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

Goal state before:
pp: case inr
ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ Tendsto (fun n ↦ ∫ (x : ℝ) in a..b, F n x ∂μ) l (𝓝 (∫ (x : ℝ) in a..b, f x ∂μ))

Tactic:
apply tendsto_integral_filter_of_dominated_convergence (bound := fun _ ↦ C + 1)

Goal state after:
case inr.hF_meas
ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ ∀ᶠ (n : ι) in l, AEStronglyMeasurable (F n) (μ.restrict (Ι a b))

case inr.h_bound
ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ ∀ᶠ (n : ι) in l, ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → ‖F n x‖ ≤ C + 1

case inr.bound_integrable
ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ IntervalIntegrable (fun x ↦ C + 1) μ a b

case inr.h_lim
ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → Tendsto (fun n ↦ F n x) l (𝓝 (f x))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case neg
ε : Type u_3
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
h : IntervalIntegrable f μ a b
hab : ¬a ≤ b
⊢ AEStronglyMeasurable f (μ.restrict (Ι a b))

Tactic:
rw [uIoc_of_ge (by linarith)]

Goal state after:
case neg
ε : Type u_3
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
h : IntervalIntegrable f μ a b
hab : ¬a ≤ b
⊢ AEStronglyMeasurable f (μ.restrict (Ioc b a))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E
ca cb : E
a b : ℝ
hf : IntervalIntegrable f volume a b
hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) volume
hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) volume
ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 ca)
hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 cb)
⊢ HasStrictFDerivAt (fun p ↦ ∫ (x : ℝ) in p.1..p.2, f x) ((snd ℝ ℝ ℝ).smulRight cb - (fst ℝ ℝ ℝ).smulRight ca) (a, b)

Tactic:
have :=
    integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hf hmeas_a hmeas_b ha hb
      (continuous_snd.fst.tendsto ((a, b), (a, b)))
      (continuous_fst.fst.tendsto ((a, b), (a, b)))
      (continuous_snd.snd.tendsto ((a, b), (a, b)))
      (continuous_fst.snd.tendsto ((a, b), (a, b)))
  refine .of_isLittleO <| (this.congr_left ?_).trans_isBigO ?_
  · simp [sub_smul]
  · exact isBigO_fst_prod.norm_left.add isBigO_snd_prod.norm_left

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

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
s_closed : IsClosed s
main : Icc a b ⊆ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal}
⊢ ∫ (y : ℝ) in a..b, (G' y).toReal ≤ (∫ (y : ℝ) in a..b, φ y) + ε

Tactic:
convert! hG'.le <;>
        · rw [intervalIntegral.integral_of_le hab]
          simp only [integral_Icc_eq_integral_Ioc', Real.volume_singleton]

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

Goal state before:
pp: G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : MeasurableSpace G
ν : Measure G
Γ : Subgroup G
μ : Measure (G ⧸ Γ)
inst✝⁷ : QuotientMeasureEqMeasurePreimage ν μ
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : PolishSpace G
inst✝² : T2Space (G ⧸ Γ)
inst✝¹ : SecondCountableTopology (G ⧸ Γ)
inst✝ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
g : G
A : Set (G ⧸ Γ)
hA : MeasurableSet A
meas_π : Measurable QuotientGroup.mk
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_translate_fundom : IsFundamentalDomain (↥Γ.op) (g • 𝓕) ν
⊢ μ ((fun x ↦ g • x) ⁻¹' A) = μ A

Tactic:
h𝓕.projection_respects_measure_apply (μ := μ)
      (meas_π (measurableSet_preimage (measurable_const_smul g) hA)),

Goal state after:
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : MeasurableSpace G
ν : Measure G
Γ : Subgroup G
μ : Measure (G ⧸ Γ)
inst✝⁷ : QuotientMeasureEqMeasurePreimage ν μ
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : PolishSpace G
inst✝² : T2Space (G ⧸ Γ)
inst✝¹ : SecondCountableTopology (G ⧸ Γ)
inst✝ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
g : G
A : Set (G ⧸ Γ)
hA : MeasurableSet A
meas_π : Measurable QuotientGroup.mk
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_translate_fundom : IsFundamentalDomain (↥Γ.op) (g • 𝓕) ν
⊢ ν (Quotient.mk (MulAction.orbitRel (↥Γ.op) G) ⁻¹' (fun x ↦ g • x) ⁻¹' A ∩ 𝓕) = μ A
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: case neg.hV
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
meas_s_ne_zero : ¬ν s = 0
⊢ μ univ = ν (QuotientGroup.mk ⁻¹' univ ∩ s)

pp: case neg.neTopV
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
meas_s_ne_zero : ¬ν s = 0
⊢ μ univ ≠ ∞

Tactic:
· rw [← h]
    simp

Goal state after:
case neg.neTopV
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
meas_s_ne_zero : ¬ν s = 0
⊢ μ univ ≠ ∞
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case pos
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
x : ℝ
hx : x < a₀
⊢ (fun x_1 ↦ if x < a₀ then f x else 0) =ᶠ[𝓝[≤] a₀] fun n ↦ if x < n then f x else 0

pp: case neg
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
x : ℝ
hx : ¬x < a₀
⊢ (fun x_1 ↦ if x < a₀ then f x else 0) =ᶠ[𝓝[≤] a₀] fun n ↦ if x < n then f x else 0

Tactic:
· filter_upwards [mem_nhdsWithin_of_mem_nhds (Ioi_mem_nhds hx)] with a ha using by grind

Goal state after:
case neg
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
x : ℝ
hx : ¬x < a₀
⊢ (fun x_1 ↦ if x < a₀ then f x else 0) =ᶠ[𝓝[≤] a₀] fun n ↦ if x < n then f x else 0
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:

Tactic:
∫⁻ z, ‖op (f z.1) (g z.2)‖ₑ ∂μ.prod ν ≤ ∫⁻ z, .ofReal C * ‖f z.1‖ₑ * ‖g z.2‖ₑ ∂μ.prod ν := by
      gcongr with z
      simp only [enorm_eq_nnnorm, ENNReal.ofReal, ← ENNReal.coe_mul, ENNReal.coe_le_coe,
        ← NNReal.coe_le_coe, NNReal.coe_mul, coe_nnnorm]
      refine (hC _ _).trans ?_
      gcongr
      apply le_coe_toNNReal
    _ ≤ ∫⁻ x, ∫⁻ y, .ofReal C * ‖f x‖ₑ * ‖g y‖ₑ ∂ν ∂μ := lintegral_prod_le _
    _ ≤ .ofReal C * (∫⁻ x, ‖f x‖ₑ ∂μ) * ∫⁻ y, ‖g y‖ₑ ∂ν := by
      simp [lintegral_const_mul', lintegral_mul_const', hg.2.ne, mul_assoc]
    _ < ∞ := by apply_rules [ENNReal.mul_lt_top, hf.2, hg.2, ENNReal.ofReal_lt_top]

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

Goal state before:
pp: case inl
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)‖

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_mod_cast [hR, mul_zero]
    exact mul_nonneg (inv_nonneg.2 Real.two_pi_pos.le)
      (intervalIntegral.integral_nonneg Real.two_pi_pos.le fun _ _ => norm_nonneg _)

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)⁻¹ ^ n * ↑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: 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
⊢ HasSum (fun n ↦ (cauchyPowerSeries f c R n) fun x ↦ w) ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - (c + w))⁻¹ • f z)

Tactic:
by
  simp only [cauchyPowerSeries_apply]
  exact (hasSum_two_pi_I_cauchyPowerSeries_integral hf hw).const_smul _

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

Goal state before:
pp: case inl
α : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
a b : ℝ
f : ℝ → α → E
h_int : Integrable (uncurry f) ((volume.restrict (uIoc a b)).prod μ)
hab : a ≤ b
⊢ ∫ (x : ℝ) in Ioc a b, ∫ (y : α), f x y ∂μ ∂volume = ∫ (y : α), ∫ (x : ℝ) in Ioc a b, f x y ∂volume ∂μ

Tactic:
simp only [hab, Set.uIoc_of_le] at h_int

Goal state after:
case inl
α : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
a b : ℝ
f : ℝ → α → E
hab : a ≤ b
h_int : Integrable (uncurry f) ((volume.restrict (Ioc a b)).prod μ)
⊢ ∫ (x : ℝ) in Ioc a b, ∫ (y : α), f x y ∂μ ∂volume = ∫ (y : α), ∫ (x : ℝ) in Ioc a b, f x y ∂volume ∂μ
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
ε' : ℝ
ε'0 : 0 < ε'
hlt : 2 * ↑c * ε' < ε
δ : ℝ
δ0 : 0 < δ
Hδ : closedBall x δ ∩ Box.Icc I ⊆ {x_1 | (fun x_2 ↦ ‖f x_2 - f x - (f' x) (x_2 - x)‖ ≤ ε' * ‖x_2 - x‖) x_1}
J : Box (Fin (n + 1))
hle : J ≤ I
hJδ : Box.Icc J ⊆ closedBall x δ
hxJ : x ∈ Box.Icc J
hJc : GP.bDistortion = true → J.distortion ≤ c
⊢ 0 ≤ volume.toBoxAdditive J

Tactic:
by dsimp; positivity

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) → 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
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
⊢ HasIntegral I GP (fun x ↦ ∑ i, (f' x) (Pi.single i 1) i) BoxAdditiveMap.volume
    (∑ i,
      (integral (I.face i) GP (fun x ↦ f (i.insertNth (I.upper i) x) i) BoxAdditiveMap.volume -
        integral (I.face i) GP (fun x ↦ f (i.insertNth (I.lower i) x) i) BoxAdditiveMap.volume))

Tactic:
refine HasIntegral.sum fun i _ => ?_

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) → 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
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)
x✝ : i ∈ Finset.univ
⊢ HasIntegral I GP (fun x ↦ (f' x) (Pi.single i 1) i) BoxAdditiveMap.volume
    (integral (I.face i) GP (fun x ↦ f (i.insertNth (I.upper i) x) i) BoxAdditiveMap.volume -
      integral (I.face i) GP (fun x ↦ f (i.insertNth (I.lower i) x) i) BoxAdditiveMap.volume)
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
s : Set (ℝ × ℝ)
hs : s.Countable
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 \ s, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume
e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
⊢ ((∫ (y : ℝ) in Set.Icc a.2 b.2, f (b.1, y)) - ∫ (y : ℝ) in Set.Icc a.2 b.2, f (a.1, y)) +
      ((∫ (x : ℝ) in Set.Icc a.1 b.1, g (x, b.2)) - ∫ (x : ℝ) in Set.Icc a.1 b.1, g (x, a.2)) =
    (((∫ (x : ℝ) in a.1..b.1, g (x, b.2)) - ∫ (x : ℝ) in a.1..b.1, g (x, a.2)) + ∫ (y : ℝ) in a.2..b.2, f (b.1, y)) -
      ∫ (y : ℝ) in a.2..b.2, f (a.1, y)

Tactic:
simp only [intervalIntegral.integral_of_le hle.1, intervalIntegral.integral_of_le hle.2,
        setIntegral_congr_set (Ioc_ae_eq_Icc (α := ℝ) (μ := volume))]

Goal state after:
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
s : Set (ℝ × ℝ)
hs : s.Countable
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 \ s, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume
e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
⊢ ((∫ (y : ℝ) in Set.Icc a.2 b.2, f (b.1, y)) - ∫ (y : ℝ) in Set.Icc a.2 b.2, f (a.1, y)) +
      ((∫ (x : ℝ) in Set.Icc a.1 b.1, g (x, b.2)) - ∫ (x : ℝ) in Set.Icc a.1 b.1, g (x, a.2)) =
    (((∫ (x : ℝ) in Set.Icc a.1 b.1, g (x, b.2)) - ∫ (x : ℝ) in Set.Icc a.1 b.1, g (x, a.2)) +
        ∫ (y : ℝ) in Set.Icc a.2 b.2, f (b.1, y)) -
      ∫ (y : ℝ) in Set.Icc a.2 b.2, f (a.1, y)
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
⊢ IsPrincipalIdealRing O

Tactic:
have := bijective_algebraMap_of_subsingleton_units_mrange hv
    exact .of_surjective _ (RingEquiv.ofBijective _ this).symm.surjective

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 : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(toLex d) = φ.lexOrder
hφ : φ ≠ 0
ne : (⇑toLex '' Function.support φ).Nonempty
hφ' : toLex d = ⋯.min (⇑toLex '' Function.support φ) ne
⊢ toLex d ∈ ⇑toLex '' Function.support φ

Tactic:
rw [hφ']

Goal state after:
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 : ↑(toLex d) = φ.lexOrder
hφ : φ ≠ 0
ne : (⇑toLex '' Function.support φ).Nonempty
hφ' : toLex d = ⋯.min (⇑toLex '' Function.support φ) ne
⊢ ⋯.min (⇑toLex '' Function.support φ) ne ∈ ⇑toLex '' Function.support φ
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
⊢ ∃ x, 1 < ‖x‖

Tactic:
obtain ⟨x, hx⟩ := Valuation.RankOne.nontrivial val.v

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
hx : v x ≠ 0 ∧ v x ≠ 1
⊢ ∃ x, 1 < ‖x‖
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
⊢ (eval P) W'.polynomial =
    P y ^ 2 * P z + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 2 -
      (P x ^ 3 + W'.a₂ * P x ^ 2 * P z + W'.a₄ * P x * P z ^ 2 + W'.a₆ * P z ^ 3)

Tactic:
rw [polynomial]
  simp

Goal state after:
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
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_1, t.IsAmalgamation t_1

Tactic:
obtain ⟨t', ht', hunique⟩ := (Presieve.isSheafFor_arrows_iff _ _).mp h₁.isSheafFor_presieve₀ _ hr

Goal state after:
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_1, t.IsAmalgamation t_1
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: case neg.refine_1
F : Type u
inst✝¹ : Field F
W : Projective F
K : Type v
inst✝ : Field K
f : F →+* K
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
h : ⇑f ∘ P ≈ ⇑f ∘ Q
hz : ¬f (P z) = 0
⊢ f (P x * Q z) = f (Q x * P z)

Tactic:
map_simp

Goal state after:
case neg.refine_1
F : Type u
inst✝¹ : Field F
W : Projective F
K : Type v
inst✝ : Field K
f : F →+* K
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
h : ⇑f ∘ P ≈ ⇑f ∘ Q
hz : ¬f (P z) = 0
⊢ f (P x) * f (Q z) = f (Q x) * f (P z)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: case neg.refine_2
F : Type u
inst✝¹ : Field F
W : Projective F
K : Type v
inst✝ : Field K
f : F →+* K
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
h : ⇑f ∘ P ≈ ⇑f ∘ Q
hz : ¬f (P z) = 0
⊢ f (P y * Q z) = f (Q y * P z)

Tactic:
map_simp

Goal state after:
case neg.refine_2
F : Type u
inst✝¹ : Field F
W : Projective F
K : Type v
inst✝ : Field K
f : F →+* K
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
h : ⇑f ∘ P ≈ ⇑f ∘ Q
hz : ¬f (P z) = 0
⊢ f (P y) * f (Q z) = f (Q y) * f (P z)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Projective R
S : Type s
inst✝ : CommRing S
f : R →+* S
⊢ «X» y ^ 2 * «X» z + C (W'.map f).a₁ * «X» x * «X» y * «X» z + C (W'.map f).a₃ * «X» y * «X» z ^ 2 -
      («X» x ^ 3 + C (W'.map f).a₂ * «X» x ^ 2 * «X» z + C (W'.map f).a₄ * «X» x * «X» z ^ 2 +
        C (W'.map f).a₆ * «X» z ^ 3) =
    (MvPolynomial.map f)
      (Y ^ 2 * Z + C W'.a₁ * X * Y * Z + C W'.a₃ * Y * Z ^ 2 -
        (X ^ 3 + C W'.a₂ * X ^ 2 * Z + C W'.a₄ * X * Z ^ 2 + C W'.a₆ * Z ^ 3))

Tactic:
map_simp

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

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Projective R
f : R →+* S
P : Fin 3 → R
⊢ ![(⇑f ∘ P) x, f (W'.negY P), (⇑f ∘ P) z] = ![f (P x), f (W'.negY P), f (P z)]

Tactic:
map_simp

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

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Bijective ⇑f
⊢ f.FormallySmooth ∧ f.FinitePresentation

Tactic:
exact ⟨.of_bijective hf, .of_bijective hf⟩

Goal state after: