mathlib-initiative/mathlib-tactics huggingface dataset:
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 := 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
A : HasFiniteFPowerSeriesOnBall g q x (n + 1) r
⊢ (iteratedFDerivWithin 𝕜 n f s x) v = ∑ σ, (p n) fun i ↦ v (σ i)

Tactic:
have B : HasFPowerSeriesWithinOnBall g q s x r :=
    A.toHasFPowerSeriesOnBall.hasFPowerSeriesWithinOnBall

Goal state after:
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
A : HasFiniteFPowerSeriesOnBall g q x (n + 1) r
B : HasFPowerSeriesWithinOnBall g q s x r
⊢ (iteratedFDerivWithin 𝕜 n f s x) v = ∑ σ, (p n) fun i ↦ v (σ i)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: p : ℝ
n : ℕ
h : ↑n ≤ p
⊢ ContDiff ℝ ↑n fun x ↦ x ^ p

Tactic:
induction n generalizing p with
  | zero => exact contDiff_zero.2 (continuous_id.rpow_const fun x => Or.inr <| by simpa using h)
  | succ n ihn =>
    have h1 : 1 ≤ p := le_trans (by simp) h
    rw [Nat.cast_add_one, ← le_sub_iff_add_le] at h
    rw [Nat.cast_add_one, contDiff_succ_iff_deriv, deriv_rpow_const' p]
    simp only [WithTop.natCast_ne_top, analyticOn_univ, IsEmpty.forall_iff, true_and]
    exact ⟨differentiable_rpow_const h1, contDiff_const.mul (ihn h)⟩

Goal state after:
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
s : Set E
x : E
h : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
σ : Perm (Fin n)
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
hp : HasFPowerSeriesWithinOnBall f p s x r
| ∑ σ, (p n) fun i ↦ v (σ i)

Tactic:
← Equiv.sum_comp (Equiv.mulLeft σ)

Goal state after:
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
s : Set E
x : E
h : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
σ : Perm (Fin n)
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
hp : HasFPowerSeriesWithinOnBall f p s x r
| ∑ i, (p n) fun i_1 ↦ v (((Equiv.mulLeft σ) i) i_1)
Module: Mathlib.Analysis.AperiodicOrder.Delone.Basic

Goal state before:
pp: X : Type u_1
inst✝ : MetricSpace X
D : DeloneSet X
x y : X
hy : y ∈ D.carrier
hdist : (x, y) ∈ {(x, y) | edist x y ≤ ↑D.coveringRadius}
⊢ dist x y ≤ ↑D.coveringRadius

Tactic:
simpa [edist_dist] using hdist

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

Goal state before:
pp: case coe
y : EReal
a✝ : ℝ
h : ↑a✝ < y
⊢ (↑a✝).exp < y.exp

Tactic:
induction y

Goal state after:
case coe.bot
a✝ : ℝ
h : ↑a✝ < ⊥
⊢ (↑a✝).exp < ⊥.exp

case coe.coe
a✝¹ a✝ : ℝ
h : ↑a✝¹ < ↑a✝
⊢ (↑a✝¹).exp < (↑a✝).exp

case coe.top
a✝ : ℝ
h : ↑a✝ < ⊤
⊢ (↑a✝).exp < ⊤.exp
Module: Mathlib.Analysis.SpecialFunctions.Log.ERealExp

Goal state before:
pp: case coe
y : EReal
a✝ : ℝ
⊢ (↑a✝ + y).exp = (↑a✝).exp * y.exp

Tactic:
induction y

Goal state after:
case coe.bot
a✝ : ℝ
⊢ (↑a✝ + ⊥).exp = (↑a✝).exp * ⊥.exp

case coe.coe
a✝¹ a✝ : ℝ
⊢ (↑a✝¹ + ↑a✝).exp = (↑a✝¹).exp * (↑a✝).exp

case coe.top
a✝ : ℝ
⊢ (↑a✝ + ⊤).exp = (↑a✝).exp * ⊤.exp
Module: Mathlib.Analysis.SpecialFunctions.Log.ERealExp

Goal state before:
pp: case top
y : EReal
⊢ (⊤ + y).exp = ⊤.exp * y.exp

Tactic:
induction y

Goal state after:
case top.bot
⊢ (⊤ + ⊥).exp = ⊤.exp * ⊥.exp

case top.coe
a✝ : ℝ
⊢ (⊤ + ↑a✝).exp = ⊤.exp * (↑a✝).exp

case top.top
⊢ (⊤ + ⊤).exp = ⊤.exp * ⊤.exp
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: case coe
a✝ : ℝ
⊢ (↑a✝).exp.log = ↑a✝

Tactic:
rw [exp_coe, log_ofReal, if_neg (not_le.mpr (Real.exp_pos _)), Real.log_exp]

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

Goal state before:
pp: u : ℕ → EReal
h : 0 < linearGrowthInf u
M a : ℝ
a_0 : 0 < a
a_v : ↑a < linearGrowthInf u
n : ℕ
hn : M / a ≤ ↑n
⊢ ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → ↑M < ↑a * ↑b

Tactic:
refine ⟨n + 1, fun k k_n ↦ ?_⟩

Goal state after:
u : ℕ → EReal
h : 0 < linearGrowthInf u
M a : ℝ
a_0 : 0 < a
a_v : ↑a < linearGrowthInf u
n : ℕ
hn : M / a ≤ ↑n
k : ℕ
k_n : n + 1 ≤ k
⊢ ↑M < ↑a * ↑k
Module: Mathlib.MeasureTheory.Covering.VitaliFamily

Goal state before:
pp: X : Type u_1
inst✝¹ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
f : X → Set (Set X)
s : Set X
h : v.FineSubfamilyOn f s
inst✝ : SecondCountableTopology X
ρ : Measure X
hρ : ρ ≪ μ
⊢ ρ (s \ ⋃ p ∈ h.index, h.covering p) + ρ (⋃ p ∈ h.index, h.covering p) = ∑' (p : ↑h.index), ρ (h.covering ↑p)

Tactic:
by
      rw [hρ h.measure_sdiff_biUnion, zero_add,
        measure_biUnion h.index_countable h.covering_disjoint fun x hx => h.measurableSet_u hx]

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

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
s : Set X
f : X → Set (Set X)
fset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : X → Set (Set X) := fun x ↦ f x ∩ v.setsAt x
x : X
hx : x ∈ s
ε : ℝ
εpos : ε > 0
⊢ ∃ t ∈ g x, t ⊆ closedBall x ε

Tactic:
obtain ⟨t, tf, ht⟩ : ∃ t ∈ f x, t ⊆ closedBall x (min ε δ) :=
        ffine x hx (min ε δ) (lt_min εpos δpos)

Goal state after:
X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
s : Set X
f : X → Set (Set X)
fset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : X → Set (Set X) := fun x ↦ f x ∩ v.setsAt x
x : X
hx : x ∈ s
ε : ℝ
εpos : ε > 0
t : Set X
tf : t ∈ f x
ht : t ⊆ closedBall x (min ε δ)
⊢ ∃ t ∈ g x, t ⊆ closedBall x ε
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case neg.inr.refine_1
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
a_0 : 0 < a
b_0 : 0 < b
a_top : a ≠ ⊤
a' : EReal
a_a' : a' < a
a_0' : 0 < a'
a_a_inv' : a⁻¹ < a'⁻¹
v_a' : ∀ᶠ (n : ℕ) in atTop, ↑(v n) ≤ a' * ↑n
u_b : ∃ᶠ (x : ℕ) in atTop, (u x ≤ b * ↑x ∧ 0 < x) ∧ ∃ m, a⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a'⁻¹ * ↑x
M M' : ℕ
aM_M' : a * ↑M ≤ ↑M'
n : ℕ
n_M' : M' ≤ n
un_bn : u n ≤ b * ↑n
right✝ : 0 < n
k : ℕ
an_k : a⁻¹ * ↑n ≤ ↑k
k_an' : ↑k ≤ a'⁻¹ * ↑n
⊢ M ≤ k

Tactic:
mul_comm a,

Goal state after:
case neg.inr.refine_1
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
a_0 : 0 < a
b_0 : 0 < b
a_top : a ≠ ⊤
a' : EReal
a_a' : a' < a
a_0' : 0 < a'
a_a_inv' : a⁻¹ < a'⁻¹
v_a' : ∀ᶠ (n : ℕ) in atTop, ↑(v n) ≤ a' * ↑n
u_b : ∃ᶠ (x : ℕ) in atTop, (u x ≤ b * ↑x ∧ 0 < x) ∧ ∃ m, a⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a'⁻¹ * ↑x
M M' : ℕ
aM_M' : ↑M * a ≤ ↑M'
n : ℕ
n_M' : M' ≤ n
un_bn : u n ≤ b * ↑n
right✝ : 0 < n
k : ℕ
an_k : a⁻¹ * ↑n ≤ ↑k
k_an' : ↑k ≤ a'⁻¹ * ↑n
⊢ M ≤ k
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
p : ↑s
this : Countable ↑(s ∩ Ioi ↑p)
⊢ μ (⋃ q, u' ↑p ∩ v ↑p ↑q) ≤ ∑' (q : ↑(s ∩ Ioi ↑p)), μ (u' ↑p ∩ v ↑p ↑q)

Tactic:
apply measure_iUnion_le

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
h₁ : MeasurableSet s
h₂ : ν ≤ μ
measure_sub : Measure α := ofMeasurable (fun t x ↦ μ t - ν t) ⋯ ⋯
t : Set α
h_t_measurable_set : MeasurableSet t
⊢ ν t + measure_sub t = μ t

Tactic:
rw [MeasureTheory.Measure.ofMeasurable_apply _ h_t_measurable_set, add_comm,
      tsub_add_cancel_of_le (h₂ t)]

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

Goal state before:
pp: case pos.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0
hr : r ≠ 0
hl : μ.HaveLebesgueDecomposition ν
⊢ μ = μ.singularPart ν + (r • ν).withDensity (r⁻¹ • μ.rnDeriv ν)

Tactic:
rw [ENNReal.smul_def r, withDensity_smul_measure, ← withDensity_smul]
      swap; · exact (measurable_rnDeriv _ _).const_smul _
      convert! haveLebesgueDecomposition_add μ ν
      ext x
      simp only [Pi.smul_apply]
      rw [← ENNReal.smul_def, smul_inv_smul₀ hr]

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
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n ↦ s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n ↦ s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ∞
J : (ρ + μ) (w n) ≠ ∞
A :
  ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
⊢ ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ↑q * μ (w n ∩ toMeasurable (ρ + μ) (u m))

Tactic:
rw [inter_comm, measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) J]

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
h : LocallyBoundedVariationOn f s
⊢ ∃ p q,
    MonotoneOn p s ∧ MonotoneOn q s ∧ f = p - q ∧ ∀ x ∈ s, ∀ y ∈ s, p y - p x + (q y - q x) = variationOnFromTo f s x y

Tactic:
rcases eq_empty_or_nonempty s with (rfl | ⟨c, cs⟩)
  · refine ⟨f, 0, subsingleton_empty.monotoneOn _, subsingleton_empty.monotoneOn _,
      (sub_zero f).symm, fun x hx y hy ↦ by simp at hx⟩
  refine ⟨fun x ↦ (variationOnFromTo f s c x + f x) / 2,
    fun x ↦ (variationOnFromTo f s c x - f x) / 2, ?_, ?_, ?_, ?_⟩
  · intro x hx y hy hxy
    dsimp
    gcongr 1
    simpa using variationOnFromTo.add_self_monotoneOn h cs hx hy hxy
  · intro x hx y hy hxy
    dsimp
    gcongr 1
    simpa using variationOnFromTo.sub_self_monotoneOn h cs hx hy hxy
  · ext
    simp
    ring
  · intro x hx y hy
    rw [← variationOnFromTo.add h hx cs hy, variationOnFromTo.eq_neg_swap]
    ring

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 t : Set α
x✝ : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s ∪ t }
n : ℕ
u : ℕ → α
hu : Monotone u
ust : ∀ (i : ℕ), u i ∈ s ∪ t
v : ℕ → α
m : ℕ
hv : Monotone v
vst : ∀ (i : ℕ), v i ∈ s ∪ t
huv : ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j))
N : ℕ
hN : N ∈ Iio m
hs : IsGreatest s (v N)
ht : IsLeast t (v N)
⊢ ∑ i ∈ Finset.Ico 0 N, edist (f (v (i + 1))) (f (v i)) + ∑ i ∈ Finset.Ico N m, edist (f (v (i + 1))) (f (v i)) ≤
    eVariationOn f s + eVariationOn f t

Tactic:
apply add_le_add <;> refine sum_le_of_monotoneOn_Icc (hv.monotoneOn _) fun i hi ↦ ?_

Goal state after:
case h₁
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
x✝ : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s ∪ t }
n : ℕ
u : ℕ → α
hu : Monotone u
ust : ∀ (i : ℕ), u i ∈ s ∪ t
v : ℕ → α
m : ℕ
hv : Monotone v
vst : ∀ (i : ℕ), v i ∈ s ∪ t
huv : ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j))
N : ℕ
hN : N ∈ Iio m
hs : IsGreatest s (v N)
ht : IsLeast t (v N)
i : ℕ
hi : i ∈ Icc 0 N
⊢ v i ∈ s

case h₂
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
x✝ : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s ∪ t }
n : ℕ
u : ℕ → α
hu : Monotone u
ust : ∀ (i : ℕ), u i ∈ s ∪ t
v : ℕ → α
m : ℕ
hv : Monotone v
vst : ∀ (i : ℕ), v i ∈ s ∪ t
huv : ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j))
N : ℕ
hN : N ∈ Iio m
hs : IsGreatest s (v N)
ht : IsLeast t (v N)
i : ℕ
hi : i ∈ Icc N m
⊢ v i ∈ t
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: case e'_3
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
t : Measure α := μ.withDensity (ρ.rnDeriv μ)
eq_add : ρ = ρ.singularPart μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x))
a✝ : α
Ax : Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt a✝) (𝓝 0)
Bx : t.rnDeriv μ a✝ = ρ.rnDeriv μ a✝
Cx : Tendsto (fun a ↦ t a / μ a) (v.filterAt a✝) (𝓝 (t.rnDeriv μ a✝))
a : Set α
⊢ ρ a / μ a = (ρ.singularPart μ) a / μ a + t a / μ a

Tactic:
conv_lhs => rw [eq_add]

Goal state after:
case e'_3
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
t : Measure α := μ.withDensity (ρ.rnDeriv μ)
eq_add : ρ = ρ.singularPart μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x))
a✝ : α
Ax : Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt a✝) (𝓝 0)
Bx : t.rnDeriv μ a✝ = ρ.rnDeriv μ a✝
Cx : Tendsto (fun a ↦ t a / μ a) (v.filterAt a✝) (𝓝 (t.rnDeriv μ a✝))
a : Set α
⊢ (ρ.singularPart μ + t) a / μ a = (ρ.singularPart μ) a / μ a + t a / μ a
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case pos
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
h : (𝓝[s ∩ Iio a] a).NeBot
h' : ContinuousWithinAt f (s ∩ Iic a) a
ha : a ∈ s
this : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 (f a))
⊢ eVariationOn f (s ∩ Iio a) = eVariationOn f (s ∩ Iic a)

Tactic:
simp [eVariationOn_on_inter_Iic_eq_Iio_add_edist h ha this]

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

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
a : A
ha : p a
⊢ cfc 0 a = 0

Tactic:
exact cfc_apply (0 : R → R) a ▸ map_zero (cfcHom ha)

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
I : ∀ (n : ℕ), v n ∈ s
⊢ False

Tactic:
have J (n : ℕ) : n * δ ≤ eVariationOn f s := calc
    n * δ
    _ = ∑ i ∈ Finset.range n, δ := by simp
    _ ≤ ∑ i ∈ Finset.range n, eVariationOn f (s ∩ Icc (v i) (v (i + 1))) := by
      gcongr with i hi
      simp only [Function.iterate_succ', Function.comp_apply, v]
      grind
    _ = eVariationOn f (s ∩ Icc (v 0) (v n)) := by
      apply eVariationOn.sum
      · apply monotone_nat_of_le_succ (fun n ↦ ?_)
        simp only [Function.iterate_succ', Function.comp_apply, v]
        exact (y_mem _ (I n)).2
      · grind
    _ ≤ eVariationOn f s := mono _ inter_subset_left

Goal state after:
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
I : ∀ (n : ℕ), v n ∈ s
J : ∀ (n : ℕ), ↑n * δ ≤ eVariationOn f s
⊢ False
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case refine_2
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
inst✝ : CompleteSpace 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
h : L.NeBot
ε : ℝ≥0∞
εpos : ε > 0
y : α
y_mem : y ∈ s
hy : eVariationOn f (s ∩ Ici y) < ε
⊢ ∀ x ∈ f '' (s ∩ Ici y), ∀ y_1 ∈ f '' (s ∩ Ici y), edist x y_1 < ε

Tactic:
rintro - ⟨a, ha, rfl⟩ - ⟨b, hb, rfl⟩

Goal state after:
case refine_2
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
inst✝ : CompleteSpace 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
h : L.NeBot
ε : ℝ≥0∞
εpos : ε > 0
y : α
y_mem : y ∈ s
hy : eVariationOn f (s ∩ Ici y) < ε
a : α
ha : a ∈ s ∩ Ici y
b : α
hb : b ∈ s ∩ Ici y
⊢ edist (f a) (f b) < ε
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring R
inst✝⁹ : ContinuousStar R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMul S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ (cfcHom ha) { toFun := (spectrum R a).domRestrict fun x ↦ s • f x, continuous_toFun := ⋯ } =
    s • (cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }

Tactic:
simp_rw [← Pi.smul_def, ← smul_one_smul R s _]

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring R
inst✝⁹ : ContinuousStar R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMul S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ (cfcHom ha) { toFun := (spectrum R a).domRestrict ((s • 1) • f), continuous_toFun := ⋯ } =
    (s • 1) • (cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : p a ∧ ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

Tactic:
obtain ⟨ha, hf⟩ := h
    rw [cfc_apply f a, ← map_star, cfc_apply _ a]
    congr

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

Goal state before:
pp: case f'
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
ha : p a
hg : ContinuousOn g (f '' spectrum R a)
hf : ContinuousOn f (spectrum R a)
this : ContinuousOn (g ∘ f) (spectrum R a)
sp_eq : spectrum R ((cfcHom ⋯) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }) = f '' spectrum R a
⊢ C(↑(spectrum R a), ↑(spectrum R ((cfcHom ⋯) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })))

Tactic:
exact ContinuousMap.mk _ <| hf.domRestrict.codRestrict fun x ↦ by rw [sp_eq]; use x.1; simp

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✝¹⁰ : Semifield R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousInv₀ R
f : R → R
a : A
hf' : ∀ x ∈ spectrum R a, f x ≠ 0
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ cfc (fun x ↦ (f x)⁻¹) a = (cfc f a)⁻¹ʳ

Tactic:
← val_inv_cfcUnits f a hf',

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Semifield R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousInv₀ R
f : R → R
a : A
hf' : ∀ x ∈ spectrum R a, f x ≠ 0
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ ↑(cfcUnits f a hf' hf ha)⁻¹ = (cfc f a)⁻¹ʳ