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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ∫⁻ (a : α), (⨅ n, aeSeq h_meas p n) a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

Tactic:
iInf_apply

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ∫⁻ (a : α), ⨅ i, aeSeq h_meas p i a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
hf₀ : ∫⁻ (a : α), f a ∂μ = 0
⊢ ∃ s, MeasurableSet s ∧ μ s < ∞ ∧ ∫⁻ (a : α) in sᶜ, f a ∂μ < ε

Tactic:
exact ⟨∅, .empty, by simp, by simpa [hf₀, pos_iff_ne_zero]⟩

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s t : Set α
h : ∃ s, MeasurableSet s ∧ SigmaFinite (μ.restrict s) ∧ ∀ t ⊆ sᶜ, ν t ≠ 0 → μ t = ∞
⊢ SigmaFinite (μ.restrict h.choose)

Tactic:
exact h.choose_spec.2.1

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : MeasurableSingletonClass α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
h : SecondCountableTopologyEither α β
f : α → β
s : Set α
hf : ContinuousOn f s
hs : sᶜ.Countable
h's : MeasurableSet s
this : f = fun x ↦ if hx : x ∈ s then f ↑⟨x, hx⟩ else f ↑⟨x, hx⟩
⊢ SecondCountableTopologyEither (↑s) β

Tactic:
cases h.out <;> infer_instance

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
hs : MeasurableSet s
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ
hνs : ν s ≠ 0
hsσ : SigmaFinite (μ.restrict s)
⊢ ν (μ.sigmaFiniteSetWRT' ν) < ν (μ.sigmaFiniteSetWRT' ν) + ν s

Tactic:
exact ENNReal.lt_add_right (measure_ne_top _ _) hνs

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

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : Nonempty Z
Φ : Z → X
Ψ : Z → Y
ε : ℝ
p : Z
⊢ ⨅ q, dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) = 0

Tactic:
have A : ∀ q, 0 ≤ dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) := fun _ => by positivity

Goal state after:
X : Type u
Y : Type v
Z : Type w
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : Nonempty Z
Φ : Z → X
Ψ : Z → Y
ε : ℝ
p : Z
A : ∀ (q : Z), 0 ≤ dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q)
⊢ ⨅ q, dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) = 0
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
x : X
y : Y
⊢ 1 ≤ Sum.dist (Sum.inl x) (Sum.inr y)

Tactic:
by
  grw [Sum.dist, ← le_add_of_nonneg_right dist_nonneg, ← le_add_of_nonneg_left dist_nonneg]

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

Goal state before:
pp: case mpr.inr
E : ℕ → Type u_1
α : Type u_2
inst✝ : PseudoMetricSpace α
f : ((n : ℕ) → E n) → α
H : ∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n
x y : (n : ℕ) → E n
hne : x ≠ y
⊢ dist (f x) (f y) ≤ dist x y

Tactic:
dist_eq_of_ne hne

Goal state after:
case mpr.inr
E : ℕ → Type u_1
α : Type u_2
inst✝ : PseudoMetricSpace α
f : ((n : ℕ) → E n) → α
H : ∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n
x y : (n : ℕ) → E n
hne : x ≠ y
⊢ dist (f x) (f y) ≤ (1 / 2) ^ firstDiff x y
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
hij : i ≠ j
z : E j
hik : i ≠ j
⊢ dist x ⋯.some + 1 + dist ⋯.some z ≤ dist x ⋯.some + 1 + (dist ⋯.some y + dist y z)

Tactic:
by
            apply_rules [add_le_add, le_rfl, dist_triangle]

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

Goal state before:
pp: β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
⊢ (inducedMap A).fst = univ

Tactic:
eq_univ_iff_forall

Goal state after:
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
⊢ ∀ (x : ℕ → β), x ∈ (inducedMap A).fst
Module: Mathlib.Dynamics.Ergodic.MeasurePreserving

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μa : Measure α
μb : Measure β
R : Type u_5
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
f : α → β
hf : MeasurePreserving f μa μb
c : R
⊢ map f (c • μa) = c • μb

Tactic:
rw [Measure.map_smul, hf.map_eq]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
⊢ Disjoint μ ν

Tactic:
by
  have h_bot_iff (ξ : Measure α) : ξ ≤ ⊥ ↔ ξ = 0 := by
    rw [le_bot_iff]
    rfl
  intro ξ hξμ hξν
  rw [h_bot_iff]
  ext s hs
  simp only [Measure.coe_zero, Pi.zero_apply]
  rw [← inter_union_compl s h.nullSet, measure_union, add_eq_zero]
  · exact ⟨measure_inter_null_of_null_right _ <| absolutelyContinuous_of_le hξμ h.measure_nullSet,
      measure_inter_null_of_null_right _ <| absolutelyContinuous_of_le hξν h.measure_compl_nullSet⟩
  · exact Disjoint.mono inter_subset_right inter_subset_right disjoint_compl_right
  · exact hs.inter h.measurableSet_nullSet.compl

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

Goal state before:
pp: α : Type u_1
δ : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α
f : α → δ
⊢ f =ᵐ[dirac a] const α (f a)

Tactic:
simp [Filter.EventuallyEq]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → Measure β
⊢ bind 0 f = 0

Tactic:
simp [bind]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
ν : Measure β
⊢ (m.bind fun x ↦ ν) = m univ • ν

Tactic:
simp [bind]

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

Goal state before:
pp: case inr.inl
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
h : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑(Classical.choose ⋯) * μ (closedBall x ε)
R : ℝ
Rpos : R ∈ Ioi 0
hR :
  Ioc 0 R ⊆ {x | (fun ε ↦ ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑(Classical.choose ⋯) * μ (closedBall x ε)) x}
x : α
t : ℝ
ht : t ∈ Ioc 0 K
hr : 0 ≤ R
⊢ μ (closedBall x (t * 0)) ≤ ↑(scalingConstantOf μ K) * μ (closedBall x 0)

Tactic:
simp only [mul_zero]
    refine le_mul_of_one_le_of_le ?_ le_rfl
    apply ENNReal.one_le_coe_iff.2 (le_max_right _ _)

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
t : Set β
⊢ (μ.prod ν) (s ×ˢ t) ≤ μ s * ν t

Tactic:
set S := toMeasurable μ s

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
t : Set β
S : Set α := toMeasurable μ s
⊢ (μ.prod ν) (s ×ˢ t) ≤ μ s * ν t
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝⁴ : SFinite ν✝
X : Type u_4
Y : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
m : MeasurableSpace X
μ : Measure X
inst✝¹ : IsLocallyFiniteMeasure μ
m' : MeasurableSpace Y
ν : Measure Y
inst✝ : IsLocallyFiniteMeasure ν
x : X × Y
⊢ (μ.prod ν).FiniteAtFilter (𝓝 x)

Tactic:
by
    rw [nhds_prod_eq]
    exact μ.finiteAt_nhds _ |>.prod <| ν.finiteAt_nhds _

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹⁰ : MeasurableSpace α
inst✝⁹ : MeasurableSpace β
inst✝⁸ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝⁷ : SFinite ν✝
X : Type u_4
Y : Type u_5
inst✝⁶ : PseudoMetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : PseudoMetricSpace Y
inst✝³ : MeasurableSpace Y
μ : Measure X
ν : Measure Y
inst✝² : SFinite ν
inst✝¹ : IsUnifLocDoublingMeasure μ
inst✝ : IsUnifLocDoublingMeasure ν
r : ℝ
hμr : ∀ (x : X), μ (Metric.closedBall x (2 * r)) ≤ ↑(doublingConstant μ) * μ (Metric.closedBall x r)
hνr : ∀ (x : Y), ν (Metric.closedBall x (2 * r)) ≤ ↑(doublingConstant ν) * ν (Metric.closedBall x r)
x : X × Y
⊢ ↑(doublingConstant μ) * μ (Metric.closedBall x.1 r) * ν (Metric.closedBall x.2 (2 * r)) ≤
    ↑(doublingConstant μ * doublingConstant ν) * (μ (Metric.closedBall x.1 r) * ν (Metric.closedBall x.2 r))

Tactic:
hνr,

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹⁰ : MeasurableSpace α
inst✝⁹ : MeasurableSpace β
inst✝⁸ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝⁷ : SFinite ν✝
X : Type u_4
Y : Type u_5
inst✝⁶ : PseudoMetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : PseudoMetricSpace Y
inst✝³ : MeasurableSpace Y
μ : Measure X
ν : Measure Y
inst✝² : SFinite ν
inst✝¹ : IsUnifLocDoublingMeasure μ
inst✝ : IsUnifLocDoublingMeasure ν
r : ℝ
hμr : ∀ (x : X), μ (Metric.closedBall x (2 * r)) ≤ ↑(doublingConstant μ) * μ (Metric.closedBall x r)
hνr : ∀ (x : Y), ν (Metric.closedBall x (2 * r)) ≤ ↑(doublingConstant ν) * ν (Metric.closedBall x r)
x : X × Y
⊢ ↑(doublingConstant μ) * μ (Metric.closedBall x.1 r) * (↑(doublingConstant ν) * ν (Metric.closedBall x.2 r)) ≤
    ↑(doublingConstant μ * doublingConstant ν) * (μ (Metric.closedBall x.1 r) * ν (Metric.closedBall x.2 r))
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝¹ : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b✝ : Set (Set γ)
b_count : b✝.Countable
b_nonempty : ∅ ∉ b✝
hb : IsTopologicalBasis b✝
this✝ : Encodable ↑b✝
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b✝ → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
this : range f = ⋂ n, F n
b s : ↑b✝
⊢ MeasurableSet (⋂ (ht : Disjoint ↑b ↑s), q ⟨(b, s), ht⟩ \ q ⟨(s, b), ⋯⟩)

Tactic:
exact MeasurableSet.iInter fun hs => (q_meas _).diff (q_meas _)

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S T : Set G
hst : IsComplement S T
a : G
ha : a ∈ S
b : G
hb : b ∈ S
hab : a ≠ b
c : G
hc : c ∈ T
d : G
hd : d ∈ T
⊢ (fun x ↦ a • x) c ≠ (fun x ↦ b • x) d

Tactic:
exact hst.1.ne (a₁ := (⟨a, ha⟩, ⟨c, hc⟩)) (a₂ := (⟨b, hb⟩, ⟨d, hd⟩)) (by simp [hab])

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

Goal state before:
pp: case inl
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : μ.OuterRegular
hx : 0 ≠ ∞
⊢ OuterRegular 0

Tactic:
exact OuterRegular.zero

Goal state after:
Module: Mathlib.GroupTheory.Complement

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

Tactic:
intro h
    rw [← h]
    exact Subtype.prop _

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case mp
G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ S
⊢ ↑(hST.equiv g).2 = g → g ∈ T

Tactic:
intro h
    rw [← h]
    exact Subtype.prop _

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
h : IsComplement (↑H) T
⊢ Finite ↑T ↔ H.FiniteIndex

Tactic:
rw [← h.rightQuotientEquiv.finite_iff,
    (QuotientGroup.quotientRightRelEquivQuotientLeftRel H).finite_iff]

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
h : IsComplement (↑H) T
⊢ Finite (G ⧸ H) ↔ H.FiniteIndex
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: X : Type u_6
E✝ : Type u_7
ι : Type u_8
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : CommMonoid E✝
inst✝⁸ : TopologicalSpace E✝
inst✝⁷ : IsCompletelyPseudoMetrizableSpace E✝
inst✝⁶ : SecondCountableTopology E✝
inst✝⁵ : MeasurableSpace E✝
inst✝⁴ : BorelSpace E✝
inst✝³ : MeasurableMul₂ E✝
inst✝² : Countable ι
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
f : ι → X → E✝
h : ∀ (i : ι), Measurable (f i)
E : Set X := {x | Multipliable (fun x_1 ↦ f x_1 x) L}
⊢ Measurable fun x ↦ ∏'[L] (i : ι), f i x

Tactic:
have hE : MeasurableSet E := measurableSet_exists_tendsto (by fun_prop)

Goal state after:
X : Type u_6
E✝ : Type u_7
ι : Type u_8
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : CommMonoid E✝
inst✝⁸ : TopologicalSpace E✝
inst✝⁷ : IsCompletelyPseudoMetrizableSpace E✝
inst✝⁶ : SecondCountableTopology E✝
inst✝⁵ : MeasurableSpace E✝
inst✝⁴ : BorelSpace E✝
inst✝³ : MeasurableMul₂ E✝
inst✝² : Countable ι
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
f : ι → X → E✝
h : ∀ (i : ι), Measurable (f i)
E : Set X := {x | Multipliable (fun x_1 ↦ f x_1 x) L}
hE : MeasurableSet E
⊢ Measurable fun x ↦ ∏'[L] (i : ι), f i x
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case 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
hr : r < ν s
⊢ ∃ K ⊆ s, IsCompact K ∧ r < (μ + ν) K

Tactic:
rcases MeasurableSet.exists_lt_isCompact hs hr with ⟨K, Ks, hK, h'K⟩

Goal state after:
case 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
hr : r < ν s
K : Set α
Ks : K ⊆ s
hK : IsCompact K
h'K : r < ν K
⊢ ∃ K ⊆ s, IsCompact K ∧ r < (μ + ν) K
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∞
h : μ s ≠ 0
h' : ν s = 0
hr : r < μ s
⊢ ∃ K ⊆ s, IsCompact K ∧ r < (μ + ν) K

Tactic:
rcases MeasurableSet.exists_lt_isCompact hs hr with ⟨K, Ks, hK, h'K⟩

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 : Set α
Ks : K ⊆ s
hK : IsCompact K
h'K : r < μ K
⊢ ∃ 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
f g : α → ℝ≥0∞
hfg : f ≤ᵐ[μ] g
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, f a ∂μ ≤ ∫⁻ (a : α) in s, g a ∂μ

Tactic:
refine setLIntegral_mono_ae' hs ?_

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hfg : f ≤ᵐ[μ] g
s : Set α
hs : MeasurableSet s
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
⊢ ((μ + ν).withDensity f) s = (μ.withDensity f + ν.withDensity f) s

Tactic:
simp only [withDensity_apply f hs, restrict_add, lintegral_add_measure, Measure.add_apply]

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

Goal state before:
pp: G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed[inst✝⁷] K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n ↦ (fun T ↦ T ∪ g T • K)^[n] K
Lcompact : ∀ (n : ℕ), IsCompact (L n)
Lclosed : ∀ (n : ℕ), IsClosed[inst✝⁷] (L n)
M : ∀ (n : ℕ), μ (L n) = ↑(n + 1) * μ K
N : Tendsto (fun n ↦ μ (L n)) atTop (𝓝 (∞ * μ K))
⊢ μ univ = ∞

Tactic:
simp only [ENNReal.top_mul', K_pos.ne', if_false] at N

Goal state after:
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed[inst✝⁷] K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n ↦ (fun T ↦ T ∪ g T • K)^[n] K
Lcompact : ∀ (n : ℕ), IsCompact (L n)
Lclosed : ∀ (n : ℕ), IsClosed[inst✝⁷] (L n)
M : ∀ (n : ℕ), μ (L n) = ↑(n + 1) * μ K
N : Tendsto (fun n ↦ μ (L n)) atTop (𝓝 ∞)
⊢ μ univ = ∞
Module: Mathlib.MeasureTheory.Function.Egorov

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝² : PseudoEMetricSpace β
μ : Measure α
s : Set α
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)
hsm : MeasurableSet s
hs : μ s ≠ ∞
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
n : ℕ
h : IsEmpty ι
⊢ (fun j ↦ μ (s ∩ notConvergentSeq f g n j)) = fun j ↦ 0

Tactic:
simp only [eq_iff_true_of_subsingleton]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
hy : 0 < y
⊢ ↑(NNReal.sqrt x.toNNReal) < ↑(NNReal.sqrt y.toNNReal) ↔ x < y

Tactic:
NNReal.coe_lt_coe,

Goal state after:
x y : ℝ
hy : 0 < y
⊢ NNReal.sqrt x.toNNReal < NNReal.sqrt y.toNNReal ↔ x < y
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ f * g
A : (fun x ↦ (f x)⁻¹ * i x) ≤ g
⊢ ∫⁻ (a : α), i a ∂μ ≤ ∫⁻ (a : α), (fun x ↦ (f x)⁻¹ * i x) a ∂μ.withDensity f

Tactic:
lintegral_withDensity_eq_lintegral_mul _ f_meas (f_meas.fun_inv.fun_mul i_meas)

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ f * g
A : (fun x ↦ (f x)⁻¹ * i x) ≤ g
⊢ ∫⁻ (a : α), i a ∂μ ≤ ∫⁻ (a : α), (f * fun i_1 ↦ (f i_1)⁻¹ * i i_1) a ∂μ
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z : ℂ
hz : ‖z‖ = 1
this : z.im * z.im = 1 - z.re * z.re
⊢ ‖z - 1‖ ^ 2 = 2 * (1 - z.re)

Tactic:
simp [Complex.sq_norm, normSq_apply, this]

Goal state after:
z : ℂ
hz : ‖z‖ = 1
this : z.im * z.im = 1 - z.re * z.re
⊢ (z.re - 1) * (z.re - 1) + (1 - z.re * z.re) = 2 * (1 - z.re)
Module: Mathlib.Analysis.Complex.Order

Goal state before:
pp: z : ℂ
⊢ -z.re = ‖z‖ ↔ z ≤ 0

Tactic:
rw [← neg_re, ← norm_neg z, re_eq_norm]
  exact neg_nonneg.and <| eq_comm.trans neg_eq_zero

Goal state after:
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: z : ℂ
⊢ (∃ r, z = ↑r) → ↑z.re = z

Tactic:
by rintro ⟨r, rfl⟩; simp [ofReal]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V : Type u_1
V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f✝ g : NormedAddGroupHom V₁ V₂
f : V₁ →+ V₂
K : ℝ≥0
h : LipschitzWith K ⇑f
x : V₁
⊢ ‖f x‖ ≤ ↑K * ‖x‖

Tactic:
simpa only [map_zero, dist_zero_right] using h.dist_le_mul x 0

Goal state after:
Module: Mathlib.Algebra.Star.Unitary

Goal state before:
pp: G : Type u_2
inst✝¹ : Group G
inst✝ : StarMul G
a b : G
⊢ (star b)⁻¹ * (star a * (a * b⁻¹)) = 1 ↔ star a * a = star b * b

Tactic:
inv_mul_eq_iff_eq_mul,

Goal state after:
G : Type u_2
inst✝¹ : Group G
inst✝ : StarMul G
a b : G
⊢ star a * (a * b⁻¹) = star b * 1 ↔ star a * a = star b * b
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
⊢ im 1 = 0

Tactic:
rw [← ofReal_one, ofReal_im]

Goal state after:
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ z - (starRingEnd K) z = ↑(re z) + ↑(im z) * I - (↑(re z) - ↑(im z) * I)

Tactic:
by rw [re_add_im, ← conj_eq_re_sub_im]

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

Goal state before:
pp: z : ℂ
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
𝕜 : Type u_2
inst✝ : RCLike 𝕜
h : RCLike.im RCLike.I = 1
x y : 𝕜
⊢ ↑(RCLike.re (x * y)) + ↑(RCLike.im (x * y)) * I =
    (↑(RCLike.re x) + ↑(RCLike.im x) * I) * (↑(RCLike.re y) + ↑(RCLike.im y) * I)

Tactic:
by
    simp only [RCLike.mul_re, ofReal_sub, ofReal_mul, RCLike.mul_im, ofReal_add]
    ring_nf
    rw [I_sq]
    ring

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Continuity

Goal state before:
pp: X : Type u_1
Y : Type u_3
Z : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
e : OpenPartialHomeomorph X Y
f : Z → X
s : Set Z
x : Z
hx : f x ∈ e.source
h : f ⁻¹' e.source ∈ 𝓝[s] x
fe_cont : ContinuousWithinAt (↑e ∘ f) (s ∩ f ⁻¹' e.source) x
⊢ ContinuousWithinAt f (s ∩ f ⁻¹' e.source) x

Tactic:
have : ContinuousWithinAt (e.symm ∘ e ∘ f) (s ∩ f ⁻¹' e.source) x :=
    haveI : ContinuousWithinAt e.symm univ (e (f x)) :=
      (e.continuousAt_symm (e.map_source hx)).continuousWithinAt
    ContinuousWithinAt.comp this fe_cont (subset_univ _)

Goal state after:
X : Type u_1
Y : Type u_3
Z : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
e : OpenPartialHomeomorph X Y
f : Z → X
s : Set Z
x : Z
hx : f x ∈ e.source
h : f ⁻¹' e.source ∈ 𝓝[s] x
fe_cont : ContinuousWithinAt (↑e ∘ f) (s ∩ f ⁻¹' e.source) x
this : ContinuousWithinAt (↑e.symm ∘ ↑e ∘ f) (s ∩ f ⁻¹' e.source) x
⊢ ContinuousWithinAt f (s ∩ f ⁻¹' e.source) x
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x y : K
⊢ ‖im x - im y‖ₑ = ‖im (x - y)‖ₑ

Tactic:
by rw [map_sub im x y]

Goal state after:
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
hk : NormedField 𝕜
h'' : hk = h.toNormedField
⊢ ∀ {z w : 𝕜},
    z ≤ w ↔
      (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z ≤
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            w ∧
        (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            z =
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            w

Tactic:
by subst h''; exact h.le_iff_re_im

Goal state after: