mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝⁵ : Ring α
inst✝⁴ : LinearOrder α
inst✝³ : FloorRing α
inst✝² : TopologicalSpace α
inst✝¹ : IsStrictOrderedRing α
inst✝ : OrderClosedTopology α
x : α
h₁ : ↑(⌈x⌉ - 1) < x
⊢ x ≤ ↑(⌈x⌉ - 1) + 1

Tactic:
by rw [cast_sub, cast_one, sub_add_cancel]; exact le_ceil _

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:

Tactic:
∫ (x : G), g (π x) * (f x) ∂μ =
        ∫ (x : G ⧸ Γ), QuotientGroup.automorphize ((g ∘ π) * f) x ∂μ_𝓕 := ?_
    _ = ∫ (x : G ⧸ Γ), g x * (QuotientGroup.automorphize f x) ∂μ_𝓕 := by simp [H₀]

Goal state after:
G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝⁵ : Countable ↥Γ
inst✝⁴ : MeasurableSpace (G ⧸ Γ)
inst✝³ : BorelSpace (G ⧸ Γ)
K : Type u_2
inst✝² : NormedField K
inst✝¹ : NormedSpace ℝ K
inst✝ : μ.IsMulRightInvariant
f : G → K
f_ℒ_1 : Integrable f μ
g : G ⧸ Γ → K
hg : AEStronglyMeasurable g (Measure.map mk (μ.restrict 𝓕))
g_ℒ_infinity : essSup (fun x ↦ ‖g x‖ₑ) (Measure.map mk (μ.restrict 𝓕)) ≠ ∞
F_ae_measurable : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ.restrict 𝓕))
π : G → G ⧸ Γ := mk
meas_π : Measurable π
H₀ : automorphize (g ∘ π * f) = g * automorphize f
⊢ ∫ (x : G), g (π x) * f x ∂μ = ∫ (x : G ⧸ Γ), automorphize (g ∘ π * f) x ∂Measure.map mk (μ.restrict 𝓕)
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case neg.refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntegrableOn (fun θ ↦ (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume
⊢ AEStronglyMeasurable (fun θ ↦ f (circleMap c R θ)) (volume.restrict (Ι 0 (2 * π)))

Tactic:
have H : ∀ {θ}, circleMap 0 R θ * I ≠ 0 := fun {θ} => by simp [h₀, I_ne_zero]
    simpa only [inv_smul_smul₀ H]
      using ((continuous_circleMap 0 R).aestronglyMeasurable.mul_const
        I).aemeasurable.fun_inv.aestronglyMeasurable.fun_smul h.aestronglyMeasurable

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 ≤ R
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
this : ‖(2 * ↑π * I)⁻¹‖ = (2 * π)⁻¹
⊢ ‖∮ (z : ℂ) in C(c, R), f z‖ / (2 * π) ≤ R * C

Tactic:
div_le_iff₀ Real.two_pi_pos,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 ≤ R
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
this : ‖(2 * ↑π * I)⁻¹‖ = (2 * π)⁻¹
⊢ ‖∮ (z : ℂ) in C(c, R), f z‖ ≤ R * C * (2 * π)
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
⊢ ‖cauchyPowerSeries f c R n‖ = (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖

Tactic:
simp [cauchyPowerSeries, norm_smul, Real.pi_pos.le]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
x : Fin (n + 1) → ℝ
hx : x ∈ Box.Icc I ∩ s
ε : ℝ
ε0 : ε > 0
δ₁ : ℝ
δ₁0 : 0 < δ₁
hδ₁ : ∀ x_1 ∈ closedBall x δ₁ ∩ Box.Icc I, f x_1 ∈ closedBall (f x) (ε / 2 / 2)
δ : ℝ
hδ : δ ∈ Set.Ioc 0 δ₁
y₁ : Fin (n + 1) → ℝ
⊢ y₁ ∈ closedBall x δ ∩ Box.Icc I → ∀ y₂ ∈ closedBall x δ ∩ Box.Icc I, ‖f y₁ - f y₂‖ ≤ ε / 2

Tactic:
hy₁

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
x : Fin (n + 1) → ℝ
hx : x ∈ Box.Icc I ∩ s
ε : ℝ
ε0 : ε > 0
δ₁ : ℝ
δ₁0 : 0 < δ₁
hδ₁ : ∀ x_1 ∈ closedBall x δ₁ ∩ Box.Icc I, f x_1 ∈ closedBall (f x) (ε / 2 / 2)
δ : ℝ
hδ : δ ∈ Set.Ioc 0 δ₁
y₁ : Fin (n + 1) → ℝ
hy₁ : y₁ ∈ closedBall x δ ∩ Box.Icc I
⊢ ∀ y₂ ∈ closedBall x δ ∩ Box.Icc I, ‖f y₁ - f y₂‖ ≤ ε / 2
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SFinite ν
inst✝³ : NormedSpace ℝ E
inst✝² : SFinite μ
𝕜 : Type u_5
inst✝¹ : RCLike 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hE : CompleteSpace E
h : Integrable (fun z ↦ f z.1 • g z.2) (μ.prod ν)
⊢ ∫ (x : α), ∫ (y : β), f (x, y).1 • g (x, y).2 ∂ν ∂μ = (∫ (x : α), f x ∂μ) • ∫ (y : β), g y ∂ν

Tactic:
simp_rw [integral_smul, integral_smul_const]

Goal state after:
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
⊢ ∮ (z : ℂ) in C(c, rexp b), (z - c)⁻¹ • f z = ∮ (z : ℂ) in C(c, rexp a), (z - c)⁻¹ • f z

Tactic:
suffices
    (∫ θ in 0..2 * π, I • f (circleMap c (Real.exp b) θ)) =
      ∫ θ in 0..2 * π, I • f (circleMap c (Real.exp a) θ) by
    simpa only [circleIntegral, add_sub_cancel_left, ofReal_exp, ← exp_add, smul_smul, ←
      div_eq_mul_inv, mul_div_cancel_left₀ _ (circleMap_ne_center (Real.exp_pos _).ne'),
      circleMap_sub_center, deriv_circleMap]

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
⊢ ∫ (θ : ℝ) in 0..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0..2 * π, I • f (circleMap c (rexp a) θ)
Module: Mathlib.Analysis.Complex.Liouville

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
n : ℕ
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
hp : ∀ z ∈ sphere c R, ‖(z - c)⁻¹ ^ (n + 1) • f z‖ ≤ C / (R ^ n * R)
hq : iteratedDeriv n f c = n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z
⊢ ‖iteratedDeriv n f c‖ ≤ ↑n.factorial * C / R ^ n

Tactic:
calc
    ‖iteratedDeriv n f c‖ = ‖n.factorial • (2 * π * I)⁻¹ •
      ∮ z in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z‖ := by rw [hq]
    _ ≤ n.factorial * (R * (C / (R ^ (n + 1)))) := by
      rw [RCLike.norm_nsmul (K := ℂ), nsmul_eq_mul, mul_le_mul_iff_right₀ (by positivity)]
      exact circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const hR.le hp
    _ = n.factorial * C / R ^ n := by
      grind

Goal state after:
Module: Mathlib.Analysis.Complex.Liouville

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
hf : Differentiable ℂ f
c : F
hb : Tendsto f (cocompact E) (𝓝 c)
⊢ f = const E c

Tactic:
have h_bdd : Bornology.IsBounded (Set.range f) := by
    obtain ⟨s, hs, hs_bdd⟩ := Metric.exists_isBounded_image_of_tendsto hb
    obtain ⟨t, ht, hts⟩ := mem_cocompact.mp hs
    apply ht.image hf.continuous |>.isBounded.union hs_bdd |>.subset
    simpa [Set.image_union, Set.image_univ] using! Set.image_mono <| calc
      Set.univ = t ∪ tᶜ := t.union_compl_self.symm
      _        ⊆ t ∪ s  := by gcongr
  obtain ⟨c', hc'⟩ := hf.exists_eq_const_of_bounded h_bdd
  convert hc'
  exact tendsto_nhds_unique hb (by simpa [hc'] using! tendsto_const_nhds)

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case hf
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : (map (algebraMap F q₁.SplittingField) p₁).Splits
h₂ : (map (algebraMap F q₂.SplittingField) p₂).Splits
⊢ (map (algebraMap F (q₁ * q₂).SplittingField) p₁).Splits

Tactic:
rw [←
      (SplittingField.lift q₁
          ((SplittingField.splits _).of_dvd (map_ne_zero (mul_ne_zero hq₁ hq₂))
             ((map_dvd_map' _).mpr (dvd_mul_right q₁ q₂)))).comp_algebraMap, ← map_map]

Goal state after:
case hf
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : (map (algebraMap F q₁.SplittingField) p₁).Splits
h₂ : (map (algebraMap F q₂.SplittingField) p₂).Splits
⊢ (map (↑(SplittingField.lift q₁ ⋯)) (map (algebraMap F q₁.SplittingField) p₁)).Splits
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
g : ℝ → ℂ := fun x ↦ w + ↑x
this : Tendsto g (𝓝 0) (𝓝 w)
l u : ℝ
hlu₀ : 0 ∈ Ioo l u
hlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)
⊢ (Ioo l u \ g ⁻¹' s).Nonempty

Tactic:
refine sdiff_nonempty.2 fun hsub => ?_
    have : (Ioo l u).Countable :=
      (hs.preimage ((add_right_injective w).comp ofReal_injective)).mono hsub
    rw [← Cardinal.le_aleph0_iff_set_countable, Cardinal.mk_Ioo_real (hlu₀.1.trans hlu₀.2)] at this
    exact this.not_gt Cardinal.aleph0_lt_continuum

Goal state after:
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
f : ℂ → E
c : ℂ
s : Set ℂ
h0 : 0 < R
n : ℕ
hs : s.Countable
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
this : (n.factorial • (cauchyPowerSeries f c (↑(NNReal.mk R ⋯)) n) fun x ↦ 1) = iteratedDeriv n f c
⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ (n + 1)) • f z =
    (2 * ↑π * I / ↑n.factorial) • n.factorial • (cauchyPowerSeries f c (↑(NNReal.mk R ⋯)) n) fun x ↦ 1

Tactic:
cauchyPowerSeries_apply,

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
f : ℂ → E
c : ℂ
s : Set ℂ
h0 : 0 < R
n : ℕ
hs : s.Countable
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
this : (n.factorial • (cauchyPowerSeries f c (↑(NNReal.mk R ⋯)) n) fun x ↦ 1) = iteratedDeriv n f c
⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ (n + 1)) • f z =
    (2 * ↑π * I / ↑n.factorial) •
      n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑(NNReal.mk R ⋯)), (1 / (z - c)) ^ n • (z - c)⁻¹ • f z
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings

Goal state before:
pp: case refine_2
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
B : ℝ
C : ℕ := ⌈max B 1 ^ finrank ℚ K * ↑((finrank ℚ K).choose (finrank ℚ K / 2))⌉₊
this :
  (⋃ f,
      ⋃ (_ : f.natDegree ≤ finrank ℚ K ∧ ∀ (i : ℕ), f.coeff i ∈ Icc (-↑C) ↑C),
        ↑(map (algebraMap ℤ K) f).roots.toFinset).Finite
x : K
hx : x ∈ {x | IsIntegral ℤ x ∧ ∀ (φ : K →+* A), ‖φ x‖ ≤ B}
h_map_ℚ_minpoly : minpoly ℚ x = map (algebraMap ℤ ℚ) (minpoly ℤ x)
i : ℕ
⊢ ↑|(minpoly ℤ x).coeff i| = ‖(algebraMap ℤ ℚ) ((minpoly ℤ x).coeff i)‖

Tactic:
eq_intCast,

Goal state after:
case refine_2
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
B : ℝ
C : ℕ := ⌈max B 1 ^ finrank ℚ K * ↑((finrank ℚ K).choose (finrank ℚ K / 2))⌉₊
this :
  (⋃ f,
      ⋃ (_ : f.natDegree ≤ finrank ℚ K ∧ ∀ (i : ℕ), f.coeff i ∈ Icc (-↑C) ↑C),
        ↑(map (algebraMap ℤ K) f).roots.toFinset).Finite
x : K
hx : x ∈ {x | IsIntegral ℤ x ∧ ∀ (φ : K →+* A), ‖φ x‖ ≤ B}
h_map_ℚ_minpoly : minpoly ℚ x = map (algebraMap ℤ ℚ) (minpoly ℤ x)
i : ℕ
⊢ ↑|(minpoly ℤ x).coeff i| = ‖↑((minpoly ℤ x).coeff i)‖
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: case mpr
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
hb : ∀ (z : ℂ), z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0
hc0 :
  ∀ (w : ↑(p.rootSet ℂ)), ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))) w = w ↔ (↑w).im = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im ≠ 0
⊢ ∃ a ∈ ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support, ↑a = z

Tactic:
exact ⟨⟨z, mem_rootSet.mpr ⟨hp, hz1⟩⟩, Equiv.Perm.mem_support.mpr (mt (hc0 _).mp hz2), rfl⟩

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: case neg.refine_2.refine_2.refine_2
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
hb : ∀ (z : ℂ), z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0
hc0 :
  ∀ (w : ↑(p.rootSet ℂ)), ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))) w = w ↔ (↑w).im = 0
hc : ∀ (z : ℂ), z ∈ c ↔ (aeval z) p = 0 ∧ z.im ≠ 0
⊢ ∀ (a_1 : ℂ), a_1 ∈ a ↔ a_1 ∈ b ∪ c

Tactic:
Finset.mem_union,

Goal state after:
case neg.refine_2.refine_2.refine_2
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
hb : ∀ (z : ℂ), z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0
hc0 :
  ∀ (w : ↑(p.rootSet ℂ)), ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))) w = w ↔ (↑w).im = 0
hc : ∀ (z : ℂ), z ∈ c ↔ (aeval z) p = 0 ∧ z.im ≠ 0
⊢ ∀ (a_1 : ℂ), a_1 ∈ a ↔ a_1 ∈ b ∨ a_1 ∈ c
Module: Mathlib.Data.Real.Embedding

Goal state before:
pp: case mp.refine_3
M : Type u_1
inst✝⁶ : AddCommGroup M
inst✝⁵ : LinearOrder M
inst✝⁴ : IsOrderedAddMonoid M
inst✝³ : One M
inst✝² : ZeroLEOneClass M
inst✝¹ : NeZero 1
inst✝ : Archimedean M
x y : M
a : ℚ
h : a.num • 1 < a.den • (x + y)
k : ℕ
hk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)
hk0 : k ≠ 0
hka0 : k * a.den ≠ 0
m : ℤ
right✝ : ∀ (y : ℤ), (fun m ↦ 0 + m • 1 ∈ Set.Ico (k • a.den • x - 1) (k • a.den • x - 1 + 1)) y → y = m
hm1 : k • a.den • x - 1 ≤ 0 + m • 1
hm2 : 0 + m • 1 < k • a.den • x - 1 + 1
⊢ mkRat a.num a.den = a

Tactic:
Rat.mkRat_num_den'

Goal state after:
case mp.refine_3
M : Type u_1
inst✝⁶ : AddCommGroup M
inst✝⁵ : LinearOrder M
inst✝⁴ : IsOrderedAddMonoid M
inst✝³ : One M
inst✝² : ZeroLEOneClass M
inst✝¹ : NeZero 1
inst✝ : Archimedean M
x y : M
a : ℚ
h : a.num • 1 < a.den • (x + y)
k : ℕ
hk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)
hk0 : k ≠ 0
hka0 : k * a.den ≠ 0
m : ℤ
right✝ : ∀ (y : ℤ), (fun m ↦ 0 + m • 1 ∈ Set.Ico (k • a.den • x - 1) (k • a.den • x - 1 + 1)) y → y = m
hm1 : k • a.den • x - 1 ≤ 0 + m • 1
hm2 : 0 + m • 1 < k • a.den • x - 1 + 1
⊢ a = a
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: case pos
R : Type u_3
inst✝¹ : Ring R
inst✝ : ValuativeRel R
h : MulArchimedean (ValueGroupWithZero R)
H : (valuation R).IsNontrivial
f : (valuation R).RankOne
⊢ IsRankLeOne R

Tactic:
exact isRankLeOne_of_rankOne

Goal state after:
Module: Mathlib.Data.Real.Embedding

Goal state before:
pp: M : Type u_1
inst✝⁶ : AddCommGroup M
inst✝⁵ : LinearOrder M
inst✝⁴ : IsOrderedAddMonoid M
inst✝³ : One M
inst✝² : ZeroLEOneClass M
inst✝¹ : NeZero 1
inst✝ : Archimedean M
x y : M
h : x < y
hyz : 0 < y - x
hy : y = y - x + x
⊢ 0 < embedRealFun (y - x)

Tactic:
obtain ⟨n, hn⟩ := Archimedean.arch 1 hyz

Goal state after:
M : Type u_1
inst✝⁶ : AddCommGroup M
inst✝⁵ : LinearOrder M
inst✝⁴ : IsOrderedAddMonoid M
inst✝³ : One M
inst✝² : ZeroLEOneClass M
inst✝¹ : NeZero 1
inst✝ : Archimedean M
x y : M
h : x < y
hyz : 0 < y - x
hy : y = y - x + x
n : ℕ
hn : 1 ≤ n • (y - x)
⊢ 0 < embedRealFun (y - x)
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
n : ℕ
a : R
φ : R⟦X⟧
⊢ (if single () n = 0 then a
    else
      -a *
        ∑ x ∈ antidiagonal (single () n),
          if x.2 < single () n then (MvPowerSeries.coeff x.1) φ * (MvPowerSeries.coeff x.2) (MvPowerSeries.inv.aux a φ)
          else 0) =
    if n = 0 then a
    else -a * ∑ x ∈ antidiagonal n, if x.2 < n then (coeff x.1) φ * (coeff x.2) (MvPowerSeries.inv.aux a φ) else 0

Tactic:
simp only [Finsupp.single_eq_zero]

Goal state after:
R : Type u_1
inst✝ : Ring R
n : ℕ
a : R
φ : R⟦X⟧
⊢ (if n = 0 then a
    else
      -a *
        ∑ x ∈ antidiagonal (single () n),
          if x.2 < single () n then (MvPowerSeries.coeff x.1) φ * (MvPowerSeries.coeff x.2) (MvPowerSeries.inv.aux a φ)
          else 0) =
    if n = 0 then a
    else -a * ∑ x ∈ antidiagonal n, if x.2 < n then (coeff x.1) φ * (coeff x.2) (MvPowerSeries.inv.aux a φ) else 0
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: R : Type u_1
k : Type u_2
inst✝ : Field k
a✝ b✝ : k⟦X⟧
hf : a✝ ≠ 0
hg : b✝ ≠ 0
⊢ (a✝ * b✝).Unit_of_divided_by_X_pow_order⁻¹ = a✝.Unit_of_divided_by_X_pow_order⁻¹ * b✝.Unit_of_divided_by_X_pow_order⁻¹

Tactic:
by
    simp only [← mul_inv, inv_inj]
    simp only [Unit_of_divided_by_X_pow_order_nonzero (mul_ne_zero hf hg),
      Unit_of_divided_by_X_pow_order_nonzero hf, Unit_of_divided_by_X_pow_order_nonzero hg,
      Units.ext_iff, Units.val_mul, ← divXPowOrder_mul]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Projective F
P : PointClass F
hP : W.NonsingularLift P
⊢ W.NonsingularLift (W.negMap P)

Tactic:
rcases P with ⟨_⟩

Goal state after:
case mk
F : Type u
inst✝ : Field F
W : Projective F
P : PointClass F
a✝ : Fin 3 → F
hP : W.NonsingularLift (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝)
⊢ W.NonsingularLift (W.negMap (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝))
Module: Mathlib.CategoryTheory.Sites.Preserves

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
α : Type u_1
inst✝ : Small.{w, u_1} α
X : α → C
c : Cofan X
hc : IsColimit c
this : HasCoproduct X
h : (Pi.lift fun i ↦ F.map (c.inj i).op) = F.map (Pi.lift fun i ↦ (c.inj i).op) ≫ piComparison F fun i ↦ op (X i)
⊢ (piComparison F fun x ↦ op (X x)) =
    (F.map (opCoproductIsoProduct' hc (productIsProduct fun x ↦ op (X x))).inv ≫ F.map (Pi.lift fun i ↦ (c.inj i).op)) ≫
      piComparison F fun i ↦ op (X i)

Tactic:
← Functor.map_comp

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
α : Type u_1
inst✝ : Small.{w, u_1} α
X : α → C
c : Cofan X
hc : IsColimit c
this : HasCoproduct X
h : (Pi.lift fun i ↦ F.map (c.inj i).op) = F.map (Pi.lift fun i ↦ (c.inj i).op) ≫ piComparison F fun i ↦ op (X i)
⊢ (piComparison F fun x ↦ op (X x)) =
    F.map ((opCoproductIsoProduct' hc (productIsProduct fun x ↦ op (X x))).inv ≫ Pi.lift fun i ↦ (c.inj i).op) ≫
      piComparison F fun i ↦ op (X i)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hP : ¬W.Nonsingular P
⊢ toAffine W P = 0

Tactic:
by
  rw [toAffine, dif_neg <| not_and_of_not_left _ hP]

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.BigZariski

Goal state before:
pp: case h₁
this : (precoverage IsOpenImmersion).PullbacksPreservedBy forgetToTop
⊢ jointlySurjectivePrecoverage ≤
    Precoverage.comap forgetToTop (Precoverage.comap (CategoryTheory.forget TopCat) Types.jointlySurjectivePrecoverage)

Tactic:
rw [← Precoverage.comap_comp, forgetToTop_comp_forget]

Goal state after:
Module: Mathlib.RingTheory.Smooth.Pi

Goal state before:
pp: case g.map_one
R : Type u_1
I : Type u_2
A : I → Type u_3
inst✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : FormallySmooth R ((i : I) → A i)
i : I
⊢ (Ideal.Quotient.mk (RingHom.ker (Pi.evalAlgHom R A i).toRingHom ^ 2)) (Pi.single i 1) -
      (Ideal.Quotient.mk (RingHom.ker (Pi.evalAlgHom R A i).toRingHom ^ 2)) 1 =
    0

Tactic:
← map_sub,

Goal state after:
case g.map_one
R : Type u_1
I : Type u_2
A : I → Type u_3
inst✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : FormallySmooth R ((i : I) → A i)
i : I
⊢ (Ideal.Quotient.mk (RingHom.ker (Pi.evalAlgHom R A i).toRingHom ^ 2)) (Pi.single i 1 - 1) = 0
Module: Mathlib.RingTheory.Unramified.Pi

Goal state before:
pp: case map_one
R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H✝ : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e✝ : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
x : I
hf : ∀ (x : (i : I) → f i), f₁ x - f₂ x ∈ J
e : (i : I) → f i := Pi.single x 1
he : IsIdempotentElem e
h₁ : f₁ e * (1 - f₂ e) = 0
h₂ : f₂ e * (1 - f₁ e) = 0
H : f₁ e = f₂ e
J' : Ideal B := Ideal.span {1 - f₁ e}
⊢ (Ideal.Quotient.mk J') 1 - (Ideal.Quotient.mk J') (f₁ (Pi.single x 1)) = 0

Tactic:
← map_sub,

Goal state after:
case map_one
R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H✝ : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e✝ : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
x : I
hf : ∀ (x : (i : I) → f i), f₁ x - f₂ x ∈ J
e : (i : I) → f i := Pi.single x 1
he : IsIdempotentElem e
h₁ : f₁ e * (1 - f₂ e) = 0
h₂ : f₂ e * (1 - f₁ e) = 0
H : f₁ e = f₂ e
J' : Ideal B := Ideal.span {1 - f₁ e}
⊢ (Ideal.Quotient.mk J') (1 - f₁ (Pi.single x 1)) = 0