mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

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
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : WithTop ℕ∞
g : F → G
f : E → F
hg : HasFTaylorSeriesUpToOn n g q t
hf : HasFTaylorSeriesUpToOn n f p s
h : MapsTo f s t
m : ℕ
hm : ↑m ≤ n
c : OrderedFinpartition m
x✝ : c ∈ Finset.univ
B : (F [×c.length]→L[𝕜] G) →L[𝕜] (E [×c.partSize i]→L[𝕜] F) [×c.length]→L[𝕜] E [×m]→L[𝕜] G :=
  compAlongOrderedFinpartitionL 𝕜 E F G c
i : Fin c.length
⊢ ContinuousOn (fun y ↦ p y (c.partSize i)) s

Tactic:
have : (c.partSize i : WithTop ℕ∞) ≤ m := by
        exact_mod_cast OrderedFinpartition.partSize_le c i

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : WithTop ℕ∞
g : F → G
f : E → F
hg : HasFTaylorSeriesUpToOn n g q t
hf : HasFTaylorSeriesUpToOn n f p s
h : MapsTo f s t
m : ℕ
hm : ↑m ≤ n
c : OrderedFinpartition m
x✝ : c ∈ Finset.univ
B : (F [×c.length]→L[𝕜] G) →L[𝕜] (E [×c.partSize i]→L[𝕜] F) [×c.length]→L[𝕜] E [×m]→L[𝕜] G :=
  compAlongOrderedFinpartitionL 𝕜 E F G c
i : Fin c.length
this : ↑(c.partSize i) ≤ ↑m
⊢ ContinuousOn (fun y ↦ p y (c.partSize i)) s
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
hdiff' : DifferentiableOn ℝ f (Ioc a b)
z : ℝ
hz'' : z ∈ Ioc a b
⊢ Ioc a b ∩ Ioi a = Icc a b ∩ Ioi a

Tactic:
simp only [Ioc_inter_Ioi, le_refl, sup_of_le_left]

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
hdiff' : DifferentiableOn ℝ f (Ioc a b)
z : ℝ
hz'' : z ∈ Ioc a b
⊢ Ioc a b = Icc a b ∩ Ioi a
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
ι : Sort u_7
f : ι → ConvexCone R M
⊢ ↑(iInf f) = ⋂ i, ↑(f i)

Tactic:
by
  simp [iInf]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: 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≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x ↦ f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x ↦ ∞) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x ↦ ∞) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), (fun x ↦ g0 x + g1 x) x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Tactic:
calc
      ∫⁻ x, g0 x + g1 x ∂μ = (∫⁻ x, g0 x ∂μ) + ∫⁻ x, g1 x ∂μ :=
        lintegral_add_left g0_cont.measurable _
      _ ≤ (∫⁻ x, f x ∂μ) + ε / 2 + (0 + ε / 2) := by
        refine add_le_add ?_ ?_
        · convert! g0_int using 2
          exact lintegral_congr_ae (fmeas.ae_eq_mk.fun_comp _)
        · convert! g1_int
          simp only [smeas, μs, lintegral_const, Set.univ_inter, MeasurableSet.univ,
            lintegral_indicator, mul_zero, restrict_apply]
      _ = (∫⁻ x, f x ∂μ) + ε := by simp only [add_assoc, ENNReal.add_halves, zero_add]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
i : ℕ
⊢ |x| ^ (i + 1) / (0 + 1) ≤ |x| ^ i

Tactic:
simpa [pow_succ] using mul_le_of_le_one_right (by positivity) h.le

Goal state after:
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x : E
a : ℝ
absorbs : Absorbent ℝ s
h : gauge s x < a
⊢ ∃ b, 0 < b ∧ b < a ∧ x ∈ b • s

Tactic:
obtain ⟨b, ⟨hb, hx⟩, hba⟩ := exists_lt_of_csInf_lt absorbs.gauge_set_nonempty h

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x : E
a : ℝ
absorbs : Absorbent ℝ s
h : gauge s x < a
b : ℝ
hba : b < a
hb : 0 < b
hx : x ∈ b • s
⊢ ∃ b, 0 < b ∧ b < a ∧ x ∈ b • s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
inst✝² : RCLike 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsScalarTower ℝ 𝕜 E
hs : Balanced 𝕜 s
r : 𝕜
x : E
θ : ℝ
hθ : 0 < θ
⊢ ‖↑‖r‖‖ = ‖r‖

Tactic:
rw [RCLike.norm_ofReal, abs_norm]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: case inr.inr
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
ht : Convex ℝ t
disj : Disjoint s t
a₀ : E
ha₀ : a₀ ∈ s
b₀ : E
hb₀ : b₀ ∈ t
x₀ : E := b₀ - a₀
C : Set E := x₀ +ᵥ s - t
this✝ : 0 ∈ C
this : Convex ℝ C
⊢ ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b

Tactic:
have : x₀ ∉ C := by
    intro hx₀
    rw [← add_zero x₀] at hx₀
    exact disj.zero_notMem_sub_set (vadd_mem_vadd_set_iff.1 hx₀)

Goal state after:
case inr.inr
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
ht : Convex ℝ t
disj : Disjoint s t
a₀ : E
ha₀ : a₀ ∈ s
b₀ : E
hb₀ : b₀ ∈ t
x₀ : E := b₀ - a₀
C : Set E := x₀ +ᵥ s - t
this✝¹ : 0 ∈ C
this✝ : Convex ℝ C
this : x₀ ∉ C
⊢ ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup V
inst✝⁴ : TopologicalSpace V
inst✝³ : TopologicalSpace R
inst✝² : Module R V
inst✝¹ : SeparatingDual R V
inst✝ : T2Space R
⊢ T2Space V

Tactic:
by
  apply (t2Space_iff _).2 (fun {x} {y} hxy ↦ ?_)
  rcases exists_separating_of_ne (R := R) hxy with ⟨f, hf⟩
  exact separated_by_continuous f.continuous hf

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.CompleteCodomain

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : SeparatingDual 𝕜 E
inst✝¹ : Nontrivial E
inst✝ : CompleteSpace (E →L[𝕜] F)
f : ℕ → F
hf : CauchySeq f
v : E
hv : v ≠ 0
⊢ ∃ a, Tendsto f atTop (𝓝 a)

Tactic:
obtain ⟨φ, hφ⟩ : ∃ φ : StrongDual 𝕜 E, φ v = 1 := exists_eq_one hv

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : SeparatingDual 𝕜 E
inst✝¹ : Nontrivial E
inst✝ : CompleteSpace (E →L[𝕜] F)
f : ℕ → F
hf : CauchySeq f
v : E
hv : v ≠ 0
φ : StrongDual 𝕜 E
hφ : φ v = 1
⊢ ∃ a, Tendsto f atTop (𝓝 a)
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
x y : E
inst✝⁶ : RCLike 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : T1Space E
hxy : x ≠ y
⊢ ∃ f, re (f x) < re (f y)

Tactic:
obtain ⟨f, s, t, hs, st, ht⟩ :=
    geometric_hahn_banach_compact_closed (𝕜 := 𝕜) (convex_singleton x) isCompact_singleton
      (convex_singleton y) isClosed_singleton (disjoint_singleton.2 hxy)
  exact ⟨f, by linarith [hs x rfl, ht y rfl]⟩

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case neg.inr
X : Type u_1
E : Type u_3
inst✝⁶ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace ℝ E
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
φ : X → H →L[𝕜] E
φ_int : Integrable φ μ
v : H
hE : ¬CompleteSpace E
hH : Nontrivial H
⊢ (∫ (x : X), φ x ∂μ) v = ∫ (x : X), (φ x) v ∂μ

Tactic:
have : ¬(CompleteSpace (H →L[𝕜] E)) := by
        rwa [SeparatingDual.completeSpace_continuousLinearMap_iff]

Goal state after:
case neg.inr
X : Type u_1
E : Type u_3
inst✝⁶ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace ℝ E
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
φ : X → H →L[𝕜] E
φ_int : Integrable φ μ
v : H
hE : ¬CompleteSpace E
hH : Nontrivial H
this : ¬CompleteSpace (H →L[𝕜] E)
⊢ (∫ (x : X), φ x ∂μ) v = ∫ (x : X), (φ x) v ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case pos
X : Type u_1
E : Type u_3
inst✝⁵ : MeasurableSpace X
μ : Measure X
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
𝕜 : Type u_8
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : X → 𝕜
c : E
hf : Integrable f μ
⊢ ∫ (x : X), f x • c ∂μ = (∫ (x : X), f x ∂μ) • c

pp: case neg
X : Type u_1
E : Type u_3
inst✝⁵ : MeasurableSpace X
μ : Measure X
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
𝕜 : Type u_8
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : X → 𝕜
c : E
hf : ¬Integrable f μ
⊢ ∫ (x : X), f x • c ∂μ = (∫ (x : X), f x ∂μ) • c

Tactic:
· exact ((1 : 𝕜 →L[𝕜] 𝕜).smulRight c).integral_comp_comm hf

Goal state after:
case neg
X : Type u_1
E : Type u_3
inst✝⁵ : MeasurableSpace X
μ : Measure X
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
𝕜 : Type u_8
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : X → 𝕜
c : E
hf : ¬Integrable f μ
⊢ ∫ (x : X), f x • c ∂μ = (∫ (x : X), f x ∂μ) • c
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
inst✝ : PseudoMetrizableSpace ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
⊢ IntegrableOn f (Ι a b) μ ↔ IntegrableOn f (Ioc a b) μ

Tactic:
uIoc_of_le hab

Goal state after:
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
inst✝ : PseudoMetrizableSpace ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
⊢ IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioc a b) μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case pos
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:
uIoc_of_le hab

Goal state after:
case pos
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 a b))
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
x t : ℝ
⊢ t ∈ Ι b₁ b₂ → ‖if t ≤ x then f t else 0‖ ≤ ‖f t‖

Tactic:
split_ifs <;> simp

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
μ : Measure ℝ
f : ℝ → E
a b : ℝ
⊢ (if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, f x ∂μ ∈ {∫ (x : ℝ) in Ι a b, f x ∂μ, -∫ (x : ℝ) in Ι a b, f x ∂μ}

Tactic:
split_ifs <;> simp

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
f : ℝ → E
μ : Measure ℝ
a₁ a₂ a₃ : ℝ
h : a₂ ∈ Icc a₁ a₃
this : {x | x ≤ a₂} ∩ Ioc a₁ a₃ = Ioc a₁ a₂
⊢ MeasurableSet {x | x ≤ a₂}

Tactic:
exact measurableSet_Iic

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
π_preA : Set G := QuotientGroup.mk ⁻¹' A
this : QuotientGroup.mk ⁻¹' (fun x ↦ g • x) ⁻¹' A = (fun x ↦ g * x) ⁻¹' π_preA
⊢ ν ((fun x ↦ g * x) ⁻¹' π_preA ∩ 𝓕) = ν (π_preA ∩ (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕)

Tactic:
trans ν ((g * ·) ⁻¹' (π_preA ∩ (g⁻¹ * ·) ⁻¹' 𝓕))

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
π_preA : Set G := ⋯
this : QuotientGroup.mk ⁻¹' (fun x ↦ g • x) ⁻¹' A = (fun x ↦ g * x) ⁻¹' π_preA
⊢ ν ((fun x ↦ g * x) ⁻¹' π_preA ∩ 𝓕) = ν ((fun x ↦ g * x) ⁻¹' (π_preA ∩ (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕))

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
π_preA : Set G := ⋯
this : QuotientGroup.mk ⁻¹' (fun x ↦ g • x) ⁻¹' A = (fun x ↦ g * x) ⁻¹' π_preA
⊢ ν ((fun x ↦ g * x) ⁻¹' (π_preA ∩ (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕)) = ν (π_preA ∩ (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕)
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: a : ℝ
⊢ ¬IntegrableOn (fun x ↦ x⁻¹) (Ioi a) volume

Tactic:
simpa only [IntegrableOn, restrict_Ioi_eq_restrict_Ici] using not_integrableOn_Ici_inv

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)
ν : Measure G
inst✝⁵ : ν.IsMulLeftInvariant
inst✝⁴ : Countable ↥Γ
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
s : Set G
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
V : Set (G ⧸ Γ)
meas_V : MeasurableSet V
neZeroV : μ V ≠ 0
hV : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ s)
neTopV : μ V ≠ ∞
U : Set (G ⧸ Γ)
a✝ : MeasurableSet U
meas_π : Measurable QuotientGroup.mk
μ' : Measure (G ⧸ Γ) := map QuotientGroup.mk (ν.restrict s)
has_fund : HasFundamentalDomain (↥Γ.op) G ν
i : QuotientMeasureEqMeasurePreimage ν μ'
this✝ : μ'.IsMulLeftInvariant
this : μ = μ'
⊢ μ U = (map (Quotient.mk (MulAction.orbitRel (↥Γ.op) G)) (ν.restrict s)) U

Tactic:
rw [this]
    rfl

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

Goal state before:
pp: case h₁
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)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
x : ℝ
hx : x ∈ Icc (b₀ - δ) (b₀ + δ)
⊢ ‖f q x‖ ≤ M

Tactic:
apply hM
              change (fun x ↦ ‖Function.uncurry f x‖) (q, x) ∈ _
              apply mem_image_of_mem
              simp only [singleton_prod, mem_image, Prod.mk.injEq, true_and, exists_eq_right]
              exact h'δ hx

Goal state after:
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 (Ioi a₀) μ
x : ℝ
hx : a₀ < x
⊢ (fun x_1 ↦ if a₀ < x then f x else 0) =ᶠ[𝓝[≥] a₀] fun n ↦ if n < x 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 (Ioi a₀) μ
x : ℝ
hx : ¬a₀ < x
⊢ (fun x_1 ↦ if a₀ < x then f x else 0) =ᶠ[𝓝[≥] a₀] fun n ↦ if n < x then f x else 0

Tactic:
· filter_upwards [mem_nhdsWithin_of_mem_nhds (Iio_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 (Ioi a₀) μ
x : ℝ
hx : ¬a₀ < x
⊢ (fun x_1 ↦ if a₀ < x then f x else 0) =ᶠ[𝓝[≥] a₀] fun n ↦ if n < x then f x else 0
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
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
Hc : ContinuousOn f (Box.Icc I)
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc I) volume
⊢ ∫ (x : Fin (n + 1) → ℝ) in Box.Icc I, ∑ i, (f' x) (e i) i =
    ∑ i,
      ((∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -
        ∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i)

Tactic:
wlog hE : CompleteSpace E generalizing

Goal state after:
case inr
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
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
Hc : ContinuousOn f (Box.Icc I)
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc I) volume
this :
  CompleteSpace E →
    ∫ (x : Fin (n + 1) → ℝ) in Box.Icc I, ∑ i, (f' x) (e i) i =
      ∑ i,
        ((∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -
          ∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i)
hE : ¬CompleteSpace E
⊢ ∫ (x : Fin (n + 1) → ℝ) in Box.Icc I, ∑ i, (f' x) (e i) i =
    ∑ i,
      ((∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -
        ∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i)

E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
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
Hc : ContinuousOn f (Box.Icc I)
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc I) volume
hE : CompleteSpace E
⊢ ∫ (x : Fin (n + 1) → ℝ) in Box.Icc I, ∑ i, (f' x) (e i) i =
    ∑ i,
      ((∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -
        ∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i)
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: 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
f : α → E
⊢ ∫ (z : α × β), f z.1 ∂μ.prod ν = ν.real univ • ∫ (x : α), f x ∂μ

Tactic:
rw [← integral_prod_swap]
  apply integral_fun_snd

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

Goal state before:
pp: 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
⊢ ∮ (z : ℂ) in C(c, R), f z = 0

Tactic:
wlog hE : CompleteSpace E generalizing

Goal state after:
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

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
hE : CompleteSpace E
⊢ ∮ (z : ℂ) in C(c, R), f z = 0
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
p : F[X]
inst✝ : CharZero F
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
hp : p.degree ≠ 0
⊢ (map (algebraMap F p.SplittingField) p).degree ≠ 0

Tactic:
by rwa [degree_map]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: case e'_3
R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
h : k ≤ s.card
⊢ (-1) ^ (s.card - k) * s.esymm (s.card - k) = (map (fun t ↦ -t) s).esymm ((map (fun t ↦ -t) s).card - k)

Tactic:
rw [esymm_neg, card_map]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
k : Type u_2
inst✝ : Field k
φ : K →+* ℂ
hφ : IsReal φ
⊢ ∀ (x y : K), (φ (x * y)).re = (φ x).re * (φ y).re

Tactic:
by
    simp only [Complex.conj_eq_iff_im.mp (RingHom.congr_fun hφ _), map_mul, mul_re,
      mul_zero, tsub_zero, forall_const]

Goal state after: