mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
⊢ ∀
    x ∈
      {p |
        ∃ r',
          (r / 2 < r' ∧ r' ≤ r) ∧
            ∀ (y : E), dist y p.2 < r' → ∀ (z : E), dist z p.2 < r' → ‖f p.1 z - f p.1 y - (L z - L y)‖ < s * r},
    {p |
        ∃ r',
          (r / 2 < r' ∧ r' ≤ r) ∧
            ∀ (y : E), dist y p.2 < r' → ∀ (z : E), dist z p.2 < r' → ‖f p.1 z - f p.1 y - (L z - L y)‖ < s * r} ∈
      𝓝 x

Tactic:
rintro ⟨a, x⟩ ⟨r', ⟨Irr', Ir'r⟩, hr⟩

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
a : α
x : E
r' : ℝ
hr :
  ∀ (y : E),
    dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
⊢ {p |
      ∃ r',
        (r / 2 < r' ∧ r' ≤ r) ∧
          ∀ (y : E), dist y p.2 < r' → ∀ (z : E), dist z p.2 < r' → ‖f p.1 z - f p.1 y - (L z - L y)‖ < s * r} ∈
    𝓝 (a, x)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
ca cb : E
la la' lb lb' : Filter ℝ
lt : Filter ι
μ : Measure ℝ
ua va ub vb : ι → ℝ
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : FTCFilter a la la'
inst✝ : FTCFilter b lb lb'
hab : IntervalIntegrable f μ a b
hmeas_a : StronglyMeasurableAtFilter f la' μ
hmeas_b : StronglyMeasurableAtFilter f lb' μ
ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)
hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)
hua : Tendsto ua lt la
hva : Tendsto va lt la
hub : Tendsto ub lt lb
hvb : Tendsto vb lt lb
this✝ : la'.IsMeasurablyGenerated
this : lb'.IsMeasurablyGenerated
A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)
A' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)
B : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)
B' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)
a✝ : ι
ua_va : IntervalIntegrable f μ (ua a✝) (va a✝)
⊢ IntervalIntegrable f μ a (ua a✝) →
    IntervalIntegrable f μ (ub a✝) (vb a✝) →
      IntervalIntegrable f μ b (ub a✝) →
        -(∫ (x : ℝ) in ua a✝..va a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ) +
            (∫ (x : ℝ) in ub a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ) =
          ∫ (x : ℝ) in va a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..ub a✝, f x ∂μ -
            (∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ)

Tactic:
a_ua

Goal state after:
ι : Type u_1
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
ca cb : E
la la' lb lb' : Filter ℝ
lt : Filter ι
μ : Measure ℝ
ua va ub vb : ι → ℝ
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : FTCFilter a la la'
inst✝ : FTCFilter b lb lb'
hab : IntervalIntegrable f μ a b
hmeas_a : StronglyMeasurableAtFilter f la' μ
hmeas_b : StronglyMeasurableAtFilter f lb' μ
ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)
hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)
hua : Tendsto ua lt la
hva : Tendsto va lt la
hub : Tendsto ub lt lb
hvb : Tendsto vb lt lb
this✝ : la'.IsMeasurablyGenerated
this : lb'.IsMeasurablyGenerated
A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)
A' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)
B : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)
B' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)
a✝ : ι
ua_va : IntervalIntegrable f μ (ua a✝) (va a✝)
a_ua : IntervalIntegrable f μ a (ua a✝)
⊢ IntervalIntegrable f μ (ub a✝) (vb a✝) →
    IntervalIntegrable f μ b (ub a✝) →
      -(∫ (x : ℝ) in ua a✝..va a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ) +
          (∫ (x : ℝ) in ub a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ) =
        ∫ (x : ℝ) in va a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..ub a✝, f x ∂μ -
          (∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
⊢ ∫ (x : ℝ) in a..b, -f x ∂μ = -∫ (x : ℝ) in a..b, f x ∂μ

Tactic:
by
  simp only [intervalIntegral, integral_neg]; abel

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
d : ℝ
⊢ ∫ (x : ℝ) in a..b, f (d - x) = ∫ (x : ℝ) in d - b..d - a, f x

Tactic:
simpa only [one_mul, one_smul, inv_one] using integral_comp_sub_mul f one_ne_zero d

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case refine_2
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
⊢ ((∀ 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) →
    (∀ k ∈ Ico m (p + 1), IntervalIntegrable f μ (a k) (a (k + 1))) →
      ∑ k ∈ Finset.Ico m (p + 1), ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a (p + 1), f x ∂μ

Tactic:
IH

Goal state after:
case refine_2
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
⊢ (∀ k ∈ Ico m (p + 1), IntervalIntegrable f μ (a k) (a (k + 1))) →
    ∑ k ∈ Finset.Ico m (p + 1), ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a (p + 1), f x ∂μ
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, δ ∈ Ioo 0 1
B : ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
⊢ Tendsto (fun x ↦ μ.real (Icc (b₀ - x) (b₀ + x))) (𝓝 0) (𝓝 (ENNReal.toReal 0))

Tactic:
exact (tendsto_toReal zero_ne_top).comp (tendsto_measure_Icc _ _)

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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd✝ : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k volume
c : ℝ
hc : c ∈ s
d : ℝ
hfd : ∀ z ∈ [[c, d]], DifferentiableAt ℝ f z
hsub : [[c, d]] ⊆ k
hd : d ∈ s
hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖
hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖
hg_ae : ∀ᵐ (x : ℝ) ∂volume.restrict (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖
hsub' : Ι c d ⊆ k
hfi : IntervalIntegrable (deriv f) volume c d
⊢ ∫ (x : ℝ) in Ι c d, C * ‖g x‖ ≤ ∫ (x : ℝ) in k, C * ‖g x‖

Tactic:
apply setIntegral_mono_set hgi
        (ae_of_all _ fun x => mul_nonneg hC₀ (norm_nonneg _)) hsub'.eventuallyLE

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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
h'd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x
h'f : Tendsto (fun x ↦ ‖f' x‖) l atTop
⊢ deriv f' =O[l] g

Tactic:
apply IsBigO.trans _ hfg

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := ⋯
f' : ℝ → UniformSpace.Completion E := ⋯
h'd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x
h'f : Tendsto (fun x ↦ ‖f' x‖) l atTop
⊢ deriv f' =O[l] deriv f
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
f g : ℝ → E
μ : Measure ℝ
inst✝ : NullSingletonClass μ
h : EqOn f g (uIoo a b)
x : ℝ
a✝ : x ≠ max a b
hx : x ∈ Ι a b
⊢ f x = g x

Tactic:
exact h ⟨hx.left, lt_of_le_of_ne hx.right ‹_›⟩

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

Goal state before:
pp: G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : LocallyCompactSpace G
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
i : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : IsFiniteMeasure μ
K : PositiveCompacts G
K' : PositiveCompacts (G ⧸ Γ) := PositiveCompacts.map QuotientGroup.mk ⋯ ⋯ K
this : μ.IsMulLeftInvariant
finiteCovol : covolume (↥Γ.op) G ν ≠ ∞
⊢ (μ ↑K' • haarMeasure K').IsHaarMeasure

Tactic:
obtain ⟨s, fund_dom_s⟩ := i

Goal state after:
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : LocallyCompactSpace G
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
K : PositiveCompacts G
K' : PositiveCompacts (G ⧸ Γ) := PositiveCompacts.map QuotientGroup.mk ⋯ ⋯ K
this : μ.IsMulLeftInvariant
finiteCovol : covolume (↥Γ.op) G ν ≠ ∞
s : Set G
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
⊢ (μ ↑K' • haarMeasure K').IsHaarMeasure
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: case e'_3
T : ℝ
hT : Fact (0 < T)
t : ℝ
f : AddCircle T → ℝ≥0∞
m : MeasurableSet (Ioc t (t + T))
this : ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x ↦ ↑↑x) volume = ∫⁻ (a : ↑(Ioc t (t + T))), f ↑↑a
⊢ ∫⁻ (b : AddCircle T), f b ∂Measure.map QuotientAddGroup.mk (volume.restrict (Ioc t (t + T))) =
    ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x ↦ ↑↑x) volume

Tactic:
congr 1
    have : ((↑) : Ioc t (t + T) → AddCircle T) = ((↑) : ℝ → AddCircle T) ∘ ((↑) : _ → ℝ) := by
      ext1 x; rfl
    simp_rw [this]
    rw [← map_map AddCircle.measurable_mk' measurable_subtype_coe, ← map_comap_subtype_coe m]
    rfl

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: E✝ : Type u_1
inst✝¹ : NormedAddCommGroup E✝
f✝ : ℝ → E✝
T✝ : ℝ
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℝ → E
T t₁ t₂ : ℝ
hf : Periodic f T
hT : T ≠ 0
⊢ IntervalIntegrable f volume t₁ (t₁ + T) ↔ IntervalIntegrable f volume t₂ (t₂ + T)

Tactic:
exact ⟨(hf.intervalIntegrable hT · t₂ (t₂ + T)), (hf.intervalIntegrable hT · t₁ (t₁ + T))⟩

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

Goal state before:
pp: case hF_meas
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
a₀ : ℝ
hf : IntegrableOn f (Iio a₀) μ
⊢ ∀ᶠ (n : ℝ) in 𝓝[≤] a₀, AEStronglyMeasurable ((Iio n).indicator f) μ

Tactic:
filter_upwards [self_mem_nhdsWithin] with a ha
    rw [aestronglyMeasurable_indicator_iff measurableSet_Iio]
    exact (hf.mono_set (Iio_subset_Iio ha)).aestronglyMeasurable

Goal state after:
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 this : ↑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)‖

Tactic:
rw [inv_mul_cancel_right₀ this]

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

Goal state before:
pp: c w : ℂ
R : ℝ
hw : w ∈ ball c R
hR : 0 < R
H : ∀ (n : ℕ), n ≠ 0 → ∮ (z : ℂ) in C(c, R), (z - c) ^ (-↑n - 1) = 0
this : ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)⁻¹ = 2 * ↑π * I
n : ℕ
hn : n ≠ 0
⊢ ∀ z ∈ sphere c R, (z - c)⁻¹ ^ n * (z - c)⁻¹ = (z - c) ^ (-↑n - 1)

Tactic:
intro z _

Goal state after:
c w : ℂ
R : ℝ
hw : w ∈ ball c R
hR : 0 < R
H : ∀ (n : ℕ), n ≠ 0 → ∮ (z : ℂ) in C(c, R), (z - c) ^ (-↑n - 1) = 0
this : ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)⁻¹ = 2 * ↑π * I
n : ℕ
hn : n ≠ 0
z : ℂ
hz✝ : z ∈ sphere c R
⊢ (z - c)⁻¹ ^ n * (z - c)⁻¹ = (z - c) ^ (-↑n - 1)
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
n : ℕ
⊢ ∑ t ∈ powersetCard n univ, ∏ i ∈ t, X i = ∑ x ∈ powersetCard n univ, ∏ x ∈ x, X x ^ 1

Tactic:
pow_one

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

Goal state before:
pp: case inr
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
h0 : 0 ≤ R
f : ℂ → E
c : ℂ
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
this : CompleteSpace E → ∮ (z : ℂ) in C(c, R), f z = 0
hE : ¬CompleteSpace E
⊢ ∮ (z : ℂ) in C(c, R), f z = 0

Tactic:
simp [circleIntegral, intervalIntegral, integral, hE]

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

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a b : ℝ × ℝ
hle : a ≤ b
Hcf : ContinuousOn f (Set.Icc a b)
Hcg : ContinuousOn g (Set.Icc a b)
Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume
⊢ ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ ∅, HasFDerivAt f (f' x) x

Tactic:
simpa only [sdiff_empty]

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

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a b : ℝ × ℝ
hle : a ≤ b
Hcf : ContinuousOn f (Set.Icc a b)
Hcg : ContinuousOn g (Set.Icc a b)
Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume
⊢ ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ ∅, HasFDerivAt g (g' x) x

Tactic:
simpa only [sdiff_empty]

Goal state after:
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
y : ℚ
hy : y.num • 1 < 0
hyx : ↑y = x
⊢ y.num ≤ 0

Tactic:
exact (neg_of_smul_neg_right hy zero_le_one).le

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 : ℝ
h : ∀ (y : ℚ), y.num • 1 < ↑y.den • 1 → ↑y ≤ x
y : ℚ
hy : y < 1
⊢ y.num < ↑y.den

Tactic:
simpa using (Rat.lt_iff _ _).mp hy

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
⊢ lexOrder 0 = ⊤

Tactic:
unfold lexOrder
  rw [dif_pos rfl]

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
⊢ min φ.lexOrder ψ.lexOrder ≤ (φ + ψ).lexOrder

Tactic:
by
  rw [le_lexOrder_iff]
  intro d hd
  simp only [lt_min_iff] at hd
  rw [map_add, coeff_eq_zero_of_lt_lexOrder hd.1, coeff_eq_zero_of_lt_lexOrder hd.2, add_zero]

Goal state after:
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 : constantCoeff φ ≠ 0
⊢ φ * φ⁻¹ = 1

Tactic:
φ.mul_inv_cancel h

Goal state after:
σ : Type u_1
k : Type u_3
inst✝ : Field k
φ : MvPowerSeries σ k
h : constantCoeff φ ≠ 0
⊢ 1 = 1
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
P Q : Fin 3 → R
u v : R
hu : IsUnit u
hv : IsUnit v
h : ¬P ≈ Q
⊢ (fun m ↦ m • W'.add P Q) ((hu.unit * hv.unit) ^ 2) = W'.add (u • P) (v • Q)

Tactic:
by convert! (add_smul_of_not_equiv h hu hv).symm

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
α : Type u_1
inst✝² : Small.{w, u_1} α
X : α → C
c : Cofan X
hc : IsColimit c
inst✝¹ : (ofArrows X c.inj).HasPairwisePullbacks
inst✝ : PreservesLimit (Discrete.functor fun x ↦ op (X x)) F
this : HasCoproduct X
hi :
  IsIso
    (F.map (opCoproductIsoProduct' hc (productIsProduct fun x ↦ op (X x))).inv ≫
      Equalizer.Presieve.Arrows.forkMap F X c.inj)
⊢ ∀ (y : Equalizer.Presieve.Arrows.FirstObj F X),
    (ConcreteCategory.hom (Equalizer.Presieve.Arrows.firstMap F X c.inj)) y =
        (ConcreteCategory.hom (Equalizer.Presieve.Arrows.secondMap F X c.inj)) y →
      ∃! x, (ConcreteCategory.hom (Equalizer.Presieve.Arrows.forkMap F X c.inj)) x = y

Tactic:
isIso_iff_bijective,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
α : Type u_1
inst✝² : Small.{w, u_1} α
X : α → C
c : Cofan X
hc : IsColimit c
inst✝¹ : (ofArrows X c.inj).HasPairwisePullbacks
inst✝ : PreservesLimit (Discrete.functor fun x ↦ op (X x)) F
this : HasCoproduct X
hi :
  Function.Bijective
    ⇑(ConcreteCategory.hom
        (F.map (opCoproductIsoProduct' hc (productIsProduct fun x ↦ op (X x))).inv ≫
          Equalizer.Presieve.Arrows.forkMap F X c.inj))
⊢ ∀ (y : Equalizer.Presieve.Arrows.FirstObj F X),
    (ConcreteCategory.hom (Equalizer.Presieve.Arrows.firstMap F X c.inj)) y =
        (ConcreteCategory.hom (Equalizer.Presieve.Arrows.secondMap F X c.inj)) y →
      ∃! x, (ConcreteCategory.hom (Equalizer.Presieve.Arrows.forkMap F X c.inj)) x = y
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
hc : IsColimit c
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)
hF' : IsSheafFor F (ofArrows X c.inj)
this : HasCoproduct X
⊢ IsIso (Equalizer.Presieve.Arrows.forkMap F X c.inj)

Tactic:
isIso_iff_bijective,

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
hc : IsColimit c
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)
hF' : IsSheafFor F (ofArrows X c.inj)
this : HasCoproduct X
⊢ Function.Bijective ⇑(ConcreteCategory.hom (Equalizer.Presieve.Arrows.forkMap F X c.inj))
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Projective R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).neg (⇑↑f ∘ P) = ⇑↑f ∘ (W'⁄A).neg P

Tactic:
← WeierstrassCurve.Projective.map_neg,

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Projective R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).neg (⇑↑f ∘ P) = ((W'⁄A).map ↑f).neg (⇑↑f ∘ P)
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])
Af : Type u_2 := Localization.Away f
this✝ : FinitePresentation A (Localization.Away f)
this : FinitePresentation R (Localization.Away f)
⊢ IsOpen (smoothLocus R Af)

Tactic:
have := IsLocalizedModule.iso (.powers f)
      (KaehlerDifferential.map R R A (Localization.Away f))

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])
Af : Type u_2 := Localization.Away f
this✝¹ : FinitePresentation A (Localization.Away f)
this✝ : FinitePresentation R (Localization.Away f)
this : LocalizedModule (Submonoid.powers f) Ω[A⁄R] ≃ₗ[A] Ω[Localization.Away f⁄R]
⊢ IsOpen (smoothLocus R Af)
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
s : S
a : A
x✝ : S
y✝ : Ω[A⁄R]
⊢ (s • x✝) ⊗ₜ[R] (a • y✝) = a • (s • x✝) ⊗ₜ[R] y✝

Tactic:
mulActionBaseChange_smul_tmul

Goal state after:
case tmul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
s : S
a : A
x✝ : S
y✝ : Ω[A⁄R]
⊢ (s • x✝) ⊗ₜ[R] (a • y✝) = (s • x✝) ⊗ₜ[R] (a • y✝)