mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.QuadraticForm.Signature

Goal state before:
pp: 𝕜 : Type u_4
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
ι : Type u_5
inst✝¹ : Fintype ι
w : ι → 𝕜
inst✝ : IsStrictOrderedRing 𝕜
⊢ ({i | 0 < w i} ∪ {i | w i ≤ 0}).ncard = Set.univ.ncard

pp: 𝕜 : Type u_4
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
ι : Type u_5
inst✝¹ : Fintype ι
w : ι → 𝕜
inst✝ : IsStrictOrderedRing 𝕜
⊢ Disjoint {i | 0 < w i} {i | w i ≤ 0}

Tactic:
· congr! 1
      ext
      grind [le_iff_lt_or_eq]

Goal state after:
𝕜 : Type u_4
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
ι : Type u_5
inst✝¹ : Fintype ι
w : ι → 𝕜
inst✝ : IsStrictOrderedRing 𝕜
⊢ Disjoint {i | 0 < w i} {i | w i ≤ 0}
Module: Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R N
P : RootPairing ι R M N
inst✝⁵ : P.IsCrystallographic
b : P.Base
inst✝⁴ : Finite ι
inst✝³ : IsDomain R
inst✝² : CharZero R
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
i j : ↥b.support
⊢ ⁅e i, ⁅f i, u j⁆⁆ = |b.cartanMatrix i j| • u i

Tactic:
ext (k | k)
  · simp [e, f, Matrix.mulVec, dotProduct, Pi.single_apply]
  · simp [e, f, Matrix.mulVec, dotProduct, P.ne_zero]

Goal state after:
Module: Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic

Goal state before:
pp: case mem.inr.inl
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R N
P : RootPairing ι R M N
inst✝⁵ : P.IsCrystallographic
b : P.Base
inst✝⁴ : Finite ι
inst✝³ : IsDomain R
inst✝² : CharZero R
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
x : Matrix (↥b.support ⊕ ι) (↥b.support ⊕ ι) R
i : ↥b.support
hu : e i ∈ range h ∪ range e ∪ range f
⊢ (ωConj b) (e i) ∈ lieAlgebra b

Tactic:
exact LieSubalgebra.subset_lieSpan <| by simp [ω_mul_e, mul_assoc]

Goal state after:
Module: Mathlib.LinearAlgebra.RootSystem.Finite.G2

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : CommRing 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.EmbeddedG2
inst✝² : Finite ι
inst✝¹ : CharZero R
inst✝ : IsDomain R
⊢ twoShortAddLongRoot P = 2 • shortRoot P + longRoot P

Tactic:
simp [twoShortAddLongRoot, twoShortAddLong, reflection_apply_root]
  module

Goal state after:
Module: Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple

Goal state before:
pp: case h₁
ι : Type u_1
K : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : CharZero K
inst✝⁸ : DecidableEq ι
inst✝⁷ : Fintype ι
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module K M
inst✝⁴ : AddCommGroup N
inst✝³ : Module K N
P : RootPairing ι K M N
inst✝² : P.IsCrystallographic
b : P.Base
inst✝¹ : P.IsReduced
inst✝ : P.IsIrreducible
i : ι
_i : InvolutiveNeg ι := ⋯
hωu : ∀ (i : ↥b.support), ω b *ᵥ u i = u i
hωv : ∀ (i : ι), ω b *ᵥ v b i = v b (-i)
⊢ ∀ i ∈ b.support, ∀ {U : LieSubmodule K (↥(lieAlgebra b)) (↥b.support ⊕ ι → K)}, v b i ∈ U → ∃ j, u j ∈ U

Tactic:
intro j hj U hj'
      let f' : lieAlgebra b := ⟨f ⟨j, hj⟩, f_mem_lieAlgebra _⟩
      have : ⁅f', v b j⁆ = u ⟨j, hj⟩ := f_lie_v_same ⟨j, hj⟩
      replace this : u ⟨j, hj⟩ ∈ U := by
        rw [← this]
        exact U.lie_mem (x := f') hj'
      exact ⟨⟨j, hj⟩, this⟩

Goal state after:
Module: Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basis

Goal state before:
pp: ι : Type u_1
K : Type u_2
M : Type u_3
N : Type u_4
inst✝¹¹ : Fintype ι
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : Field K
inst✝⁸ : CharZero K
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module K M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module K N
P : RootPairing ι K M N
inst✝³ : P.IsReduced
inst✝² : P.IsCrystallographic
inst✝¹ : P.IsIrreducible
inst✝ : P.IsRootSystem
b : P.Base
i : ↥b.support
t : IsSl2Triple (h i) (e i) (f i) := isSl2Triple i
⊢ ⁅⟨h i, ⋯⟩, ⟨e i, ⋯⟩⁆ = 2 • ⟨e i, ⋯⟩

Tactic:
simp [Subtype.ext_iff, t.lie_h_e_nsmul]

Goal state after:
Module: Mathlib.LinearAlgebra.RootSystem.OfBilinear

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : M →ₗ[R] M →ₗ[R] R
x : M
hx : B.IsReflective x
a b : M
⊢ 2 * (B x) (a + b) = (B x) x * Exists.choose ⋯ + (B x) x * Exists.choose ⋯

Tactic:
← (hx.2 a).choose_spec,

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : M →ₗ[R] M →ₗ[R] R
x : M
hx : B.IsReflective x
a b : M
⊢ 2 * (B x) (a + b) = 2 * (B x) a + (B x) x * Exists.choose ⋯
Module: Mathlib.LinearAlgebra.RootSystem.OfBilinear

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : M →ₗ[R] M →ₗ[R] R
x : M
hx : B.IsReflective x
r : R
a : M
⊢ 2 * (B x) (r • a) = r * ((B x) x * Exists.choose ⋯)

Tactic:
← (hx.2 a).choose_spec,

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : M →ₗ[R] M →ₗ[R] R
x : M
hx : B.IsReflective x
r : R
a : M
⊢ 2 * (B x) (r • a) = r * (2 * (B x) a)
Module: Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations

Goal state before:
pp: 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
x : ι
hx : P.root x = P.root ↑i + P.root l
hx' : P.root k = P.root x - P.root ↑j
⊢ P.root k = P.root l + P.root ↑i - P.root ↑j

Tactic:
rw [hx', hx]

Goal state after:
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
x : ι
hx : P.root x = P.root ↑i + P.root l
hx' : P.root k = P.root x - P.root ↑j
⊢ P.root ↑i + P.root l - P.root ↑j = P.root l + P.root ↑i - P.root ↑j
Module: Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations

Goal state before:
pp: 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
aux₃ : ∀ x ∈ Finset.univ, ¬(P.root x = P.root ↑i + P.root l ∧ P.root k = P.root x - P.root ↑j)
x : ι
hx : P.root x = P.root l - P.root ↑j
hx' : P.root k = P.root ↑i + P.root x
⊢ P.root k = P.root l + P.root ↑i - P.root ↑j

Tactic:
rw [hx', hx]

Goal state after:
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
aux₃ : ∀ x ∈ Finset.univ, ¬(P.root x = P.root ↑i + P.root l ∧ P.root k = P.root x - P.root ↑j)
x : ι
hx : P.root x = P.root l - P.root ↑j
hx' : P.root k = P.root ↑i + P.root x
⊢ P.root ↑i + (P.root l - P.root ↑j) = P.root l + P.root ↑i - P.root ↑j
Module: Mathlib.LinearAlgebra.SymmetricAlgebra.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : M →ₗ[R] A
e : SymmetricAlgebra R M ≃ₐ[R] A
he : ↑↑e ∘ₗ SymmetricAlgebra.ι R M = f
x : M
⊢ (↑↑e ∘ₗ SymmetricAlgebra.ι R M) x = (↑(SymmetricAlgebra.lift f) ∘ₗ SymmetricAlgebra.ι R M) x

Tactic:
simpa using congr($he x)

Goal state after:
Module: Mathlib.LinearAlgebra.TensorAlgebra.Grading

Goal state before:
pp: case add
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
i✝ : ℕ
x✝ x y : TensorAlgebra R M
i : ℕ
hx : x ∈ (ι R).range ^ i
hy : y ∈ (ι R).range ^ i
ihx : ((lift R) (GradedAlgebra.ι R M)) x = (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) i) ⟨x, hx⟩
ihy : ((lift R) (GradedAlgebra.ι R M)) y = (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) i) ⟨y, hy⟩
⊢ ((lift R) (GradedAlgebra.ι R M)) (x + y) = (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) i) ⟨x + y, ⋯⟩

Tactic:
| add x y i hx hy ihx ihy =>

Goal state after:
Module: Mathlib.LinearAlgebra.TensorPower.Symmetric

Goal state before:
pp: case of
R ι : Type u
inst✝² : CommSemiring R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
r : R
x✝ y✝ x y : ⨂[R] (x : ι), M
h : Rel R ι M x y
⊢ (addConGen (Rel R ι M)) (r • x) (r • y)

Tactic:
| of x y h =>

Goal state after:
Module: Mathlib.MeasureTheory.Category.MeasCat

Goal state before:
pp: case hf
μ : MeasureTheory.Measure (MeasureTheory.Measure ℝ≥0∞)
⊢ Measurable fun x ↦ x

Tactic:
apply_rules [Measurable.aemeasurable, measurable_id, Measure.measurable_lintegral]

Goal state after:
Module: Mathlib.MeasureTheory.Category.MeasCat

Goal state before:
pp: case hg
μ : MeasureTheory.Measure (MeasureTheory.Measure ℝ≥0∞)
⊢ Measurable fun m ↦ ∫⁻ (x : ℝ≥0∞), x ∂m

Tactic:
apply_rules [Measurable.aemeasurable, measurable_id, Measure.measurable_lintegral]

Goal state after:
Module: Mathlib.MeasureTheory.Category.MeasCat

Goal state before:
pp: μ : MeasureTheory.Measure (MeasureTheory.Measure ℝ≥0∞)
⊢ AEMeasurable (fun x ↦ x) μ.join

Tactic:
apply_rules [Measurable.aemeasurable, measurable_id, Measure.measurable_lintegral]

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: case hv
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f : Dual K V
u : V
hf : (↑e).fixedSubmodule ⊔ K ∙ (e u - u) ≤ ker f
hfu : f u = 1
hu' : u ∈ ker f
⊢ False

Tactic:
rw [mem_ker] at hu'

Goal state after:
case hv
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f : Dual K V
u : V
hf : (↑e).fixedSubmodule ⊔ K ∙ (e u - u) ≤ ker f
hfu : f u = 1
hu' : f u = 0
⊢ False
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f : Dual K V
u : V
hf : (↑e).fixedSubmodule ⊔ K ∙ (e u - u) ≤ ker f
hfu : f u = 1
hu : u ∉ (↑e).fixedSubmodule
⊢ finrank K (V ⧸ (↑(auxTransvection hf * e)).fixedSubmodule) + 1 = finrank K (V ⧸ (↑e).fixedSubmodule)

Tactic:
auxTransvection_mul_fixed hfu

Goal state after:
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f : Dual K V
u : V
hf : (↑e).fixedSubmodule ⊔ K ∙ (e u - u) ≤ ker f
hfu : f u = 1
hu : u ∉ (↑e).fixedSubmodule
⊢ finrank K (V ⧸ (↑e).fixedSubmodule ⊔ K ∙ u) + 1 = finrank K (V ⧸ (↑e).fixedSubmodule)
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
hC : IsSetSemiring C
m : AddContent G C
s : Set α
J : Finpartition s
hJ : ↑J.parts ⊆ C
⊢ ⋃ x ∈ J.parts, id x = ⋃₀ ↑J.parts

Tactic:
sUnion_eq_biUnion

Goal state after:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
hC : IsSetSemiring C
m : AddContent G C
s : Set α
J : Finpartition s
hJ : ↑J.parts ⊆ C
⊢ ⋃ x ∈ J.parts, id x = ⋃ i ∈ ↑J.parts, i
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ v (A \ B) + v (A ∩ B) = v (A \ B) + v (A ∩ B ∪ B \ A)

Tactic:
rw [of_union, h', add_zero]
      · exact Set.disjoint_of_subset_left A.inter_subset_left disjoint_sdiff_self_right
      · exact hA.inter hB
      · exact hB.diff hA

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : VectorMeasure α ℝ≥0∞
s : Set α
hs : MeasurableSet s
⊢ μ.ennrealToMeasure.toENNRealVectorMeasure s = μ s

Tactic:
by
  rw [toENNRealVectorMeasure_apply_measurable hs, ennrealToMeasure_apply hs]

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

Goal state before:
pp: case refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ MeasurableSet (a ∩ disjointed f n)

Tactic:
exact ha₁.inter (MeasurableSet.disjointed hf₁ n)

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

Goal state before:
pp: case refine_3.refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ MeasurableSet (a ∩ disjointed f n)

Tactic:
exact ha₁.inter (MeasurableSet.disjointed hf₁ n)

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

Goal state before:
pp: case refine_4.refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ MeasurableSet (a ∩ disjointed f n)

Tactic:
exact ha₁.inter (MeasurableSet.disjointed hf₁ n)

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
u v : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : 0 ≤[u] -s
hsv : 0 ≤[v] -s
hs : s (u ∆ v) = 0
this : s (u ∆ v) = 0 → s (u \ v) = 0 ∧ s (v \ u) = 0
⊢ s (u \ v) = 0 ∧ s (v \ u) = 0

Tactic:
exact this hs

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ MeasurableSet (a ∩ disjointed f n)

Tactic:
exact ha₁.inter (MeasurableSet.disjointed hf₁ n)

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hw : MeasurableSet w
hsu : 0 ≤[u] -s
hsv : 0 ≤[v] -s
hs : s (u ∆ v) = 0
this : s (u ∆ v) = 0 → s (w ∩ u) = s (w ∩ v)
⊢ s (w ∩ u) = s (w ∩ v)

Tactic:
exact this hs

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
⊢ Measurable (s.rnDeriv μ)

Tactic:
rw [rnDeriv_def]

Goal state after:
α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
μ : Measure α
⊢ Measurable fun x ↦
    (s.toJordanDecomposition.posPart.rnDeriv μ x).toReal - (s.toJordanDecomposition.negPart.rnDeriv μ x).toReal
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ∞
hμ : ¬μ = 0
hε' : 0 < ε.toReal / 3
hdivp : 0 ≤ 1 / p.toReal
hpow : 0 < measureUnivNNReal μ ^ (1 / p.toReal)
δ₁ : ℝ
hδ₁ : 0 < δ₁
heLpNorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
heLpNorm₂ :
  ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
t : Set α
htm : MeasurableSet t
ht₁ : μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hN : ∀ x ∈ tᶜ, dist (g x) (f n x) < ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
hn : n ≥ N
hnf : eLpNorm (t.indicator (f n)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
hng : eLpNorm (t.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
this✝ : 0 ≤ ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
this :
  eLpNorm (tᶜ.indicator (f n - g)) p μ ≤
    ENNReal.ofReal (ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))) * μ tᶜ ^ (1 / p.toReal)
⊢ 0 ≤ ε.toReal / 3 * (1 / ↑(measureUnivNNReal μ) ^ (1 / p.toReal))

Tactic:
· positivity

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✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : TendstoInMeasure μ f atTop g
ns : ℕ → ℕ
hns : Tendsto ns atTop atTop
⊢ ∃ ms, Tendsto (fun n ↦ eLpNorm (f (ns (ms n)) - g) p μ) atTop (𝓝 0)

Tactic:
obtain ⟨ms, _, hms'⟩ := TendstoInMeasure.exists_seq_tendsto_ae fun ε hε => (hfg ε hε).comp hns

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : TendstoInMeasure μ f atTop g
ns : ℕ → ℕ
hns : Tendsto ns atTop atTop
ms : ℕ → ℕ
left✝ : StrictMono ms
hms' : ∀ᵐ (x : α) ∂μ, Tendsto (fun i ↦ f (ns (ms i)) x) atTop (𝓝 (g x))
⊢ ∃ ms, Tendsto (fun n ↦ eLpNorm (f (ns (ms n)) - g) p μ) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (μ.trim hm)
f : α → ℝ
hf : Integrable f μ
s : Set α
hs : MeasurableSet s
a✝ : μ s < ∞
⊢ ∫ (x : α) in s, SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) x ∂μ = ∫ (x : α) in s, f x ∂μ

Tactic:
conv_rhs => rw [← hf.withDensityᵥ_trim_eq_integral hm hs,
      ← SignedMeasure.withDensityᵥ_rnDeriv_eq ((μ.withDensityᵥ f).trim hm) (μ.trim hm)
        (hf.withDensityᵥ_trim_absolutelyContinuous hm)]

Goal state after:
case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (μ.trim hm)
f : α → ℝ
hf : Integrable f μ
s : Set α
hs : MeasurableSet s
a✝ : μ s < ∞
⊢ ∫ (x : α) in s, SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) x ∂μ =
    ((μ.trim hm).withDensityᵥ (SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm))) s
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

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 → ProbabilityMeasure Ω
μ : ProbabilityMeasure Ω
⊢ Tendsto (toFiniteMeasure ∘ μs) F (𝓝 μ.toFiniteMeasure) ↔
    ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i ↦ ∫⁻ (ω : Ω), ↑(f ω) ∂↑(μs i)) F (𝓝 (∫⁻ (ω : Ω), ↑(f ω) ∂↑μ))

Tactic:
exact FiniteMeasure.tendsto_iff_forall_lintegral_tendsto

Goal state after:
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✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
hf : ∀ (i : ι), MemLp (f i) p μ
⊢ UniformIntegrable f p μ

Tactic:
cases nonempty_fintype ι

Goal state after:
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
hf : ∀ (i : ι), MemLp (f i) p μ
val✝ : Fintype ι
⊢ UniformIntegrable f p μ
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✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i ↦ Exists.choose ⋯
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgeq : ∀ (i : ι), g i =ᵐ[μ] f i
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖g i x‖₊}.indicator (g i)) p μ ≤ ENNReal.ofReal ε

Tactic:
refine ⟨C, fun i => le_trans (le_of_eq <| eLpNorm_congr_ae ?_) (hC i)⟩

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i ↦ Exists.choose ⋯
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgeq : ∀ (i : ι), g i =ᵐ[μ] f i
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
⊢ {x | C ≤ ‖g i x‖₊}.indicator (g i) =ᵐ[μ] {x | C ≤ ‖f i x‖₊}.indicator (f i)
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i ↦ (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
⊢ 1 - μ E ≤ liminf ((fun x ↦ 1 - x) ∘ fun i ↦ (μs i) E) L

Tactic:
have key := antitone_const_tsub.map_limsup_of_continuousAt (F := L)
    (fun i ↦ μs i E) (ENNReal.continuous_sub_left ENNReal.one_ne_top).continuousAt

Goal state after:
case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i ↦ (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
key : 1 - limsup (fun i ↦ (μs i) E) L = liminf ((fun x ↦ 1 - x) ∘ fun i ↦ (μs i) E) L
⊢ 1 - μ E ≤ liminf ((fun x ↦ 1 - x) ∘ fun i ↦ (μs i) E) L