mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations

Goal state before:
pp: case pos
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹¹ : Finite ι
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
inst✝⁸ : CharZero R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
inst✝³ : P.IsCrystallographic
b : P.Base
inst✝² : Fintype ι
i j : ↥b.support
hij : i ≠ j
inst✝¹ : P.IsReduced
inst✝ : P.IsIrreducible
hij' : ↑i ≠ ↑j
this : InvolutiveNeg ι := P.indexNeg
k l : ι
h₃ : l ≠ ↑j
h₄ : l ≠ -↑i
h₅ : P.root l + P.root ↑i - P.root ↑j ∈ range ⇑P.root
m : ι
hm : P.root m = P.root l - P.root ↑j
l' : ι
hl' : P.root l' = P.root l + P.root ↑i
hk : P.root k = P.root l + P.root ↑i - P.root ↑j
aux₃ : ∀ (x : ι), x ≠ m → ¬(P.root x = P.root l - P.root ↑j ∧ P.root k = P.root ↑i + P.root x)
aux₄ : ∀ (x : ι), x ≠ l' → ¬(P.root x = P.root ↑i + P.root l ∧ P.root k = P.root x - P.root ↑j)
⊢ (∑ x,
      if P.root x = P.root l - P.root ↑j ∧ P.root k = P.root ↑i + P.root x then
        (↑(chainBotCoeff (↑i) x) + 1) * (↑(chainTopCoeff (↑j) l) + 1)
      else 0) =
    ∑ x,
      if P.root x = P.root ↑i + P.root l ∧ P.root k = P.root x - P.root ↑j then
        (↑(chainTopCoeff (↑j) x) + 1) * (↑(chainBotCoeff (↑i) l) + 1)
      else 0

Tactic:
Finset.sum_eq_single_of_mem m (Finset.mem_univ _) (by rintro x - h; rw [if_neg (aux₃ _ h)]),

Goal state after:
case pos
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹¹ : Finite ι
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
inst✝⁸ : CharZero R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
inst✝³ : P.IsCrystallographic
b : P.Base
inst✝² : Fintype ι
i j : ↥b.support
hij : i ≠ j
inst✝¹ : P.IsReduced
inst✝ : P.IsIrreducible
hij' : ↑i ≠ ↑j
this : InvolutiveNeg ι := P.indexNeg
k l : ι
h₃ : l ≠ ↑j
h₄ : l ≠ -↑i
h₅ : P.root l + P.root ↑i - P.root ↑j ∈ range ⇑P.root
m : ι
hm : P.root m = P.root l - P.root ↑j
l' : ι
hl' : P.root l' = P.root l + P.root ↑i
hk : P.root k = P.root l + P.root ↑i - P.root ↑j
aux₃ : ∀ (x : ι), x ≠ m → ¬(P.root x = P.root l - P.root ↑j ∧ P.root k = P.root ↑i + P.root x)
aux₄ : ∀ (x : ι), x ≠ l' → ¬(P.root x = P.root ↑i + P.root l ∧ P.root k = P.root x - P.root ↑j)
⊢ (if P.root m = P.root l - P.root ↑j ∧ P.root k = P.root ↑i + P.root m then
      (↑(chainBotCoeff (↑i) m) + 1) * (↑(chainTopCoeff (↑j) l) + 1)
    else 0) =
    ∑ x,
      if P.root x = P.root ↑i + P.root l ∧ P.root k = P.root x - P.root ↑j then
        (↑(chainTopCoeff (↑j) x) + 1) * (↑(chainBotCoeff (↑i) l) + 1)
      else 0
Module: Mathlib.MeasureTheory.Constructions.Projective

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
h : IsEmpty ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
I : Finset ι
i : ι
hi : IsEmpty (α i)
⊢ IsEmpty ((j : ↥(insert i I)) → α ↑j)

Tactic:
simp [hi]

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

Goal state before:
pp: case refine_2
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set (α i)
h₁ : t₁ ∈ univ.pi C
s₂ : Finset ι
t₂ : (i : ι) → Set (α i)
h₂ : t₂ ∈ univ.pi C
t₁' : (i : ι) → Set (α i) := s₁.piecewise t₁ fun i ↦ univ
t₂' : (i : ι) → Set (α i) := s₂.piecewise t₂ fun i ↦ univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
i : ι
f : (i : ι) → α i
hf : (∀ i ∈ ↑s₁, f i ∈ t₁' i) ∧ ∀ i ∈ ↑s₂, f i ∈ t₂' i
⊢ f i ∈ t₂' i

Tactic:
· by_cases hi₂ : i ∈ s₂
        · exact hf.2 i hi₂
        · rw [h2' i hi₂]
          exact mem_univ _

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

Goal state before:
pp: case refine_1
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set (α i)
h₁ : t₁ ∈ univ.pi C
s₂ : Finset ι
t₂ : (i : ι) → Set (α i)
h₂ : t₂ ∈ univ.pi C
hst_nonempty : ((↑s₁).pi t₁ ∩ (↑s₂).pi t₂).Nonempty
t₁' : (i : ι) → Set (α i) := s₁.piecewise t₁ fun i ↦ univ
t₂' : (i : ι) → Set (α i) := s₂.piecewise t₂ fun i ↦ univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
⊢ (fun i ↦ t₁' i ∩ t₂' i) ∈ univ.pi C

Tactic:
rw [mem_univ_pi]
    intro i
    have : (t₁' i ∩ t₂' i).Nonempty := by
      obtain ⟨f, hf⟩ := hst_nonempty
      rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, mem_inter_iff, mem_pi, mem_pi] at hf
      refine ⟨f i, ⟨?_, ?_⟩⟩
      · by_cases hi₁ : i ∈ s₁
        · exact hf.1 i hi₁
        · rw [h1' i hi₁]
          exact mem_univ _
      · by_cases hi₂ : i ∈ s₂
        · exact hf.2 i hi₂
        · rw [h2' i hi₂]
          exact mem_univ _
    refine hC i _ ?_ _ ?_ this
    · by_cases hi₁ : i ∈ s₁
      · rw [← h1 i hi₁]
        exact h₁ i (mem_univ _)
      · rw [h1' i hi₁]
        exact hC_univ i
    · by_cases hi₂ : i ∈ s₂
      · rw [← h2 i hi₂]
        exact h₂ i (mem_univ _)
      · rw [h2' i hi₂]
        exact hC_univ i

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
hC : IsSetRing C
s : ℕ → Set α
hs : ∀ (n : ℕ), s n ∈ C
n : ℕ
⊢ ⋃ i, ⋃ (_ : i ≤ n), s i ∈ C

Tactic:
induction n with
  | zero => simp [hs 0]
  | succ n hn => rw [biUnion_le_succ]; exact hC.union_mem hn (hs _)

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ (↑⋯.choose.parts).PairwiseDisjoint id

Tactic:
← supIndep_iff_pairwiseDisjoint

Goal state after:
α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ⋯.choose.parts.SupIndep id
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u v : α
huv✝ : u ≤ v
h'uv : ⋃₀ ↑I = Set.Ioc u v
huv : u < v
this : v ∈ ⋃₀ ↑I
u' : α
uu' : u ≤ u'
hu'v' : u' ≤ v
tI : Set.Ioc u' v ∈ I
ht : v ∈ Set.Ioc u' v
left✝ : v ≤ v
I' : Finset (Set α) := I.erase (Set.Ioc u' v)
I'I : I' ⊆ I
I_eq_insert : I = insert (Set.Ioc u' v) I'
⊢ ∀ t ∈ ↑I, ¬t = Set.Ioc u' v → Disjoint (Set.Ioc u' v) t

Tactic:
intro u

Goal state after:
α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u✝ v : α
huv✝ : u✝ ≤ v
h'uv : ⋃₀ ↑I = Set.Ioc u✝ v
huv : u✝ < v
this : v ∈ ⋃₀ ↑I
u' : α
uu' : u✝ ≤ u'
hu'v' : u' ≤ v
tI : Set.Ioc u' v ∈ I
ht : v ∈ Set.Ioc u' v
left✝ : v ≤ v
I' : Finset (Set α) := I.erase (Set.Ioc u' v)
I'I : I' ⊆ I
I_eq_insert : I = insert (Set.Ioc u' v) I'
u : Set α
⊢ u ∈ ↑I → ¬u = Set.Ioc u' v → Disjoint (Set.Ioc u' v) u
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetRing C
m_iUnion :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) = ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
⊢ ⨆ i, (partialSups f) i ∈ C

Tactic:
iSup_partialSups_eq

Goal state after:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetRing C
m_iUnion :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) = ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
⊢ ⨆ i, f i ∈ C
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
E : Set α
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ_pos : 0 < δ
hδε :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ (s ∩ E) ≤ ENNReal.ofReal δ
⊢ eLpNorm (s.indicator (f i)) p (μ.restrict E) ≤ ENNReal.ofReal ε

Tactic:
← measure_toMeasurable

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
E : Set α
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ_pos : 0 < δ
hδε :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ (toMeasurable μ (s ∩ E)) ≤ ENNReal.ofReal δ
⊢ eLpNorm (s.indicator (f i)) p (μ.restrict E) ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
i : Set α
hi : MeasurableSet i
⊢ (toSignedMeasure 0) i = 0 i

Tactic:
simp [hi]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hs : s.totalVariation i = 0
⊢ s i = 0

Tactic:
rw [totalVariation, Measure.coe_add, Pi.add_apply, add_eq_zero] at hs

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hs : s.toJordanDecomposition.posPart i = 0 ∧ s.toJordanDecomposition.negPart i = 0
⊢ s i = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hs : s.toJordanDecomposition.posPart i = 0 ∧ s.toJordanDecomposition.negPart i = 0
hi : ¬MeasurableSet i
⊢ s i = 0

Tactic:
simp [hi]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ (v.trim hle) (j ∩ i) = v (j ∩ i)

pp: case hi
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ MeasurableSet i

pp: case hj
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ MeasurableSet j

pp: case hi
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ MeasurableSet i

pp: case hj
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ MeasurableSet j

Tactic:
trim_measurableSet_eq

Goal state after:
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ v (j ∩ i) = v (j ∩ i)

case hi
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ MeasurableSet (j ∩ i)

case hi
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ MeasurableSet i

case hj
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ MeasurableSet j

case hi
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ MeasurableSet i

case hj
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ MeasurableSet j
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensity

Goal state before:
pp: case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
i : Set α
hi : MeasurableSet i
⊢ Integrable f (μ.restrict i)

Tactic:
exact hf.integrableOn

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
f : α → E
hf : ¬AEStronglyMeasurable f μ
hp₀ : p ≠ 0
⊢ p.toReal⁻¹ ≠ 0

Tactic:
simp [ENNReal.toReal_eq_zero_iff, *]

Goal state after:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : Tendsto (fun n ↦ eLpNorm (f n) p μ) atTop (𝓝 0)
⊢ UnifIntegrable f p μ

Tactic:
intro ε hε
  rw [ENNReal.tendsto_atTop_zero] at hf_tendsto
  obtain ⟨N, hN⟩ := hf_tendsto (ENNReal.ofReal ε) (by simpa)
  let F : Fin N → α → β := fun n => f n
  have hF : ∀ n, MemLp (F n) p μ := fun n => hf n
  obtain ⟨δ₁, hδpos₁, hδ₁⟩ := unifIntegrable_fin hp hp' hF hε
  refine ⟨δ₁, hδpos₁, fun n s hs hμs => ?_⟩
  by_cases! hn : n < N
  · exact hδ₁ ⟨n, hn⟩ s hs hμs
  · exact (eLpNorm_indicator_le _).trans (hN n hn)

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
μ : Measure α
r : ℝ≥0
| ((r • s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure -
    ((r • s).toJordanDecomposition.negPart.singularPart μ).toSignedMeasure

Tactic:
congr
    · congr
      · rw [toJordanDecomposition_smul, JordanDecomposition.smul_posPart, singularPart_smul]
    · congr
      rw [toJordanDecomposition_smul, JordanDecomposition.smul_negPart, singularPart_smul]

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

Goal state before:
pp: Ω : Type u_1
inst✝ : MeasurableSpace Ω
μ ν : ProbabilityMeasure Ω
h : (fun μ s ↦ (↑μ s).toNNReal) μ = (fun μ s ↦ (↑μ s).toNNReal) ν
s : Set Ω
x✝ : MeasurableSet s
⊢ ↑μ s = ↑ν s

Tactic:
simpa [ENNReal.toNNReal_eq_toNNReal_iff, measure_ne_top] using congr_fun h s

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

Goal state before:
pp: Ω : Type u_1
inst✝ : MeasurableSpace Ω
s✝ t : Set Ω
μ ν : FiniteMeasure Ω
h : (fun μ s ↦ (↑μ s).toNNReal) μ = (fun μ s ↦ (↑μ s).toNNReal) ν
s : Set Ω
x✝ : MeasurableSet s
⊢ ↑μ s = ↑ν s

Tactic:
simpa [ENNReal.toNNReal_eq_toNNReal_iff, measure_ne_top] using congr_fun h s

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

Goal state before:
pp: Ω : Type u_1
inst✝⁵ : MeasurableSpace Ω
R : Type u_2
inst✝⁴ : SMul R ℝ≥0
inst✝³ : SMul R ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0 ℝ≥0
c : R
μ : FiniteMeasure Ω
s : Set Ω
⊢ (c • μ) s = c • μ s

Tactic:
coeFn_smul,

Goal state after:
Ω : Type u_1
inst✝⁵ : MeasurableSpace Ω
R : Type u_2
inst✝⁴ : SMul R ℝ≥0
inst✝³ : SMul R ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0 ℝ≥0
c : R
μ : FiniteMeasure Ω
s : Set Ω
⊢ (c • ⇑μ) s = c • μ s
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ^ p ∂μ ≤ ∫ (x : α) in s, μ[fun x ↦ ‖f x‖ ^ p | m] x ∂μ

Tactic:
by
    refine integral_mono_of_nonneg ?_ ?_ ?_
    · filter_upwards with a using by positivity
    · exact integrable_condExp.congr (condExp_restrict_ae_eq_restrict hm hs hf)
    · exact ae_restrict_le (Integrable.norm_condExp_rpow_le hp hf)

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

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i ↦ ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
lip : LipschitzWith 1 NNReal.toReal
f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f
_def_f₀ : f₀ = comp NNReal.toReal lip f
f₀_eq : ⇑f₀ = NNReal.toReal ∘ ⇑f
f₀_nn : 0 ≤ ⇑f₀
f₀_ae_nn : 0 ≤ᵐ[↑μ] ⇑f₀
f₀_ae_nns : ∀ (i : γ), 0 ≤ᵐ[↑(μs i)] ⇑f₀
aux : ∫ (a : Ω), ↑(f a) ∂↑μ = (∫⁻ (a : Ω), ↑(f a) ∂↑μ).toReal
auxs : ∀ (i : γ), ∫ (a : Ω), ↑(f a) ∂↑(μs i) = (∫⁻ (a : Ω), ↑(f a) ∂↑(μs i)).toReal
⊢ Tendsto (fun n ↦ (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)

Tactic:
simpa only [← aux, ← auxs] using! h f₀

Goal state after:
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
β : ι → Type u_10
γ : ι → Type u_11
mβ : (i : ι) → MeasurableSpace (β i)
mγ : (i : ι) → MeasurableSpace (γ i)
f : (i : ι) → Ω → β i
h : iIndepFun f μ
g : (i : ι) → β i → γ i
hf : ∀ (i : ι), AEMeasurable (f i) μ
hg : ∀ (i : ι), AEMeasurable (g i) (Measure.map (f i) μ)
⊢ ∀ (i : ι), AEMeasurable (f i) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())

Tactic:
by simp [hf]

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
f' : Ω → β
g' : Ω → β'
hfg : IndepFun f g κ μ
hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[κ a] f'
hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[κ a] g'
⊢ IndepFun f' g' κ μ

Tactic:
rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩

Goal state after:
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
f' : Ω → β
g' : Ω → β'
hfg : IndepFun f g κ μ
hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[κ a] f'
hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[κ a] g'
A : Set β
hA : MeasurableSet A
B : Set β'
hB : MeasurableSet B
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (f' ⁻¹' A ∩ g' ⁻¹' B) = (κ a) (f' ⁻¹' A) * (κ a) (g' ⁻¹' B)
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
β : Type u_6
β' : Type u_7
γ : Type u_8
γ' : Type u_9
_mΩ : MeasurableSpace Ω
μ : Measure Ω
f : Ω → β
g : Ω → β'
_mβ : MeasurableSpace β
_mβ' : MeasurableSpace β'
_mγ : MeasurableSpace γ
_mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : f ⟂ᵢ[μ] g
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hφ : AEMeasurable φ (Measure.map f μ)
hψ : AEMeasurable ψ (Measure.map g μ)
⊢ AEMeasurable f (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())

Tactic:
by simp [hf]

Goal state after:
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
β : ι → Type u_10
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
h_indep : iIndepFun f μ
hf : ∀ (i : ι), AEMeasurable (f i) μ
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
⊢ ∀ (i : ι), AEMeasurable (f i) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())

Tactic:
by simp [hf]

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hφ : Measurable φ
hψ : Measurable ψ
⊢ IndepFun (φ ∘ f) (ψ ∘ g) κ μ

Tactic:
rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩

Goal state after:
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hφ : Measurable φ
hψ : Measurable ψ
A : Set γ
hA : MeasurableSet A
B : Set γ'
hB : MeasurableSet B
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (φ ∘ f ⁻¹' A ∩ ψ ∘ g ⁻¹' B) = (κ a) (φ ∘ f ⁻¹' A) * (κ a) (ψ ∘ g ⁻¹' B)
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case a
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hφ : Measurable φ
hψ : Measurable ψ
A : Set γ
hA : MeasurableSet A
B : Set γ'
hB : MeasurableSet B
⊢ ψ ∘ g ⁻¹' B ∈ {s | MeasurableSet s}

Tactic:
exact ⟨ψ ⁻¹' B, hψ hB, Set.preimage_comp.symm⟩

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case refine_1
α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
h :
  ∀ (s : Set β) (t : Set γ),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
⊢ ∀ {s : Set α} {t : Set β} {u : Set γ},
    MeasurableSet s →
      MeasurableSet t →
        MeasurableSet u → (μ ⊗ₘ κ.map fun ω ↦ (f ω, g ω)) (s ×ˢ t ×ˢ u) = (μ ⊗ₘ (κ.map f ×ₖ κ.map g)) (s ×ˢ t ×ˢ u)

Tactic:
intro u

Goal state after:
case refine_1
α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
h :
  ∀ (s : Set β) (t : Set γ),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
u : Set α
⊢ ∀ {t : Set β} {u_1 : Set γ},
    MeasurableSet u →
      MeasurableSet t →
        MeasurableSet u_1 →
          (μ ⊗ₘ κ.map fun ω ↦ (f ω, g ω)) (u ×ˢ t ×ˢ u_1) = (μ ⊗ₘ (κ.map f ×ₖ κ.map g)) (u ×ˢ t ×ˢ u_1)
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: case refine_2.hf
ι : Type u_11
inst✝² : Fintype ι
Ω : ι → Type u_12
mΩ : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝¹ : ∀ (i : ι), IsProbabilityMeasure (μ i)
𝓧 : ι → Type u_13
inst✝ : (i : ι) → MeasurableSpace (𝓧 i)
X : (i : ι) → Ω i → 𝓧 i
mX : ∀ (i : ι), AEMeasurable (X i) (μ i)
i : ι
⊢ AEMeasurable (Function.eval i) (Measure.pi μ)

Tactic:
· exact (measurable_pi_apply i).aemeasurable

Goal state after: