mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: case neg
a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
h : ¬↑z - ↑x = 0
⊢ 0 ≤ (↑y - ↑x) / (↑z - ↑x)

Tactic:
replace hxy : (x : ℝ) ≤ (y : ℝ) := hxy

Goal state after:
case neg
a b : ℝ
x y z : ↑(Icc a b)
hyz : y ≤ z
h : ¬↑z - ↑x = 0
hxy : ↑x ≤ ↑y
⊢ 0 ≤ (↑y - ↑x) / (↑z - ↑x)
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: case neg
a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
h : ¬↑z - ↑x = 0
⊢ (↑y - ↑x) / (↑z - ↑x) ≤ 1

Tactic:
replace hxy : (x : ℝ) ≤ (y : ℝ) := hxy

Goal state after:
case neg
a b : ℝ
x y z : ↑(Icc a b)
hyz : y ≤ z
h : ¬↑z - ↑x = 0
hxy : ↑x ≤ ↑y
⊢ (↑y - ↑x) / (↑z - ↑x) ≤ 1
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: case pos
a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
h : ↑z - ↑x = 0
⊢ ↑y = ↑x

Tactic:
replace hxy : (x : ℝ) ≤ (y : ℝ) := hxy

Goal state after:
case pos
a b : ℝ
x y z : ↑(Icc a b)
hyz : y ≤ z
h : ↑z - ↑x = 0
hxy : ↑x ≤ ↑y
⊢ ↑y = ↑x
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
n : ℕ
h : PseudoMetricSpace ((n : ℕ) × X n) := inductivePremetric I
x✝ : TopologicalSpace ((n : ℕ) × X n) := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
x : X n
⊢ dist ⟨n.succ, f n x⟩ ⟨n, x⟩ = 0

Tactic:
change inductiveLimitDist f ⟨n.succ, f n x⟩ ⟨n, x⟩ = 0

Goal state after:
X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
n : ℕ
h : PseudoMetricSpace ((n : ℕ) × X n) := inductivePremetric I
x✝ : TopologicalSpace ((n : ℕ) × X n) := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
x : X n
⊢ inductiveLimitDist f ⟨n.succ, f n x⟩ ⟨n, x⟩ = 0
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: case refine_1
α : Type u_1
m0 : MeasurableSpace α
ν : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ (sum μ) (⋂ i, s i) = 0

pp: case refine_2
α : Type u_1
m0 : MeasurableSpace α
ν : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ ν (⋂ i, s i)ᶜ = 0

Tactic:
· rw [sum_apply _ (MeasurableSet.iInter hsm), ENNReal.tsum_eq_zero]
    exact fun i => measure_mono_null (iInter_subset _ _) (hsμ i)

Goal state after:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
ν : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ ν (⋂ i, s i)ᶜ = 0
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
⊢ AnalyticSet (⋂ n, s n)

Tactic:
let F : t → α := fun x => f i₀ ((x : γ) i₀)

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)
⊢ AnalyticSet (⋂ n, s n)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
inst✝ : IsProbabilityMeasure (map f μ)
hf : AEMeasurable f μ
⊢ μ univ = 1

Tactic:
← Set.preimage_univ (f := f),

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
inst✝ : IsProbabilityMeasure (map f μ)
hf : AEMeasurable f μ
⊢ μ (f ⁻¹' univ) = 1
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:
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.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
⊢ ∃ s, μ = sum fun x ↦ μ (measurableAtom ↑x) • dirac ↑x

Tactic:
have h_nonempty (s : measurableAtoms) : Set.Nonempty s.1 := by
    obtain ⟨y, _, hy⟩ := s.2
    rw [← hy]
    exact ⟨y, mem_measurableAtom_self y⟩

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : ↑measurableAtoms), (↑s).Nonempty
⊢ ∃ s, μ = sum fun x ↦ μ (measurableAtom ↑x) • dirac ↑x
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_3
β : Type u_4
t : TopologicalSpace α
inst✝⁵ : PolishSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
tβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
f : α → β
inst✝ : SecondCountableTopology ↑(range f)
hf : Measurable f
b : Set (Set ↑(range f))
b_count : b.Countable
hb : IsTopologicalBasis b
⊢ ∃ t' ≤ t, Continuous[_, tβ] f ∧ PolishSpace α

Tactic:
have : Countable b := b_count.to_subtype

Goal state after:
α : Type u_3
β : Type u_4
t : TopologicalSpace α
inst✝⁵ : PolishSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
tβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
f : α → β
inst✝ : SecondCountableTopology ↑(range f)
hf : Measurable f
b : Set (Set ↑(range f))
b_count : b.Countable
hb : IsTopologicalBasis b
this : Countable ↑b
⊢ ∃ t' ≤ t, Continuous[_, tβ] f ∧ PolishSpace α
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: case pos
α : Type u_1
inst✝² : MeasurableSpace α
s : Set α
a : α
inst✝¹ : MeasurableSingletonClass α
inst✝ : Decidable (a ∈ s)
has : a ∈ s
⊢ ∀ᵐ (x : α) ∂dirac a, x ∈ s

Tactic:
ae_dirac_eq

Goal state after:
case pos
α : Type u_1
inst✝² : MeasurableSpace α
s : Set α
a : α
inst✝¹ : MeasurableSingletonClass α
inst✝ : Decidable (a ∈ s)
has : a ∈ s
⊢ ∀ᶠ (x : α) in pure a, x ∈ s
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x ↦ Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
y : α
n j : ℕ
hj : dist y (u j) < (1 / 2) ^ n
⊢ ∃ j, y ∈ closedBall (u j) ((1 / 2) ^ n)

Tactic:
exact ⟨j, hj.le⟩

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

Goal state before:
pp: α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x ↦ Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
y : α
x : ℕ → ℕ
hx : ∀ (n : ℕ), y ∈ closedBall (u (x n)) ((1 / 2) ^ n)
I : (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty
⊢ ∃ a, g a = y

Tactic:
refine ⟨⟨x, I⟩, ?_⟩

Goal state after:
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x ↦ Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
y : α
x : ℕ → ℕ
hx : ∀ (n : ℕ), y ∈ closedBall (u (x n)) ((1 / 2) ^ n)
I : (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty
⊢ g ⟨x, I⟩ = y
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: ι : Type u_2
X : Type u_3
Y : ι → Type u_4
f : (i : ι) → X → Y i
x y : PiNatEmbed X Y f
hxy : x.ofPiNat = y.ofPiNat
⊢ x = y

Tactic:
cases x; congr!

Goal state after:
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
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
h : (μ.prod ν) s = 0
t : Set (α × β)
hst : s ⊆ t
mt : MeasurableSet t
ht : (μ.prod ν) t = 0
⊢ (fun x ↦ ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0

Tactic:
rw [measure_prod_null mt] at ht

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
h : (μ.prod ν) s = 0
t : Set (α × β)
hst : s ⊆ t
mt : MeasurableSet t
ht : (fun x ↦ ν (Prod.mk x ⁻¹' t)) =ᵐ[μ] 0
⊢ (fun x ↦ ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Group.MeasurableEquiv

Goal state before:
pp: case hg
G : Type u_1
A : Type u_2
inst✝³ : Group G
inst✝² : MulAction G A
inst✝¹ : MeasurableSpace A
inst✝ : MeasurableConstSMul G A
μ✝ ν : Measure A
g✝ : G
g g' : Gᵈᵐᵃ
μ : Measure A
⊢ Measurable fun x ↦ DomMulAct.mk.symm g⁻¹ • x

pp: case hf
G : Type u_1
A : Type u_2
inst✝³ : Group G
inst✝² : MulAction G A
inst✝¹ : MeasurableSpace A
inst✝ : MeasurableConstSMul G A
μ✝ ν : Measure A
g✝ : G
g g' : Gᵈᵐᵃ
μ : Measure A
⊢ Measurable fun x ↦ DomMulAct.mk.symm g'⁻¹ • x

Tactic:
· exact measurable_const_smul ..

Goal state after:
case hf
G : Type u_1
A : Type u_2
inst✝³ : Group G
inst✝² : MulAction G A
inst✝¹ : MeasurableSpace A
inst✝ : MeasurableConstSMul G A
μ✝ ν : Measure A
g✝ : G
g g' : Gᵈᵐᵃ
μ : Measure A
⊢ Measurable fun x ↦ DomMulAct.mk.symm g'⁻¹ • x
Module: Mathlib.MeasureTheory.Group.MeasurableEquiv

Goal state before:
pp: case hf
G : Type u_1
A : Type u_2
inst✝³ : Group G
inst✝² : MulAction G A
inst✝¹ : MeasurableSpace A
inst✝ : MeasurableConstSMul G A
μ✝ ν : Measure A
g✝ : G
g g' : Gᵈᵐᵃ
μ : Measure A
⊢ Measurable fun x ↦ DomMulAct.mk.symm g'⁻¹ • x

Tactic:
· exact measurable_const_smul ..

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
⊢ (hHT.equiv g₁).2 = (hHT.equiv g₂).2 ↔ RightCosetEquivalence (↑H) g₁ g₂

Tactic:
rw [RightCosetEquivalence, rightCoset_eq_iff]

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
⊢ (hHT.equiv g₁).2 = (hHT.equiv g₂).2 ↔ g₂ * g₁⁻¹ ∈ H
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 : Set α
⊢ s ∈ c • ae μ ↔ s ∈ ae μ

Tactic:
simp only [mem_smul_filter, preimage_smul, smul_mem_ae]

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : Set α → Prop
h : μ.InnerRegularWRT p fun s ↦ MeasurableSet s ∧ μ s ≠ ∞
A s : Set α
s_meas : MeasurableSet s
hs : μ (s ∩ A) ≠ ∞
r : ℝ≥0∞
hr : r < (μ.restrict A) s
K : Set α
K_subs : K ⊆ toMeasurable μ (s ∩ A) ∩ s
pK : p K
rK : r < μ K
⊢ ∃ K ⊆ s, p K ∧ r < (μ.restrict A) K

Tactic:
refine ⟨K, K_subs.trans inter_subset_right, pK, ?_⟩

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : Set α → Prop
h : μ.InnerRegularWRT p fun s ↦ MeasurableSet s ∧ μ s ≠ ∞
A s : Set α
s_meas : MeasurableSet s
hs : μ (s ∩ A) ≠ ∞
r : ℝ≥0∞
hr : r < (μ.restrict A) s
K : Set α
K_subs : K ⊆ toMeasurable μ (s ∩ A) ∩ s
pK : p K
rK : r < μ K
⊢ r < (μ.restrict A) K
Module: Mathlib.Topology.ContinuousMap.CocompactMap

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : CocompactMap α β
s : Set β
hs : IsCompact s
h's : IsClosed s
t : Set α
ht : IsCompact t
hts : (⇑f ⁻¹' s)ᶜᶜ ⊆ t
⊢ ⇑f ⁻¹' s ⊆ t

Tactic:
by simpa using hts

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

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H :
  ∀ (i : ℕ) (ε : ℝ≥0∞),
    ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed[inst✝²] (F n)
hUo : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
this : Tendsto (fun t ↦ ∑ k ∈ t, μ (s k) + ε / 2) atTop (𝓝 (μ (⋃ n, s n) + ε / 2))
t : Finset ℕ
ht : μ (⋃ n, s n) < ∑ k ∈ t, μ (s k) + ε / 2
⊢ ∑ k ∈ t, μ (F k) + ∑ k ∈ t, δ k + ε / 2 ≤ ∑ k ∈ t, μ (F k) + ε / 2 + ε / 2

Tactic:
by
          gcongr
          exact (ENNReal.sum_le_tsum _).trans hδε.le

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

Goal state before:
pp: α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : μ K < r
V : Set α
hKV : V ⊇ K
hVo : IsOpen[inst✝⁴] V
hμV : μ V < ∞
this : Fact (μ V < ∞)
U : Set α
hKU : K ⊆ U
hUo : IsOpen[inst✝⁴] U
hμU : (μ.restrict V) U < r
⊢ μ (U ∩ V) < r

Tactic:
restrict_apply hUo.measurableSet

Goal state after:
α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : μ K < r
V : Set α
hKV : V ⊇ K
hVo : IsOpen[inst✝⁴] V
hμV : μ V < ∞
this : Fact (μ V < ∞)
U : Set α
hKU : K ⊆ U
hUo : IsOpen[inst✝⁴] U
hμU : μ (U ∩ V) < r
⊢ μ (U ∩ V) < r
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace α
h : μ.InnerRegularCompactLTTop
c : ℝ≥0∞
⊢ (c • μ).InnerRegularCompactLTTop

Tactic:
by
  by_cases hc : c = 0
  · simp only [hc, zero_smul]
    infer_instance
  by_cases h'c : c = ∞
  · constructor
    intro s hs r hr
    by_cases h's : μ s = 0
    · simp [h's] at hr
    · simp [h'c, h's] at hs
  · constructor
    convert! InnerRegularWRT.smul h.innerRegular c using 2 with s
    have : (c • μ) s ≠ ∞ ↔ μ s ≠ ∞ := by simp [ENNReal.mul_eq_top, hc, h'c]
    simp only [this]

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

Goal state before:
pp: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
s : Set G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
h2s : ν' s ≠ 0
h3s : ν' s ≠ ∞
t : Set G
⊢ μ' t = μ' s * ν' t / ν' s

Tactic:
measure_mul_measure_eq μ' ν' s t h2s h3s,

Goal state after:
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
s : Set G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
h2s : ν' s ≠ 0
h3s : ν' s ≠ ∞
t : Set G
⊢ μ' t = ν' s * μ' t / ν' s
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case htc
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := ⋯
hg : AEMeasurable g (μ.withDensity f')
this : μ.withDensity f = μ.withDensity f'
g' : α → ℝ≥0∞ := ⋯
⊢ ∀ᵐ (x : α) ∂μ.restrict {x | f' x ≠ 0}ᶜ, (f' * g') x = (f' * g) x

Tactic:
· have M : MeasurableSet { x : α | f' x ≠ 0 }ᶜ :=
          (hf.measurable_mk (measurableSet_singleton 0).compl).compl
        filter_upwards [ae_restrict_mem M]
        intro x hx
        simp only [Classical.not_not, mem_ofPred_eq, mem_compl_iff] at hx
        simp only [hx, zero_mul, Pi.mul_apply]

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

Goal state before:
pp: case mpr
x y : ℝ
h : 0 ≤ y
⊢ |x| ≤ √y → |x| ^ 2 ≤ y

Tactic:
exact (le_sqrt (abs_nonneg x) h).mp

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

Goal state before:
pp: z : ℂ
⊢ z.re * z.re - z.im * z.im ≤ 0 ∧ z.re * z.im * 2 = 0 ↔ z.re = 0

Tactic:
mul_eq_zero_iff_right
    two_ne_zero,

Goal state after:
z : ℂ
⊢ z.re * z.re - z.im * z.im ≤ 0 ∧ z.re * z.im = 0 ↔ z.re = 0
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ≥0∞
this : ∀ {f : α → ℝ≥0∞}, Measurable f → SFinite (μ.withDensity f)
hfm : ¬Measurable f
⊢ SFinite (μ.withDensity f)

pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ≥0∞
hfm : Measurable f
⊢ SFinite (μ.withDensity f)

Tactic:
· rcases exists_measurable_le_withDensity_eq μ f with ⟨g, hgm, -, h⟩
    exact h ▸ this hgm

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ≥0∞
hfm : Measurable f
⊢ SFinite (μ.withDensity f)
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: ⊢ (↑0)⁻¹ = ↑0

Tactic:
← ofReal_inv,

Goal state after:
⊢ ↑0⁻¹ = ↑0
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: z : ℂ
x : ℝ
⊢ (↑x)⁻¹ * z = { re := x⁻¹ * z.re, im := x⁻¹ * z.im }

Tactic:
← ofReal_inv,

Goal state after:
z : ℂ
x : ℝ
⊢ ↑x⁻¹ * z = { re := x⁻¹ * z.re, im := x⁻¹ * z.im }
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V : Type u_1
W : Type u_2
V₁ : Type u_3
V₂ : Type u_4
V₃ : Type u_5
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : SeminormedAddCommGroup W
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
s : AddSubgroup V
v : ↥s
⊢ ‖↑v‖ ≤ ‖v‖

Tactic:
AddSubgroup.coe_norm

Goal state after:
V : Type u_1
W : Type u_2
V₁ : Type u_3
V₂ : Type u_4
V₃ : Type u_5
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : SeminormedAddCommGroup W
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
s : AddSubgroup V
v : ↥s
⊢ ‖↑v‖ ≤ ‖↑v‖
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V : Type u_1
W : Type u_2
V₁ : Type u_3
V₂ : Type u_4
V₃ : Type u_5
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : SeminormedAddCommGroup W
inst✝⁵ : SeminormedAddCommGroup V₁
inst✝⁴ : SeminormedAddCommGroup V₂
inst✝³ : SeminormedAddCommGroup V₃
f : NormedAddGroupHom V W
W₁ : Type u_6
W₂ : Type u_7
W₃ : Type u_8
inst✝² : SeminormedAddCommGroup W₁
inst✝¹ : SeminormedAddCommGroup W₂
inst✝ : SeminormedAddCommGroup W₃
g : NormedAddGroupHom V W
f₁ g₁ : NormedAddGroupHom V₁ W₁
f₂ g₂ : NormedAddGroupHom V₂ W₂
f₃ g₃ : NormedAddGroupHom V₃ W₃
ψ : NormedAddGroupHom V₁ ↥(f.equalizer g)
⊢ (f.comp (ι f g)).comp ψ = (g.comp (ι f g)).comp ψ

Tactic:
comp_ι_eq

Goal state after:
V : Type u_1
W : Type u_2
V₁ : Type u_3
V₂ : Type u_4
V₃ : Type u_5
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : SeminormedAddCommGroup W
inst✝⁵ : SeminormedAddCommGroup V₁
inst✝⁴ : SeminormedAddCommGroup V₂
inst✝³ : SeminormedAddCommGroup V₃
f : NormedAddGroupHom V W
W₁ : Type u_6
W₂ : Type u_7
W₃ : Type u_8
inst✝² : SeminormedAddCommGroup W₁
inst✝¹ : SeminormedAddCommGroup W₂
inst✝ : SeminormedAddCommGroup W₃
g : NormedAddGroupHom V W
f₁ g₁ : NormedAddGroupHom V₁ W₁
f₂ g₂ : NormedAddGroupHom V₂ W₂
f₃ g₃ : NormedAddGroupHom V₃ W₃
ψ : NormedAddGroupHom V₁ ↥(f.equalizer g)
⊢ (g.comp (ι f g)).comp ψ = (g.comp (ι f g)).comp ψ
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ -↑(im z) = -(I * ((starRingEnd K) z - z) / 2)

Tactic:
← ofReal_neg,

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ ↑(-im z) = -(I * ((starRingEnd K) z - z) / 2)
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
r : ℝ
⊢ re ((↑r)⁻¹ * z) = r⁻¹ * re z

Tactic:
← ofReal_inv,

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
r : ℝ
⊢ re (↑r⁻¹ * z) = r⁻¹ * re z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
z : K
r : ℤ
x : K
⊢ |↑r| * ‖x‖ = ‖r‖ * ‖x‖

Tactic:
Int.norm_eq_abs

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
z : K
r : ℤ
x : K
⊢ |↑r| * ‖x‖ = |↑r| * ‖x‖
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ z < 0 ↔ ∃ x < 0, ↑x = z

Tactic:
neg_iff (K := K),

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ re z < 0 ∧ im z = 0 ↔ ∃ x < 0, ↑x = z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
x : ℝ
hx : x > 0
hx' : ↑x = z
⊢ 0 < (↑x)⁻¹

Tactic:
← ofReal_inv,

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
x : ℝ
hx : x > 0
hx' : ↑x = z
⊢ 0 < ↑x⁻¹
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
s : Set X
t : Set Z
x : X
f : X → Z
hf : ContinuousWithinAt f s x
hxe : x ∈ e.source
ht : t ∈ 𝓝 (f x)
y : X
hy : y ∈ e.source
hyu : y ∈ s → y ∈ f ⁻¹' t
⊢ ↑e.symm (↑e y) ∈ s ↔ ↑e.symm (↑e y) ∈ s ∧ ↑e.symm (↑e y) ∈ f ⁻¹' t

Tactic:
iff_self_and,

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
s : Set X
t : Set Z
x : X
f : X → Z
hf : ContinuousWithinAt f s x
hxe : x ∈ e.source
ht : t ∈ 𝓝 (f x)
y : X
hy : y ∈ e.source
hyu : y ∈ s → y ∈ f ⁻¹' t
⊢ ↑e.symm (↑e y) ∈ s → ↑e.symm (↑e y) ∈ f ⁻¹' t
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: case inr
K : Type u_1
inst✝ : RCLike K
z : K
h : im I = 1
this : I ≠ 0
⊢ ‖z‖ ≤ im z ↔ -I * z = -I * (I * ↑‖z‖)

Tactic:
convert! norm_le_re_iff_eq_norm (z := -I * z) using 2

Goal state after:
case e'_1.e'_3
K : Type u_1
inst✝ : RCLike K
z : K
h : im I = 1
this : I ≠ 0
⊢ ‖z‖ = ‖-I * z‖

case e'_1.e'_4
K : Type u_1
inst✝ : RCLike K
z : K
h : im I = 1
this : I ≠ 0
⊢ im z = re (-I * z)

case e'_2
K : Type u_1
inst✝ : RCLike K
z : K
h : im I = 1
this : I ≠ 0
⊢ -I * (I * ↑‖z‖) = ↑‖-I * z‖
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 : 𝕜),
    { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }
          ((⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z) +
        { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }
            ((⋯ ▸
                let __spread.0 := h.toNormedField;
                im)
              z) *
          I =
      z

Tactic:
subst h''; exact h.re_add_im_ax

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
hk : NormedField 𝕜
h : IsRCLikeNormedField 𝕜
⊢ RCLike 𝕜

Tactic:
by
  choose p hp using h.out
  exact p.copy_of_normedField hk hp

Goal state after: