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

Goal state before:
pp: case pos
α : Type u_1
inst✝⁵ : TopologicalSpace α
s : Set α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
y : γ
hy : (fun z ↦ f z + g z) x > y
u v : Set γ
u_open : IsOpen[inst✝¹] u
xu : f x ∈ u
v_open : IsOpen[inst✝¹] v
xv : g x ∈ v
h : u ×ˢ v ⊆ {p | y < p.1 + p.2}
hx₁ : ∃ l, l < f x
z₁ : γ
z₁lt : z₁ < f x
h₁ : Ioc z₁ (f x) ⊆ u
hx₂ : ∀ (x_1 : γ), g x ≤ x_1
z : α
h₁z : f z > z₁
H : f z ≤ f x
⊢ min (f z) (f x) ∈ u

Tactic:
simpa using

Goal state after:
case pos
α : Type u_1
inst✝⁵ : TopologicalSpace α
s : Set α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
y : γ
hy : (fun z ↦ f z + g z) x > y
u v : Set γ
u_open : IsOpen[inst✝¹] u
xu : f x ∈ u
v_open : IsOpen[inst✝¹] v
xv : g x ∈ v
h : u ×ˢ v ⊆ {p | y < p.1 + p.2}
hx₁ : ∃ l, l < f x
z₁ : γ
z₁lt : z₁ < f x
h₁ : Ioc z₁ (f x) ⊆ u
hx₂ : ∀ (x_1 : γ), g x ≤ x_1
z : α
h₁z : f z > z₁
H : f z ≤ f x
⊢ f z ∈ u
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
r C : ℝ≥0∞
f : ℕ → α
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n
a : α
ha : Tendsto f atTop (𝓝 a)
⊢ edist (f 0) a ≤ C / (1 - r)

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
r C : ℝ≥0∞
f : ℕ → α
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n
a : α
ha : Tendsto f atTop (𝓝 a)
⊢ edist (f 0) a ≤ C / (1 - r)
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
C : ℝ≥0∞
f : ℕ → α
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C / 2 ^ n
a : α
ha : Tendsto f atTop (𝓝 a)
⊢ edist (f 0) a ≤ 2 * C

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
C : ℝ≥0∞
f : ℕ → α
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C / 2 ^ n
a : α
ha : Tendsto f atTop (𝓝 a)
⊢ edist (f 0) a ≤ 2 * C
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: case pos
α : Type u_1
inst✝⁵ : TopologicalSpace α
s : Set α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
y : γ
hy : (fun z ↦ f z + g z) x > y
u v : Set γ
u_open : IsOpen[inst✝¹] u
xu : f x ∈ u
v_open : IsOpen[inst✝¹] v
xv : g x ∈ v
h : u ×ˢ v ⊆ {p | y < p.1 + p.2}
hx₁ : ∀ (x_1 : γ), f x ≤ x_1
hx₂ : ∃ l, l < g x
z₂ : γ
z₂lt : z₂ < g x
h₂ : Ioc z₂ (g x) ⊆ v
z : α
h₂z : g z > z₂
H : g z ≤ g x
⊢ min (g z) (g x) ∈ v

Tactic:
simpa using

Goal state after:
case pos
α : Type u_1
inst✝⁵ : TopologicalSpace α
s : Set α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
y : γ
hy : (fun z ↦ f z + g z) x > y
u v : Set γ
u_open : IsOpen[inst✝¹] u
xu : f x ∈ u
v_open : IsOpen[inst✝¹] v
xv : g x ∈ v
h : u ×ˢ v ⊆ {p | y < p.1 + p.2}
hx₁ : ∀ (x_1 : γ), f x ≤ x_1
hx₂ : ∃ l, l < g x
z₂ : γ
z₂lt : z₂ < g x
h₂ : Ioc z₂ (g x) ⊆ v
z : α
h₂z : g z > z₂
H : g z ≤ g x
⊢ g z ∈ v
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: case inr
α : Type u_1
inst✝ : PseudoMetricSpace α
r C : ℝ
f : ℕ → α
hr : r < 1
hu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n
left✝ : 0 < C
r₀ : 0 ≤ r
⊢ HasSum (HPow.hPow r) (1 - r)⁻¹

Tactic:
simpa using

Goal state after:
case inr
α : Type u_1
inst✝ : PseudoMetricSpace α
r C : ℝ
f : ℕ → α
hr : r < 1
hu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n
left✝ : 0 < C
r₀ : 0 ≤ r
⊢ HasSum (HPow.hPow r) (1 - r)⁻¹
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: case neg
α : Type u_1
inst✝⁵ : TopologicalSpace α
s : Set α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
y : γ
hy : (fun z ↦ f z + g z) x > y
u v : Set γ
u_open : IsOpen[inst✝¹] u
xu : f x ∈ u
v_open : IsOpen[inst✝¹] v
xv : g x ∈ v
h : u ×ˢ v ⊆ {p | y < p.1 + p.2}
hx₁ : ∀ (x_1 : γ), f x ≤ x_1
hx₂ : ∃ l, l < g x
z₂ : γ
z₂lt : z₂ < g x
h₂ : Ioc z₂ (g x) ⊆ v
z : α
h₂z : g z > z₂
H : g x < g z
⊢ min (g z) (g x) ∈ v

Tactic:
simpa using

Goal state after:
case neg
α : Type u_1
inst✝⁵ : TopologicalSpace α
s : Set α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
y : γ
hy : (fun z ↦ f z + g z) x > y
u v : Set γ
u_open : IsOpen[inst✝¹] u
xu : f x ∈ u
v_open : IsOpen[inst✝¹] v
xv : g x ∈ v
h : u ×ˢ v ⊆ {p | y < p.1 + p.2}
hx₁ : ∀ (x_1 : γ), f x ≤ x_1
hx₂ : ∃ l, l < g x
z₂ : γ
z₂lt : z₂ < g x
h₂ : Ioc z₂ (g x) ⊆ v
z : α
h₂z : g z > z₂
H : g x < g z
⊢ g x ∈ v
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝⁶ : TopologicalSpace α
s : Set α
x : α
ι : Type u_4
γ : Type u_5
inst✝⁵ : AddCommMonoid γ
inst✝⁴ : LinearOrder γ
inst✝³ : IsOrderedAddMonoid γ
inst✝² : TopologicalSpace γ
inst✝¹ : OrderTopology γ
inst✝ : ContinuousAdd γ
f : ι → α → γ
a : Finset ι
ha : ∀ i ∈ a, LowerSemicontinuousWithinAt (f i) s x
⊢ LowerSemicontinuousWithinAt (fun z ↦ ∑ i ∈ a, f i z) s x

Tactic:
classical
    induction a using Finset.induction_on with
    | empty => exact lowerSemicontinuousWithinAt_const
    | insert _ _ ia IH =>
      simp only [ia, Finset.sum_insert, not_false_iff]
      exact
        LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self ..))
          (IH fun j ja => ha j (Finset.mem_insert_of_mem ja))

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

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : TopologicalSpace α
s : Set α
x : α
ι : Sort u_4
δ' : Type u_6
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i ↦ f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
h✝ : IsEmpty ι
⊢ LowerSemicontinuousWithinAt (fun x' ↦ ⨆ i, f i x') s x

Tactic:
simpa using

Goal state after:
case inl
α : Type u_1
inst✝¹ : TopologicalSpace α
s : Set α
x : α
ι : Sort u_4
δ' : Type u_6
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i ↦ f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
h✝ : IsEmpty ι
⊢ LowerSemicontinuousWithinAt (fun x' ↦ sSup ∅) s x
Module: Mathlib.MeasureTheory.OuterMeasure.OfFunction

Goal state before:
pp: case refine_2
α : Type u_1
ι : Sort u_2
β : Type u_3
inst✝ : Nonempty ι
f : α → β
m : ι → OuterMeasure β
s : Set β
t : ℕ → Set α
ht : f ⁻¹' s ⊆ iUnion t
n : ℕ
i : ι
⊢ f '' f ⁻¹' (fun n ↦ f '' t n ∪ (range f)ᶜ) n ⊆ f '' t n

Tactic:
simpa using

Goal state after:
case refine_2
α : Type u_1
ι : Sort u_2
β : Type u_3
inst✝ : Nonempty ι
f : α → β
m : ι → OuterMeasure β
s : Set β
t : ℕ → Set α
ht : f ⁻¹' s ⊆ iUnion t
n : ℕ
i : ι
⊢ f ⁻¹' f '' t n ⊆ f ⁻¹' f '' t n
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: k : ℕ
hn : 0 < k.succ
⊢ ↑(∏ i ∈ Finset.range k.succ, (i + 1)) * (∏ _k ∈ Finset.range k.succ, ↑k.succ)⁻¹ ≤ 1 / ↑k.succ

Tactic:
← inv_eq_one_div,

Goal state after:
k : ℕ
hn : 0 < k.succ
⊢ ↑(∏ i ∈ Finset.range k.succ, (i + 1)) * (∏ _k ∈ Finset.range k.succ, ↑k.succ)⁻¹ ≤ (↑k.succ)⁻¹
Module: Mathlib.Order.SetDissipate

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : Preorder α
s : α → Set β
x : α
⊢ ⋂ y, ⋂ (_ : y ≤ x), dissipate s y = dissipate s x

Tactic:
apply Subset.antisymm

Goal state after:
case h₁
α : Type u_1
β : Type u_2
inst✝ : Preorder α
s : α → Set β
x : α
⊢ ⋂ y, ⋂ (_ : y ≤ x), dissipate s y ⊆ dissipate s x

case h₂
α : Type u_1
β : Type u_2
inst✝ : Preorder α
s : α → Set β
x : α
⊢ dissipate s x ⊆ ⋂ y, ⋂ (_ : y ≤ x), dissipate s y
Module: Mathlib.Order.SetDissipate

Goal state before:
pp: α : Type u_1
β : Type u_2
s : α → Set β
inst✝ : Preorder α
⊢ ⋂ x, dissipate s x = ⋂ x, s x

Tactic:
apply Subset.antisymm

Goal state after:
case h₁
α : Type u_1
β : Type u_2
s : α → Set β
inst✝ : Preorder α
⊢ ⋂ x, dissipate s x ⊆ ⋂ x, s x

case h₂
α : Type u_1
β : Type u_2
s : α → Set β
inst✝ : Preorder α
⊢ ⋂ x, s x ⊆ ⋂ x, dissipate s x
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case zero
α : Type u_1
s : ℕ → Set α
C : Set (Set α)
hC : IsPiSystem C
h : ∀ (n : ℕ), s n ∈ C
h' : (dissipate s 0).Nonempty
⊢ dissipate s 0 ∈ C

Tactic:
simpa using

Goal state after:
case zero
α : Type u_1
s : ℕ → Set α
C : Set (Set α)
hC : IsPiSystem C
h : ∀ (n : ℕ), s n ∈ C
h' : (dissipate s 0).Nonempty
⊢ s 0 ∈ C
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: R : Type u_4
inst✝⁵ : TopologicalSpace R
inst✝⁴ : Field R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
⊢ Tendsto (fun x ↦ ↑⌊x⌋₊ / x) atTop (𝓝 1)

Tactic:
simpa using

Goal state after:
R : Type u_4
inst✝⁵ : TopologicalSpace R
inst✝⁴ : Field R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
⊢ Tendsto (fun x ↦ ↑⌊x⌋₊ / x) atTop (𝓝 1)
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case singleton
α : Type u_1
S : Set (Set α)
h_pi : IsPiSystem S
t : Finset (Set α)
a : Set α
ht : ∀ s ∈ {a}, s ∈ S
h' : (⋂ s ∈ {a}, s).Nonempty
⊢ ⋂ s ∈ {a}, s ∈ S

Tactic:
simpa using

Goal state after:
case singleton
α : Type u_1
S : Set (Set α)
h_pi : IsPiSystem S
t : Finset (Set α)
a : Set α
ht : ∀ s ∈ {a}, s ∈ S
h' : (⋂ s ∈ {a}, s).Nonempty
⊢ a ∈ S
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: R : Type u_4
inst✝⁵ : TopologicalSpace R
inst✝⁴ : Field R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
⊢ Tendsto (fun x ↦ ↑⌈x⌉₊ / x) atTop (𝓝 1)

Tactic:
simpa using

Goal state after:
R : Type u_4
inst✝⁵ : TopologicalSpace R
inst✝⁴ : Field R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
⊢ Tendsto (fun x ↦ ↑⌈x⌉₊ / x) atTop (𝓝 1)
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α : Type u_1
ι : Sort u_3
ι' : Sort u_4
inst✝ : LinearOrder α
Ixx : α → α → Set α
p : α → α → Prop
Hne : ∀ {a b : α}, (Ixx a b).Nonempty → p a b
Hi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)
f : ι → α
g : ι' → α
⊢ IsPiSystem {S | ∃ i j, p (f i) (g j) ∧ Ixx (f i) (g j) = S}

Tactic:
simpa using

Goal state after:
α : Type u_1
ι : Sort u_3
ι' : Sort u_4
inst✝ : LinearOrder α
Ixx : α → α → Set α
p : α → α → Prop
Hne : ∀ {a b : α}, (Ixx a b).Nonempty → p a b
Hi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)
f : ι → α
g : ι' → α
⊢ IsPiSystem {S | ∃ i j, p (f i) (g j) ∧ Ixx (f i) (g j) = S}
Module: Mathlib.MeasureTheory.OuterMeasure.Caratheodory

Goal state before:
pp: α : Type u
m : OuterMeasure α
s : Set α
h : m.IsCaratheodory sᶜ
⊢ m.IsCaratheodory s

Tactic:
simpa using

Goal state after:
α : Type u
m : OuterMeasure α
s : Set α
h : m.IsCaratheodory sᶜ
⊢ m.IsCaratheodory s
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case base
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s : Set α
b : β
h_s_in_t' : s ∈ (fun b ↦ g b) b
⊢ s = ⋂ b_1 ∈ {b}, (fun x ↦ s) b_1 ∧ ∀ b_1 ∈ {b}, (fun x ↦ s) b_1 ∈ g b_1

Tactic:
simpa using

Goal state after:
case base
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s : Set α
b : β
h_s_in_t' : s ∈ (fun b ↦ g b) b
⊢ s ∈ g b
Module: Mathlib.MeasureTheory.OuterMeasure.Caratheodory

Goal state before:
pp: α : Type u
m : OuterMeasure α
ι : Type u_1
inst✝¹ : Preorder ι
inst✝ : LocallyFiniteOrderBot ι
s : ι → Set α
h : ∀ (i : ι), m.IsCaratheodory (s i)
i : ι
⊢ m.IsCaratheodory ((partialSups s) i)

Tactic:
simpa using

Goal state after:
α : Type u
m : OuterMeasure α
ι : Type u_1
inst✝¹ : Preorder ι
inst✝ : LocallyFiniteOrderBot ι
s : ι → Set α
h : ∀ (i : ι), m.IsCaratheodory (s i)
i : ι
⊢ m.IsCaratheodory (⋃ x ∈ Iic i, s x)
Module: Mathlib.Topology.Instances.EReal.Lemmas

Goal state before:
pp: α : Type u_2
f : Filter α
m : α → EReal
a b : EReal
hm : Tendsto m f (𝓝 a)
h₁ : a ≠ 0 ∨ b ≠ ⊥
h₂ : a ≠ 0 ∨ b ≠ ⊤
⊢ Tendsto (fun x ↦ m x * b) f (𝓝 (a * b))

Tactic:
simpa using

Goal state after:
α : Type u_2
f : Filter α
m : α → EReal
a b : EReal
hm : Tendsto m f (𝓝 a)
h₁ : a ≠ 0 ∨ b ≠ ⊥
h₂ : a ≠ 0 ∨ b ≠ ⊤
⊢ Tendsto (fun x ↦ b * m x) f (𝓝 (a * b))
Module: Mathlib.MeasureTheory.OuterMeasure.Caratheodory

Goal state before:
pp: α : Type u
m : OuterMeasure α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
t : Set α
n : ℕ
a : α
⊢ a ∈ s n ∩ ⋃ k, ⋃ (_ : k < n), s k → a ∈ ∅

Tactic:
simpa using

Goal state after:
α : Type u
m : OuterMeasure α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
t : Set α
n : ℕ
a : α
⊢ a ∈ s n → ∀ x < n, a ∉ s x
Module: Mathlib.MeasureTheory.OuterMeasure.AE

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : FunLike F (Set α) ℝ≥0∞
inst✝ : OuterMeasureClass F α
μ : F
h : ∀ (s : Set α), μ sᶜ = 0 ↔ s = univ
a : α
ha : μ {a} = 0
⊢ False

Tactic:
simpa using

Goal state after:
α : Type u_1
F : Type u_3
inst✝¹ : FunLike F (Set α) ℝ≥0∞
inst✝ : OuterMeasureClass F α
μ : F
h : ∀ (s : Set α), μ sᶜ = 0 ↔ s = univ
a : α
ha : μ {a} = 0
⊢ False
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α : Type u_3
d : DynkinSystem α
a : Set α
h : d.Has aᶜ
⊢ d.Has a

Tactic:
simpa using

Goal state after:
α : Type u_3
d : DynkinSystem α
a : Set α
h : d.Has aᶜ
⊢ d.Has a
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α : Type u_3
d : DynkinSystem α
⊢ d.Has univ

Tactic:
simpa using

Goal state after:
α : Type u_3
d : DynkinSystem α
⊢ d.Has univ
Module: Mathlib.MeasureTheory.Measure.MeasureSpaceDef

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
hs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s
h's : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)
⊢ μ h's.choose = μ s

Tactic:
simpa using

Goal state after:
case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
hs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s
h's : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)
⊢ μ h's.choose = μ s
Module: Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
⊢ MeasurableSet (limsup s atTop)

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
⊢ MeasurableSet (blimsup s atTop fun x ↦ True)
Module: Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
⊢ MeasurableSet (liminf s atTop)

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
⊢ MeasurableSet (bliminf s atTop fun x ↦ True)
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
s : Set α
mα : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
h : MeasurableSet (f '' s)
⊢ MeasurableSet s

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
s : Set α
mα : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
h : MeasurableSet (f '' s)
⊢ MeasurableSet s
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 = ∞
r : ℝ≥0∞
hs : r > ms
⊢ ∃ t, MeasurableSet t ∧ s ⊆ t ∧ m t < r

Tactic:
simpa using

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 = ∞
r : ℝ≥0∞
hs : r > ms
⊢ ∃ t, MeasurableSet t ∧ s ⊆ t ∧ m t < r
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
e : α ≃ᵐ β
s : Set β
h : MeasurableSet (⇑e ⁻¹' s)
⊢ MeasurableSet s

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
e : α ≃ᵐ β
s : Set β
h : MeasurableSet (⇑e ⁻¹' s)
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
hf : Injective f ∧ MeasurableSpace.comap f inst✝ = inst✝¹ ∧ MeasurableSet (range f)
s : Set β
hs : MeasurableSet s
⊢ MeasurableSet (f '' f ⁻¹' s)

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
hf : Injective f ∧ MeasurableSpace.comap f inst✝ = inst✝¹ ∧ MeasurableSet (range f)
s : Set β
hs : MeasurableSet s
⊢ MeasurableSet (s ∩ range f)
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: 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
s : Set α
hf : MeasurableEmbedding f
u : Set α
hu : MeasurableSet u
⊢ MeasurableSet (⇑(Equiv.Set.image f s ⋯).symm ⁻¹' Subtype.val ⁻¹' u)

Tactic:
simpa using

Goal state after:
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
s : Set α
hf : MeasurableEmbedding f
u : Set α
hu : MeasurableSet u
⊢ MeasurableSet (Subtype.val ⁻¹' f '' u)
Module: Mathlib.Order.Interval.Set.Monotone

Goal state before:
pp: case neg.inl
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 : k ≤ φ k
hφ : StrictMonoOn φ (Iic (succ k))
hk : ¬IsMax k
hm : succ k ≤ succ k
⊢ succ (φ k) ≤ φ (succ k)

Tactic:
refine succ_le_of_lt (hφ (le_succ _) le_rfl ?_)

Goal state after:
case neg.inl
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 : k ≤ φ k
hφ : StrictMonoOn φ (Iic (succ k))
hk : ¬IsMax k
hm : succ k ≤ succ k
⊢ k < succ k
Module: Mathlib.MeasureTheory.Measure.AEDisjoint

Goal state before:
pp: α : Type u_2
m : MeasurableSpace α
μ : Measure α
s t : Set α
h : AEDisjoint μ s t
x : α
hx : x ∈ s \ toMeasurable μ (s ∩ t)
⊢ x ∈ s \ t

Tactic:
simpa using

Goal state after:
α : Type u_2
m : MeasurableSpace α
μ : Measure α
s t : Set α
h : AEDisjoint μ s t
x : α
hx : x ∈ s \ toMeasurable μ (s ∩ t)
⊢ x ∈ s ∧ x ∉ t
Module: Mathlib.MeasureTheory.Measure.NullMeasurable

Goal state before:
pp: case refine_2
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
htm : MeasurableSet t
hst : s =ᵐ[μ] t
this : toMeasurable μ (s \ t) =ᵐ[μ] ∅
⊢ t ∪ toMeasurable μ (s \ t) =ᵐ[μ] s

Tactic:
simpa using

Goal state after:
case refine_2
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
htm : MeasurableSet t
hst : s =ᵐ[μ] t
this : toMeasurable μ (s \ t) =ᵐ[μ] ∅
⊢ t ∪ toMeasurable μ (s \ t) =ᵐ[μ] s
Module: Mathlib.MeasureTheory.Measure.NullMeasurable

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : μ sᶜ = 0
⊢ μ s = μ univ

Tactic:
simpa using

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : μ sᶜ = 0
⊢ μ s = μ univ
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f g : α → β
h : f =ᵐ[μ] g
hf : ¬AEMeasurable f μ
⊢ ¬AEMeasurable g μ

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f g : α → β
h : f =ᵐ[μ] g
hf : ¬AEMeasurable f μ
⊢ ¬AEMeasurable f μ
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
c : R
μ : Measure α
f : α → β
this : ∀ (c : ℝ≥0∞), map f (c • μ) = c • map f μ
⊢ map f (c • μ) = c • map f μ

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
c : R
μ : Measure α
f : α → β
this : ∀ (c : ℝ≥0∞), map f (c • μ) = c • map f μ
⊢ map f (c • μ) = c • map f μ
Module: Mathlib.MeasureTheory.Measure.AbsolutelyContinuous

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
h : μ ≪ ν
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (map f ν) s = 0 → (map f μ) s = 0

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
h : μ ≪ ν
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ ν (f ⁻¹' s) = 0 → μ (f ⁻¹' s) = 0
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
e : α ≃ᵐ β
μ₁ μ₂ : Measure α
hμ : map (⇑e.symm) (map (⇑e) μ₁) = map (⇑e.symm) (map (⇑e) μ₂)
⊢ μ₁ = μ₂

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
e : α ≃ᵐ β
μ₁ μ₂ : Measure α
hμ : map (⇑e.symm) (map (⇑e) μ₁) = map (⇑e.symm) (map (⇑e) μ₂)
⊢ μ₁ = μ₂
Module: Mathlib.MeasureTheory.Measure.AbsolutelyContinuous

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
s : Set α
hs0 : ν s = 0
⊢ μ₁ s = 0 ∧ μ₂ s = 0

Tactic:
simpa using

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
s : Set α
hs0 : ν s = 0
⊢ μ₁ s = 0 ∧ μ₂ s = 0
Module: Mathlib.MeasureTheory.OuterMeasure.BorelCantelli

Goal state before:
pp: case e_6
α : Type u_1
ι : Type u_2
F : Type u_3
inst✝² : FunLike F (Set α) ℝ≥0∞
inst✝¹ : OuterMeasureClass F α
inst✝ : Countable ι
μ : F
s : ι → Set α
h : ∑' (i : ι), μ (s i) ≠ ∞
x : α
⊢ x ∈ {a | ¬{i | a ∈ s i}.Finite} ↔ x ∈ limsup s cofinite

Tactic:
simp [mem_limsup_iff_frequently_mem, Filter.Frequently]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
n : ℕ
⊢ (preimage f)^[n] s =ᵐ[μ] s

Tactic:
simpa using

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
n : ℕ
⊢ (preimage f)^[n] s =ᵐ[μ] s
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
c : ℝ≥0∞
hc : c ≠ 0
h : Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s : Set α
hs : MeasurableSet s
⊢ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) (c • μ)

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
c : ℝ≥0∞
hc : c ≠ 0
h : Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s : Set α
hs : MeasurableSet s
⊢ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ