mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: f : ℝ → ℝ
f' x p : ℝ
s : Set ℝ
hf : HasDerivWithinAt f f' s x
hx : f x ≠ 0 ∨ 1 ≤ p
⊢ HasDerivWithinAt (fun y ↦ f y ^ p) (f' * p * f x ^ (p - 1)) s x

Tactic:
convert! (hasDerivAt_rpow_const hx).comp_hasDerivWithinAt x hf using 1
  ring

Goal state after:
Module: Mathlib.Topology.MetricSpace.MetricSeparated

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
s : Set X
ε K₁ : ℝ≥0
f : X → Y
hs : IsSeparated (↑ε) s
hf : AntilipschitzWith K₁ f
hK₁ : 0 < K₁
⊢ IsSeparated (↑(ε / K₁)) (f '' s)

Tactic:
rintro x' ⟨x, hx, rfl⟩ y' ⟨y, hy, rfl⟩ hne
  have hmul : (↑ε : ℝ≥0∞) < edist (f x) (f y) * ↑K₁ :=
    lt_of_lt_of_le (hs hx hy (by grind)) (by rw [mul_comm]; exact hf x y)
  exact ENNReal.coe_div hK₁.ne' ▸ ENNReal.div_lt_of_lt_mul hmul

Goal state after:
Module: Mathlib.Analysis.AperiodicOrder.Delone.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
D : DeloneSet X
carrier : Set X
packingRadius coveringRadius : ℝ≥0
h_carrier : carrier = D.carrier
h_packing : packingRadius = D.packingRadius
h_covering : coveringRadius = D.coveringRadius
⊢ IsSeparated (↑packingRadius) carrier

Tactic:
by
    simpa [h_carrier, h_packing] using D.isSeparated_packingRadius

Goal state after:
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
    · simp
    · simp only [← EReal.coe_add, exp_coe]
      rw [← ENNReal.ofReal_mul (Real.exp_nonneg _), Real.exp_add]
    · simp only [EReal.coe_add_top, exp_top, exp_coe]
      rw [ENNReal.mul_top]
      simp [Real.exp_pos]

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

Goal state before:
pp: b : ℝ≥0∞
hb : 1 < b
⊢ Filter.Tendsto (fun x ↦ ↑x * b.log) Filter.atBot (𝓝 ⊥)

Tactic:
convert! EReal.Tendsto.mul_const tendsto_coe_atBot _ _

Goal state after:
case e'_5
b : ℝ≥0∞
hb : 1 < b
⊢ ⊥ = ⊥ * b.log

case convert_2
b : ℝ≥0∞
hb : 1 < b
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊥

case convert_3
b : ℝ≥0∞
hb : 1 < b
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊤
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: b : ℝ≥0∞
hb : b < 1
⊢ Filter.Tendsto (fun x ↦ ↑x * b.log) Filter.atBot (𝓝 ⊤)

Tactic:
convert! EReal.Tendsto.mul_const tendsto_coe_atBot _ _

Goal state after:
case e'_5
b : ℝ≥0∞
hb : b < 1
⊢ ⊤ = ⊥ * b.log

case convert_2
b : ℝ≥0∞
hb : b < 1
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊥

case convert_3
b : ℝ≥0∞
hb : b < 1
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊤
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inl
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : Tendsto v atTop atTop
m : ℕ
hum : ¬u m = ⊥ m
um_uvn : ∀ᶠ (n : ℕ) in atTop, u m ≤ (u ∘ v) n
hum' : u m = ⊤
⊢ 0 ≤ linearGrowthInf fun x ↦ u m

Tactic:
hum',

Goal state after:
case inl
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : Tendsto v atTop atTop
m : ℕ
hum : ¬u m = ⊥ m
um_uvn : ∀ᶠ (n : ℕ) in atTop, u m ≤ (u ∘ v) n
hum' : u m = ⊤
⊢ 0 ≤ linearGrowthInf fun x ↦ ⊤
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
ε N : ℝ≥0
p : ℕ
εpos : 0 < ε
s : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p
s_lt_top : μ s < ∞
⊢ ∫⁻ (x : α) in s, g x ∂μ + ↑ε * μ s = ∫⁻ (x : α) in s, g x ∂μ + ∫⁻ (x : α) in s, ↑ε ∂μ

Tactic:
simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]

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

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
hf₀ : 0 ≤ᵐ[μ] f
hμ : μ ≠ 0
⊢ ENNReal.ofReal (μ univ)⁻¹.toReal * ENNReal.ofReal (∫ (x : α), f x ∂μ) = (∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ) / μ univ

Tactic:
ofReal_toReal (inv_ne_top.2 <| measure_univ_ne_zero.2 hμ),

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
hf₀ : 0 ≤ᵐ[μ] f
hμ : μ ≠ 0
⊢ (μ univ)⁻¹ * ENNReal.ofReal (∫ (x : α), f x ∂μ) = (∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ) / μ univ
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
k : Set α := ⋃ a ∈ w, B ↑a
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
⊢ z ∉ k

Tactic:
simp only [k, not_exists, exists_prop, mem_iUnion, forall_exists_index,
        SetCoe.exists, not_and, exists_and_right]

Goal state after:
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
k : Set α := ⋃ a ∈ w, B ↑a
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
⊢ ∀ (x : ι) (x_1 : x ∈ v), ⟨x, ⋯⟩ ∈ w → z ∉ B x
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
this : μ.HaveLebesgueDecomposition ν
⊢ μ.singularPart ν = μ

Tactic:
conv_rhs => rw [← singularPart_add_rnDeriv μ ν]

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
this : μ.HaveLebesgueDecomposition ν
⊢ μ.singularPart ν = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
A : Set α
hA : MeasurableSet A
⊢ (s.restrict (S ∩ T)ᶜ) A = ((μ.singularPart ν).restrict (S ∩ T)ᶜ) A

Tactic:
have hf : ν.withDensity f (A ∩ (S ∩ T)ᶜ) = 0 := by
      refine withDensity_absolutelyContinuous ν _ ?_
      rw [← nonpos_iff_eq_zero]
      exact hνinter ▸ measure_mono inter_subset_right

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf✝ : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
A : Set α
hA : MeasurableSet A
hf : (ν.withDensity f) (A ∩ (S ∩ T)ᶜ) = 0
⊢ (s.restrict (S ∩ T)ᶜ) A = ((μ.singularPart ν).restrict (S ∩ T)ᶜ) A
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : μ.HaveLebesgueDecomposition ν
hμs : (μ.singularPart ν) sᶜ = 0
hνs : (ν.withDensity (μ.rnDeriv ν)) s = 0
⊢ μ.singularPart ν = μ.restrict s

Tactic:
conv_rhs => rw [← singularPart_add_rnDeriv μ ν]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : μ.HaveLebesgueDecomposition ν
hμs : (μ.singularPart ν) sᶜ = 0
hνs : (ν.withDensity (μ.rnDeriv ν)) s = 0
⊢ μ.singularPart ν = (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).restrict s
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
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ρ o = 0
μo : μ oᶜ = 0
⊢ (↑ε)⁻¹ * ρ o = 0

Tactic:
ρo,

Goal state after:
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
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ρ o = 0
μo : μ oᶜ = 0
⊢ (↑ε)⁻¹ * 0 = 0
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν
n : ℕ
⊢ ⨆ k, ⨆ (_ : k ≤ n), f k ∈ measurableLE μ ν

Tactic:
convert! iSup_mem_measurableLE f hf n

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν
n : ℕ
x✝ : α
⊢ (⨆ k, ⨆ (_ : k ≤ n), f k) x✝ = ⨆ k, ⨆ (_ : k ≤ n), f k x✝
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: case hgf
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
this :
  Tendsto (fun y ↦ (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[<] x)
    (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x, (↑hf.stieltjesFunction b - f x) / (b - x) ≤ (f b - f x) / (b - x)

pp: case hfh
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
this :
  Tendsto (fun y ↦ (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[<] x)
    (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x, (f b - f x) / (b - x) ≤ (↑hf.stieltjesFunction (b + -1 * (b - x) ^ 2) - f x) / (b - x)

Tactic:
· filter_upwards [self_mem_nhdsWithin]
      rintro y hy
      rw [mem_Iio, ← sub_neg] at hy
      apply div_le_div_of_nonpos_of_le hy.le
      exact (sub_le_sub_iff_right _).2 (hf.le_rightLim (le_refl _))

Goal state after:
case hfh
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
this :
  Tendsto (fun y ↦ (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[<] x)
    (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x, (f b - f x) / (b - x) ≤ (↑hf.stieltjesFunction (b + -1 * (b - x) ^ 2) - f x) / (b - x)
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

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 : LocallyBoundedVariationOn f s
a b c : α
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
⊢ variationOnFromTo f s a c = variationOnFromTo f s a b + variationOnFromTo f s b c

Tactic:
refine additive_of_total (· ≤ · : α → α → Prop) (variationOnFromTo f s) (· ∈ s) ?_ ?_ ha hb hc

Goal state after:
case refine_1
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a b c : α
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
⊢ ∀ {a b : α}, a ∈ s → b ∈ s → variationOnFromTo f s a b + variationOnFromTo f s b a = 0

case refine_2
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a b c : α
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
⊢ ∀ {a b c : α},
    a ≤ b →
      b ≤ c → a ∈ s → b ∈ s → c ∈ s → variationOnFromTo f s a c = variationOnFromTo f s a b + variationOnFromTo f s b c
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
x y : α
hx : x ∈ s
hy : y ∈ s
⊢ edist (f x) (f y) ≤ eVariationOn f s

Tactic:
wlog hxy : y ≤ x generalizing x y

Goal state after:
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

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
hxy : y ≤ x
⊢ edist (f x) (f y) ≤ eVariationOn f s
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
ι : Type u_3
F : ι → α → E
p : Filter ι
f : α → E
s : Set α
Ffs : ∀ x ∈ s, Tendsto (fun i ↦ F i x) p (𝓝 (f x))
v : ℝ≥0∞
hv : v < eVariationOn f s
n : ℕ
u : ℕ → α
um : Monotone u
us : ∀ (i : ℕ), u i ∈ s
hlt : v < ∑ i ∈ Finset.range (n, ⟨u, ⋯⟩).1, edist (f (↑(n, ⟨u, ⋯⟩).2 (i + 1))) (f (↑(n, ⟨u, ⋯⟩).2 i))
this :
  Tendsto (fun j ↦ ∑ i ∈ Finset.range n, edist (F j (u (i + 1))) (F j (u i))) p
    (𝓝 (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))))
⊢ ∀ᶠ (n : ι) in p, v < eVariationOn (F n) s

Tactic:
exact (this.eventually_const_lt hlt).mono fun i h => h.trans_le (sum_le um us)

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
s : Set α
t : Set α := toMeasurable μ s
A : ∀ᵐ (x : α) ∂μ.restrict s, Tendsto (fun a ↦ μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 (t.indicator 1 x))
a✝ : α
hx : a✝ ∈ toMeasurable μ s
⊢ t.indicator 1 a✝ = 1

Tactic:
simp only [t, hx, Pi.one_apply, indicator_of_mem]

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

Goal state before:
pp: case pos.e_6
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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : p a ∧ ContinuousOn g (spectrum R a)
⊢ (spectrum R a).domRestrict f = (spectrum R a).domRestrict g

Tactic:
exact Set.domRestrict_eq_iff.mpr hfg

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

Goal state before:
pp: case mp
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✝ : StarModule R A
a : A
ha : p a
f : R → R
⊢ (fun x ↦ cfc x a) f ∈ ↑(cfcHom ha).range

Tactic:
exact cfc_cases _ a f (zero_mem _) fun hf ha ↦ ⟨_, rfl⟩

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

Goal state before:
pp: case ha
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
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
H : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x) + edist (f x) l) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))
y : α
hy : y ∈ s ∩ Iio x
N : 𝓝[s ∩ Ici y ∩ Iio x] x = 𝓝[s ∩ Iio x] x
⊢ x ∈ s ∩ Ici y

Tactic:
exact ⟨hx, hy.2.le⟩

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
a : Aˣ
ha : p ↑a
hx : 0 ∈ spectrum R ↑a
⊢ False

Tactic:
exact spectrum.zero_notMem R a.isUnit hx

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✝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
f g : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ g x
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : p a
⊢ cfc f a ≤ cfc g a

pp: case neg
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
f g : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ g x
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : ¬p a
⊢ cfc f a ≤ cfc g a

Tactic:
· rw [cfc_apply f a, cfc_apply g a]
    exact cfcHom_mono ha fun x ↦ h x.1 x.2

Goal state after:
case neg
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
f g : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ g x
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : ¬p a
⊢ cfc f a ≤ cfc g a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos.e_6
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h : p a ∧ ContinuousOn g (σₙ R a) ∧ g 0 = 0
⊢ (σₙ R a).domRestrict f = (σₙ R a).domRestrict g

Tactic:
exact Set.domRestrict_eq_iff.mpr hfg

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

Goal state before:
pp: case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
hg : ¬ContinuousOn g (σₙ R a)
⊢ cfcₙ f a = cfcₙ g a

Tactic:
rw [cfcₙ_apply_of_not_continuousOn a hg, cfcₙ_apply_of_not_continuousOn]
      exact fun hf ↦ hg (hf.congr hfg.symm)

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✝¹⁴ : CommRing R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalRing R
inst✝⁹ : ContinuousStar R
inst✝⁸ : ContinuousSqrt R
inst✝⁷ : StarOrderedRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ cfc f a ≤ 1 ↔ ∀ x ∈ spectrum R a, f x ≤ 1

Tactic:
by
  simpa using cfc_le_algebraMap_iff f 1 a

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ bernsteinPolynomial ℤ 3 2 = 3 * X ^ 2 - 3 * X ^ 3

Tactic:
norm_num [bernsteinPolynomial, choose]

Goal state after:
R : Type u_1
inst✝ : CommRing R
⊢ 3 * X ^ 2 * (1 - X) = 3 * X ^ 2 - 3 * X ^ 3