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

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
g : α → EReal
g_lt_f : ∀ (x : α), ↑(-f x) < g x
gcont : LowerSemicontinuous g
g_integrable : Integrable (fun x ↦ (g x).toReal) μ
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
gint : ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), -f x ∂μ + ε
⊢ ∃ g,
    (∀ (x : α), g x < ↑(f x)) ∧
      UpperSemicontinuous g ∧
        Integrable (fun x ↦ (g x).toReal) μ ∧
          (∀ᵐ (x : α) ∂μ, ⊥ < g x) ∧ ∫ (x : α), f x ∂μ < ∫ (x : α), (g x).toReal ∂μ + ε

Tactic:
refine ⟨fun x => -g x, ?_, ?_, ?_, ?_, ?_⟩

Goal state after:
case refine_1
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
g : α → EReal
g_lt_f : ∀ (x : α), ↑(-f x) < g x
gcont : LowerSemicontinuous g
g_integrable : Integrable (fun x ↦ (g x).toReal) μ
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
gint : ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), -f x ∂μ + ε
⊢ ∀ (x : α), (fun x ↦ -g x) x < ↑(f x)

case refine_2
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
g : α → EReal
g_lt_f : ∀ (x : α), ↑(-f x) < g x
gcont : LowerSemicontinuous g
g_integrable : Integrable (fun x ↦ (g x).toReal) μ
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
gint : ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), -f x ∂μ + ε
⊢ UpperSemicontinuous fun x ↦ -g x

case refine_3
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
g : α → EReal
g_lt_f : ∀ (x : α), ↑(-f x) < g x
gcont : LowerSemicontinuous g
g_integrable : Integrable (fun x ↦ (g x).toReal) μ
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
gint : ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), -f x ∂μ + ε
⊢ Integrable (fun x ↦ ((fun x ↦ -g x) x).toReal) μ

case refine_4
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
g : α → EReal
g_lt_f : ∀ (x : α), ↑(-f x) < g x
gcont : LowerSemicontinuous g
g_integrable : Integrable (fun x ↦ (g x).toReal) μ
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
gint : ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), -f x ∂μ + ε
⊢ ∀ᵐ (x : α) ∂μ, ⊥ < (fun x ↦ -g x) x

case refine_5
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
g : α → EReal
g_lt_f : ∀ (x : α), ↑(-f x) < g x
gcont : LowerSemicontinuous g
g_integrable : Integrable (fun x ↦ (g x).toReal) μ
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
gint : ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), -f x ∂μ + ε
⊢ ∫ (x : α), f x ∂μ < ∫ (x : α), ((fun x ↦ -g x) x).toReal ∂μ + ε
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 ↦ ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
    ‖f z - f x - (z - x) • derivWithin f (Ici x) x - (f y - f x - (y - x) • derivWithin f (Ici x) x)‖

Tactic:
by
      congr 1; simp only [sub_smul]; abel

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
r x : ℝ
hr : 0 < r
ε : ℝ
L₁ L₂ : F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
H : r / 2 * ‖L₁ - L₂‖ ≤ r / 2 * (4 * ε)
⊢ ‖L₁ - L₂‖ ≤ 4 * ε

Tactic:
mul_le_mul_iff_right₀ (half_pos hr)

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
r x : ℝ
hr : 0 < r
ε : ℝ
L₁ L₂ : F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
H : ‖L₁ - L₂‖ ≤ 4 * ε
⊢ ‖L₁ - L₂‖ ≤ 4 * ε
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case pos.refine_1.hfi
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ Integrable (fun x ↦ ↑(f x)) (μ.restrict s)

Tactic:
refine ⟨f_meas.coe_nnreal_real.aemeasurable.aestronglyMeasurable, ?_⟩
      simpa [withDensity_apply _ s_meas, hasFiniteIntegral_iff_enorm] using hs

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

Goal state before:
pp: case e'_6
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
inst✝ : PseudoMetrizableSpace ε
hf : IntegrableOn f [[a, b]] volume
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
h' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞
hc : c ≠ 0
A : MeasurableEmbedding fun x ↦ x * c⁻¹
⊢ ((fun x ↦ f (c * x)) ∘ fun x ↦ x * c⁻¹) = f

Tactic:
ext; simp only [comp_apply]; congr 1; field

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

Goal state before:
pp: case pos.refine_3
ι : Type u_1
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : ℝ → E
c : E
l l' : Filter ℝ
lt : Filter ι
μ : Measure ℝ
u v : ι → ℝ
inst✝¹ : l'.IsMeasurablyGenerated
inst✝ : TendstoIxxClass Ioc l l'
hfm : StronglyMeasurableAtFilter f l' μ
hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)
hl : μ.FiniteAtFilter l'
hu : Tendsto u lt l
hv : Tendsto v lt l
hE : CompleteSpace E
A :
  (fun i ↦ ∫ (x : ℝ) in Ioc (u i) (v i), f x ∂μ - μ.real (Ioc (u i) (v i)) • c) =o[lt] fun i ↦ μ.real (Ioc (u i) (v i))
B :
  (fun i ↦ ∫ (x : ℝ) in Ioc (v i) (u i), f x ∂μ - μ.real (Ioc (v i) (u i)) • c) =o[lt] fun i ↦ μ.real (Ioc (v i) (u i))
t : ι
⊢ ∫ (x : ℝ) in Ioc (u t) (v t), f x ∂μ - μ.real (Ioc (u t) (v t)) • c -
      (∫ (x : ℝ) in Ioc (v t) (u t), f x ∂μ - μ.real (Ioc (v t) (u t)) • c) =
    ∫ (x : ℝ) in u t..v t, f x ∂μ - (μ.real (Ioc (u t) (v t)) • c - μ.real (Ioc (v t) (u t)) • c)

Tactic:
· simp_rw [intervalIntegral]
    abel

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

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
inst✝ : PseudoMetrizableSpace ε
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ IntervalIntegrable (fun x ↦ f (x + c)) volume (a - c) (b - c) ↔ IntervalIntegrable f volume a b

Tactic:
rw [IntervalIntegrable.comp_add_right_iff (by grind)]

Goal state after:
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
inst✝ : PseudoMetrizableSpace ε
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ IntervalIntegrable f volume (a - c + c) (b - c + c) ↔ IntervalIntegrable f volume a b
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
inst✝ : PseudoMetrizableSpace ε
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ IntervalIntegrable (fun x ↦ f (x + -c)) volume (a + c) (b + c) ↔ IntervalIntegrable f volume a b

Tactic:
rw [IntervalIntegrable.comp_add_right_iff (by grind)]

Goal state after:
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
inst✝ : PseudoMetrizableSpace ε
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ IntervalIntegrable f volume (a + c + -c) (b + c + -c) ↔ IntervalIntegrable f volume a b
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: case hbc
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
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
I1 : ∀ᶠ (u : ℝ) in 𝓝[>] t, (u - t) * y ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I2 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ (u - t) * y
I3 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I4 : ∀ᶠ (u : ℝ) in 𝓝[>] t, u ∈ Ioc t (min v b)
x : ℝ
hx : g x - g t ≤ ∫ (w : ℝ) in t..x, (G' w).toReal
h'x : x ∈ Ioc t (min v b)
⊢ IntegrableOn (fun x ↦ (G' x).toReal) (Ioc t x) volume

Tactic:
apply IntegrableOn.mono_set G'int

Goal state after:
case hbc
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 := ⋯
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
I1 : ∀ᶠ (u : ℝ) in 𝓝[>] t, (u - t) * y ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I2 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ (u - t) * y
I3 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I4 : ∀ᶠ (u : ℝ) in 𝓝[>] t, u ∈ Ioc t (min v b)
x : ℝ
hx : g x - g t ≤ ∫ (w : ℝ) in t..x, (G' w).toReal
h'x : x ∈ Ioc t (min v b)
⊢ Ioc t x ⊆ Icc a b
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hα : SecondCountableTopology α
this : ProperSpace 𝕜
p : α × 𝕜
A : deriv (f p.1) p.2 ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(Submodule.span 𝕜 (range (f p.1)))
x : 𝕜
⊢ f p.1 x ∈ range (Function.uncurry f)

Tactic:
exact mem_range_self (p.1, x)

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

Goal state before:
pp: case hle
f g : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hgc : ContinuousOn g (Icc a b)
hle : ∀ x ∈ Ioc a b, f x ≤ g x
hlt : ∃ c ∈ Icc a b, f c < g c
⊢ f ≤ᵐ[volume.restrict (Ioc a b)] g

Tactic:
simpa only [measurableSet_Ioc, ae_restrict_eq]
      using! (ae_restrict_mem measurableSet_Ioc).mono hle

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

Goal state before:
pp: a : ℝ
⊢ ∀ᶠ (x : ℝ) in atTop, HasDerivAt (fun x ↦ Real.log x) x⁻¹ x

Tactic:
filter_upwards [Ioi_mem_atTop 0] with x hx using Real.hasDerivAt_log (ne_of_gt hx)

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

Goal state before:
pp: case hf
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ IntegrableOn f (Ioo a b) μ

Tactic:
apply hf.1.mono Ioo_subset_Ioc_self le_rfl

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : LinearOrder α
inst✝¹ : FloorRing α
inst✝ : IsStrictOrderedRing α
b : ℤ
⊢ b ≤ ⌊↑(b + 1)⌋

Tactic:
rw [floor_intCast]; exact (lt_add_one _).le

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

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
c : ℂ
R : ℝ
f : ℂ → E
⊢ CircleIntegrable f c (-R) ↔ CircleIntegrable f c R

Tactic:
by
  unfold CircleIntegrable
  rw [intervalIntegrable_congr (f := fun θ ↦ f (circleMap c (-R) θ))
    (g := fun θ ↦ (f ∘ (circleMap c R)) (θ + π)) (fun _ _ ↦ by simp [circleMap_neg_radius]),
    IntervalIntegrable.comp_add_right_iff (c := π), add_comm (2 * π) π]
  simpa using! ((periodic_circleMap c R).comp f).intervalIntegrable_iff (t₂ := 0)

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Iic a₀) μ
⊢ ContinuousOn (fun b ↦ ∫ (x : ℝ) in Iio b, f x ∂μ) (Iic a₀)

Tactic:
exact (hf.mono_set Iio_subset_Iic_self).continuousOn_Iic_primitive_Iio

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
⊢ (2 * π)⁻¹ *
      ∫ (θ : ℝ) in 0..2 * π,
        ‖deriv (circleMap c R) θ • (circleMap c R θ - c)⁻¹ ^ n • (circleMap c R θ - c)⁻¹ • f (circleMap c R θ)‖ =
    (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0..2 * π, ‖f (circleMap c R x)‖)))

Tactic:
simp [norm_smul, mul_left_comm |R|]

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

Goal state before:
pp: case pos.h_add
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
hE : CompleteSpace E
f g : α × β → E
i_f : Integrable f (μ.prod ν)
i_g : Integrable g (μ.prod ν)
hf : ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ
hg : ∫ (z : α × β), g z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ
⊢ ∫ (z : α × β), (f + g) z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), (f + g) (x, y) ∂ν ∂μ

Tactic:
integral_add' i_f i_g,

Goal state after:
case pos.h_add
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
hE : CompleteSpace E
f g : α × β → E
i_f : Integrable f (μ.prod ν)
i_g : Integrable g (μ.prod ν)
hf : ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ
hg : ∫ (z : α × β), g z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ
⊢ ∫ (a : α × β), f a ∂μ.prod ν + ∫ (a : α × β), g a ∂μ.prod ν = ∫ (x : α), ∫ (y : β), (f + g) (x, y) ∂ν ∂μ
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x ↦ c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
h_maps : MapsTo g R A
hc : ContinuousOn (f ∘ g) R
⊢ ∀ z ∈ Ioo (min a b) (max a b) ×ℂ Ioo (min 0 (2 * π)) (max 0 (2 * π)) \ g ⁻¹' s, DifferentiableAt ℂ (f ∘ g) z

Tactic:
refine fun z hz => (hd (g z) ⟨?_, hz.2⟩).comp z (hdg _)

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x ↦ c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
h_maps : MapsTo g R A
hc : ContinuousOn (f ∘ g) R
z : ℂ
hz : z ∈ Ioo (min a b) (max a b) ×ℂ Ioo (min 0 (2 * π)) (max 0 (2 * π)) \ g ⁻¹' s
⊢ g z ∈ ball c (rexp b) \ closedBall c (rexp a)
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case right
F : Type u_1
inst✝² : Field F
p : F[X]
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
h : Fact (map (algebraMap F E) p).Splits
y : ↑(p.rootSet E)
key :
  (map (↑(IsScalarTower.toAlgHom F p.SplittingField E)) (map (algebraMap F p.SplittingField) p)).roots =
    Multiset.map (⇑↑(IsScalarTower.toAlgHom F p.SplittingField E)) (map (algebraMap F p.SplittingField) p).roots
⊢ ∃ a, mapRoots p E a = y

Tactic:
map_map,

Goal state after:
case right
F : Type u_1
inst✝² : Field F
p : F[X]
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
h : Fact (map (algebraMap F E) p).Splits
y : ↑(p.rootSet E)
key :
  (map ((↑(IsScalarTower.toAlgHom F p.SplittingField E)).comp (algebraMap F p.SplittingField)) p).roots =
    Multiset.map (⇑↑(IsScalarTower.toAlgHom F p.SplittingField E)) (map (algebraMap F p.SplittingField) p).roots
⊢ ∃ a, mapRoots p E a = y
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : Monoid S
inst✝² : DistribMulAction S R
inst✝¹ : IsScalarTower S R R
inst✝ : SMulCommClass S R R
s : S
n : ℕ
m : Multiset R
⊢ (map (fun x ↦ (map (fun x ↦ s • x) x).prod) (powersetCard n m)).sum =
    (map (fun x ↦ x.prod) (map (map fun x ↦ s • x) (powersetCard n m))).sum

Tactic:
map_map,

Goal state after:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : Monoid S
inst✝² : DistribMulAction S R
inst✝¹ : IsScalarTower S R R
inst✝ : SMulCommClass S R R
s : S
n : ℕ
m : Multiset R
⊢ (map (fun x ↦ (map (fun x ↦ s • x) x).prod) (powersetCard n m)).sum =
    (map ((fun x ↦ x.prod) ∘ map fun x ↦ s • x) (powersetCard n m)).sum
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits
r : F[X]
hr : Irreducible r
hr' : ¬r.natDegree = 0
h : (map (algebraMap F (r.comp q).SplittingField) (r.comp q)).degree = 0
⊢ (r.comp q).degree = ↑0

Tactic:
by rwa [degree_map] at h

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

Goal state before:

Tactic:
(∮ z in C(c, R), f z) = ∮ z in C(c, R), (z - c)⁻¹ • (z - c) • f z :=
      (circleIntegral.integral_sub_inv_smul_sub_smul _ _ _ _).symm
    _ = (2 * ↑π * I : ℂ) • (c - c) • f c :=
      (circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable h0 hs
        ((continuousOn_id.sub continuousOn_const).smul hc) fun z hz =>
        (differentiableAt_id.sub_const _).smul (hd z hz))
    _ = 0 := by rw [sub_self, zero_smul, smul_zero]

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

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : (insert w s).Countable
⊢ ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = (2 * ↑π * I) • f w

Tactic:
have hcF : ContinuousOn F (closedBall c R) :=
    (continuousOn_dslope <| closedBall_mem_nhds_of_mem hw.1).2 ⟨hc, hd _ hw⟩

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
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : (insert w s).Countable
hcF : ContinuousOn F (closedBall c R)
⊢ ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = (2 * ↑π * I) • f w
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
s : Multiset R
⊢ (map (fun p ↦ (map (fun r ↦ X) p.1).prod * (map (⇑C) p.2).prod) (map (fun t ↦ (s - t, t)) s.powerset)).sum =
    ∑ j ∈ Finset.range (s.card + 1), C (s.esymm j) * X ^ (s.card - j)

Tactic:
map_map,

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
s : Multiset R
⊢ (map ((fun p ↦ (map (fun r ↦ X) p.1).prod * (map (⇑C) p.2).prod) ∘ fun t ↦ (s - t, t)) s.powerset).sum =
    ∑ j ∈ Finset.range (s.card + 1), C (s.esymm j) * X ^ (s.card - j)
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
⊢ ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)

Tactic:
eq_one_of_roots_le hB h1 h2 h3,

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
⊢ ‖(map f 1).coeff i‖ ≤ B ^ (natDegree 1 - i) * ↑((natDegree 1).choose i)
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: case inr.inr
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 : 0 ≤ B
hi : i ≤ (map f p).natDegree
⊢ ‖(map f p).coeff i‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)

Tactic:
rw [coeff_eq_esymm_roots_of_splits h2 hi, (h1.map _).leadingCoeff,
    one_mul, norm_mul, norm_pow, norm_neg, norm_one, one_pow, one_mul]

Goal state after:
case inr.inr
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 : 0 ≤ B
hi : i ≤ (map f p).natDegree
⊢ ‖(map f p).roots.esymm ((map f p).natDegree - i)‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: Γ₀ : Type u_2
inst✝² : LinearOrderedCommGroupWithZero Γ₀
K : Type u_4
inst✝¹ : DivisionRing K
v : Valuation K Γ₀
inst✝ : v.RankLeOne
⊢ Subsingleton (ofClass v).ValueGroup₀ˣ ∨ ∀ {γ : ℝ≥0}, γ ≠ 0 → ∃ x, x ≠ 0 ∧ (hom' v) (v.restrict x) < γ

Tactic:
simp only [ne_eq, or_iff_not_imp_left, not_subsingleton_iff_nontrivial]

Goal state after:
Γ₀ : Type u_2
inst✝² : LinearOrderedCommGroupWithZero Γ₀
K : Type u_4
inst✝¹ : DivisionRing K
v : Valuation K Γ₀
inst✝ : v.RankLeOne
⊢ Nontrivial (ofClass v).ValueGroup₀ˣ → ∀ {γ : ℝ≥0}, ¬γ = 0 → ∃ x, ¬x = 0 ∧ (hom' v) (v.restrict x) < γ
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
⊢ ‖∏' (i : ι), f i‖₊ ≤ ⨆ i, ‖f i‖₊

Tactic:
simpa only [← NNReal.coe_le_coe, coe_nnnorm', coe_iSup] using norm_tprod_le f

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 (0, n).2, if x.2 < (0, n).2 then (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) else 0) +
      ∑ x ∈ (antidiagonal n).erase (0, n), (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) =
    0

Tactic:
←
        Finset.insert_erase this,

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 ∈ insert (0, n) ((antidiagonal n).erase (0, n)),
          if x.2 < (0, n).2 then (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) else 0) +
      ∑ x ∈ (insert (0, n) ((antidiagonal n).erase (0, n))).erase (0, n), (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) =
    0