mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case refine_2.hab
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a : ℕ → ℝ
m n : ℕ
hmn : m ≤ n
p : ℕ
hmp : m ≤ p
IH :
  (∀ k ∈ Ico m p, IntervalIntegrable f μ (a k) (a (k + 1))) →
    ∑ k ∈ Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a p, f x ∂μ
h : ∀ k ∈ Ico m (p + 1), IntervalIntegrable f μ (a k) (a (k + 1))
⊢ IntervalIntegrable f μ (a m) (a p)

Tactic:
refine IntervalIntegrable.trans_iterate_Ico hmp fun k hk => h k ?_

Goal state after:
case refine_2.hab
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a : ℕ → ℝ
m n : ℕ
hmn : m ≤ n
p : ℕ
hmp : m ≤ p
IH :
  (∀ k ∈ Ico m p, IntervalIntegrable f μ (a k) (a (k + 1))) →
    ∑ k ∈ Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a p, f x ∂μ
h : ∀ k ∈ Ico m (p + 1), IntervalIntegrable f μ (a k) (a (k + 1))
k : ℕ
hk : k ∈ Ico m p
⊢ k ∈ Ico m (p + 1)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hab : a < b
fa fb : E
hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hint : IntervalIntegrable f' volume a b
ha : Tendsto f (𝓝[>] a) (𝓝 fa)
hb : Tendsto f (𝓝[<] b) (𝓝 fb)
⊢ ∫ (y : ℝ) in a..b, f' y = fb - fa

Tactic:
set F : ℝ → E := update (update f a fa) b fb

Goal state after:
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hab : a < b
fa fb : E
hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hint : IntervalIntegrable f' volume a b
ha : Tendsto f (𝓝[>] a) (𝓝 fa)
hb : Tendsto f (𝓝[<] b) (𝓝 fb)
F : ℝ → E := update (update f a fa) b fb
⊢ ∫ (y : ℝ) in a..b, f' y = fb - fa
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hab : a < b
fa fb : E
hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hint : IntervalIntegrable f' volume a b
ha : Tendsto f (𝓝[>] a) (𝓝 fa)
hb : Tendsto f (𝓝[<] b) (𝓝 fb)
F : ℝ → E := update (update f a fa) b fb
Fderiv : ∀ x ∈ Ioo a b, HasDerivAt F (f' x) x
hcont : ContinuousOn F (Icc a b)
⊢ ∫ (y : ℝ) in a..b, f' y = fb - fa

Tactic:
simpa [F, hab.ne, hab.ne'] using integral_eq_sub_of_hasDerivAt_of_le hab.le hcont Fderiv hint

Goal state after:
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
n : ℤ
⊢ pure (IntCast.intCast (n + 1)) ≤ 𝓝[≥] (↑n + ↑1)

Tactic:
← cast_add

Goal state after:
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
n : ℤ
⊢ pure (IntCast.intCast (n + 1)) ≤ 𝓝[≥] ↑(n + 1)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t : ℝ
f : AddCircle T → E
m : MeasurableSet (Ioc t (t + T))
⊢ ∫ (a : ℝ) in Ioc t (t + T), f ↑a = ∫ (b : AddCircle T), f b

Tactic:
have := integral_map_equiv (μ := volume) (measurableEquivIoc T t).symm f

Goal state after:
T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t : ℝ
f : AddCircle T → E
m : MeasurableSet (Ioc t (t + T))
this :
  ∫ (y : AddCircle T), f y ∂Measure.map (⇑(measurableEquivIoc T t).symm) volume =
    ∫ (x : ↑(Ioc t (t + T))), f ((measurableEquivIoc T t).symm x)
⊢ ∫ (a : ℝ) in Ioc t (t + T), f ↑a = ∫ (b : AddCircle T), f b
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: case e'_3
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
c : ℝ≥0
hc : I.distortion ≤ c
e : ℝ → (Fin n → ℝ) → Fin (n + 1) → ℝ := i.insertNth
Hl : I.lower i ∈ Set.Icc (I.lower i) (I.upper i)
Hu : I.upper i ∈ Set.Icc (I.lower i) (I.upper i)
y : Fin n → ℝ
hy : y ∈ Box.Icc (I.face i)
g : (Fin (n + 1) → ℝ) → E
hε : ∀ y ∈ Box.Icc I, ‖g y‖ ≤ ε * ‖y - x‖
hfc : ContinuousOn (fun y ↦ a + f' (y - x) + g y) (Box.Icc I)
Hi :
  ∀ x_1 ∈ Set.Icc (I.lower i) (I.upper i),
    Integrable (I.face i) ⊥ ((fun y ↦ a + f' (y - x) + g y) ∘ e x_1) BoxAdditiveMap.volume
hg : g = fun y ↦ (fun y ↦ a + f' (y - x) + g y) y - a - f' (y - x)
⊢ f' (Pi.single i (I.upper i - I.lower i)) -
      ((fun y ↦ a + f' (y - x) + g y) (e (I.upper i) y) - (fun y ↦ a + f' (y - x) + g y) (e (I.lower i) y)) =
    g (e (I.lower i) y) - g (e (I.upper i) y)

Tactic:
have := Fin.insertNth_sub_same (α := fun _ ↦ ℝ) i (I.upper i) (I.lower i) y

Goal state after:
case e'_3
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
c : ℝ≥0
hc : I.distortion ≤ c
e : ℝ → (Fin n → ℝ) → Fin (n + 1) → ℝ := i.insertNth
Hl : I.lower i ∈ Set.Icc (I.lower i) (I.upper i)
Hu : I.upper i ∈ Set.Icc (I.lower i) (I.upper i)
y : Fin n → ℝ
hy : y ∈ Box.Icc (I.face i)
g : (Fin (n + 1) → ℝ) → E
hε : ∀ y ∈ Box.Icc I, ‖g y‖ ≤ ε * ‖y - x‖
hfc : ContinuousOn (fun y ↦ a + f' (y - x) + g y) (Box.Icc I)
Hi :
  ∀ x_1 ∈ Set.Icc (I.lower i) (I.upper i),
    Integrable (I.face i) ⊥ ((fun y ↦ a + f' (y - x) + g y) ∘ e x_1) BoxAdditiveMap.volume
hg : g = fun y ↦ (fun y ↦ a + f' (y - x) + g y) y - a - f' (y - x)
this : i.insertNth (I.upper i) y - i.insertNth (I.lower i) y = Pi.single i (I.upper i - I.lower i)
⊢ f' (Pi.single i (I.upper i - I.lower i)) -
      ((fun y ↦ a + f' (y - x) + g y) (e (I.upper i) y) - (fun y ↦ a + f' (y - x) + g y) (e (I.lower i) y)) =
    g (e (I.lower i) y) - g (e (I.upper i) y)
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case pos
E : Type u_1
inst✝ : NormedAddCommGroup E
c : ℂ
R : ℝ
ι : Type u_3
f : ι → ℂ → E
h : ∀ (i : ι), CircleIntegrable (f i) c R
h₁ : (support f).Finite
⊢ CircleIntegrable (∑ᶠ (i : ι), f i) c R

Tactic:
finsum_eq_sum f h₁

Goal state after:
case pos
E : Type u_1
inst✝ : NormedAddCommGroup E
c : ℂ
R : ℝ
ι : Type u_3
f : ι → ℂ → E
h : ∀ (i : ι), CircleIntegrable (f i) c R
h₁ : (support f).Finite
⊢ CircleIntegrable (∑ i ∈ h₁.toFinset, f i) c R
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case h_lim
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
ι : Type u_2
f : ι → ℂ → E
g : ℂ → E
c : ℂ
R : ℝ
l : Filter ι
inst✝ : l.IsCountablyGenerated
hR : 0 ≤ R
hf : ∀ᶠ (i : ι) in l, ContinuousOn (f i) (sphere c R)
h : ∀ ε > 0, ∀ᶠ (n : ι) in l, ∀ x ∈ sphere c R, dist (g x) (f n x) < ε
ε : ℝ
hε : ε > 0
δ : ℝ
hδ₀ : 0 < δ
hRδ : R * δ < ε
i : ι
hi : ∀ x ∈ sphere c R, dist (g x) (f i x) < δ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ R * dist (g (circleMap c R x)) (f i (circleMap c R x)) < ε

Tactic:
← hRδ,

Goal state after:
case h_lim
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
ι : Type u_2
f : ι → ℂ → E
g : ℂ → E
c : ℂ
R : ℝ
l : Filter ι
inst✝ : l.IsCountablyGenerated
hR : 0 ≤ R
hf : ∀ᶠ (i : ι) in l, ContinuousOn (f i) (sphere c R)
h : ∀ ε > 0, ∀ᶠ (n : ι) in l, ∀ x ∈ sphere c R, dist (g x) (f n x) < ε
ε : ℝ
hε : ε > 0
δ : ℝ
hδ₀ : 0 < δ
hRδ : R * δ < ε
i : ι
hi : ∀ x ∈ sphere c R, dist (g x) (f i x) < δ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ R * dist (g (circleMap c R x)) (f i (circleMap c R x)) ≤ R * δ
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞
x : α
hx : ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞
⊢ ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν ≠ ∞

Tactic:
rw [← lt_top_iff_ne_top]

Goal state after:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞
x : α
hx : ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞
⊢ ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞
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)
this : 2 * ↑π * I / ↑n.factorial ≠ 0
⊢ iteratedDeriv n f c = n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z

Tactic:
rw [← inv_smul_smul₀ this (iteratedDeriv n f c), inv_div, div_eq_inv_mul, mul_comm,
      ← nsmul_eq_mul, smul_assoc]

Goal state after:
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)
this : 2 * ↑π * I / ↑n.factorial ≠ 0
⊢ n.factorial • (2 * ↑π * I)⁻¹ • (2 * ↑π * I / ↑n.factorial) • iteratedDeriv n f c =
    n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : p ∣ q
hq : q ≠ 0
⊢ Function.Surjective ⇑(restrictDvd hpq)

Tactic:
classical
  have := Fact.mk <|
    (SplittingField.splits q).of_dvd (map_ne_zero hq) ((map_dvd_map' _).mpr hpq)
  simpa only [restrictDvd_def, dif_neg hq] using! restrict_surjective _ _

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

Goal state before:
pp: case mp
R : Type u_1
Γ₀ : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.IsNontrivial
⊢ Nonempty v.RankOne → MulArchimedean (ofClass v).ValueGroup₀

pp: case mpr
R : Type u_1
Γ₀ : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.IsNontrivial
⊢ MulArchimedean (ofClass v).ValueGroup₀ → Nonempty v.RankOne

Tactic:
· intro h
    obtain hv := Nonempty.some h
    exact MulArchimedean.comap hv.hom'.toMonoidHom hv.strictMono'

Goal state after:
case mpr
R : Type u_1
Γ₀ : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.IsNontrivial
⊢ MulArchimedean (ofClass v).ValueGroup₀ → Nonempty v.RankOne
Module: Mathlib.Data.Real.Embedding

Goal state before:
pp: case a
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 : ℝ
h : ∀ (y : ℚ), y.num • 1 < 0 → ↑y ≤ x
h' : x < 0
y : ℚ
hxy : x < ↑y
hy : ↑y < 0
⊢ ∃ y < 0, x < ↑y

Tactic:
exact ⟨y, by simpa using hy, hxy⟩

Goal state after:
Module: Mathlib.Topology.MetricSpace.Ultra.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x y : X
r : ℝ
h : y ∈ ball x r
a : X
⊢ a ∈ ball x r ↔ a ∈ ball y r

Tactic:
mem_ball

Goal state after:
X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x y : X
r : ℝ
a : X
h : dist y x < r
⊢ dist a x < r ↔ dist a y < r
Module: Mathlib.Topology.MetricSpace.Ultra.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x y : X
r s : ℝ
⊢ ball x r ⊆ ball y s ∨ ball y s ⊆ ball x r ∨ Disjoint (ball x r) (ball y s)

Tactic:
by
  wlog! hrs : r ≤ s generalizing x y r s
  · rw [disjoint_comm, ← or_assoc, or_comm (b := (_ : Set X) ⊆ _), or_assoc]
    exact this y x s r hrs.le
  · refine Set.disjoint_or_nonempty_inter (ball x r) (ball y s) |>.symm.imp (fun h ↦ ?_) (Or.inr ·)
    obtain ⟨hxz, hyz⟩ := (Set.mem_inter_iff _ _ _).mp h.some_mem
    have hx := ball_subset_ball hrs (x := x)
    rwa [ball_eq_of_mem hyz |>.trans (ball_eq_of_mem <| hx hxz).symm]

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
⊢ U ∈ nhds 0 ↔ ∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U

Tactic:
by
      rw [Metric.mem_nhds_iff]
      refine ⟨?_, ?_⟩
      · rintro ⟨ε, hε, h⟩
        rcases RankLeOne.exists_val_lt (valuation (K := K)) with H | H
        · use Units.mk0 (valuation.restrict 1) (by simp)
          intro x hx
          simp only [Units.val_mk0, mem_ofPred_eq, map_one] at hx
          by_cases hx0 : x = 0
          · exact h (hx0 ▸ Metric.mem_ball_self hε)
          · exfalso
            rw [← (valuation (K := K)).restrict.zero_iff, ← ne_eq, ← isUnit_iff_ne_zero] at hx0
            apply not_le.mpr hx
            apply le_of_eq
            rw [eq_comm]
            simpa only [Units.ext_iff, hx0.unit_spec, Units.val_one,
              Submonoid.mk_eq_one] using! H.elim hx0.unit 1
        · obtain ⟨x, hx, hxy⟩ := H (γ := ⟨ε, le_of_lt hε⟩) (pos_iff_ne_zero.mp hε)
          use Units.mk0 (valuation.restrict x) (by simp [Valuation.restrict_def, hx])
          intro y hy
          apply h
          simp only [Metric.mem_ball, dist_zero_right]
          simp only [Units.val_mk0, mem_ofPred_eq, restrict_lt_iff, ← NNReal.coe_lt_coe] at hy
          apply lt_trans hy
          simpa [RankLeOne.hom', valuation.restrict_def] using! hxy
      · rintro ⟨ε, hε⟩
        refine ⟨(embedding ε.1 : ℝ≥0), ?_, fun x hx ↦ hε ?_⟩
        · exact NNReal.coe_pos.mpr <| embedding_strictMono.lt_iff_lt.mpr ε.zero_lt
        · simpa [restrict_lt_iff_lt_embedding] using! (mem_ball_zero_iff.mp hx)

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
h : constantCoeff φ = ↑u
n : σ →₀ ℕ
this✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)
H : ¬n = 0
this : (0, n) ∈ antidiagonal n
⊢ (∑ x ∈ (antidiagonal n).erase (0, n), (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) -
      ∑ x ∈ (antidiagonal n).erase (0, n), if x.2 < (0, n).2 then (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) else 0) =
    0

Tactic:
sub_eq_zero,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
h : constantCoeff φ = ↑u
n : σ →₀ ℕ
this✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)
H : ¬n = 0
this : (0, n) ∈ antidiagonal n
⊢ ∑ x ∈ (antidiagonal n).erase (0, n), (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) =
    ∑ x ∈ (antidiagonal n).erase (0, n), if x.2 < (0, n).2 then (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) else 0
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g < ⊤
p : ℕ := (weightedOrder w f).toNat
hp : ↑p = weightedOrder w f
q : ℕ := (weightedOrder w g).toNat
hq : ↑q = weightedOrder w g
⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g

Tactic:
have : f.weightedHomogeneousComponent w p * g.weightedHomogeneousComponent w q ≠ 0 := by
        simp only [ne_eq, mul_eq_zero]
        intro H
        rcases H with H | H <;>
        · refine weightedHomogeneousComponent_of_weightedOrder ?_ H
          simp only [ENat.natCast_toNat_eq_self, ne_eq, weightedOrder_eq_top_iff, p, q]
          rw [← ne_eq, ne_zero_iff_weightedOrder_finite w]
          exact ENat.natCast_toNat (ne_top_of_lt (by simpa))

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g < ⊤
p : ℕ := (weightedOrder w f).toNat
hp : ↑p = weightedOrder w f
q : ℕ := (weightedOrder w g).toNat
hq : ↑q = weightedOrder w g
this : (weightedHomogeneousComponent w p) f * (weightedHomogeneousComponent w q) g ≠ 0
⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
k : Type u_3
inst✝ : Field k
φ : MvPowerSeries σ k
h : φ⁻¹ = 0
⊢ constantCoeff φ = 0

Tactic:
simpa using congr_arg constantCoeff h

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
hfX : ¬constantCoeff f = 0
hfI : f ∈ I
this : C (f 0) = f - (f - C (f 0))
⊢ (coeff 0) f - f 0 = 0

Tactic:
sub_eq_zero,

Goal state after:
k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
hfX : ¬constantCoeff f = 0
hfI : f ∈ I
this : C (f 0) = f - (f - C (f 0))
⊢ (coeff 0) f = f 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
P : Fin 3 → R
⊢ ![P x, P y, P z] = P

Tactic:
by
  ext n; fin_cases n <;> rfl

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Limits.HasPullbacks C
K : Precoverage C
inst✝ : K.IsStableUnderBaseChange
S : C
F : Cᵒᵖ ⥤ Type u_2
𝒰 : K.ZeroHypercover S
𝒱 : K.ZeroHypercover S
f : Hom K 𝒰 𝒱
H₁ : Presieve.IsSheafFor F (Presieve.ofArrows 𝒰.X 𝒰.f)
H₂ : ∀ {X : C} (f : X ⟶ S), Presieve.IsSeparatedFor F (Presieve.ofArrows (pullback₂ f 𝒰).X (pullback₂ f 𝒰).f)
⊢ Presieve.IsSheafFor F (Presieve.ofArrows 𝒱.X 𝒱.f)

Tactic:
Presieve.isSheafFor_iff_generate

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Limits.HasPullbacks C
K : Precoverage C
inst✝ : K.IsStableUnderBaseChange
S : C
F : Cᵒᵖ ⥤ Type u_2
𝒰 : K.ZeroHypercover S
𝒱 : K.ZeroHypercover S
f : Hom K 𝒰 𝒱
H₁ : Presieve.IsSheafFor F (Presieve.ofArrows 𝒰.X 𝒰.f)
H₂ : ∀ {X : C} (f : X ⟶ S), Presieve.IsSeparatedFor F (Presieve.ofArrows (pullback₂ f 𝒰).X (pullback₂ f 𝒰).f)
⊢ Presieve.IsSheafFor F (Sieve.generate (Presieve.ofArrows 𝒱.X 𝒱.f)).arrows
Module: Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : C
E : PreOneHypercover X
F : Cᵒᵖ ⥤ Type u_2
h : E.IsStronglySeparatedFor F
x : (i : E.I₀) → F.obj (op (E.X i))
hc :
  ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j),
    (ConcreteCategory.hom (F.map (E.p₁ k).op)) (x i) = (ConcreteCategory.hom (F.map (E.p₂ k).op)) (x j)
i₁ i₂ : E.I₀
Z : C
g₁ : Z ⟶ E.X i₁
g₂ : Z ⟶ E.X i₂
heq : g₁ ≫ E.f i₁ = g₂ ≫ E.f i₂
W : C
f : W ⟶ Z
x✝ : (E.sieve₁ g₁ g₂).arrows f
T : E.I₁ i₁ i₂
u : W ⟶ E.Y T
h₁ : f ≫ g₁ = u ≫ E.p₁ T
h₂ : f ≫ g₂ = u ≫ E.p₂ T
⊢ (ConcreteCategory.hom (F.map f.op)) ((ConcreteCategory.hom (F.map g₁.op)) (x i₁)) =
    (ConcreteCategory.hom (F.map f.op)) ((ConcreteCategory.hom (F.map g₂.op)) (x i₂))

Tactic:
← comp_apply,

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : C
E : PreOneHypercover X
F : Cᵒᵖ ⥤ Type u_2
h : E.IsStronglySeparatedFor F
x : (i : E.I₀) → F.obj (op (E.X i))
hc :
  ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j),
    (ConcreteCategory.hom (F.map (E.p₁ k).op)) (x i) = (ConcreteCategory.hom (F.map (E.p₂ k).op)) (x j)
i₁ i₂ : E.I₀
Z : C
g₁ : Z ⟶ E.X i₁
g₂ : Z ⟶ E.X i₂
heq : g₁ ≫ E.f i₁ = g₂ ≫ E.f i₂
W : C
f : W ⟶ Z
x✝ : (E.sieve₁ g₁ g₂).arrows f
T : E.I₁ i₁ i₂
u : W ⟶ E.Y T
h₁ : f ≫ g₁ = u ≫ E.p₁ T
h₂ : f ≫ g₂ = u ≫ E.p₂ T
⊢ (ConcreteCategory.hom (F.map g₁.op ≫ F.map f.op)) (x i₁) =
    (ConcreteCategory.hom (F.map f.op)) ((ConcreteCategory.hom (F.map g₂.op)) (x i₂))
Module: Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : C
E : PreOneHypercover X
F : Cᵒᵖ ⥤ Type u_2
h : E.IsStronglySeparatedFor F
x : (i : E.I₀) → F.obj (op (E.X i))
hc :
  ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j),
    (ConcreteCategory.hom (F.map (E.p₁ k).op)) (x i) = (ConcreteCategory.hom (F.map (E.p₂ k).op)) (x j)
i₁ i₂ : E.I₀
Z : C
g₁ : Z ⟶ E.X i₁
g₂ : Z ⟶ E.X i₂
heq : g₁ ≫ E.f i₁ = g₂ ≫ E.f i₂
W : C
f : W ⟶ Z
x✝ : (E.sieve₁ g₁ g₂).arrows f
T : E.I₁ i₁ i₂
u : W ⟶ E.Y T
h₁ : f ≫ g₁ = u ≫ E.p₁ T
h₂ : f ≫ g₂ = u ≫ E.p₂ T
| (ConcreteCategory.hom (F.map f.op)) ((ConcreteCategory.hom (F.map g₂.op)) (x i₂))

Tactic:
← comp_apply,

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : C
E : PreOneHypercover X
F : Cᵒᵖ ⥤ Type u_2
h : E.IsStronglySeparatedFor F
x : (i : E.I₀) → F.obj (op (E.X i))
hc :
  ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j),
    (ConcreteCategory.hom (F.map (E.p₁ k).op)) (x i) = (ConcreteCategory.hom (F.map (E.p₂ k).op)) (x j)
i₁ i₂ : E.I₀
Z : C
g₁ : Z ⟶ E.X i₁
g₂ : Z ⟶ E.X i₂
heq : g₁ ≫ E.f i₁ = g₂ ≫ E.f i₂
W : C
f : W ⟶ Z
x✝ : (E.sieve₁ g₁ g₂).arrows f
T : E.I₁ i₁ i₂
u : W ⟶ E.Y T
h₁ : f ≫ g₁ = u ≫ E.p₁ T
h₂ : f ≫ g₂ = u ≫ E.p₂ T
| (ConcreteCategory.hom (F.map g₂.op ≫ F.map f.op)) (x i₂)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ ⟦W.neg P⟧ = ⟦![P x / P z, W.toAffine.negY (P x / P z) (P y / P z), 1]⟧

Tactic:
neg_of_Z_ne_zero hPz,

Goal state after:
F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ ⟦P z • ![P x / P z, W.toAffine.negY (P x / P z) (P y / P z), 1]⟧ =
    ⟦![P x / P z, W.toAffine.negY (P x / P z) (P y / P z), 1]⟧
Module: Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes

Goal state before:
pp: case refine_1
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : C
E : PreOneHypercover X
F : Cᵒᵖ ⥤ Type u_2
h₁ : E.IsStronglySheafFor F
h₂ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor F (Sieve.pullback f E.sieve₀).arrows
S : Sieve X
H : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
H' : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
t : Presieve.FamilyOfElements F S.arrows
ht : t.Compatible
s : (i : E.I₀) → F.obj (op (E.X i))
hs : ∀ (i : E.I₀), (fun t_1 ↦ (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation t_1) (s i)
huniq :
  ∀ (i : E.I₀) (y : F.obj (op (E.X i))),
    (fun t_1 ↦ (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation t_1) y → y = s i
hr : Presieve.Arrows.Compatible F E.f s
s' : Presieve.FamilyOfElements F (Presieve.ofArrows E.X E.f)
hs' : ∀ (i : E.I₀), s' (E.f i) ⋯ = s i
t' : F.obj (op X)
ht' : ∀ (i : E.I₀), (ConcreteCategory.hom (F.map (E.f i).op)) t' = s i
hunique :
  ∀ (y : (fun X ↦ X) (F.obj (op X))),
    (fun t ↦ ∀ (i : E.I₀), (ConcreteCategory.hom (F.map (E.f i).op)) t = s i) y → y = t'
T : C
f : T ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ T
Y : C
i : E.I₀
w : Z ⟶ E.X i
heq : w ≫ E.f i = g ≫ f
⊢ (ConcreteCategory.hom (F.map g.op)) ((ConcreteCategory.hom (F.map f.op)) t') =
    (ConcreteCategory.hom (F.map g.op)) (t f hf)

Tactic:
← comp_apply,

Goal state after:
case refine_1
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : C
E : PreOneHypercover X
F : Cᵒᵖ ⥤ Type u_2
h₁ : E.IsStronglySheafFor F
h₂ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor F (Sieve.pullback f E.sieve₀).arrows
S : Sieve X
H : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
H' : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
t : Presieve.FamilyOfElements F S.arrows
ht : t.Compatible
s : (i : E.I₀) → F.obj (op (E.X i))
hs : ∀ (i : E.I₀), (fun t_1 ↦ (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation t_1) (s i)
huniq :
  ∀ (i : E.I₀) (y : F.obj (op (E.X i))),
    (fun t_1 ↦ (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation t_1) y → y = s i
hr : Presieve.Arrows.Compatible F E.f s
s' : Presieve.FamilyOfElements F (Presieve.ofArrows E.X E.f)
hs' : ∀ (i : E.I₀), s' (E.f i) ⋯ = s i
t' : F.obj (op X)
ht' : ∀ (i : E.I₀), (ConcreteCategory.hom (F.map (E.f i).op)) t' = s i
hunique :
  ∀ (y : (fun X ↦ X) (F.obj (op X))),
    (fun t ↦ ∀ (i : E.I₀), (ConcreteCategory.hom (F.map (E.f i).op)) t = s i) y → y = t'
T : C
f : T ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ T
Y : C
i : E.I₀
w : Z ⟶ E.X i
heq : w ≫ E.f i = g ≫ f
⊢ (ConcreteCategory.hom (F.map f.op ≫ F.map g.op)) t' = (ConcreteCategory.hom (F.map g.op)) (t f hf)