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

Goal state before:
pp: α : Type u_1
β : Type u_2
C : Set (Set α)
D : Set (Set β)
hC : IsCountablySpanning C
hD : IsCountablySpanning D
⊢ IsCountablySpanning (image2 (fun x1 x2 ↦ x1 ×ˢ x2) C D)

Tactic:
by
  rcases hC, hD with ⟨⟨s, h1s, h2s⟩, t, h1t, h2t⟩
  refine ⟨fun n => s n.unpair.1 ×ˢ t n.unpair.2, fun n => mem_image2_of_mem (h1s _) (h1t _), ?_⟩
  rw [iUnion_unpair_prod, h2s, h2t, univ_prod_univ]

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

Goal state before:
pp: 𝕜✝ : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
F : Type u_4
α : Type u_5
𝕜 : Type u_6
E : Type u_7
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Norm E
core : NormedSpace.Core 𝕜 E
this : SeminormedAddCommGroup E := SeminormedAddCommGroup.ofCore ⋯
x y : E
h : dist x y = 0
⊢ x = y

Tactic:
rw [← sub_eq_zero, ← core.norm_eq_zero_iff, ← norm_neg_add]

Goal state after:
𝕜✝ : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
F : Type u_4
α : Type u_5
𝕜 : Type u_6
E : Type u_7
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Norm E
core : NormedSpace.Core 𝕜 E
this : SeminormedAddCommGroup E := SeminormedAddCommGroup.ofCore ⋯
x y : E
h : dist x y = 0
⊢ ‖-x + y‖ = 0
Module: Mathlib.Analysis.Normed.Module.Basic

Goal state before:
pp: 𝕜✝ : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
F : Type u_4
α : Type u_5
𝕜 : Type u_6
E : Type u_7
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Norm E
U : UniformSpace E
core : NormedSpace.Core 𝕜 E
H : 𝓤 E = 𝓤 E
this : SeminormedAddCommGroup E := SeminormedAddCommGroup.ofCore ⋯
x y : E
h : dist x y = 0
⊢ x = y

Tactic:
rw [← sub_eq_zero, ← core.norm_eq_zero_iff, ← norm_neg_add]

Goal state after:
𝕜✝ : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
F : Type u_4
α : Type u_5
𝕜 : Type u_6
E : Type u_7
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Norm E
U : UniformSpace E
core : NormedSpace.Core 𝕜 E
H : 𝓤 E = 𝓤 E
this : SeminormedAddCommGroup E := SeminormedAddCommGroup.ofCore ⋯
x y : E
h : dist x y = 0
⊢ ‖-x + y‖ = 0
Module: Mathlib.Analysis.Normed.Module.Basic

Goal state before:
pp: 𝕜✝ : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
F : Type u_4
α : Type u_5
𝕜 : Type u_6
E : Type u_7
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Norm E
T : TopologicalSpace E
core : NormedSpace.Core 𝕜 E
H : T = PseudoEMetricSpace.toUniformSpace.toTopologicalSpace
this : SeminormedAddCommGroup E := SeminormedAddCommGroup.ofCore ⋯
x y : E
h : dist x y = 0
⊢ x = y

Tactic:
rw [← sub_eq_zero, ← core.norm_eq_zero_iff, ← norm_neg_add]

Goal state after:
𝕜✝ : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
F : Type u_4
α : Type u_5
𝕜 : Type u_6
E : Type u_7
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Norm E
T : TopologicalSpace E
core : NormedSpace.Core 𝕜 E
H : T = PseudoEMetricSpace.toUniformSpace.toTopologicalSpace
this : SeminormedAddCommGroup E := SeminormedAddCommGroup.ofCore ⋯
x y : E
h : dist x y = 0
⊢ ‖-x + y‖ = 0
Module: Mathlib.Analysis.Normed.Module.Basic

Goal state before:
pp: 𝕜✝ : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
F : Type u_4
α : Type u_5
𝕜 : Type u_6
E : Type u_7
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Norm E
U : UniformSpace E
B : Bornology E
core : NormedSpace.Core 𝕜 E
HU : 𝓤 E = 𝓤 E
HB : ∀ (s : Set E), Bornology.IsBounded s ↔ Bornology.IsBounded s
this : SeminormedAddCommGroup E := SeminormedAddCommGroup.ofCore ⋯
x y : E
h : dist x y = 0
⊢ x = y

Tactic:
rw [← sub_eq_zero, ← core.norm_eq_zero_iff, ← norm_neg_add]

Goal state after:
𝕜✝ : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
F : Type u_4
α : Type u_5
𝕜 : Type u_6
E : Type u_7
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Norm E
U : UniformSpace E
B : Bornology E
core : NormedSpace.Core 𝕜 E
HU : 𝓤 E = 𝓤 E
HB : ∀ (s : Set E), Bornology.IsBounded s ↔ Bornology.IsBounded s
this : SeminormedAddCommGroup E := SeminormedAddCommGroup.ofCore ⋯
x y : E
h : dist x y = 0
⊢ ‖-x + y‖ = 0
Module: Mathlib.MeasureTheory.OuterMeasure.Basic

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝² : FunLike F (Set α) ℝ≥0∞
inst✝¹ : OuterMeasureClass F α
ι : Type u_4
μ : F
s : ι → Set α
l : Filter ι
inst✝ : l.NeBot
S : Set α := ⋃ n, s n
h0 : Tendsto (fun k ↦ μ (S \ s k)) l (𝓝 0)
M : ℝ≥0∞ := ⨆ n, μ (s n)
A : ∀ (k : ι), μ S ≤ M + μ (S \ s k)
B : Tendsto (fun k ↦ M + μ (S \ s k)) l (𝓝 M)
⊢ μ S ≤ M

Tactic:
exact ge_of_tendsto' B A

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

Goal state before:
pp: δ : Type u_4
X : δ → Type u_6
inst✝ : (a : δ) → MeasurableSpace (X a)
s : Set δ
t : (i : δ) → Set (X i)
hs : s.Countable
ht : ∀ i ∈ s, MeasurableSet (t i)
⊢ MeasurableSet (s.pi t)

Tactic:
rw [pi_def]

Goal state after:
δ : Type u_4
X : δ → Type u_6
inst✝ : (a : δ) → MeasurableSpace (X a)
s : Set δ
t : (i : δ) → Set (X i)
hs : s.Countable
ht : ∀ i ∈ s, MeasurableSet (t i)
⊢ MeasurableSet (⋂ a ∈ s, eval a ⁻¹' t a)
Module: Mathlib.MeasureTheory.MeasurableSpace.Constructions

Goal state before:
pp: α : Type u_1
ι : Sort uι
inst✝¹ : MeasurableSpace α
inst✝ : Countable ι
p : ι → α → Prop
hp : ∀ (i : ι), Measurable (p i)
⊢ MeasurableSet {a | ∀ (i : ι), p i a}

Tactic:
rw [ofPred_forall]; exact MeasurableSet.iInter fun i ↦ (hp i).setOf

Goal state after:
Module: Mathlib.Data.Sign.Defs

Goal state before:
pp: ⊢ ∀ (x : SignType), - -x = x

Tactic:
by rintro ⟨_⟩ <;> rfl

Goal state after:
Module: Mathlib.Data.EReal.Inv

Goal state before:
pp: case neg_bot
x : ℝ
x_neg : x < 0
⊢ (↑x * ⊥)⁻¹ = (↑x)⁻¹ * ⊥⁻¹

Tactic:
| neg_bot x x_neg =>

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

Goal state before:
pp: α : Type u_4
G : Type u_5
inst✝ : GroupWithZero G
f g : α → G
l : Filter α
hg : Tendsto g l (𝓟 {0}ᶜ)
⊢ (fun x ↦ f x / g x * g x) =ᶠ[l] fun x ↦ f x

Tactic:
by
  filter_upwards [hg.le_comap <| preimage_mem_comap (m := g) (mem_principal_self {0}ᶜ)] with x hx
  simp_all

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

Goal state before:
pp: u : ℕ → ℝ
c : ℝ
hc : 0 ≤ c
n : ℕ
hn : 0 < n
h : ∀ k < n, c * u k < u (k + 1)
⊢ ∀ k < n, c ^ (k + 1) * u 0 ≤ c * (c ^ k * u 0)

Tactic:
simp [_root_.pow_succ', mul_assoc]

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

Goal state before:
pp: u : ℕ → ℝ
c : ℝ
hc : 0 ≤ c
n : ℕ
h : ∀ k < n, c * u k ≤ u (k + 1)
⊢ c ^ 0 * u 0 ≤ u 0

Tactic:
simp [_root_.pow_succ', mul_assoc]

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

Goal state before:
pp: u : ℕ → ℝ
c : ℝ
hc : 0 ≤ c
n : ℕ
h : ∀ k < n, c * u k ≤ u (k + 1)
⊢ ∀ k < n, c ^ (k + 1) * u 0 ≤ c * (c ^ k * u 0)

Tactic:
simp [_root_.pow_succ', mul_assoc]

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

Goal state before:
pp: u : ℕ → ℝ
c : ℝ
hc : 0 ≤ c
n : ℕ
hn : 0 < n
h : ∀ k < n, u (k + 1) < c * u k
⊢ ∀ k < n, c * (c ^ k * u 0) ≤ c ^ (k + 1) * u 0

Tactic:
simp [_root_.pow_succ', mul_assoc]

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

Goal state before:
pp: u : ℕ → ℝ
c : ℝ
hc : 0 ≤ c
n : ℕ
h : ∀ k < n, u (k + 1) ≤ c * u k
⊢ u 0 ≤ c ^ 0 * u 0

Tactic:
simp [_root_.pow_succ', mul_assoc]

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

Goal state before:
pp: u : ℕ → ℝ
c : ℝ
hc : 0 ≤ c
n : ℕ
h : ∀ k < n, u (k + 1) ≤ c * u k
⊢ ∀ k < n, c * (c ^ k * u 0) ≤ c ^ (k + 1) * u 0

Tactic:
simp [_root_.pow_succ', mul_assoc]

Goal state after:
Module: Mathlib.Topology.Instances.EReal.Lemmas

Goal state before:
pp: case bot.top
h : (⊥, ⊤).1 ≠ ⊤ ∨ (⊥, ⊤).2 ≠ ⊥
h' : (⊥, ⊤).1 ≠ ⊥ ∨ (⊥, ⊤).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (⊥, ⊤)

pp: case coe.bot
a✝ : ℝ
h : (↑a✝, ⊥).1 ≠ ⊤ ∨ (↑a✝, ⊥).2 ≠ ⊥
h' : (↑a✝, ⊥).1 ≠ ⊥ ∨ (↑a✝, ⊥).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (↑a✝, ⊥)

pp: case coe.coe
a✝¹ a✝ : ℝ
h : (↑a✝¹, ↑a✝).1 ≠ ⊤ ∨ (↑a✝¹, ↑a✝).2 ≠ ⊥
h' : (↑a✝¹, ↑a✝).1 ≠ ⊥ ∨ (↑a✝¹, ↑a✝).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (↑a✝¹, ↑a✝)

pp: case coe.top
a✝ : ℝ
h : (↑a✝, ⊤).1 ≠ ⊤ ∨ (↑a✝, ⊤).2 ≠ ⊥
h' : (↑a✝, ⊤).1 ≠ ⊥ ∨ (↑a✝, ⊤).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (↑a✝, ⊤)

pp: case top.bot
h : (⊤, ⊥).1 ≠ ⊤ ∨ (⊤, ⊥).2 ≠ ⊥
h' : (⊤, ⊥).1 ≠ ⊥ ∨ (⊤, ⊥).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (⊤, ⊥)

pp: case top.coe
a✝ : ℝ
h : (⊤, ↑a✝).1 ≠ ⊤ ∨ (⊤, ↑a✝).2 ≠ ⊥
h' : (⊤, ↑a✝).1 ≠ ⊥ ∨ (⊤, ↑a✝).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (⊤, ↑a✝)

pp: case top.top
h : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ ⊥
h' : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (⊤, ⊤)

Tactic:
· simp at h'

Goal state after:
case coe.bot
a✝ : ℝ
h : (↑a✝, ⊥).1 ≠ ⊤ ∨ (↑a✝, ⊥).2 ≠ ⊥
h' : (↑a✝, ⊥).1 ≠ ⊥ ∨ (↑a✝, ⊥).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (↑a✝, ⊥)

case coe.coe
a✝¹ a✝ : ℝ
h : (↑a✝¹, ↑a✝).1 ≠ ⊤ ∨ (↑a✝¹, ↑a✝).2 ≠ ⊥
h' : (↑a✝¹, ↑a✝).1 ≠ ⊥ ∨ (↑a✝¹, ↑a✝).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (↑a✝¹, ↑a✝)

case coe.top
a✝ : ℝ
h : (↑a✝, ⊤).1 ≠ ⊤ ∨ (↑a✝, ⊤).2 ≠ ⊥
h' : (↑a✝, ⊤).1 ≠ ⊥ ∨ (↑a✝, ⊤).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (↑a✝, ⊤)

case top.bot
h : (⊤, ⊥).1 ≠ ⊤ ∨ (⊤, ⊥).2 ≠ ⊥
h' : (⊤, ⊥).1 ≠ ⊥ ∨ (⊤, ⊥).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (⊤, ⊥)

case top.coe
a✝ : ℝ
h : (⊤, ↑a✝).1 ≠ ⊤ ∨ (⊤, ↑a✝).2 ≠ ⊥
h' : (⊤, ↑a✝).1 ≠ ⊥ ∨ (⊤, ↑a✝).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (⊤, ↑a✝)

case top.top
h : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ ⊥
h' : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ ⊤
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (⊤, ⊤)
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: r : ℝ≥0
hr : r < 1
⊢ HasSum (fun n ↦ r ^ n) (1 - r)⁻¹

Tactic:
by
  apply NNReal.hasSum_coe.1
  push_cast
  rw [NNReal.coe_sub (le_of_lt hr)]
  exact hasSum_geometric_of_lt_one r.coe_nonneg hr

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
I : Set ι
m : ι → OuterMeasure α
s : Set α
hs : s.Nonempty
⊢ (⨅ i ∈ I, m i) s = ⨅ t, ⨅ (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ i ∈ I, (m i) (t n)

Tactic:
simp only [← iInf_subtype'', iInf_apply' _ hs]

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

Goal state before:
pp: α : Type u_4
inst✝³ : TopologicalSpace α
β : Type u_5
f : α → β
s : Set α
a : α
inst✝² : PartialOrder β
inst✝¹ : CommGroup β
inst✝ : IsOrderedMonoid β
⊢ UpperSemicontinuousWithinAt f⁻¹ s a ↔ LowerSemicontinuousWithinAt f s a

Tactic:
by
  simp [← lowerSemicontinuousWithinAt_inv_iff]

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∞
c : ℝ≥0∞
hc : c ≠ 0
⊢ c • extend m = extend fun s h ↦ c • m s h

Tactic:
ext s; by_cases h : P s <;> simp [extend, *]

Goal state after:
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α : Type u_1
g : Set (Set α)
⊢ generateFrom (generatePiSystem g) = generateFrom g

Tactic:
by
  apply le_antisymm <;> apply generateFrom_le
  · exact fun t h_t => generateFrom_measurableSet_of_generatePiSystem t h_t
  · exact fun t h_t => measurableSet_generateFrom (generatePiSystem.base h_t)

Goal state after:
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)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s : Set α
hs : (inducedOuterMeasure m P0 m0) s ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
h : (inducedOuterMeasure m P0 m0) s < (inducedOuterMeasure m P0 m0) s + ε
| (inducedOuterMeasure m P0 m0) s < (inducedOuterMeasure m P0 m0) s + ε

Tactic:
lhs
    rw [inducedOuterMeasure_eq_iInf _ msU m_mono]

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)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s : Set α
hs : (inducedOuterMeasure m P0 m0) s ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
h : (inducedOuterMeasure m P0 m0) s < (inducedOuterMeasure m P0 m0) s + ε
| ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht
Module: Mathlib.MeasureTheory.Measure.NullMeasurable

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
t s : Set α
ht : NullMeasurableSet t μ
⊢ μ (s ∩ t) + μ (s \ t) ≤ μ s

Tactic:
rcases exists_measurable_superset μ s with ⟨s', hsub, hs'm, hs'⟩

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
t s : Set α
ht : NullMeasurableSet t μ
s' : Set α
hsub : s ⊆ s'
hs'm : MeasurableSet s'
hs' : μ s' = μ s
⊢ μ (s ∩ t) + μ (s \ t) ≤ μ s
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 : α → β
hf : Measurable f
⊢ (mapₗ f) μ = 0 ↔ μ = 0

Tactic:
mapₗ_apply_of_measurable hf,

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
⊢ map f μ = 0 ↔ μ = 0
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
s : Set X
a : X
hsa : s ⊆ Ioi a
hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s
ha : IsPredPrelimit a
⊢ (⇑ofDual ⁻¹' s).Nonempty → ∃ a_2 < toDual a, Ioo a_2 (toDual a) ⊆ ⇑ofDual ⁻¹' s

Tactic:
simpa only [OrderDual.exists, Ioo_toDual]

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
C : Set (Set α)
ht : ∀ t ∈ C, μ (s ∩ t) = 0
t : ℕ → Set α
ht1 : ∀ (n : ℕ), t n ∈ C
ht2 : ⋃ n, t n = univ
⊢ μ s = 0

Tactic:
show s = ⋃ n, s ∩ t n by rw [← inter_iUnion, ht2, inter_univ],

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
C : Set (Set α)
ht : ∀ t ∈ C, μ (s ∩ t) = 0
t : ℕ → Set α
ht1 : ∀ (n : ℕ), t n ∈ C
ht2 : ⋃ n, t n = univ
⊢ μ (⋃ n, s ∩ t n) = 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
⊢ {i | 0 < μ (As i)}.Countable

Tactic:
rw [← sum_sfiniteSeq μ] at As_disj As_mble ⊢

Goal state after:
α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) (sum (sfiniteSeq μ))
As_disj : Pairwise (AEDisjoint (sum (sfiniteSeq μ)) on As)
⊢ {i | 0 < (sum (sfiniteSeq μ)) (As i)}.Countable
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
⊢ (μ s).toReal - (μ t).toReal = (μ (s ∪ t) - μ t).toReal - (μ (t ∪ s) - μ s).toReal

Tactic:
ENNReal.toReal_sub_of_le measure_le_measure_union_right (by finiteness),

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
⊢ (μ s).toReal - (μ t).toReal = (μ (s ∪ t)).toReal - (μ t).toReal - (μ (t ∪ s) - μ s).toReal
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
⊢ (μ s).toReal - (μ t).toReal = (μ (s ∪ t)).toReal - (μ t).toReal - (μ (t ∪ s) - μ s).toReal

Tactic:
ENNReal.toReal_sub_of_le measure_le_measure_union_right (by finiteness),

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
⊢ (μ s).toReal - (μ t).toReal = (μ (s ∪ t)).toReal - (μ t).toReal - ((μ (t ∪ s)).toReal - (μ s).toReal)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
⊢ (μ s).toReal - (μ t).toReal = (μ (s \ t)).toReal - (μ (t \ s)).toReal

Tactic:
rw [measure_sdiff' s ht ht', measure_sdiff' t hs hs',
    ENNReal.toReal_sub_of_le measure_le_measure_union_right (by finiteness),
    ENNReal.toReal_sub_of_le measure_le_measure_union_right (by finiteness),
    union_comm t s]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
⊢ (μ s).toReal - (μ t).toReal = (μ (s ∪ t)).toReal - (μ t).toReal - ((μ (s ∪ t)).toReal - (μ s).toReal)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
A : ∀ (n : ℕ), ∃ D ⊆ C, D.Countable ∧ ∀ s ∈ C, s ≤ᵐ[sfiniteSeq μ n] ⋃₀ D
⊢ ∃ D ⊆ C, D.Countable ∧ ∀ s ∈ C, s ≤ᵐ[μ] ⋃₀ D

Tactic:
choose D DC D_count hD using A

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
D : ℕ → Set (Set α)
DC : ∀ (n : ℕ), D n ⊆ C
D_count : ∀ (n : ℕ), (D n).Countable
hD : ∀ (n : ℕ), ∀ s ∈ C, s ≤ᵐ[sfiniteSeq μ n] ⋃₀ D n
⊢ ∃ D ⊆ C, D.Countable ∧ ∀ s ∈ C, s ≤ᵐ[μ] ⋃₀ D
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
s : Set α
hs : MeasurableSet s
μ : Measure α
⊢ Measure.map Subtype.val (Measure.comap Subtype.val μ) = μ.restrict s

Tactic:
(MeasurableEmbedding.subtype_coe hs).map_comap,

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
s : Set α
hs : MeasurableSet s
μ : Measure α
⊢ μ.restrict (range Subtype.val) = μ.restrict s
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
x : α
A : ℕ → Set α := natGeneratingSequence α
hA : ∀ (n : ℕ), MeasurableSet (A n)
hA_gen : generateFrom (range A) = mα
⊢ measurableAtom x = countablyGeneratedAtom α fun x_1 ↦ x ∈ A x_1

Tactic:
refine subset_antisymm (measurableAtom_subset ?_ ?_) ?_

Goal state after:
case refine_1
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
x : α
A : ℕ → Set α := natGeneratingSequence α
hA : ∀ (n : ℕ), MeasurableSet (A n)
hA_gen : generateFrom (range A) = mα
⊢ MeasurableSet (countablyGeneratedAtom α fun x_1 ↦ x ∈ A x_1)

case refine_2
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
x : α
A : ℕ → Set α := natGeneratingSequence α
hA : ∀ (n : ℕ), MeasurableSet (A n)
hA_gen : generateFrom (range A) = mα
⊢ x ∈ countablyGeneratedAtom α fun x_1 ↦ x ∈ A x_1

case refine_3
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
x : α
A : ℕ → Set α := natGeneratingSequence α
hA : ∀ (n : ℕ), MeasurableSet (A n)
hA_gen : generateFrom (range A) = mα
⊢ (countablyGeneratedAtom α fun x_1 ↦ x ∈ A x_1) ⊆ measurableAtom x
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
β : Type u_2
ι✝ : Type u_3
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν : Measure α
s t : Set α
a : α
ι : Type u_4
inst✝¹ : Finite ι
μ : ι → Measure α
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
⊢ SigmaFinite (sum μ)

Tactic:
cases nonempty_fintype ι
  have : ∀ n, MeasurableSet (⋂ i : ι, spanningSets (μ i) n) := fun n =>
    MeasurableSet.iInter fun i => measurableSet_spanningSets (μ i) n
  refine ⟨⟨⟨fun n => ⋂ i, spanningSets (μ i) n, fun _ => trivial, fun n => ?_, ?_⟩⟩⟩
  · rw [sum_apply _ (this n), tsum_fintype, ENNReal.sum_lt_top]
    rintro i -
    exact (measure_mono <| iInter_subset _ i).trans_lt (measure_spanningSets_lt_top (μ i) n)
  · rw [iUnion_iInter_of_monotone]
    · simp_rw [iUnion_spanningSets, iInter_univ]
    exact fun i => monotone_spanningSets (μ i)

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

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : α → β
h : NeZero (Measure.map f μ)
h' : ¬AEMeasurable f μ
⊢ False

Tactic:
simp [h'] at h

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
t : ℕ → Set α
ht : ∀ (n : ℕ), MeasurableSet (t n)
⊢ generateFrom (⋃ n, memPartition t n) ≤ m

Tactic:
refine (generateFrom_iUnion_memPartition t).trans_le (generateFrom_le ?_)

Goal state after:
α : Type u_1
m : MeasurableSpace α
t : ℕ → Set α
ht : ∀ (n : ℕ), MeasurableSet (t n)
⊢ ∀ t_1 ∈ range t, MeasurableSet t_1
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: case refine_3
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
μ : Measure α
H : AEMeasurable f μ
t : Set β
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
h₀ : t.Nonempty
s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ
g : α → β := s.piecewise (fun x ↦ h₀.some) (mk f H)
A : μ (toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ) = 0
⊢ f =ᵐ[μ] g

Tactic:
filter_upwards [compl_mem_ae_iff.2 A] with x hx

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
μ : Measure α
H : AEMeasurable f μ
t : Set β
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
h₀ : t.Nonempty
s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ
g : α → β := s.piecewise (fun x ↦ h₀.some) (mk f H)
A : μ (toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ) = 0
x : α
hx : x ∈ (toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ)ᶜ
⊢ f x = g x
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
⊢ ∑ C ∈ Cs, M • (μ.restrict (P C)) t = M • (μ.restrict (⋃ C ∈ Cs, P C)) t

Tactic:
rw [← smul_sum, ← Cs.tsum_subtype, μ.restrict_biUnion_finset _ P_meas, Measure.sum_apply _ ht]
      refine fun C₁ hC₁ C₂ hC₂ hC ↦ Set.disjoint_iff.mpr fun x hx ↦ hC <| ?_
      exact subset_antisymm (C_subset_C hC₁ hx) (C_subset_C hC₂ (Set.inter_comm _ _ ▸ hx))

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

Goal state before:
pp: case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, Measurable f) → Measurable l.prod
hl : ∀ f_1 ∈ f :: l, Measurable f_1
⊢ Measurable (f :: l).prod

Tactic:
| cons f l ihl =>

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

Goal state before:
pp: case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
μ : Measure α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEMeasurable f μ) → AEMeasurable l.prod μ
hl : ∀ f_1 ∈ f :: l, AEMeasurable f_1 μ
⊢ AEMeasurable (f :: l).prod μ

Tactic:
| cons f l ihl =>

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x : α
E : Set α
h : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ ∃ ε, 0 < ε ∧ ENNReal.ofReal ε < infEDist x E

Tactic:
rw [← infEDist_pos_iff_notMem_closure, ENNReal.lt_iff_exists_real_btwn] at h
  rcases h with ⟨ε, ⟨_, ⟨ε_pos, ε_lt⟩⟩⟩
  exact ⟨ε, ⟨ENNReal.ofReal_pos.mp ε_pos, ε_lt⟩⟩

Goal state after: