mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
c : 𝕜
U : Set 𝕜
h : MeromorphicOn (f ∘ fun x ↦ x + c) U
y : 𝕜
hy : y ∈ U + {c}
⊢ MeromorphicAt f y

pp: case refine_2
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
c : 𝕜
U : Set 𝕜
h : MeromorphicOn f (U + {c})
y : 𝕜
hy : y ∈ U
⊢ MeromorphicAt (f ∘ fun x ↦ x + c) y

Tactic:
· rw [add_singleton, mem_image] at hy
    obtain ⟨x, h₁x, h₂x⟩ := hy
    simpa [← h₂x, ← meromorphicAt_comp_add_const_iff_meromorphicAt] using h x h₁x

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
c : 𝕜
U : Set 𝕜
h : MeromorphicOn f (U + {c})
y : 𝕜
hy : y ∈ U
⊢ MeromorphicAt (f ∘ fun x ↦ x + c) y
Module: Mathlib.Analysis.Calculus.FDeriv.Extend

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x ↦ deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
⊢ HasDerivWithinAt f e (Ici a) a

Tactic:
have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[>] a
f_lim' : Tendsto (fun x ↦ deriv f x) (𝓝[>] a) (𝓝 e)
b : ℝ
ab : a < b
sab : Ioc a b ⊆ s
t : Set ℝ := Ioo a b
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
⊢ HasDerivWithinAt f e (Ici a) a
Module: Mathlib.Analysis.Calculus.FDeriv.Extend

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[<] a
f_lim' : Tendsto (fun x ↦ deriv f x) (𝓝[<] a) (𝓝 e)
b : ℝ
ba : b ∈ Iio a
sab : Ico b a ⊆ s
t : Set ℝ := Ioo b a
ts : t ⊆ s
⊢ HasDerivWithinAt f e (Iic a) a

Tactic:
have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set ℝ
e : E
a : ℝ
f : ℝ → E
f_diff : DifferentiableOn ℝ f s
f_lim : ContinuousWithinAt f s a
hs : s ∈ 𝓝[<] a
f_lim' : Tendsto (fun x ↦ deriv f x) (𝓝[<] a) (𝓝 e)
b : ℝ
ba : b ∈ Iio a
sab : Ico b a ⊆ s
t : Set ℝ := Ioo b a
ts : t ⊆ s
t_diff : DifferentiableOn ℝ f t
⊢ HasDerivWithinAt f e (Iic a) a
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℂ
n : ℕ
B : Set ℂ := Metric.ball 0 1
this : iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B 0 = (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n)) 0
⊢ (descPochhammer ℤ n).smeval a = iteratedDeriv n (fun x ↦ (1 + x) ^ a) 0

Tactic:
iteratedDerivWithin_of_isOpen Metric.isOpen_ball (by simp)

Goal state after:
a : ℂ
n : ℕ
B : Set ℂ := Metric.ball 0 1
this : iteratedDeriv n (fun x ↦ (1 + x) ^ a) 0 = (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n)) 0
⊢ (descPochhammer ℤ n).smeval a = iteratedDeriv n (fun x ↦ (1 + x) ^ a) 0
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 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 := ⋯
⊢ ContDiff 𝕜 ω fun z x_1 ↦ z - x

Tactic:
exact contDiff_pi.2 (fun i ↦ contDiff_id.sub contDiff_const)

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

Goal state before:
pp: z : ℂ
hz : z ≠ 0
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (z - x)) (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + 1))⁻¹) 0 ‖z‖ₑ

Tactic:
by
  simpa using one_div_sub_pow_hasFPowerSeriesOnBall_zero (a := 0) hz

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

Goal state before:
pp: case a
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
p : FormalMultilinearSeries 𝕜 E F
this✝ : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)
r : ℝ≥0
hr : ↑r < p.radius
this : ∃ a ∈ Set.Ioo 0 1, ∀ᶠ (n : ℕ) in atTop, |‖p n‖ * ↑r ^ n| ≤ a ^ n
⊢ ↑r ≤ liminf (fun n ↦ 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop

Tactic:
obtain ⟨a, ha, H⟩ := this

Goal state after:
case a
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
p : FormalMultilinearSeries 𝕜 E F
this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)
r : ℝ≥0
hr : ↑r < p.radius
a : ℝ
ha : a ∈ Set.Ioo 0 1
H : ∀ᶠ (n : ℕ) in atTop, |‖p n‖ * ↑r ^ n| ≤ a ^ n
⊢ ↑r ≤ liminf (fun n ↦ 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: case hx
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
this : iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f (s ∩ Metric.eball x r) x
⊢ x ∈ s ∩ Metric.eball x r

Tactic:
exact ⟨hx, Metric.mem_eball_self h.r_pos⟩

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

Goal state before:
pp: case hx
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
this : iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f (s ∩ Metric.eball x r) x
⊢ x ∈ s ∩ Metric.eball x r

Tactic:
exact ⟨hx, Metric.mem_eball_self h.r_pos⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inl.inr.inr
y : ℝ≥0∞
y_real : 0 < y.toReal
⊢ ¬y = 0

Tactic:
exact (ENNReal.toReal_pos_iff.1 y_real).1.ne'

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ Real.log (x.toReal * y.toReal) = Real.log x.toReal + Real.log y.toReal

pp: case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ 0 < (x * y).toReal

Tactic:
· exact Real.log_mul x_real.ne' y_real.ne'

Goal state after:
case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ 0 < (x * y).toReal
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: u : ℕ → EReal
⊢ linearGrowthSup (-u) = -liminf (fun n ↦ u n / ↑n) atTop

Tactic:
← limsup_neg

Goal state after:
u : ℕ → EReal
⊢ linearGrowthSup (-u) = limsup (-fun n ↦ u n / ↑n) atTop
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : ∀ (z : ℕ), IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun a ↦ |k a ^ z * f a|
z : ℕ
m : β
hm : ∀ᶠ (x : β) in Filter.map (fun a ↦ |k a ^ (z + 1) * f a|) l, (fun x1 x2 ↦ x1 ≤ x2) x m
h1 : Tendsto (fun x ↦ 0) l (𝓝 0)
⊢ Tendsto (fun a ↦ |k a ^ z * f a|) l (𝓝 0)

Tactic:
have h2 : Tendsto (fun a : α => |(k a)⁻¹| * m) l (𝓝 0) :=
    zero_mul m ▸
      Tendsto.mul_const m ((tendsto_zero_iff_abs_tendsto_zero _).1 hk.inv_tendsto_atTop)

Goal state after:
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : ∀ (z : ℕ), IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun a ↦ |k a ^ z * f a|
z : ℕ
m : β
hm : ∀ᶠ (x : β) in Filter.map (fun a ↦ |k a ^ (z + 1) * f a|) l, (fun x1 x2 ↦ x1 ≤ x2) x m
h1 : Tendsto (fun x ↦ 0) l (𝓝 0)
h2 : Tendsto (fun a ↦ |(k a)⁻¹| * m) l (𝓝 0)
⊢ Tendsto (fun a ↦ |k a ^ z * f a|) l (𝓝 0)
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u : ℕ → ℝ≥0∞
⊢ expGrowthSup u⁻¹ = -liminf (fun n ↦ (u n).log / ↑n) atTop

Tactic:
← limsup_neg

Goal state after:
u : ℕ → ℝ≥0∞
⊢ expGrowthSup u⁻¹ = limsup (-fun n ↦ (u n).log / ↑n) atTop
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case neg.inr
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : ¬u = ⊥
v_0 : 0 < linearGrowthSup fun n ↦ ↑(v n)
a : EReal
v_a : a > linearGrowthSup fun n ↦ ↑(v n)
b : EReal
u_b : b > linearGrowthInf u
a_0 : 0 < a
b_0 : 0 < b
a_top : a ≠ ⊤
a' : EReal
v_a' : (linearGrowthSup fun n ↦ ↑(v n)) < a'
a_a' : a' < a
⊢ ∃ᶠ (n : ℕ) in atTop, (u ∘ v) n ≤ a * b * ↑n

Tactic:
have a_0' := v_0.trans v_a'

Goal state after:
case neg.inr
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : ¬u = ⊥
v_0 : 0 < linearGrowthSup fun n ↦ ↑(v n)
a : EReal
v_a : a > linearGrowthSup fun n ↦ ↑(v n)
b : EReal
u_b : b > linearGrowthInf u
a_0 : 0 < a
b_0 : 0 < b
a_top : a ≠ ⊤
a' : EReal
v_a' : (linearGrowthSup fun n ↦ ↑(v n)) < a'
a_a' : a' < a
a_0' : 0 < a'
⊢ ∃ᶠ (n : ℕ) in atTop, (u ∘ v) n ≤ a * b * ↑n
Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h :
  ∀ (p q : ℝ≥0),
    p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < ↑p} ⊆ u ∧ {x | ↑q < f x} ⊆ v ∧ μ (u ∩ v) = 0
s : Set ℝ≥0∞
s_count : s.Countable
s_dense : Dense s
left✝ : 0 ∉ s
s_top : ∞ ∉ s
⊢ AEMeasurable f μ

Tactic:
have I : ∀ x ∈ s, x ≠ ∞ := fun x xs hx => s_top (hx ▸ xs)

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h :
  ∀ (p q : ℝ≥0),
    p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < ↑p} ⊆ u ∧ {x | ↑q < f x} ⊆ v ∧ μ (u ∩ v) = 0
s : Set ℝ≥0∞
s_count : s.Countable
s_dense : Dense s
left✝ : 0 ∉ s
s_top : ∞ ∉ s
I : ∀ x ∈ s, x ≠ ∞
⊢ AEMeasurable f μ
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: case neg.inr
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 : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : r a < 0
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)

Tactic:
rcases ht with ⟨b, rb⟩
    rcases A b ⟨rb.1, rb.2⟩ with ⟨c, cu, _⟩
    exact ⟨c, cu, by simp only [closedBall_eq_empty.2 h'a, empty_subset]⟩

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : IsFiniteMeasure μ
hμ : μ ≠ 0
hf : Integrable f μ
⊢ 0 < μ {x | f x ≤ ⨍ (a : α), f a ∂μ}

Tactic:
simpa using measure_le_setAverage_pos (Measure.measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)
    hf.integrableOn

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

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
⊢ sInf ((fun μ ↦ μ.restrict s) '' {d | μ ≤ d + ν}) ≤ sInf {d | μ.restrict s ≤ d + ν.restrict s}

Tactic:
refine sInf_le_sInf_of_isCoinitialFor ?_
    intro ν' h_ν'_in
    rw [mem_ofPred_eq] at h_ν'_in
    refine ⟨ν'.restrict s, ?_, restrict_le_self⟩
    refine ⟨ν' + (⊤ : Measure α).restrict sᶜ, ?_, ?_⟩
    · rw [mem_ofPred_eq, add_right_comm, Measure.le_iff]
      intro t h_meas_t
      repeat rw [← measure_inter_add_sdiff t h_meas_s]
      refine add_le_add ?_ ?_
      · rw [add_apply, add_apply]
        apply le_add_right _
        rw [← restrict_eq_self μ inter_subset_right,
          ← restrict_eq_self ν inter_subset_right]
        apply h_ν'_in
      · rw [add_apply, restrict_apply (h_meas_t.diff h_meas_s), sdiff_eq, inter_assoc, inter_self,
          ← add_apply]
        have h_mu_le_add_top : μ ≤ ν' + ν + ⊤ := by simp only [add_top, le_top]
        exact Measure.le_iff'.1 h_mu_le_add_top _
    · ext1 t h_meas_t
      simp [restrict_apply h_meas_t, restrict_apply (h_meas_t.inter h_meas_s), inter_assoc]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
⊢ Measurable (μ.rnDeriv ν)

Tactic:
by_cases h : HaveLebesgueDecomposition μ ν
  · exact (haveLebesgueDecomposition_spec μ ν).1
  · rw [rnDeriv_of_not_haveLebesgueDecomposition h]
    exact measurable_zero

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ✝ ν : Measure α
ι : Type u_2
inst✝¹ : Countable ι
μ : ι → Measure α
inst✝ : ∀ (i : ι), (μ i).HaveLebesgueDecomposition ν
⊢ Measurable (sum fun i ↦ (μ i).singularPart ν, ∑' (i : ι), (μ i).rnDeriv ν).2

Tactic:
dsimp only; fun_prop

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ
this : Tendsto (fun x ↦ K * ⨍ (y : α) in a x, ‖f y - c‖ ∂μ) l (𝓝 0)
i : ι
hi : support (g i) ⊆ a i
h'i : ∀ (x : α), |g i x| ≤ K / μ.real (a i)
h''i : IntegrableOn f (a i) μ
hi_int : 0 < ∫ (y : α), g i y ∂μ
h : μ (a i) = ∞
⊢ False

Tactic:
simp only [h, ENNReal.toReal_top, _root_.div_zero, abs_nonpos_iff, measureReal_def] at h'i

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ
this : Tendsto (fun x ↦ K * ⨍ (y : α) in a x, ‖f y - c‖ ∂μ) l (𝓝 0)
i : ι
hi : support (g i) ⊆ a i
h''i : IntegrableOn f (a i) μ
hi_int : 0 < ∫ (y : α), g i y ∂μ
h : μ (a i) = ∞
h'i : ∀ (x : α), g i x = 0
⊢ False
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) = g n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
A : Set α
hA : MeasurableSet A
⊢ μ A = (μ - ν.withDensity ξ, ξ).1 A + (ν.withDensity (μ - ν.withDensity ξ, ξ).2) A

Tactic:
Measure.sub_apply hA hle,

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) = g n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
A : Set α
hA : MeasurableSet A
⊢ μ A = μ A - (ν.withDensity ξ) A + (ν.withDensity (μ - ν.withDensity ξ, ξ).2) A
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x y : α
hx : x ∈ s
hy : y ∈ s
this : ∀ {x y : α}, x ∈ s → y ∈ s → y ≤ x → edist (f x) (f y) ≤ eVariationOn f s
hxy : ¬y ≤ x
⊢ edist (f x) (f y) ≤ eVariationOn f s

Tactic:
rw [edist_comm]
    exact this hy hx (le_of_not_ge hxy)

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
x✝ : x ∈ {x | v.limRatioMeas hρ x = ∞}
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : ρ o < ∞
s : Set α := {x | v.limRatioMeas hρ x = ∞} ∩ o
ρs : ρ s ≠ ∞
q : ℝ≥0
hq : 1 ≤ q
y : α
hy : y ∈ s
this : v.limRatioMeas hρ y = ∞
⊢ y ∈ {x | ↑q < v.limRatioMeas hρ x}

Tactic:
simp only [this, ENNReal.coe_lt_top, mem_ofPred_eq]

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := ⋯
f : α → ℝ≥0∞ := ⋯
f_meas : Measurable f
A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0})
B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞})
n : ℤ
I : Set ℝ≥0∞ := ⋯
M : MeasurableSet (s ∩ f ⁻¹' I)
x : α
hx : x ∈ s ∩ f ⁻¹' I
⊢ ↑t ^ (n + 1) = ↑(t ^ (n + 1))

Tactic:
rw [ENNReal.coe_zpow t_ne_zero']

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

Goal state before:
pp: α : Type u_1
inst✝³ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → ℝ≥0∞
hf : Measurable f
h'f : ∫⁻ (y : α), f y ∂μ ≠ ∞
ρ : Measure α := ⋯
this : IsFiniteMeasure ρ
x : α
hx : Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))
h'x : (μ.withDensity f).rnDeriv μ x = f x
⊢ (fun a ↦ ρ a / μ a) =ᶠ[v.filterAt x] fun a ↦ (∫⁻ (y : α) in a, f y ∂μ) / μ a

Tactic:
filter_upwards [v.eventually_filterAt_measurableSet x] with a ha

Goal state after:
α : Type u_1
inst✝³ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → ℝ≥0∞
hf : Measurable f
h'f : ∫⁻ (y : α), f y ∂μ ≠ ∞
ρ : Measure α := μ.withDensity f
this : IsFiniteMeasure ρ
x : α
hx : Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))
h'x : (μ.withDensity f).rnDeriv μ x = f x
a : Set α
ha : MeasurableSet a
⊢ ρ a / μ a = (∫⁻ (y : α) in a, f y ∂μ) / μ a
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case refine_2
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
a : α
l : E
h : (𝓝[s ∩ Iio a] a).NeBot
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
b : α
hb : b ∈ s ∩ Iio a
⊢ eVariationOn f (s ∩ Iio a) + edist (f a) l ≤ eVariationOn f (s ∩ Iic a)

Tactic:
have : Nonempty ((n : ℕ) × { u // StrictMonoOn u (Iic n) ∧ ∀ i ∈ Iic n, u i ∈ s ∩ Iio a }) :=
      ⟨0, ⟨fun i ↦ b, by grind [StrictMonoOn]⟩⟩

Goal state after:
case refine_2
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
a : α
l : E
h : (𝓝[s ∩ Iio a] a).NeBot
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
b : α
hb : b ∈ s ∩ Iio a
this : Nonempty ((n : ℕ) × { u // StrictMonoOn u (Iic n) ∧ ∀ i ∈ Iic n, u i ∈ s ∩ Iio a })
⊢ eVariationOn f (s ∩ Iio a) + edist (f a) l ≤ eVariationOn f (s ∩ Iic a)
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
u : ℕ → Set α
u_open : ∀ (n : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (u n)
u_univ : ⋃ n, u n = univ
hu : ∀ (n : ℕ), IntegrableOn f (u n) μ
n : ℕ
⊢ Integrable ((u n).indicator f) μ

Tactic:
exact (integrable_indicator_iff (u_open n).measurableSet).2 (hu n)

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
L : Filter α
hL : ∀ y ∈ s, s ∩ Ici y ∈ L
x₀ : α
hx₀ : x₀ ∈ s
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ < ε
H : ∃ᶠ (x : α) in L, ε ≤ eVariationOn f (s ∩ Ici x)
y : α → α
y_mem : ∀ y_1 ∈ s, y y_1 ∈ s ∩ Ici y_1
le_y : ∀ y_1 ∈ s, δ ≤ eVariationOn f (s ∩ Icc y_1 (y y_1))
v : ℕ → α := fun n ↦ y^[n] x₀
n : ℕ
⊢ v n ∈ s

Tactic:
induction n with
    | zero => simpa [v] using hx₀
    | succ n ih =>
      simp only [Function.iterate_succ', Function.comp_apply, v]
      exact (y_mem _ ih).1

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
a : A
h : ∀ x ∈ spectrum R a, 1 ≤ x
ha : p a
⊢ 1 ≤ a

Tactic:
by
  simpa using algebraMap_le_of_le_spectrum h

Goal state after: