mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
r C : ℝ≥0∞
hr : r < 1
hC : C ≠ ∞
f : ℕ → α
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n
⊢ ∑' (n : ℕ), C * r ^ n ≠ ∞

Tactic:
rw [ENNReal.tsum_mul_left, ENNReal.tsum_geometric]

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
r C : ℝ≥0∞
hr : r < 1
hC : C ≠ ∞
f : ℕ → α
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n
⊢ C * (1 - r)⁻¹ ≠ ∞
Module: Mathlib.MeasureTheory.OuterMeasure.OfFunction

Goal state before:
pp: case refine_2
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
β : Type u_2
f : α → β
hf : Injective f
s : Set β
t : ℕ → Set α
ht : f ⁻¹' s ⊆ iUnion t
⊢ ∑' (n : ℕ), m (f ⁻¹' (fun n ↦ (range f)ᶜ ∪ f '' t n) n) ≤ ∑' (n : ℕ), m (t n)

Tactic:
· refine ENNReal.tsum_le_tsum fun n => le_of_eq ?_
    simp [hf.preimage_image]

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

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
ι✝ : Type u_3
ε : ℝ
hε : 0 < ε
ι : Type ?u.11
inst✝ : Encodable ι
f : ℕ → ℝ := fun n ↦ ε / 2 / 2 ^ n
hf : HasSum f ε
f0 : ∀ (n : ℕ), 0 < f n
c : ℝ
hg : HasSum (f ∘ Encodable.encode) c
⊢ ∀ (i : ι), (f ∘ Encodable.encode) i ≤ f (Encodable.encode i)

Tactic:
intro n

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
ι✝ : Type u_3
ε : ℝ
hε : 0 < ε
ι : Type ?u.11
inst✝ : Encodable ι
f : ℕ → ℝ := fun n ↦ ε / 2 / 2 ^ n
hf : HasSum f ε
f0 : ∀ (n : ℕ), 0 < f n
c : ℝ
hg : HasSum (f ∘ Encodable.encode) c
n : ι
⊢ (f ∘ Encodable.encode) n ≤ f (Encodable.encode n)
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: α : Type u_4
inst✝³ : Semiring α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorSemiring α
⊢ Tendsto (fun x ↦ ⌈x⌉₊) atTop atTop

Tactic:
refine Nat.ceil_mono.tendsto_atTop_atTop (fun x ↦ ⟨x, ?_⟩)
  simp only [Nat.ceil_natCast, le_refl]

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
s : Set α
x : α
γ : Type u_4
inst✝² : LinearOrder γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f : α → γ
h₁ : LowerSemicontinuousWithinAt f s x
h₂ : UpperSemicontinuousWithinAt f s x
v : Set γ
hv : v ∈ 𝓝 (f x)
Hl : ∃ l, l < f x
l : γ
lfx : l < f x
hl : Ioc l (f x) ⊆ v
Hu : ∃ u, f x < u
u : γ
fxu : f x < u
hu : Ico (f x) u ⊆ v
a : α
lfa : f a > l
⊢ f a < u → a ∈ f ⁻¹' v

Tactic:
fau

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
s : Set α
x : α
γ : Type u_4
inst✝² : LinearOrder γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f : α → γ
h₁ : LowerSemicontinuousWithinAt f s x
h₂ : UpperSemicontinuousWithinAt f s x
v : Set γ
hv : v ∈ 𝓝 (f x)
Hl : ∃ l, l < f x
l : γ
lfx : l < f x
hl : Ioc l (f x) ⊆ v
Hu : ∃ u, f x < u
u : γ
fxu : f x < u
hu : Ico (f x) u ⊆ v
a : α
lfa : f a > l
fau : f a < u
⊢ a ∈ f ⁻¹' v
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_2
P : α → Prop
m : (s : α) → P s → ℝ≥0∞
s : α
h : P s
⊢ extend m s = m s h

Tactic:
by simp [extend, h]

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_2
P : α → Prop
m : (s : α) → P s → ℝ≥0∞
s : α
h : ¬P s
⊢ extend m s = ∞

Tactic:
by simp [extend, h]

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.Caratheodory

Goal state before:
pp: α : Type u
m : OuterMeasure α
s s₁ s₂ : Set α
⊢ MeasurableSpace α

Tactic:
apply MeasurableSpace.DynkinSystem.toMeasurableSpace (caratheodoryDynkin m)

Goal state after:
α : Type u
m : OuterMeasure α
s s₁ s₂ : Set α
⊢ ∀ (s₁ s₂ : Set α), m.caratheodoryDynkin.Has s₁ → m.caratheodoryDynkin.Has s₂ → m.caratheodoryDynkin.Has (s₁ ∩ s₂)
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case pos
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∉ T_s
hbt : b ∈ T_t'
⊢ f_t' b ∈ g b

pp: case neg
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∉ T_s
hbt : b ∉ T_t'
⊢ ∅ ∈ g b

Tactic:
· exact h_t' b hbt

Goal state after:
case neg
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∉ T_s
hbt : b ∉ T_t'
⊢ ∅ ∈ g b
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
s₁ s₂ : Set α
hd : Disjoint s₁ s₂
h₁ : P s₁
h₂ : P s₂
⊢ extend m (s₁ ∪ s₂) = extend m s₁ + extend m s₂

Tactic:
union_eq_iUnion,

Goal state after:
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
s₁ s₂ : Set α
hd : Disjoint s₁ s₂
h₁ : P s₁
h₂ : P s₂
⊢ extend m (⋃ b, bif b then s₁ else s₂) = extend m s₁ + extend m s₂
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case refine_1
α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
i : ι
t : Finset ι
f : ι → Set α
hfπ : ∀ x ∈ t, f x ∈ π x
hti : ∀ y ∈ t, y = i
⊢ ⋂ x ∈ t, f x ∈ π i ∨ ⋂ x ∈ t, f x ∈ {univ}

Tactic:
by_cases hi : i ∈ t

Goal state after:
case pos
α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
i : ι
t : Finset ι
f : ι → Set α
hfπ : ∀ x ∈ t, f x ∈ π x
hti : ∀ y ∈ t, y = i
hi : i ∈ t
⊢ ⋂ x ∈ t, f x ∈ π i ∨ ⋂ x ∈ t, f x ∈ {univ}

case neg
α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
i : ι
t : Finset ι
f : ι → Set α
hfπ : ∀ x ∈ t, f x ∈ π x
hti : ∀ y ∈ t, y = i
hi : i ∉ t
⊢ ⋂ x ∈ t, f x ∈ π i ∨ ⋂ x ∈ t, f x ∈ {univ}
Module: Mathlib.MeasureTheory.Function.AEMeasurableSequence

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : ι → α → β
μ : Measure α
p : α → (ι → β) → Prop
hf : ∀ (i : ι), AEMeasurable (f i) μ
x : α
hx : x ∈ aeSeqSet hf p
⊢ p x fun n ↦ aeSeq hf p n x

Tactic:
by
  simp only [aeSeq, hx, if_true]
  rw [funext fun n => mk_eq_fun_of_mem_aeSeqSet hf hx n]
  have h_ss : aeSeqSet hf p ⊆ { x | p x fun n => f n x } := by
    rw [← compl_compl { x | p x fun n => f n x }, aeSeqSet, Set.compl_subset_compl]
    refine Set.Subset.trans (Set.compl_subset_compl.mpr ?_) (subset_toMeasurable _ _)
    exact fun x hx => hx.2
  have hx' := Set.mem_of_subset_of_mem h_ss hx
  exact hx'

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s✝ t✝ : Set α
l : Filter α
inst✝ : CountableInterFilter l
s t : ℕ → Set α
ht : ∀ (i : ℕ), MeasurableSet (t i)
hts : ∀ (i : ℕ), s i =ᶠ[l] t i
⊢ ∃ t, MeasurableSet t ∧ ⋃ i, s i =ᶠ[l] t

Tactic:
exact ⟨⋃ i, t i, MeasurableSet.iUnion ht, .countable_iUnion hts⟩

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
m : OuterMeasure α
s : Set α
⊢ m.trim s = ⨅ t, m ↑t

Tactic:
simp [iInf_subtype, iInf_and, trim_eq_iInf]

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t
hs : ¬ms = ∞
this : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r
⊢ ∀ (n : ℕ), ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (↑n)⁻¹

Tactic:
intro n

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t
hs : ¬ms = ∞
this : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r
n : ℕ
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (↑n)⁻¹
Module: Mathlib.MeasureTheory.Measure.AEDisjoint

Goal state before:
pp: α : Type u_2
m : MeasurableSpace α
μ : Measure α
s t : Set α
h : Disjoint s t
⊢ μ (s ∩ t) = 0

Tactic:
disjoint_iff_inter_eq_empty.1 h,

Goal state after:
α : Type u_2
m : MeasurableSpace α
μ : Measure α
s t : Set α
h : Disjoint s t
⊢ μ ∅ = 0
Module: Mathlib.Order.Interval.Set.Monotone

Goal state before:
pp: case pos
α : Type u_1
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : OrderBot α
n : α
φ : α → α
k : α
ih : StrictMonoOn φ (Iic k) → ∀ m ≤ k, m ≤ φ m
hφ : StrictMonoOn φ (Iic (succ k))
m : α
hm : m ≤ succ k
hk : IsMax k
⊢ m ≤ φ m

Tactic:
succ_eq_iff_isMax.2 hk

Goal state after:
case pos
α : Type u_1
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : OrderBot α
n : α
φ : α → α
k : α
ih : StrictMonoOn φ (Iic k) → ∀ m ≤ k, m ≤ φ m
hφ : StrictMonoOn φ (Iic (succ k))
m : α
hm : m ≤ k
hk : IsMax k
⊢ m ≤ φ m
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: case refine_2.a
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := ⋯
Fmono : ∀ {A B : Set α}, A ⊆ B → F A ⊆ F B
X : ℕ → Set α := ⋯
⊢ ∀ (i : ℕ), F (iInter X) ⊆ F^[i] univ

Tactic:
intro n

Goal state after:
case refine_2.a
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A ↦ (g '' (f '' A)ᶜ)ᶜ
Fmono : ∀ {A B : Set α}, A ⊆ B → F A ⊆ F B
X : ℕ → Set α := fun n ↦ F^[n] univ
n : ℕ
⊢ F (iInter X) ⊆ F^[n] univ
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: case a
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A ↦ (g '' (f '' A)ᶜ)ᶜ
Fmono : ∀ {A B : Set α}, A ⊆ B → F A ⊆ F B
X : ℕ → Set α := fun n ↦ F^[n] univ
y : β
hy : y ∈ (f '' iInter X)ᶜ
hx : ∀ (i : ℕ), g y ∈ F^[i] univ
⊢ ∀ (i : ℕ), y ∈ f '' F^[i] univ

Tactic:
intro n

Goal state after:
case a
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A ↦ (g '' (f '' A)ᶜ)ᶜ
Fmono : ∀ {A B : Set α}, A ⊆ B → F A ⊆ F B
X : ℕ → Set α := fun n ↦ F^[n] univ
y : β
hy : y ∈ (f '' iInter X)ᶜ
hx : ∀ (i : ℕ), g y ∈ F^[i] univ
n : ℕ
⊢ y ∈ f '' F^[n] univ
Module: Mathlib.MeasureTheory.Measure.NullMeasurable

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hd : AEDisjoint μ s t
⊢ μ (s ∪ t) = μ s + μ t

Tactic:
union_eq_iUnion,

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hd : AEDisjoint μ s t
⊢ μ (⋃ b, bif b then s else t) = μ s + μ t
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α ≃ᵐ β
μ : Measure α
s : Set β
⊢ ⇑f ⁻¹' s ∈ ae μ ↔ s ∈ ae (map (⇑f) μ)

Tactic:
mem_ae_iff,

Goal state after:
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α ≃ᵐ β
μ : Measure α
s : Set β
⊢ μ (⇑f ⁻¹' s)ᶜ = 0 ↔ (map (⇑f) μ) sᶜ = 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s₁ s₂ t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ s₂ ≤ μ s₁
ht : t₁ ⊆ t₂
htμ : μ t₂ ≤ μ t₁
⊢ μ (s₁ ∪ t₁) = μ (s₂ ∪ t₂)

Tactic:
union_eq_iUnion,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s₁ s₂ t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ s₂ ≤ μ s₁
ht : t₁ ⊆ t₂
htμ : μ t₂ ≤ μ t₁
⊢ μ (⋃ b, bif b then s₁ else t₁) = μ (s₂ ∪ t₂)
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: case inr.left
X : Type u_1
inst✝⁶ : LinearOrder X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : OrderTopology X
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoMaxOrder X
inst✝ : NoMinOrder X
f g : α → X
hl : Tendsto f atTop atTop
hu : Tendsto g atBot atBot
x : X
h✝ : Nonempty α
x_L : X
hx_L : x_L < x
x_R : X
hx_R : x < x_R
a_L : α
ha_L : ∀ a ≤ a_L, g a ≤ x_L
a_R : α
ha_R : ∀ a ≥ a_R, x_R ≤ f a
n : α
y : X
hf : f n ≤ y
hg : y ≤ g n
hxR : y < x_R
hxL : n < a_L
⊢ y ≤ x_L

Tactic:
exact hg.trans (ha_L n hxL.le)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
⊢ measureUnivNNReal μ = 0 ↔ μ = 0

Tactic:
rw [← MeasureTheory.Measure.measure_univ_eq_zero, ← coe_measureUnivNNReal]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
⊢ measureUnivNNReal μ = 0 ↔ ↑(measureUnivNNReal μ) = 0
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ μ.restrict s + μ.restrict sᶜ = μ

Tactic:
restrict_add_restrict_compl hs

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ μ = μ
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : ∀ᵐ (ω : α) ∂μ, Monotone fun x ↦ ω ∈ s x
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
hfin : ∃ i, μ (s i) ≠ ∞
⊢ μ (⋂ i, s i) = ⨅ i, μ (s i)

Tactic:
by
  obtain ⟨i, hi⟩ := hfin
  have : Nonempty ι := ⟨i⟩
  let t : ι → Set α := fun i ↦ s i ∩ {ω | Monotone (ω ∈ s ·)}
  have hst (i : ι) : s i =ᵐ[μ] t i := by
    filter_upwards [hs] with ω hω
    suffices ω ∈ s i ↔ ω ∈ t i from propext this
    simpa [t] using fun _ ↦ hω
  have hMono : Monotone t := fun i j hij ω hω ↦ ⟨hω.2 hij hω.1, hω.2⟩
  rw [iInf_congr <| fun i ↦ measure_congr <| hst i,
    ← hMono.measure_iInter (fun i ↦ (hsm i).congr (hst i)) ⟨i, by rwa [← measure_congr (hst i)]⟩]
  refine measure_congr ?_
  nth_rw 1 [← iInter_inter, ← inter_univ (⋂ i, s i)]
  exact ae_eq_set_inter (by rfl) (ae_eq_univ.2 hs).symm

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
⊢ IsFiniteMeasure (↑c • μ)

Tactic:
exact MeasureTheory.isFiniteMeasureSMulNNReal

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

Goal state before:
pp: α : Type u_2
t : Set α
m0 : MeasurableSpace α
m : Set (Measure α)
hm : m.Nonempty
ht : MeasurableSet t
s : Set α
hs : MeasurableSet s
⊢ (sInf (toOuterMeasure '' m)) (s ∩ t) = (sInf (⇑(OuterMeasure.restrict t) '' toOuterMeasure '' m)) s

Tactic:
← OuterMeasure.restrict_sInf_eq_sInf_restrict _ (hm.image _),

Goal state after:
α : Type u_2
t : Set α
m0 : MeasurableSpace α
m : Set (Measure α)
hm : m.Nonempty
ht : MeasurableSet t
s : Set α
hs : MeasurableSet s
⊢ (sInf (toOuterMeasure '' m)) (s ∩ t) = ((OuterMeasure.restrict t) (sInf (toOuterMeasure '' m))) s
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}
T : Set (Set α)
T_count : T.Countable
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : T.Nonempty
f : ℕ → Set α
hf : T = range f
⊢ ∀ (n : ℕ), f n ∈ S

Tactic:
intro n

Goal state after:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}
T : Set (Set α)
T_count : T.Countable
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : T.Nonempty
f : ℕ → Set α
hf : T = range f
n : ℕ
⊢ f n ∈ S
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
v : ℕ → Set α
hv : t ⊆ ⋃ n, v n
h'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ∞
w : ℕ → Set α := fun n ↦ toMeasurable μ (t ∩ v n)
⊢ ∀ (n : ℕ), μ (w n) < ∞

Tactic:
intro n

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
v : ℕ → Set α
hv : t ⊆ ⋃ n, v n
h'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ∞
w : ℕ → Set α := fun n ↦ toMeasurable μ (t ∩ v n)
n : ℕ
⊢ μ (w n) < ∞
Module: Mathlib.MeasureTheory.Measure.Trim

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
⊢ μ.trim hm = Measure.map id μ

Tactic:
refine @Measure.ext α m _ _ (fun s hs ↦ ?_)
  rw [Measure.map_apply (measurable_id'' hm) hs, trim_measurableSet_eq hm hs, Set.preimage_id]

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

Goal state before:
pp: α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
t : Set α
ht : MeasurableSet t
⊢ (μ.restrict (⋃ i, s i)) t ≤ (sum fun i ↦ μ.restrict (s i)) t

Tactic:
by simpa [ht, inter_iUnion] using measure_iUnion_le (t ∩ s ·)

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
s : Set α
t : Set β
f : α → β
hs : MeasurableSet s
⊢ f ⁻¹' t ∈ ae (μ.restrict s) ↔ μ ((f ⁻¹' t)ᶜ ∩ s) = 0

Tactic:
mem_ae_iff,

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
s : Set α
t : Set β
f : α → β
hs : MeasurableSet s
⊢ (μ.restrict s) (f ⁻¹' t)ᶜ = 0 ↔ μ ((f ⁻¹' t)ᶜ ∩ s) = 0
Module: Mathlib.Order.Filter.CountableSeparatingOn

Goal state before:
pp: case inr
α : Type u_1
l : Filter α
inst✝² : CountableInterFilter l
inst✝¹ : Nonempty α
p : Set α → Prop
s : Set α
inst✝ : HasCountableSeparatingOn α p s
hs : s ∈ l
hl : ∀ (U : Set α), p U → U ∈ l ∨ Uᶜ ∈ l
hne : s.Nonempty
⊢ ∃ a, {a} ∈ l

Tactic:
exact (exists_mem_singleton_mem_of_mem_of_nonempty_of_forall_separating p hs hne hl).imp fun _ ↦
      And.right

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
t : Set β
ht : 0 ∉ t
⊢ s.indicator f ⁻¹' t ∈ ae μ ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0

Tactic:
mem_ae_iff,

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
t : Set β
ht : 0 ∉ t
⊢ μ (s.indicator f ⁻¹' t)ᶜ = 0 ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
t : Set β
ht : 0 ∉ t
⊢ t ∈ Filter.map (s.indicator f) (ae μ) ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0

Tactic:
classical
  rw [mem_map, mem_ae_iff, Set.indicator_preimage, Set.ite, Set.compl_union, Set.compl_inter]
  change μ (((f ⁻¹' t)ᶜ ∪ sᶜ) ∩ ((fun _ => (0 : β)) ⁻¹' t \ s)ᶜ) = 0 ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0
  simp only [ht, if_false, Set.compl_empty, Set.empty_sdiff, Set.inter_univ, Set.preimage_const]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
⊢ ∀ (x : ℕ), ¬x = 1 → ¬x = 0 → min (μ (t' x)) (ν (t' x)) = 0

Tactic:
intro n

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
n : ℕ
⊢ ¬n = 1 → ¬n = 0 → min (μ (t' n)) (ν (t' n)) = 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
⊢ ∀ (x : ℕ), ¬x = 1 → ¬x = 0 → min (μ (t' x)) (ν (t' x)) = 0

Tactic:
intro n hn₁ hn₀

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
n : ℕ
hn₁ : ¬n = 1
hn₀ : ¬n = 0
⊢ min (μ (t' n)) (ν (t' n)) = 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
P : α → Prop
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ν s < ∞ → ∀ᵐ (x : α) ∂μ.restrict s, P x
⊢ ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, x ∈ spanningSets (μ + ν) n → P x

Tactic:
intro n

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
P : α → Prop
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ν s < ∞ → ∀ᵐ (x : α) ∂μ.restrict s, P x
n : ℕ
⊢ ∀ᵐ (x : α) ∂μ, x ∈ spanningSets (μ + ν) n → P x
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
⊢ MeasurableSet (t n)

Tactic:
have : t n = ⋃ u ∈ memPartition t n, u ∩ t n := by
    simp_rw [← iUnion_inter, ← sUnion_eq_biUnion, sUnion_memPartition, univ_inter]
  rw [this]
  refine MeasurableSet.biUnion (finite_memPartition _ _).countable (fun v hv ↦ ?_)
  refine measurableSet_generateFrom ?_
  rw [memPartition_succ]
  exact ⟨v, hv, Or.inl rfl⟩

Goal state after: