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

Goal state before:
pp: case e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f g : α → ℝ≥0∞
inst✝ : (j : α) → Decidable (j ∈ s)
⊢ ∫⁻ (x : α) in s, s.piecewise f g x ∂μ = ∫⁻ (a : α) in s, f a ∂μ

pp: case e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f g : α → ℝ≥0∞
inst✝ : (j : α) → Decidable (j ∈ s)
⊢ ∫⁻ (x : α) in sᶜ, s.piecewise f g x ∂μ = ∫⁻ (a : α) in sᶜ, g a ∂μ

Tactic:
· exact setLIntegral_congr_fun hs <| fun _ ↦ Set.piecewise_eq_of_mem _ _ _

Goal state after:
case e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f g : α → ℝ≥0∞
inst✝ : (j : α) → Decidable (j ∈ s)
⊢ ∫⁻ (x : α) in sᶜ, s.piecewise f g x ∂μ = ∫⁻ (a : α) in sᶜ, g a ∂μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
s : Set α
m : MeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : (x : α) → Decidable (x ∈ s)
f : ↑s → β
hf : StronglyMeasurable f
g : ↑sᶜ → β
hg : StronglyMeasurable g
hs : MeasurableSet s
x : α
hx : x ∈ s
⊢ Tendsto (fun n ↦ ((fun n ↦ SimpleFunc.dite s hs (hf.approx n) (hg.approx n)) n) x) atTop
    (𝓝 ((fun x ↦ if hx : x ∈ s then f ⟨x, hx⟩ else g ⟨x, hx⟩) x))

Tactic:
simpa [hx] using hf.tendsto_approx ⟨x, hx⟩

Goal state after:
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case left
α : Type u_1
inst✝ : TopologicalSpace α
C U : Set α
hC : Perfect C
x : α
xC : x ∈ C
xU : x ∈ U
Uop : IsOpen[inst✝] U
⊢ Preperfect (U ∩ C)

Tactic:
exact hC.acc.open_inter Uop

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
μ : Measure α
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : SubtractionMonoid β
inst✝ : ContinuousNeg β
hf : FinStronglyMeasurable f μ
⊢ FinStronglyMeasurable (-f) μ

Tactic:
refine ⟨fun n ↦ -hf.approx n, fun n ↦ ?_, fun x ↦ (hf.tendsto_approx x).neg⟩

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
μ : Measure α
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : SubtractionMonoid β
inst✝ : ContinuousNeg β
hf : FinStronglyMeasurable f μ
n : ℕ
⊢ μ (support ⇑((fun n ↦ -hf.approx n) n)) < ∞
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
⊢ (∀ e ∈ 𝓝 0, ∃ s, ∀ (t : Finset ι), Disjoint t s → ∑ b ∈ t, f b ∈ e) ↔
    ∀ ε > 0, ∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, f i‖ < ε

Tactic:
nhds_basis_ball.forall_iff

Goal state after:
ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
⊢ (∀ (i : ℝ), 0 < i → ∃ s, ∀ (t : Finset ι), Disjoint t s → ∑ b ∈ t, f b ∈ ball 0 i) ↔
    ∀ ε > 0, ∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, f i‖ < ε

ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
⊢ ∀ ⦃s t : Set E⦄,
    s ⊆ t →
      (∃ s_1, ∀ (t : Finset ι), Disjoint t s_1 → ∑ b ∈ t, f b ∈ s) →
        ∃ s, ∀ (t_1 : Finset ι), Disjoint t_1 s → ∑ b ∈ t_1, f b ∈ t
Module: Mathlib.Analysis.Normed.Group.FunctionSeries

Goal state before:
pp: β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
ι : Type u_4
f : ι → β → F
u : ι → ℝ
hu : Summable u
hfu : ∀ᶠ (n : ι) in cofinite, ∀ (x : β), ‖f n x‖ ≤ u n
⊢ TendstoUniformly (fun t x ↦ ∑ n ∈ t, f n x) (fun x ↦ ∑' (n : ι), f n x) atTop

Tactic:
rw [← tendstoUniformlyOn_univ]
  apply tendstoUniformlyOn_tsum_of_cofinite_eventually hu
  simpa using hfu

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: case h
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
⊢ LipschitzOnWith (t.sup fun i ↦ K ↑i ⋯ + 2 * K' ↑i ⋯) f s

Tactic:
intro x hx y hy

Goal state after:
case h
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
x : α
hx : x ∈ s
y : α
hy : y ∈ s
⊢ edist (f x) (f y) ≤ ↑(t.sup fun i ↦ K ↑i ⋯ + 2 * K' ↑i ⋯) * edist x y
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b c✝ d✝ : α
h : a ≤ b
c d : α
hdc : 0 ≤ c - d
⊢ min b c - min b d ≤ c - d

Tactic:
refine ((le_abs_self _).trans <| abs_min_sub_min_le_max _ _ _ _).trans (max_le ?_ ?_)

Goal state after:
case refine_1
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b c✝ d✝ : α
h : a ≤ b
c d : α
hdc : 0 ≤ c - d
⊢ |b - b| ≤ c - d

case refine_2
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b c✝ d✝ : α
h : a ≤ b
c d : α
hdc : 0 ≤ c - d
⊢ |c - d| ≤ c - d
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Map

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
⊢ ⨆ n, (eapprox f n).lintegral (Measure.map g μ) = ⨆ n, (eapprox (f ∘ g) n).lintegral μ

Tactic:
congr with n : 1

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
n : ℕ
⊢ (eapprox f n).lintegral (Measure.map g μ) = (eapprox (f ∘ g) n).lintegral μ
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Countable ι
inst✝ : T2Space α
s : ι → Set α
hs : ∀ (n : ι), AnalyticSet (s n)
i₀ : ι
β : ι → Type
hβ : (n : ι) → TopologicalSpace (β n)
h'β : ∀ (n : ι), PolishSpace (β n)
f : (n : ι) → β n → α
f_cont : ∀ (n : ι), Continuous[hβ n, inst✝²] (f n)
f_range : ∀ (n : ι), range (f n) = s n
γ : Type u_2 := (n : ι) → β n
t : Set γ := ⋂ n, {x | f n (x n) = f i₀ (x i₀)}
⊢ IsClosed[Pi.topologicalSpace] t

Tactic:
apply isClosed_iInter

Goal state after:
α : Type u_1
ι : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Countable ι
inst✝ : T2Space α
s : ι → Set α
hs : ∀ (n : ι), AnalyticSet (s n)
i₀ : ι
β : ι → Type
hβ : (n : ι) → TopologicalSpace (β n)
h'β : ∀ (n : ι), PolishSpace (β n)
f : (n : ι) → β n → α
f_cont : ∀ (n : ι), Continuous[hβ n, inst✝²] (f n)
f_range : ∀ (n : ι), range (f n) = s n
γ : Type u_2 := ⋯
t : Set γ := ⋯
⊢ ∀ (i : ι), IsClosed[Pi.topologicalSpace] {x | f i (x i) = f i₀ (x i₀)}
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Countable ι
inst✝ : T2Space α
s : ι → Set α
hs : ∀ (n : ι), AnalyticSet (s n)
i₀ : ι
β : ι → Type
hβ : (n : ι) → TopologicalSpace (β n)
h'β : ∀ (n : ι), PolishSpace (β n)
f : (n : ι) → β n → α
f_cont : ∀ (n : ι), Continuous[hβ n, inst✝²] (f n)
f_range : ∀ (n : ι), range (f n) = s n
γ : Type u_2 := (n : ι) → β n
t : Set γ := ⋂ n, {x | f n (x n) = f i₀ (x i₀)}
n : ι
⊢ IsClosed[Pi.topologicalSpace] {x | f n (x n) = f i₀ (x i₀)}

Tactic:
exact
      isClosed_eq ((f_cont n).comp (continuous_apply n)) ((f_cont i₀).comp (continuous_apply i₀))

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Countable ι
inst✝ : T2Space α
s : ι → Set α
hs : ∀ (n : ι), AnalyticSet (s n)
i₀ : ι
β : ι → Type
hβ : (n : ι) → TopologicalSpace (β n)
h'β : ∀ (n : ι), PolishSpace (β n)
f : (n : ι) → β n → α
f_cont : ∀ (n : ι), Continuous[hβ n, inst✝²] (f n)
f_range : ∀ (n : ι), range (f n) = s n
γ : Type u_2 := (n : ι) → β n
t : Set γ := ⋂ n, {x | f n (x n) = f i₀ (x i₀)}
t_closed : IsClosed[Pi.topologicalSpace] t
this : PolishSpace ↑t
F : ↑t → α := fun x ↦ f i₀ (↑x i₀)
⊢ AnalyticSet (⋂ n, s n)

Tactic:
have F_cont : Continuous F := (f_cont i₀).comp ((continuous_apply i₀).comp continuous_subtype_val)

Goal state after:
α : Type u_1
ι : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Countable ι
inst✝ : T2Space α
s : ι → Set α
hs : ∀ (n : ι), AnalyticSet (s n)
i₀ : ι
β : ι → Type
hβ : (n : ι) → TopologicalSpace (β n)
h'β : ∀ (n : ι), PolishSpace (β n)
f : (n : ι) → β n → α
f_cont : ∀ (n : ι), Continuous[hβ n, inst✝²] (f n)
f_range : ∀ (n : ι), range (f n) = s n
γ : Type u_2 := (n : ι) → β n
t : Set γ := ⋂ n, {x | f n (x n) = f i₀ (x i₀)}
t_closed : IsClosed[Pi.topologicalSpace] t
this : PolishSpace ↑t
F : ↑t → α := fun x ↦ f i₀ (↑x i₀)
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
⊢ AnalyticSet (⋂ n, s n)
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
ε : ℝ≥0
hε : 0 < ε
h₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0
n : ℕ
⊢ ∃ x ∈ {m | ∃ t, m = μ t + ν tᶜ}, x < ↑ε * (1 / 2) ^ n

Tactic:
by
      refine exists_lt_of_csInf_lt ⟨ν univ, ∅, by simp⟩ <| h₁ ▸ ENNReal.mul_pos ?_ (by simp)
      norm_cast
      exact hε.ne.symm

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
a : α
⊢ (dirac a) s = 0 ∨ (dirac a) s = 1

Tactic:
by
  rw [← measure_toMeasurable s, dirac_apply' a (measurableSet_toMeasurable ..), indicator]
  simp only [Pi.one_apply, ite_eq_right_iff, one_ne_zero, imp_false, ite_eq_left_iff, zero_ne_one,
    not_not]
  tauto

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = Ay.some
H : firstDiff x y ≤ longestPrefix x s ∧ firstDiff x y ≤ longestPrefix y s
I1 : cylinder Ax.some (firstDiff x y) = cylinder x (firstDiff x y)
I3 : cylinder y (firstDiff x y) = cylinder Ay.some (firstDiff x y)
⊢ cylinder y (firstDiff x y) = cylinder Ay.some (firstDiff x y)

Tactic:
I3

Goal state after:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = Ay.some
H : firstDiff x y ≤ longestPrefix x s ∧ firstDiff x y ≤ longestPrefix y s
I1 : cylinder Ax.some (firstDiff x y) = cylinder x (firstDiff x y)
I3 : cylinder y (firstDiff x y) = cylinder Ay.some (firstDiff x y)
⊢ cylinder Ay.some (firstDiff x y) = cylinder Ay.some (firstDiff x y)
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace.SeparatesPoints α
x y : α
x_ne_y : x ≠ y
⊢ dirac x ≠ dirac y

Tactic:
obtain ⟨A, A_mble, x_in_A, y_notin_A⟩ := exists_measurableSet_of_ne x_ne_y

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace.SeparatesPoints α
x y : α
x_ne_y : x ≠ y
A : Set α
A_mble : MeasurableSet A
x_in_A : x ∈ A
y_notin_A : y ∉ A
⊢ dirac x ≠ dirac y
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case h
E : ℕ → Type u_1
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → PseudoEMetricSpace (F i)
x y z : (i : ι) → F i
n : ι
⊢ min (2⁻¹ ^ encode n) (edist (x n) (y n) + edist (y n) (z n)) ≤
    min (2⁻¹ ^ encode n) (edist (x n) (y n)) + min (2⁻¹ ^ encode n) (edist (y n) (z n))

Tactic:
min_add_distrib,

Goal state after:
case h
E : ℕ → Type u_1
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → PseudoEMetricSpace (F i)
x y z : (i : ι) → F i
n : ι
⊢ min (2⁻¹ ^ encode n) (min (2⁻¹ ^ encode n) (edist (x n) (y n)) + min (2⁻¹ ^ encode n) (edist (y n) (z n))) ≤
    min (2⁻¹ ^ encode n) (edist (x n) (y n)) + min (2⁻¹ ^ encode n) (edist (y n) (z n))
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
S : ℕ → Set α := spanningSets (μ.trim hm)
⊢ f univ ≤ C

Tactic:
have hS_mono : Monotone S := @monotone_spanningSets _ m (μ.trim hm) _

Goal state after:
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
S : ℕ → Set α := spanningSets (μ.trim hm)
hS_mono : Monotone S
⊢ f univ ≤ C
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case hn
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : IsFiniteMeasure ν
s : Set (α × β)
f : ℕ → Set (α × β)
hfd : Pairwise (Disjoint on f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf : ∀ (i : ℕ), Measurable fun x ↦ ν (Prod.mk x ⁻¹' f i)
a : α
⊢ Pairwise (Disjoint on fun i ↦ Prod.mk a ⁻¹' f i)

pp: case h
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : IsFiniteMeasure ν
s : Set (α × β)
f : ℕ → Set (α × β)
hfd : Pairwise (Disjoint on f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf : ∀ (i : ℕ), Measurable fun x ↦ ν (Prod.mk x ⁻¹' f i)
a : α
⊢ ∀ (i : ℕ), MeasurableSet (Prod.mk a ⁻¹' f i)

Tactic:
exacts [hfd.mono fun _ _ ↦ .preimage _, fun i ↦ measurable_prodMk_left (hfm i)]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
s : Set α
hs : MeasurableSet s
h2s : μ s = 0
⊢ 0 * ν univ = 0

Tactic:
zero_mul

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
s : Set α
hs : MeasurableSet s
h2s : μ s = 0
⊢ 0 = 0
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case refine_3
G : Type u_1
inst✝ : Group G
g : G
⊢ IsComplement univ {g}

Tactic:
exact isComplement_univ_singleton

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S T : Set G
⊢ IsComplement S T ↔ ∀ (g : G), ∃! s, (↑s)⁻¹ * g ∈ T

Tactic:
by
  convert! isComplement_iff_existsUnique with g
  constructor <;> rintro ⟨x, hx, hx'⟩
  · exact ⟨(x, ⟨_, hx⟩), by simp, by aesop⟩
  · exact ⟨x.1, by simp [← hx], fun y hy ↦ (Prod.ext_iff.1 <| by simpa using hx' (y, ⟨_, hy⟩)).1⟩

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)
h'' : univ ⊆ ⋃ n, s n
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
⊢ ∃ U ⊇ A, IsOpen[inst✝¹] U ∧ μ U < r

Tactic:
have HA : μ A < ∞ := lt_of_lt_of_le hr le_top

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)
h'' : univ ⊆ ⋃ n, s n
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
HA : μ A < ∞
⊢ ∃ U ⊇ A, IsOpen[inst✝¹] U ∧ μ U < r
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ S
⊢ ↑(hST.equiv g).1 = 1 ↔ g ∈ T

Tactic:
rw [equiv_fst_eq_mul_inv, mul_inv_eq_one, eq_comm, equiv_snd_eq_self_iff_mem _ h1]

Goal state after:
Module: Mathlib.MeasureTheory.Group.Action

Goal state before:
pp: G : Type u
α : Type w
m : MeasurableSpace α
inst✝² : Group G
inst✝¹ : MulAction G α
μ : Measure α
inst✝ : SMulInvariantMeasure G α μ
c : G
s t : Set α
⊢ μ (c • (s ∪ c⁻¹ • t)) = μ (c • s ∪ t)

Tactic:
smul_set_union,

Goal state after:
G : Type u
α : Type w
m : MeasurableSpace α
inst✝² : Group G
inst✝¹ : MulAction G α
μ : Measure α
inst✝ : SMulInvariantMeasure G α μ
c : G
s t : Set α
⊢ μ (c • s ∪ c • c⁻¹ • t) = μ (c • s ∪ t)
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
x : α
⊢ (dirac x).prod ν = map (Prod.mk x) ν

Tactic:
rw [← sum_sfiniteSeq ν, prod_sum_right, map_sum measurable_prodMk_left.aemeasurable]
  congr
  ext1 i
  refine prod_eq fun s t hs ht => ?_
  simp_rw [map_apply measurable_prodMk_left (hs.prod ht), mk_preimage_prod_right_eq_if, measure_if,
    dirac_apply' _ hs, ← indicator_mul_left _ _ fun _ => sfiniteSeq ν i t, Pi.one_apply, one_mul]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_6
β : Type u_7
t : Set α
g : α → β
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : LinearOrder α
inst✝⁶ : OrderTopology α
inst✝⁵ : PolishSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : LinearOrder β
inst✝ : OrderTopology β
ht : MeasurableSet t
hg : MonotoneOn g t
h'g : ContinuousOn g t
u : Set β := {c | ∃ x y, x ∈ t ∧ y ∈ t ∧ x < y ∧ g x = c ∧ g y = c}
hu : u.Countable
t' : Set α := t ∩ g ⁻¹' u
ht' : MeasurableSet t'
⊢ g '' t = g '' (t \ t') ∪ g '' t'

Tactic:
simp [← image_union, t']

Goal state after:
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Mul G
μ : Measure G
inst✝⁴ : MeasurableMul G
H : Type u_3
inst✝³ : MeasurableSpace H
inst✝² : Mul H
inst✝¹ : MeasurableMul H
inst✝ : μ.IsMulLeftInvariant
f : G →ₙ* H
hf : Measurable ⇑f
h_surj : Surjective ⇑f
h : H
⊢ Measure.map (fun x ↦ h * x) (Measure.map (⇑f) μ) = Measure.map (⇑f) μ

Tactic:
map_map (measurable_const_mul _) hf

Goal state after:
G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Mul G
μ : Measure G
inst✝⁴ : MeasurableMul G
H : Type u_3
inst✝³ : MeasurableSpace H
inst✝² : Mul H
inst✝¹ : MeasurableMul H
inst✝ : μ.IsMulLeftInvariant
f : G →ₙ* H
hf : Measurable ⇑f
h_surj : Surjective ⇑f
h : H
⊢ Measure.map ((fun x ↦ h * x) ∘ ⇑f) μ = Measure.map (⇑f) μ
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulLeftInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
this : ⇑f '' (fun x ↦ g * x) ⁻¹' s = (fun x ↦ f g * x) ⁻¹' ⇑f '' s
⊢ MeasurableSet (⇑f '' s)

Tactic:
exact hf.measurableSet_image.mpr hs

Goal state after:
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
this : ⇑f '' (fun x ↦ x * g) ⁻¹' s = (fun x ↦ x * f g) ⁻¹' ⇑f '' s
⊢ MeasurableSet (⇑f '' s)

Tactic:
exact hf.measurableSet_image.mpr hs

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case pos
X : Type u_6
E : Type u_7
ι : Type u_8
inst✝⁸ : MeasurableSpace X
inst✝⁷ : CommMonoid E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : PseudoMetrizableSpace E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : MeasurableMul₂ E
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
f : ι → X → E
h : ∀ (i : ι), Measurable (f i)
hm : Multipliable (fun i ↦ f i) L
h✝¹ : L.HasSupport ∧ ((mulSupport fun i ↦ f i) ∩ L.support).Finite
h✝ : HasFiniteMulSupport ((L.support.mulIndicator fun i ↦ f i) ∘ PLift.down)
x✝¹ : PLift ι
x✝ : x✝¹ ∈ Finite.toFinset h✝
⊢ Measurable ↿fun a ↦ if x✝¹.down ∈ L.support then f x✝¹.down else 1

Tactic:
split_ifs <;> fun_prop

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case pos
X : Type u_6
E : Type u_7
ι : Type u_8
inst✝⁸ : MeasurableSpace X
inst✝⁷ : CommMonoid E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : PseudoMetrizableSpace E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : MeasurableMul₂ E
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
μ : Measure X
f : ι → X → E
h : ∀ (i : ι), AEMeasurable (f i) μ
hm : Multipliable (fun i ↦ f i) L
h✝¹ : L.HasSupport ∧ ((mulSupport fun i ↦ f i) ∩ L.support).Finite
h✝ : HasFiniteMulSupport ((L.support.mulIndicator fun i ↦ f i) ∘ PLift.down)
x✝¹ : PLift ι
x✝ : x✝¹ ∈ Finite.toFinset h✝
⊢ AEMeasurable (if x✝¹.down ∈ L.support then f x✝¹.down else 1) μ

Tactic:
split_ifs <;> fun_prop

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : MeasurableSpace X
μ : Measure X
⊢ μ.InnerRegularWRT IsCompact IsClosed[inst✝²]

Tactic:
by
  intro F hF r hr
  set B : ℕ → Set X := compactCovering X
  have hBc : ∀ n, IsCompact (F ∩ B n) := fun n => (isCompact_compactCovering X n).inter_left hF
  have hBU : ⋃ n, F ∩ B n = F := by rw [← inter_iUnion, iUnion_compactCovering, Set.inter_univ]
  have : μ F = ⨆ n, μ (F ∩ B n) := by
    rw [← Monotone.measure_iUnion, hBU]
    exact monotone_const.inter monotone_accumulate
  rw [this] at hr
  rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
  exact ⟨_, inter_subset_left, hBc n, hn⟩

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case inr.inl
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace α
μ ν : Measure α
inst✝¹ : μ.InnerRegular
inst✝ : ν.InnerRegular
s : Set α
hs : MeasurableSet s
r : ℝ≥0∞
hr : r < μ s + ν s
h : μ s ≠ 0
h' : ν s = 0
⊢ ∃ K ⊆ s, IsCompact K ∧ r < (μ + ν) K

Tactic:
simp only [h', add_zero] at hr

Goal state after:
case inr.inl
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace α
μ ν : Measure α
inst✝¹ : μ.InnerRegular
inst✝ : ν.InnerRegular
s : Set α
hs : MeasurableSet s
r : ℝ≥0∞
h : μ s ≠ 0
h' : ν s = 0
hr : r < μ s
⊢ ∃ K ⊆ s, IsCompact K ∧ r < (μ + ν) K
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, (r • f) a ∂μ = ∫⁻ (a : α) in s, r * f a ∂μ

Tactic:
simp only [Pi.smul_apply, smul_eq_mul]

Goal state after: