex.equivRealProd.symm (x, y)).re ∨ (Complex.equivRealProd.symm (x, y)).im ≠ 0

pp: case right.inr
x y : ℝ
hxy : (x, y) ∈ {q | q.2 ≠ 0}
⊢ 0 ≤ (Complex.equivRealProd.symm (x, y)).re ∨ (Complex.equivRealProd.symm (x, y)).im ≠ 0

Tactic:
· exact Or.inl (le_of_lt hxy)

Goal state after:
case right.inr
x y : ℝ
hxy : (x, y) ∈ {q | q.2 ≠ 0}
⊢ 0 ≤ (Complex.equivRealProd.symm (x, y)).re ∨ (Complex.equivRealProd.symm (x, y)).im ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: p s : ℝ
hs : -1 < s
hp : 0 < p
⊢ IntegrableOn (fun x ↦ x ^ s * rexp (-x ^ p)) (Ioi 0) volume

Tactic:
by
  -- Substitute `u = x ^ p`, reducing to convergence of the `Γ`-integral at `(s + 1) / p`.
  have ht : (0 : ℝ) < (s + 1) / p := div_pos (by linarith) hp
  refine ((integrableOn_Ioi_comp_rpow_iff' _ hp.ne').mpr
    (GammaIntegral_convergent ht)).congr_fun (fun x hx => ?_) measurableSet_Ioi
  simp only [smul_eq_mul]
  rw [mul_comm (exp (-x ^ p)), ← mul_assoc, ← rpow_mul hx.le, ← rpow_add hx]
  field_simp
  ring_nf

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c : ℝ
⊢ Tendsto (HMul.hMul b.re) atTop atTop

Tactic:
exact (tendsto_const_mul_atTop_of_pos hb).mpr tendsto_id

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

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

Tactic:
contrapose hs

Goal state after:
case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s + 1 = -↑m
⊢ s = -↑(1 + m)
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace ℝ V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : 𝓢(V, F)
⊢ ‖(𝓕 f).toBoundedContinuousFunction‖ ≤ ‖f.toLp 1 volume‖

Tactic:
rw [BoundedContinuousFunction.norm_le (by positivity)]

Goal state after:
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace ℝ V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : 𝓢(V, F)
⊢ ∀ (x : V), ‖(𝓕 f).toBoundedContinuousFunction x‖ ≤ ‖f.toLp 1 volume‖
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℂ F
inst✝⁴ : CompleteSpace F
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
hμ : μ.HasTemperateGrowth
inst✝¹ : FiniteDimensional ℝ E
inst✝ : IsLocallyFiniteMeasure μ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
⊢ ∀ (m : ↥(Lp F p μ)), ↑(toTemperedDistributionCLM F μ p) m = 0 → m = 0

Tactic:
ContinuousLinearMap.coe_coe

Goal state after:
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℂ F
inst✝⁴ : CompleteSpace F
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
hμ : μ.HasTemperateGrowth
inst✝¹ : FiniteDimensional ℝ E
inst✝ : IsLocallyFiniteMeasure μ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
⊢ ∀ (m : ↥(Lp F p μ)), (toTemperedDistributionCLM F μ p) m = 0 → m = 0
Module: Mathlib.Analysis.Fourier.AddCircleMulti

Goal state before:
pp: d : Type u_1
inst✝¹ : Fintype d
inst✝ : DecidableEq d
z : d → AddCircle 1
i : d
this :
  (fourier (Pi.single i 1 i)) (z i) * ∏ i_1 ∈ {i}ᶜ, (fourier (Pi.single i 1 i_1)) (z i_1) =
    ∏ i_1, (fourier (Pi.single i 1 i_1)) (z i_1)
j : d
hj : j ∈ {i}ᶜ
⊢ (fourier (Pi.single i 1 j)) (z j) = 1

Tactic:
rw [Finset.mem_compl, Finset.mem_singleton] at hj

Goal state after:
d : Type u_1
inst✝¹ : Fintype d
inst✝ : DecidableEq d
z : d → AddCircle 1
i : d
this :
  (fourier (Pi.single i 1 i)) (z i) * ∏ i_1 ∈ {i}ᶜ, (fourier (Pi.single i 1 i_1)) (z i_1) =
    ∏ i_1, (fourier (Pi.single i 1 i_1)) (z i_1)
j : d
hj : ¬j = i
⊢ (fourier (Pi.single i 1 j)) (z j) = 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
x : V →ᵇ ℂ
hx : x ∈ (charAlgHom he hL).range
⊢ star x ∈ (charAlgHom he hL).range

Tactic:
simp only [AlgHom.mem_range] at hx ⊢

Goal state after:
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
x : V →ᵇ ℂ
hx : ∃ x_1, (charAlgHom he hL) x_1 = x
⊢ ∃ x_1, (charAlgHom he hL) x_1 = star x
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ ∑' (n : ℤ), ∫ (x : ℝ) in 0..1, (e * f.comp (ContinuousMap.addRight ↑n)) x =
    ∑' (n : ℤ), ∫ (x : ℝ) in 0..1, (e.comp (ContinuousMap.addRight ↑n) * f.comp (ContinuousMap.addRight ↑n)) x

Tactic:
simp_rw [eadd]

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

Goal state before:
pp: f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ (fun n ↦
      ‖ContinuousMap.restrict (Icc 0 1) (e.comp (ContinuousMap.addRight ↑n) * f.comp (ContinuousMap.addRight ↑n))‖) =
    fun n ↦ ‖ContinuousMap.restrict (↑{ carrier := Icc 0 1, isCompact' := ⋯ }) (f.comp (ContinuousMap.addRight ↑n))‖

Tactic:
simp_rw [eadd]

Goal state after:
f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ (fun n ↦ ‖ContinuousMap.restrict (Icc 0 1) (e * f.comp (ContinuousMap.addRight ↑n))‖) = fun n ↦
    ‖ContinuousMap.restrict (↑{ carrier := Icc 0 1, isCompact' := ⋯ }) (f.comp (ContinuousMap.addRight ↑n))‖
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) * cyclotomic' n K =
      cyclotomic' n K * ∏ i ∈ n.properDivisors, cyclotomic' i K ∧
    ⊥ < (∏ i ∈ n.properDivisors, cyclotomic' i K).degree

Tactic:
refine ⟨by rw [mul_comm], ?_⟩

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
k : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
hζ : IsPrimitiveRoot ζ k
hk : 1 < k
⊢ ∑ i ∈ range k.pred, ζ ^ i + ζ ^ k.pred = 0

Tactic:
← sum_range_succ,

Goal state after:
R : Type u_4
k : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
hζ : IsPrimitiveRoot ζ k
hk : 1 < k
⊢ ∑ x ∈ range (k.pred + 1), ζ ^ x = 0
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: case left
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✝³ : CommMonoid M
inst✝² : CommMonoid N
inst✝¹ : DivisionCommMonoid G
k l : ℕ
inst✝ : CommRing R
ζ : Rˣ
h✝ h : IsPrimitiveRoot ζ k
i : ZMod k
hi :
  ↑((((Int.castAddHom (ZMod k)).liftOfRightInverse ZMod.cast ⋯)
          ⟨{ toFun := fun i ↦ Additive.ofMul ⟨(fun x ↦ ζ ^ x) i, ⋯⟩, map_zero' := ⋯, map_add' := ⋯ }, ⋯⟩)
        i) =
    ↑0
this : ↑k ∣ i.cast
⊢ i = 0

Tactic:
← (CharP.intCast_eq_zero_iff (ZMod k) k _).mpr this,

Goal state after:
case left
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✝³ : CommMonoid M
inst✝² : CommMonoid N
inst✝¹ : DivisionCommMonoid G
k l : ℕ
inst✝ : CommRing R
ζ : Rˣ
h✝ h : IsPrimitiveRoot ζ k
i : ZMod k
hi :
  ↑((((Int.castAddHom (ZMod k)).liftOfRightInverse ZMod.cast ⋯)
          ⟨{ toFun := fun i ↦ Additive.ofMul ⟨(fun x ↦ ζ ^ x) i, ⋯⟩, map_zero' := ⋯, map_add' := ⋯ }, ⋯⟩)
        i) =
    ↑0
this : ↑k ∣ i.cast
⊢ i = ↑i.cast
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: case h.inr
K : Type u_2
inst✝¹ : CommRing K
inst✝ : IsDomain K
k : ℕ
ihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)
ζ : K
h : IsPrimitiveRoot ζ k
hpos : k > 0
B : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K
Bmo : B.Monic
B₁ : ℤ[X]
hB₁ : map (Int.castRingHom K) B₁ = B
left✝ : B₁.degree = B.degree
hB₁mo : B₁.Monic
Q₁ : ℤ[X] := (X ^ k - 1) /ₘ B₁
⊢ cyclotomic' k K ∈ lifts (Int.castRingHom K)

Tactic:
have huniq : 0 + B * cyclotomic' k K = X ^ k - 1 ∧ (0 : K[X]).degree < B.degree := by
    constructor
    · rw [zero_add, mul_comm, ← prod_cyclotomic'_eq_X_pow_sub_one hpos h, ←
        Nat.cons_self_properDivisors hpos.ne', Finset.prod_cons]
    · simpa only [degree_zero, bot_lt_iff_ne_bot, Ne, degree_eq_bot] using Bmo.ne_zero

Goal state after:
case h.inr
K : Type u_2
inst✝¹ : CommRing K
inst✝ : IsDomain K
k : ℕ
ihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)
ζ : K
h : IsPrimitiveRoot ζ k
hpos : k > 0
B : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K
Bmo : B.Monic
B₁ : ℤ[X]
hB₁ : map (Int.castRingHom K) B₁ = B
left✝ : B₁.degree = B.degree
hB₁mo : B₁.Monic
Q₁ : ℤ[X] := (X ^ k - 1) /ₘ B₁
huniq : 0 + B * cyclotomic' k K = X ^ k - 1 ∧ degree 0 < B.degree
⊢ cyclotomic' k K ∈ lifts (Int.castRingHom K)
Module: Mathlib.RingTheory.RootsOfUnity.Minpoly

Goal state before:
pp: case refine_1
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
⊢ IsCoprime (map (Int.castRingHom ℚ) P) (map (Int.castRingHom ℚ) Q)

Tactic:
have aux := IsPrimitive.Int.irreducible_iff_irreducible_map_cast Pmonic.isPrimitive

Goal state after:
case refine_1
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
aux : Irreducible P ↔ Irreducible (map (Int.castRingHom ℚ) P)
⊢ IsCoprime (map (Int.castRingHom ℚ) P) (map (Int.castRingHom ℚ) Q)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
⊢ cyclotomic 1 R = X - 1

Tactic:
have hspec : map (Int.castRingHom ℂ) (X - 1) = cyclotomic' 1 ℂ := by
    simp only [cyclotomic'_one, map_X, Polynomial.map_one, Polynomial.map_sub]

Goal state after:
R : Type u_1
inst✝ : Ring R
hspec : map (Int.castRingHom ℂ) (X - 1) = cyclotomic' 1 ℂ
⊢ cyclotomic 1 R = X - 1
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: case refine_1
R : Type u_4
inst✝³ : CommRing R
inst✝² : IsDomain R
S : Type u_7
inst✝¹ : CommSemiring S
inst✝ : Algebra S R
ζ₁ ζ₂ : R
k₁ k₂ : ℕ
hζ₁ : IsPrimitiveRoot ζ₁ k₁
hζ₂ : IsPrimitiveRoot ζ₂ k₂
hk₁ : k₁ ≠ 0
hk₂ : k₂ ≠ 0
ζ : R
hζ : IsPrimitiveRoot ζ (k₁.lcm k₂)
this : NeZero (k₁.lcm k₂)
⊢ {ζ₁, ζ₂} ⊆ ↑S[ζ]

Tactic:
refine Set.pair_subset_iff.mpr ⟨?_, ?_⟩

Goal state after:
case refine_1.refine_1
R : Type u_4
inst✝³ : CommRing R
inst✝² : IsDomain R
S : Type u_7
inst✝¹ : CommSemiring S
inst✝ : Algebra S R
ζ₁ ζ₂ : R
k₁ k₂ : ℕ
hζ₁ : IsPrimitiveRoot ζ₁ k₁
hζ₂ : IsPrimitiveRoot ζ₂ k₂
hk₁ : k₁ ≠ 0
hk₂ : k₂ ≠ 0
ζ : R
hζ : IsPrimitiveRoot ζ (k₁.lcm k₂)
this : NeZero (k₁.lcm k₂)
⊢ ζ₁ ∈ ↑S[ζ]

case refine_1.refine_2
R : Type u_4
inst✝³ : CommRing R
inst✝² : IsDomain R
S : Type u_7
inst✝¹ : CommSemiring S
inst✝ : Algebra S R
ζ₁ ζ₂ : R
k₁ k₂ : ℕ
hζ₁ : IsPrimitiveRoot ζ₁ k₁
hζ₂ : IsPrimitiveRoot ζ₂ k₂
hk₁ : k₁ ≠ 0
hk₂ : k₂ ≠ 0
ζ : R
hζ : IsPrimitiveRoot ζ (k₁.lcm k₂)
this : NeZero (k₁.lcm k₂)
⊢ ζ₂ ∈ ↑S[ζ]
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 : ↑1 ↑↑μ' = ↑↑μ' ^ h1.choose
⊢ (fun x ↦ ↑↑x) μ' = (fun x ↦ ↑↑x) (μ' ^ h1.choose)

Tactic:
by simpa only [rootsOfUnity.coe_pow] using! h

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

Goal state before:

Tactic:
n.totient = (primitiveRoots n K).card := h.card_primitiveRoots.symm
    _ ≤ P_K.roots.toFinset.card := Finset.card_le_card (is_roots_of_minpoly h)
    _ ≤ Multiset.card P_K.roots := Multiset.toFinset_card_le _
    _ ≤ P_K.natDegree := card_roots' _
    _ ≤ P.natDegree := natDegree_map_le

Goal state after: