mathlib-initiative/mathlib-tactics huggingface dataset:
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
hy₁ : y₁ ∈ closedBall x δ ∩ Box.Icc I
y₂ : Fin (n + 1) → ℝ
hy₂ : y₂ ∈ closedBall x δ ∩ Box.Icc I
⊢ ‖f y₁ - f y₂‖ ≤ ε / 2

Tactic:
have : closedBall x δ ∩ (Box.Icc I) ⊆ closedBall x δ₁ ∩ (Box.Icc I) := by gcongr; exact hδ.2

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 : Fin (n + 1) → ℝ
hy₂ : y₂ ∈ closedBall x δ ∩ Box.Icc I
this : closedBall x δ ∩ Box.Icc I ⊆ closedBall x δ₁ ∩ Box.Icc I
⊢ ‖f y₁ - f y₂‖ ≤ ε / 2
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ≥0
n : ℕ
hR : ↑R ^ n ≠ 0
⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * (↑R)⁻¹ ^ n * ↑R ^ n ≤
    (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖

Tactic:
rw [inv_pow]

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ≥0
n : ℕ
hR : ↑R ^ n ≠ 0
⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * (↑R ^ n)⁻¹ * ↑R ^ n ≤
    (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
w : ℂ
hf : CircleIntegrable f c R
hw : ‖w‖ < R
hR : 0 < R
hwR : ‖w‖ / R ∈ Ico 0 1
⊢ ∀ᵐ (t : ℝ), t ∈ Ι 0 (2 * π) → Summable fun n ↦ ‖f (circleMap c R t)‖ * (‖w‖ / R) ^ n

pp: case refine_4
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
w : ℂ
hf : CircleIntegrable f c R
hw : ‖w‖ < R
hR : 0 < R
hwR : ‖w‖ / R ∈ Ico 0 1
⊢ IntervalIntegrable (fun t ↦ ∑' (n : ℕ), ‖f (circleMap c R t)‖ * (‖w‖ / R) ^ n) volume 0 (2 * π)

pp: case refine_5
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
w : ℂ
hf : CircleIntegrable f c R
hw : ‖w‖ < R
hR : 0 < R
hwR : ‖w‖ / R ∈ Ico 0 1
⊢ ∀ᵐ (t : ℝ),
    t ∈ Ι 0 (2 * π) →
      HasSum (fun n ↦ deriv (circleMap c R) t • (fun z ↦ (w / (z - c)) ^ n • (z - c)⁻¹ • f z) (circleMap c R t))
        (deriv (circleMap c R) t • (fun z ↦ (z - (c + w))⁻¹ • f z) (circleMap c R t))

Tactic:
· exact Eventually.of_forall fun _ _ => (summable_geometric_of_lt_one hwR.1 hwR.2).mul_left _

Goal state after:
case refine_4
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
w : ℂ
hf : CircleIntegrable f c R
hw : ‖w‖ < R
hR : 0 < R
hwR : ‖w‖ / R ∈ Ico 0 1
⊢ IntervalIntegrable (fun t ↦ ∑' (n : ℕ), ‖f (circleMap c R t)‖ * (‖w‖ / R) ^ n) volume 0 (2 * π)

case refine_5
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
w : ℂ
hf : CircleIntegrable f c R
hw : ‖w‖ < R
hR : 0 < R
hwR : ‖w‖ / R ∈ Ico 0 1
⊢ ∀ᵐ (t : ℝ),
    t ∈ Ι 0 (2 * π) →
      HasSum (fun n ↦ deriv (circleMap c R) t • (fun z ↦ (w / (z - c)) ^ n • (z - c)⁻¹ • f z) (circleMap c R t))
        (deriv (circleMap c R) t • (fun z ↦ (z - (c + w))⁻¹ • f z) (circleMap c R t))
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case inl
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
i : Fin (n + 1)
hi : a i = b i
hi' : Set.Ioc (a i) (b i) = ∅
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
⊢ 0 =
    ∑ i,
      ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (b i) x) i) -
        ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (a i) x) i)

Tactic:
sum_eq_zero

Goal state after:
case inl
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
i : Fin (n + 1)
hi : a i = b i
hi' : Set.Ioc (a i) (b i) = ∅
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
⊢ 0 = 0

case inl
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
i : Fin (n + 1)
hi : a i = b i
hi' : Set.Ioc (a i) (b i) = ∅
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
⊢ ∀ x ∈ Finset.univ,
    (∫ (x_1 : Fin n → ℝ) in Set.Icc (a ∘ x.succAbove) (b ∘ x.succAbove), f (x.insertNth (b x) x_1) x) -
        ∫ (x_1 : Fin n → ℝ) in Set.Icc (a ∘ x.succAbove) (b ∘ x.succAbove), f (x.insertNth (a x) x_1) x =
      0
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
x : Multiset R
hx : x ∈ powersetCard k s
⊢ (prod ∘ map Neg.neg) x = (replicate x.card (-1)).prod * x.prod

Tactic:
← Multiset.map_const

Goal state after:
R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
x : Multiset R
hx : x ∈ powersetCard k s
⊢ (prod ∘ map Neg.neg) x = (map (Function.const R (-1)) x).prod * x.prod
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
hf : DiffContOnCl ℂ f (ball c R)
hw : w ∈ ball c R
⊢ (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = f w

Tactic:
have hR : 0 < R := not_le.mp (ball_eq_empty.not.mp (Set.nonempty_of_mem hw).ne_empty)

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
hf : DiffContOnCl ℂ f (ball c R)
hw : w ∈ ball c R
hR : 0 < R
⊢ (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = f w
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: case inl
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : B < 0
⊢ ‖coeff 1 i‖ ≤ B ^ (natDegree 1 - i) * ↑((natDegree 1).choose i)

Tactic:
natDegree_one,

Goal state after:
case inl
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : B < 0
⊢ ‖coeff 1 i‖ ≤ B ^ (0 - i) * ↑(Nat.choose 0 i)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings

Goal state before:
pp: K : Type u_1
A : Type u_2
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : Field A
inst✝¹ : Algebra ℚ A
inst✝ : IsAlgClosed A
x : K
a : A
⊢ (a ∈ range fun φ ↦ φ x) ↔ a ∈ range fun ψ ↦ ψ x

Tactic:
exact ⟨fun ⟨φ, hφ⟩ => ⟨φ.toRatAlgHom, hφ⟩, fun ⟨φ, hφ⟩ => ⟨φ.toRingHom, hφ⟩⟩

Goal state after:
Module: Mathlib.RingTheory.Valuation.Archimedean

Goal state before:
pp: case inl
F : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : CommRing O
inst✝¹ : Algebra O F
v : Valuation F Γ₀
inst✝ : MulArchimedean ↥(MonoidHom.mrange v)
hv : v.Integers O
H : ¬DenselyOrdered ↑(Set.range ⇑v)
hs : Subsingleton (↥(MonoidHom.mrange v))ˣ
this : Function.Bijective ⇑(algebraMap O F)
⊢ IsPrincipalIdealRing O

Tactic:
exact .of_surjective _ (RingEquiv.ofBijective _ this).symm.surjective

Goal state after:
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: case neg.inl
R : Type u_3
inst✝¹ : Ring R
inst✝ : ValuativeRel R
h : MulArchimedean (ValueGroupWithZero R)
b : ValueGroupWithZero R
H : 0 < b
H' : 1 b ≤ 1 0
⊢ IsNontrivial R

pp: case neg.inr
R : Type u_3
inst✝¹ : Ring R
inst✝ : ValuativeRel R
h : MulArchimedean (ValueGroupWithZero R)
a b : ValueGroupWithZero R
H : a < b
H' : 1 b ≤ 1 a
ha : a ≠ 0
⊢ IsNontrivial R

Tactic:
· simp_all

Goal state after:
case neg.inr
R : Type u_3
inst✝¹ : Ring R
inst✝ : ValuativeRel R
h : MulArchimedean (ValueGroupWithZero R)
a b : ValueGroupWithZero R
H : a < b
H' : 1 b ≤ 1 a
ha : a ≠ 0
⊢ IsNontrivial R
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
d : σ →₀ ℕ
h : (coeff d) φ ≠ 0
⊢ φ.lexOrder ≤ ↑(toLex d)

Tactic:
rw [← not_lt]
  intro h'
  exact h (coeff_eq_zero_of_lt_lexOrder h')

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
a : ι
s : Finset ι
ha : a ∉ s
IH : ‖∏ i ∈ s, f i‖₊ = s.sup fun i ↦ ‖f i‖₊
hs : (↑(Finset.cons a s ha)).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊
hs' : s.Nonempty
⊢ ∃ j ∈ s, ‖∏ i ∈ s, f i‖₊ = ‖f j‖₊

Tactic:
simpa [IH] using s.exists_mem_eq_sup hs' _

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
u v : σ →₀ ℕ
h : (u, v) ∈ Finset.antidiagonal (p + q)
h' : (u, v) ≠ (p, q)
⊢ (coeff (u, v).1) φ * (coeff (u, v).2) ψ = 0

Tactic:
simp only [Finset.mem_antidiagonal] at h

Goal state after:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
u v : σ →₀ ℕ
h' : (u, v) ≠ (p, q)
h : u + v = p + q
⊢ (coeff (u, v).1) φ * (coeff (u, v).2) ψ = 0
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : (u, v) ∈ Finset.antidiagonal d
⊢ (coeff (u, v).1) φ * (coeff (u, v).2) ψ = 0

Tactic:
simp only [Finset.mem_antidiagonal] at h

Goal state after:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
⊢ (coeff (u, v).1) φ * (coeff (u, v).2) ψ = 0
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
hφ : constantCoeff φ ∈ nonZeroDivisorsLeft R
x : MvPowerSeries σ R
hx : φ * x = 0
d e : σ →₀ ℕ
he : ∀ y < e, (coeff y) x = (coeff y) 0
fst✝ u : σ →₀ ℕ
huv : (fst✝, u) ∈ antidiagonal e
a✝ : (fst✝, u) ≠ (0, e)
this : u < e
⊢ (coeff (fst✝, u).1) φ * (coeff (fst✝, u).2) x = 0

Tactic:
simp only [he u this, mul_zero, map_zero]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Completion.FinitePlace

Goal state before:
pp: A : Type u_1
inst✝⁴ : CommRing A
inst✝³ : IsDedekindDomain A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : HeightOneSpectrum A
hv : Finite (A ⧸ v.asIdeal)
⊢ IsPrincipalIdealRing ↥(adicCompletionIntegers K v)

Tactic:
unfold HeightOneSpectrum.adicCompletionIntegers

Goal state after:
A : Type u_1
inst✝⁴ : CommRing A
inst✝³ : IsDedekindDomain A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : HeightOneSpectrum A
hv : Finite (A ⧸ v.asIdeal)
⊢ IsPrincipalIdealRing ↥Valued.v.valuationSubring
Module: Mathlib.NumberTheory.NumberField.Completion.FinitePlace

Goal state before:
pp: A : Type u_1
inst✝⁴ : CommRing A
inst✝³ : IsDedekindDomain A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : HeightOneSpectrum A
hv : Finite (A ⧸ v.asIdeal)
⊢ IsLocalRing.maximalIdeal ↥(adicCompletionIntegers K v) ≠ ⊥

Tactic:
unfold HeightOneSpectrum.adicCompletionIntegers

Goal state after:
A : Type u_1
inst✝⁴ : CommRing A
inst✝³ : IsDedekindDomain A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : HeightOneSpectrum A
hv : Finite (A ⧸ v.asIdeal)
⊢ IsLocalRing.maximalIdeal ↥Valued.v.valuationSubring ≠ ⊥
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: R : Type u_1
k : Type u_2
inst✝ : Field k
u : k⟦X⟧ˣ
⊢ (↑u).Unit_of_divided_by_X_pow_order⁻¹ = u⁻¹

Tactic:
by
    set u₀ := u.1 with hu
    have h₀ : IsUnit u₀ := ⟨u, hu.symm⟩
    rw [inv_inj, Units.ext_iff, ← hu, Unit_of_divided_by_X_pow_order_nonzero h₀.ne_zero]
    exact ((eq_divided_by_X_pow_order_Iff_Unit h₀.ne_zero).mpr h₀).symm

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

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ W'.neg P = -P y • ![0, 1, 0]

Tactic:
simp [neg, X_eq_zero_of_Z_eq_zero hP hPz, negY_of_Z_eq_zero hP hPz, hPz]

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

Goal state before:
pp: F : Type u
inst✝ : Field F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ P x * ![P x / P z, P y / P z, 1] z = ![P x / P z, P y / P z, 1] x * P z

Tactic:
by linear_combination (norm := (matrix_simp; ring1)) -P x * div_self hPz

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

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hQ : W'.Equation Q
hPz : P z ≠ 0
hQz : Q z = 0
⊢ W'.addXYZ P Q = -(Q y ^ 2 * P z) • P

Tactic:
addXYZ_of_Z_eq_zero_right hQ hQz

Goal state after:
R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hQ : W'.Equation Q
hPz : P z ≠ 0
hQz : Q z = 0
⊢ -(Q y ^ 2 * P z) • P = -(Q y ^ 2 * P z) • P
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
⊢ W'.Nonsingular P ↔
    W'.Equation P ∧
      (W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z + W'.a₄ * P z ^ 2) ≠ 0 ∨
        2 * P y * P z + W'.a₁ * P x * P z + W'.a₃ * P z ^ 2 ≠ 0 ∨
          P y ^ 2 + W'.a₁ * P x * P y + 2 * W'.a₃ * P y * P z -
              (W'.a₂ * P x ^ 2 + 2 * W'.a₄ * P x * P z + 3 * W'.a₆ * P z ^ 2) ≠
            0)

Tactic:
rw [Nonsingular, eval_polynomialX, eval_polynomialY, eval_polynomialZ]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Preserves

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
I : C
F : Cᵒᵖ ⥤ Type w
hF : IsSheafFor F (ofArrows Empty.elim fun a ↦ Empty.instIsEmpty.elim a)
hI : IsInitial I
α : Type u_1
inst✝³ : Small.{w, u_1} α
X : α → C
c : Cofan X
inst✝² : (ofArrows X c.inj).HasPairwisePullbacks
inst✝¹ : HasInitial C
inst✝ : ∀ (i : α), Mono (c.inj i)
hd : Pairwise fun i j ↦ IsPullback (initial.to (X i)) (initial.to (X j)) (c.inj i) (c.inj j)
⊢ Equalizer.Presieve.Arrows.firstMap F X c.inj = Equalizer.Presieve.Arrows.secondMap F X c.inj

Tactic:
ext ⟨i, j⟩ a

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
I : C
F : Cᵒᵖ ⥤ Type w
hF : IsSheafFor F (ofArrows Empty.elim fun a ↦ Empty.instIsEmpty.elim a)
hI : IsInitial I
α : Type u_1
inst✝³ : Small.{w, u_1} α
X : α → C
c : Cofan X
inst✝² : (ofArrows X c.inj).HasPairwisePullbacks
inst✝¹ : HasInitial C
inst✝ : ∀ (i : α), Mono (c.inj i)
hd : Pairwise fun i j ↦ IsPullback (initial.to (X i)) (initial.to (X j)) (c.inj i) (c.inj j)
i j : α
a : Equalizer.Presieve.Arrows.FirstObj F X
⊢ (ConcreteCategory.hom
          (Equalizer.Presieve.Arrows.firstMap F X c.inj ≫
            Pi.π (fun ij ↦ F.obj (op (Limits.pullback (c.inj ij.1) (c.inj ij.2)))) (i, j))).toFun
      a =
    (ConcreteCategory.hom
          (Equalizer.Presieve.Arrows.secondMap F X c.inj ≫
            Pi.π (fun ij ↦ F.obj (op (Limits.pullback (c.inj ij.1) (c.inj ij.2)))) (i, j))).toFun
      a
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Projective F
inst✝ : DecidableEq F
P Q : W.Point
⊢ (P + Q).toAffineLift = P.toAffineLift + Q.toAffineLift

Tactic:
rcases P, Q with ⟨@⟨⟨_⟩, hP⟩, @⟨⟨_⟩, hQ⟩⟩
  exact toAffine_add hP hQ

Goal state after:
Module: Mathlib.Topology.Category.TopCat.GrothendieckTopology

Goal state before:
pp: case refine_3
X Y : TopCat
𝒰 : precoverage.ZeroHypercover Y
heq : ∀ (i : 𝒰.I₀) (y : ↑(𝒰.X i)), (ConcreteCategory.hom (𝒰.f i)) y = ↑⟨(ConcreteCategory.hom (𝒰.f i)) y, ⋯⟩
x : (i : 𝒰.I₀) → (yoneda.obj X).obj (Opposite.op (𝒰.X i))
hx : Presieve.Arrows.Compatible (yoneda.obj X) 𝒰.f x
f : (fun X ↦ X) ((yoneda.obj X).obj (Opposite.op Y))
hf : (fun t ↦ ∀ (i : 𝒰.I₀), (ConcreteCategory.hom ((yoneda.obj X).map (𝒰.f i).op)) t = x i) f
⊢ f =
    ofHom
      (ContinuousMap.liftCover (fun i ↦ Set.range ⇑(ConcreteCategory.hom (𝒰.f i)))
        (fun i ↦ { toFun := ⇑(Hom.hom (x i)) ∘ ⇑⋯.toHomeomorph.symm, continuous_toFun := ⋯ }) ⋯ ⋯)

Tactic:
· dsimp
    ext x
    obtain ⟨i, y, rfl⟩ := exists_mem_zeroHypercover_range 𝒰 x
    have := congr($(hf i).hom y)
    dsimp at this ⊢
    rw [this, heq, ContinuousMap.liftCover_coe]
    simp

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

Goal state before:
pp: x✝⁴ x✝³ : Type u_6
x✝² : CommRing x✝⁴
x✝¹ : CommRing x✝³
x✝ : x✝⁴ →+* x✝³
⊢ x✝.Smooth ↔ x✝.FormallySmooth ∧ x✝.FinitePresentation

Tactic:
rw [smooth_def]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : FinitePresentation R A
p : Ideal A
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
f : A
hxf : { asIdeal := p, isPrime := inst✝¹ } ∈ ↑(basicOpen f)
hf : ↑(basicOpen f) ⊆ smoothLocus R A
⊢ ∃ f ∉ p, Smooth R (Localization.Away f)

Tactic:
refine ⟨f, by simpa using hxf, ⟨?_, inferInstance⟩⟩

Goal state after:
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : FinitePresentation R A
p : Ideal A
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
f : A
hxf : { asIdeal := p, isPrime := inst✝¹ } ∈ ↑(basicOpen f)
hf : ↑(basicOpen f) ⊆ smoothLocus R A
⊢ FormallySmooth R (Localization.Away f)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hPz : P z ≠ 0
hy : P y = W.negY P
hy' : (eval P) W.polynomialY = (P y - W.negY P) * P z
⊢ W.Nonsingular P ↔ W.Equation P ∧ (eval P) W.polynomialX ≠ 0

Tactic:
rw [nonsingular_iff_of_Z_ne_zero hPz, hy', hy, sub_self, zero_mul, ne_self_iff_false, or_false]

Goal state after:
Module: Mathlib.RingTheory.Etale.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
⊢ Module.support A Ω[A⁄R] ∪ Module.support A (H1Cotangent R A) ⊆ ∅ ↔ FormallyEtale R A

Tactic:
Set.union_subset_iff,

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
⊢ Module.support A Ω[A⁄R] ⊆ ∅ ∧ Module.support A (H1Cotangent R A) ⊆ ∅ ↔ FormallyEtale R A
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
b : B
a : A
this : Algebra.IsPushout R A S B
⊢ (tensorKaehlerEquiv R S A B) (b ⊗ₜ[A] (D R A) a) = b • (D S B) ((algebraMap A B) a)

Tactic:
obtain ⟨b, rfl⟩ := (Algebra.IsPushout.equiv R A S B).surjective b

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
a : A
this : Algebra.IsPushout R A S B
b : A ⊗[R] S
⊢ (tensorKaehlerEquiv R S A B) ((Algebra.IsPushout.equiv R A S B) b ⊗ₜ[A] (D R A) a) =
    (Algebra.IsPushout.equiv R A S B) b • (D S B) ((algebraMap A B) a)
Module: Mathlib.RingTheory.AdicCompletion.Exactness

Goal state before:
pp: R : Type u
inst✝⁶ : CommRing R
I : Ideal R
M : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
k : ℕ
hk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k) • (I ^ k • ⊤ ⊓ f.range)
x : AdicCompletion I M
⊢ (map I f) x = 0 → x = 0

Tactic:
apply AdicCompletion.induction_on I M x (fun a ↦ ?_)

Goal state after:
R : Type u
inst✝⁶ : CommRing R
I : Ideal R
M : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
k : ℕ
hk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k) • (I ^ k • ⊤ ⊓ f.range)
x : AdicCompletion I M
a : AdicCauchySequence I M
⊢ (map I f) ((mk I M) a) = 0 → (mk I M) a = 0
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
m n : ℕ
hmn : m ≤ n
r : AdicCauchySequence I R
h : I ^ m = I ^ m • ⊤
⊢ (Ideal.Quotient.mk (I ^ m)) (↑r n) = (Ideal.Quotient.mk (I ^ m)) (↑r m)

Tactic:
← Ideal.Quotient.mk_eq_mk,

Goal state after:
R : Type u_1
inst✝ : CommRing R
I : Ideal R
m n : ℕ
hmn : m ≤ n
r : AdicCauchySequence I R
h : I ^ m = I ^ m • ⊤
⊢ Submodule.Quotient.mk (↑r n) = (Ideal.Quotient.mk (I ^ m)) (↑r m)
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
m n : ℕ
hmn : m ≤ n
r : AdicCauchySequence I R
h : I ^ m = I ^ m • ⊤
⊢ Submodule.Quotient.mk (↑r n) = (Ideal.Quotient.mk (I ^ m)) (↑r m)

Tactic:
← Ideal.Quotient.mk_eq_mk,

Goal state after:
R : Type u_1
inst✝ : CommRing R
I : Ideal R
m n : ℕ
hmn : m ≤ n
r : AdicCauchySequence I R
h : I ^ m = I ^ m • ⊤
⊢ Submodule.Quotient.mk (↑r n) = Submodule.Quotient.mk (↑r m)