mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
inst✝⁷ : Field R
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
inst✝¹ : OrderTopology S
ι✝ : Type u_3
inst✝ : Finite ι✝
v✝ : ι✝ → AbsoluteValue R S
this : Fintype ι✝
P : (ι : Type u_3) → [Fintype ι] → Prop :=
  fun ι [Fintype ι] ↦
    ∀ (v : ι → AbsoluteValue R S),
      (∀ (i : ι), (v i).IsNontrivial) →
        (Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)) → ∀ (i : ι), ∃ a, 1 < (v i) a ∧ ∀ (j : ι), j ≠ i → (v j) a < 1
ι : Type u_3
x✝¹ : Fintype ι
x✝ : Subsingleton ι
v : ι → AbsoluteValue R S
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
i : ι
⊢ ∃ a, 1 < (v i) a ∧ ∀ (j : ι), j ≠ i → (v j) a < 1

Tactic:
let ⟨a, ha⟩ := (h i).exists_abv_gt_one

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
inst✝⁷ : Field R
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
inst✝¹ : OrderTopology S
ι✝ : Type u_3
inst✝ : Finite ι✝
v✝ : ι✝ → AbsoluteValue R S
this : Fintype ι✝
P : (ι : Type u_3) → [Fintype ι] → Prop :=
  fun ι [Fintype ι] ↦
    ∀ (v : ι → AbsoluteValue R S),
      (∀ (i : ι), (v i).IsNontrivial) →
        (Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)) → ∀ (i : ι), ∃ a, 1 < (v i) a ∧ ∀ (j : ι), j ≠ i → (v j) a < 1
ι : Type u_3
x✝¹ : Fintype ι
x✝ : Subsingleton ι
v : ι → AbsoluteValue R S
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
i : ι
a : R
ha : 1 < (v i) a
⊢ ∃ a, 1 < (v i) a ∧ ∀ (j : ι), j ≠ i → (v j) a < 1
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: case «1»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op))
      (f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 δ₂_zero_comp_σ₂_one._proof_4 δ₂_zero_comp_σ₂_one._proof_3).op)) x)) =
    (ConcreteCategory.hom (Y.map (Hom.tr (SimplexCategory.δ 0) ⋯ f₂._proof_4 ≫ σ₂ 1 _proof_15 _proof_12).op)) (f₁ x)

Tactic:
δ₂_zero_comp_σ₂_one,

Goal state after:
case «1»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op))
      (f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 δ₂_zero_comp_σ₂_one._proof_4 δ₂_zero_comp_σ₂_one._proof_3).op)) x)) =
    (ConcreteCategory.hom
        (Y.map
          (σ₂ 0 δ₂_zero_comp_σ₂_one._proof_3 δ₂_zero_comp_σ₂_one._proof_4 ≫
              δ₂ 0 δ₂_zero_comp_σ₂_one._proof_4 δ₂_zero_comp_σ₂_one._proof_3).op))
      (f₁ x)
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
g : 𝕜 → E
z₀ : 𝕜
f : 𝕜 → 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
hf' : ¬analyticOrderAt f z₀ = ⊤
hg' : ¬analyticOrderAt g z₀ = ⊤
f' : 𝕜 → 𝕜
h₁f' : AnalyticAt 𝕜 f' z₀
h₂f' : f' z₀ ≠ 0
h₃f' : f =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt f z₀ • f' z
g' : 𝕜 → E
h₁g' : AnalyticAt 𝕜 g' z₀
h₂g' : g' z₀ ≠ 0
h₃g' : g =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt g z₀ • g' z
t : Set 𝕜
h₁t : ∀ y ∈ t, f y = (fun z ↦ (z - z₀) ^ analyticOrderNatAt f z₀ • f' z) y
h₂t : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
h₃t : z₀ ∈ t
s : Set 𝕜
h₁s : ∀ y ∈ s, g y = (fun z ↦ (z - z₀) ^ analyticOrderNatAt g z₀ • g' z) y
h₂s : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h₃s : z₀ ∈ s
y : 𝕜
hy : y ∈ t ∩ s
⊢ (f • g) y = (y - z₀) ^ (analyticOrderNatAt f z₀ + analyticOrderNatAt g z₀) • (f' • g') y

Tactic:
by simp [h₁t y hy.1, h₁s y hy.2]; module

Goal state after:
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case succ
𝕜 : Type u_3
E : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
inst✝ : CharZero 𝕜
n' : ℕ
hk : ∀ {n : ℕ}, ↑n = analyticOrderAt f z₀ → n ≠ 0 → n' ≤ n → analyticOrderAt (deriv^[n'] f) z₀ = ↑(n - n')
n : ℕ
Hn : ↑n = analyticOrderAt f z₀
Hpos : n ≠ 0
Hk : n' + 1 ≤ n
⊢ analyticOrderAt (deriv^[n' + 1] f) z₀ = ↑(n - (n' + 1))

Tactic:
rw [Function.iterate_succ']

Goal state after:
case succ
𝕜 : Type u_3
E : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
inst✝ : CharZero 𝕜
n' : ℕ
hk : ∀ {n : ℕ}, ↑n = analyticOrderAt f z₀ → n ≠ 0 → n' ≤ n → analyticOrderAt (deriv^[n'] f) z₀ = ↑(n - n')
n : ℕ
Hn : ↑n = analyticOrderAt f z₀
Hpos : n ≠ 0
Hk : n' + 1 ≤ n
⊢ analyticOrderAt ((deriv ∘ deriv^[n']) f) z₀ = ↑(n - (n' + 1))
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
hf : AnalyticAt 𝕜 f 0
n : ℕ
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F 0
hF : ∀ᶠ (z : 𝕜) in 𝓝 0, f z = ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0 + z ^ n • F z
⊢ ?m.89

Tactic:
rwa [eventually_iff_exists_mem] at hF

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : Field 𝕂
inst✝³ : Ring 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
a b c : 𝕂
n : ℕ
⊢ ((ofScalars 𝔸 (ordinaryHypergeometricCoefficient a b c) n) fun x ↦ 0) = Pi.single 0 1 n

Tactic:
ofScalars_apply_eq,

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : Field 𝕂
inst✝³ : Ring 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
a b c : 𝕂
n : ℕ
⊢ ordinaryHypergeometricCoefficient a b c n • 0 ^ n = Pi.single 0 1 n
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
f : 𝕜 → 𝕜
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
⊢ analyticOrderNatAt (f ^ n) z₀ = n • analyticOrderNatAt f z₀

Tactic:
simp [analyticOrderNatAt, analyticOrderAt_pow, hf]

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: 𝕂 : Type v
inst✝² : RCLike 𝕂
𝔸 : Type u
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a : ℕ
⊢ ∀ (k : ℕ), ↑k ≠ -1

Tactic:
by norm_cast; simp

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: 𝕂 : Type v
inst✝² : RCLike 𝕂
𝔸 : Type u
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a : 𝕂
ha : ∀ (k : ℕ), a ≠ ↑k
⊢ ∀ (k : ℕ), ↑k ≠ -1

Tactic:
by norm_cast; simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: p : ℝ × ℝ
hp : p.1 ≠ 0
n : WithTop ℕ∞
⊢ ContDiffAt ℝ n (fun p ↦ p.1 ^ p.2) p

Tactic:
by
  rcases hp.lt_or_gt with hneg | hpos
  exacts
    [(((contDiffAt_fst.log hneg.ne).mul contDiffAt_snd).exp.mul
          (contDiffAt_snd.mul contDiffAt_const).cos).congr_of_eventuallyEq
      ((continuousAt_fst.eventually (gt_mem_nhds hneg)).mono fun p hp => rpow_def_of_neg hp _),
    ((contDiffAt_fst.log hpos.ne').mul contDiffAt_snd).exp.congr_of_eventuallyEq
      ((continuousAt_fst.eventually (lt_mem_nhds hpos)).mono fun p hp => rpow_def_of_pos hp _)]

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: case h₀
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
y : E
hy : y ∈ Metric.eball 0 r
i : ℕ
hi : i ∈ Finset.range (n + 1)
h'i : i ≠ n
⊢ ((q i) fun x ↦ y) = 0

Tactic:
simp [q, h'i.symm]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
k f g g' : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : CommSemiring β
inst✝² : PartialOrder β
inst✝¹ : IsOrderedRing β
inst✝ : OrderTopology β
hk : 0 ≤ᶠ[l] k
hg : SuperpolynomialDecay l k g
hg' : SuperpolynomialDecay l k g'
hfg : g ≤ᶠ[l] f
hfg' : f ≤ᶠ[l] g'
z : ℕ
⊢ ∀ᶠ (b : α) in l, k b ^ z * f b ≤ k b ^ z * g' b

Tactic:
by filter_upwards [hfg', hk] with x hx (hx' : 0 ≤ k x) using by gcongr

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
  {u |
    u ⊆ t ∧
      u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c).Nonempty ∧ δ a ≤ τ * δ c}
U : Set (Set ι)
UT : U ⊆ T
hU : IsChain (fun x1 x2 ↦ x1 ⊆ x2) U
⊢ ∃ ub ∈ T, ∀ s ∈ U, s ⊆ ub

Tactic:
refine ⟨⋃₀ U, ?_, fun s hs => subset_sUnion_of_mem hs⟩

Goal state after:
α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
  {u |
    u ⊆ t ∧
      u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c).Nonempty ∧ δ a ≤ τ * δ c}
U : Set (Set ι)
UT : U ⊆ T
hU : IsChain (fun x1 x2 ↦ x1 ⊆ x2) U
⊢ ⋃₀ U ∈ T
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: α : Type u_1
u : α → ℕ → ℝ≥0∞
s : Set α
hs : s.Finite
⊢ expGrowthInf (⨅ x ∈ s, u x) = ⨅ x ∈ s, expGrowthInf (u x)

Tactic:
have := map_finset_inf expGrowthInfTopHom hs.toFinset u

Goal state after:
α : Type u_1
u : α → ℕ → ℝ≥0∞
s : Set α
hs : s.Finite
this : expGrowthInfTopHom (hs.toFinset.inf u) = hs.toFinset.inf (⇑expGrowthInfTopHom ∘ u)
⊢ expGrowthInf (⨅ x ∈ s, u x) = ⨅ x ∈ s, expGrowthInf (u x)
Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0
this✝ : Encodable ↑s
u v : β → β → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x ↦ ⨅ i, (u' ↑i).piecewise (fun x ↦ ↑i) (fun x ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ p, ⋃ q, u' ↑p ∩ v ↑p ↑q
μt : μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, x ∉ t
x : α
hx : x ∉ t
q : β
hq : f x < q
r : β
rs : r ∈ s
xr : f x < r
rq : r < q
⊢ (u' ↑⟨r, rs⟩).piecewise (fun x ↦ ↑⟨r, rs⟩) (fun x ↦ ⊤) x < q

Tactic:
have A : x ∈ u' r := mem_biInter fun i _ => (huv r i).2.2.1 xr

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0
this✝ : Encodable ↑s
u v : β → β → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x ↦ ⨅ i, (u' ↑i).piecewise (fun x ↦ ↑i) (fun x ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ p, ⋃ q, u' ↑p ∩ v ↑p ↑q
μt : μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, x ∉ t
x : α
hx : x ∉ t
q : β
hq : f x < q
r : β
rs : r ∈ s
xr : f x < r
rq : r < q
A : x ∈ u' r
⊢ (u' ↑⟨r, rs⟩).piecewise (fun x ↦ ↑⟨r, rs⟩) (fun x ↦ ⊤) x < q
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)

Tactic:
rcases hu a ha with ⟨b, bu, hb, rb⟩

Goal state after:
α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
b : ι
bu : b ∈ u
hb : (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty
rb : r a ≤ (τ - 1) / 2 * r b
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
f : α → ℝ≥0∞
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
⊢ ⨍⁻ (x : α) in s ∪ t, f x ∂μ =
    μ s / (μ s + μ t) * ⨍⁻ (x : α) in s, f x ∂μ + μ t / (μ s + μ t) * ⨍⁻ (x : α) in t, f x ∂μ

Tactic:
by
  rw [restrict_union₀ hd ht, laverage_add_measure, restrict_apply_univ, restrict_apply_univ]

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
rcases hu a ha with ⟨b, bu, hb, rb⟩

Goal state after:
α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
b : ι
bu : b ∈ u
hb : (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty
rb : r a ≤ (τ - 1) / 2 * r b
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ∞
hν : ∀ (s : Set α), ν s ≠ ∞
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) = γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1)))

Tactic:
by
          rw [pow_succ, mul_one_div, _root_.sub_half]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
hμν : μ ⟂ₘ ν
⊢ μ.rnDeriv ν =ᵐ[ν] 0

Tactic:
by_cases h : μ.HaveLebesgueDecomposition ν
  · rw [rnDeriv_eq_zero]
    exact hμν
  · rw [rnDeriv_of_not_haveLebesgueDecomposition h]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
hμν : μ ⟂ₘ ν
⊢ μ.rnDeriv ν =ᵐ[ν] 0

Tactic:
by
  by_cases h : μ.HaveLebesgueDecomposition ν
  · rw [rnDeriv_eq_zero]
    exact hμν
  · rw [rnDeriv_of_not_haveLebesgueDecomposition h]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
s : Set α
hs : MeasurableSet s
⊢ (μ.restrict s).singularPart ν = (μ.singularPart ν).restrict s

Tactic:
refine (Measure.eq_singularPart (f := s.indicator (μ.rnDeriv ν)) ?_ ?_ ?_).symm

Goal state after:
case refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
s : Set α
hs : MeasurableSet s
⊢ Measurable (s.indicator (μ.rnDeriv ν))

case refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
s : Set α
hs : MeasurableSet s
⊢ (μ.singularPart ν).restrict s ⟂ₘ ν

case refine_3
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
s : Set α
hs : MeasurableSet s
⊢ μ.restrict s = (μ.singularPart ν).restrict s + ν.withDensity (s.indicator (μ.rnDeriv ν))
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0
hr : r ≠ 0
⊢ ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ

Tactic:
by
  refine (absolutelyContinuous_smul <| ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  rotate_left
  · exact (measurable_rnDeriv _ _).aemeasurable
  · exact (measurable_rnDeriv _ _).aemeasurable.const_smul _
  · exact (lintegral_rnDeriv_lt_top ν _).ne
  · simp_rw [ENNReal.smul_def]
    rw [withDensity_smul _ (measurable_rnDeriv _ _)]
    suffices ν.singularPart (r • μ) + withDensity (r • μ) (rnDeriv ν (r • μ))
        = ν.singularPart (r • μ) + r⁻¹ • withDensity (r • μ) (rnDeriv ν μ) by
      rwa [add_right_inj] at this
    rw [← ν.haveLebesgueDecomposition_add (r • μ), singularPart_smul_right _ _ _ hr,
      ENNReal.smul_def r, withDensity_smul_measure, ← ENNReal.smul_def, ← smul_assoc,
      smul_eq_mul, inv_mul_cancel₀ hr, one_smul]
    exact ν.haveLebesgueDecomposition_add μ

Goal state after:
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : ℝ → ℝ
x a c d : ℝ
l : Filter ℝ
hl : l ≤ 𝓝[≠] x
hf : Tendsto (fun y ↦ (f y - d) / (y - x)) l (𝓝 a)
h' : Tendsto (fun y ↦ y + c * (y - x) ^ 2) l l
L : Tendsto (fun y ↦ (y + c * (y - x) ^ 2 - x) / (y - x)) l (𝓝 1)
Z :
  Tendsto
    (fun x_1 ↦
      ((fun y ↦ (f y - d) / (y - x)) ∘ fun y ↦ y + c * (y - x) ^ 2) x_1 * ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x)))
    l (𝓝 a)
⊢ (fun x_1 ↦
      ((fun y ↦ (f y - d) / (y - x)) ∘ fun y ↦ y + c * (y - x) ^ 2) x_1 *
        ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x))) =ᶠ[l]
    fun y ↦ (f (y + c * (y - x) ^ 2) - d) / (y - x)

Tactic:
have : ∀ᶠ y in l, y + c * (y - x) ^ 2 ≠ x := by apply Tendsto.mono_right h' hl self_mem_nhdsWithin

Goal state after:
f : ℝ → ℝ
x a c d : ℝ
l : Filter ℝ
hl : l ≤ 𝓝[≠] x
hf : Tendsto (fun y ↦ (f y - d) / (y - x)) l (𝓝 a)
h' : Tendsto (fun y ↦ y + c * (y - x) ^ 2) l l
L : Tendsto (fun y ↦ (y + c * (y - x) ^ 2 - x) / (y - x)) l (𝓝 1)
Z :
  Tendsto
    (fun x_1 ↦
      ((fun y ↦ (f y - d) / (y - x)) ∘ fun y ↦ y + c * (y - x) ^ 2) x_1 * ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x)))
    l (𝓝 a)
this : ∀ᶠ (y : ℝ) in l, y + c * (y - x) ^ 2 ≠ x
⊢ (fun x_1 ↦
      ((fun y ↦ (f y - d) / (y - x)) ∘ fun y ↦ y + c * (y - x) ^ 2) x_1 *
        ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x))) =ᶠ[l]
    fun y ↦ (f (y + c * (y - x) ^ 2) - d) / (y - x)
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx : HasDerivAt (↑hf.stieltjesFunction) (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x
h'x : ¬¬ContinuousAt f x
⊢ ↑hf.stieltjesFunction x = f x

Tactic:
Classical.not_not,

Goal state after:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx : HasDerivAt (↑hf.stieltjesFunction) (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x
h'x : ContinuousAt f x
⊢ ↑hf.stieltjesFunction x = f x