mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.FDeriv.Extend

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
x : ℝ
f_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y
hf : ContinuousAt f x
hg : ContinuousAt g x
diff : DifferentiableOn ℝ f (Ioi x)
this : Tendsto g (𝓝[>] x) (𝓝 (g x))
⊢ g =ᶠ[𝓝[>] x] fun x ↦ deriv f x

Tactic:
apply mem_of_superset self_mem_nhdsWithin fun y hy => _

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
x : ℝ
f_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y
hf : ContinuousAt f x
hg : ContinuousAt g x
diff : DifferentiableOn ℝ f (Ioi x)
this : Tendsto g (𝓝[>] x) (𝓝 (g x))
⊢ ∀ y ∈ Ioi x, y ∈ {x | (fun x ↦ g x = (fun x ↦ deriv f x) x) x}
Module: Mathlib.Analysis.Calculus.FDeriv.Extend

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
x : ℝ
f_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y
hf : ContinuousAt f x
hg : ContinuousAt g x
A : HasDerivWithinAt f (g x) (Ici x) x
diff : DifferentiableOn ℝ f (Iio x)
this : Tendsto g (𝓝[<] x) (𝓝 (g x))
⊢ g =ᶠ[𝓝[<] x] fun x ↦ deriv f x

Tactic:
apply mem_of_superset self_mem_nhdsWithin fun y hy => _

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
x : ℝ
f_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y
hf : ContinuousAt f x
hg : ContinuousAt g x
A : HasDerivWithinAt f (g x) (Ici x) x
diff : DifferentiableOn ℝ f (Iio x)
this : Tendsto g (𝓝[<] x) (𝓝 (g x))
⊢ ∀ y ∈ Iio x, y ∈ {x | (fun x ↦ g x = (fun x ↦ deriv f x) x) x}
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: case succ
R : Type u_1
inst✝ : Ring R
r s : R
h : Commute r s
k : ℕ
ih :
  (descPochhammer ℤ k).smeval (r + s) =
    ∑ ij ∈ antidiagonal k, ↑(k.choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s)
⊢ ((r + s) ^ 1 - k • 1) * (descPochhammer ℤ k).smeval (r + s) =
    ∑ x ∈ antidiagonal k,
      (↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ (x.2 + 1)).smeval s) +
        ↑(k.choose x.2) * ((descPochhammer ℤ (x.1 + 1)).smeval r * (descPochhammer ℤ x.2).smeval s))

Tactic:
pow_one,

Goal state after:
case succ
R : Type u_1
inst✝ : Ring R
r s : R
h : Commute r s
k : ℕ
ih :
  (descPochhammer ℤ k).smeval (r + s) =
    ∑ ij ∈ antidiagonal k, ↑(k.choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s)
⊢ (r + s - k • 1) * (descPochhammer ℤ k).smeval (r + s) =
    ∑ x ∈ antidiagonal k,
      (↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ (x.2 + 1)).smeval s) +
        ↑(k.choose x.2) * ((descPochhammer ℤ (x.1 + 1)).smeval r * (descPochhammer ℤ x.2).smeval s))
Module: Mathlib.Analysis.Analytic.Binomial

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

Tactic:
have := one_div_one_sub_pow_hasFPowerSeriesOnBall_zero a

Goal state after:
a : ℕ
z : ℂ
hz : z ≠ 0
this :
  HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)) 0 1
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (z - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)) 0 ‖z‖ₑ
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case succ
n : ℕ
ihn : ∀ {p : ℝ}, ↑n ≤ p → ContDiff ℝ ↑n fun x ↦ x ^ p
p : ℝ
h : ↑n + 1 ≤ p
h1 : 1 ≤ p
⊢ ContDiff ℝ ↑(n + 1) fun x ↦ x ^ p

Tactic:
← le_sub_iff_add_le

Goal state after:
case succ
n : ℕ
ihn : ∀ {p : ℝ}, ↑n ≤ p → ContDiff ℝ ↑n fun x ↦ x ^ p
p : ℝ
h : ↑n ≤ p - 1
h1 : 1 ≤ p
⊢ ContDiff ℝ ↑(n + 1) fun x ↦ x ^ p
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : E → ℝ
f' g' : StrongDual ℝ E
x : E
s : Set E
hf : HasFDerivWithinAt f f' s x
hg : HasFDerivWithinAt g g' s x
h : 0 < f x
⊢ HasFDerivWithinAt (fun x ↦ f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Real.log (f x)) • g') s x

Tactic:
by
  -- `by exact` to deal with tricky unification.
  exact (hasStrictFDerivAt_rpow_of_pos (f x, g x) h).hasFDerivAt.comp_hasFDerivWithinAt x
    (hf.prodMk hg)

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

Goal state before:
pp: case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ (linearGrowthInf fun x ↦ v x + b) ≤ linearGrowthInf v

Tactic:
· apply (linearGrowthInf_add_le' _ _).trans_eq <;> rw [linearGrowthSup_const b_bot.ne' hb]
    · exact add_zero (linearGrowthInf v)
    · exact Or.inr EReal.zero_ne_top
    · exact Or.inr EReal.zero_ne_bot

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

Goal state before:
pp: u v : ℕ → ℝ≥0∞
⊢ expGrowthSup (u + v) = max (expGrowthSup u) (expGrowthSup v)

Tactic:
rw [← expGrowthSup_sup]
  apply le_antisymm
  · refine expGrowthSup_le_of_eventually_le (b := 2) ofNat_ne_top (Eventually.of_forall fun n ↦ ?_)
    rw [Pi.sup_apply u v n, Pi.add_apply u v n, two_mul]
    exact add_le_add (le_max_left (u n) (v n)) (le_max_right (u n) (v n))
  · refine expGrowthSup_monotone fun n ↦ ?_
    exact sup_le (self_le_add_right (u n) (v n)) (self_le_add_left (v n) (u n))

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

Goal state before:
pp: u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
⊢ linearGrowthInf (u ∘ v) = a * linearGrowthInf u

Tactic:
have v_top := tendsto_atTop_of_linearGrowthInf_natCast_pos hv₁.ne.symm

Goal state after:
u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
v_top : Tendsto v atTop atTop
⊢ linearGrowthInf (u ∘ v) = a * linearGrowthInf u
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
⊢ linearGrowthSup (u ∘ v) = a * linearGrowthSup u

Tactic:
have v_top := tendsto_atTop_of_linearGrowthInf_natCast_pos hv₁.ne.symm

Goal state after:
u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
v_top : Tendsto v atTop atTop
⊢ linearGrowthSup (u ∘ v) = a * linearGrowthSup u
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (x : α), f x ∂μ ≠ ∞
hμ : μ ≠ 0
⊢ ⨍⁻ (x : α), f x ∂μ < ∞

Tactic:
rw [laverage_eq]
    finiteness [measure_univ_ne_zero.2 hμ]

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✝ : IsProbabilityMeasure μ
hf : Integrable f μ
⊢ 0 < μ {x | ∫ (a : α), f a ∂μ ≤ f x}

Tactic:
by
  simpa only [average_eq_integral] using
    measure_average_le_pos (IsProbabilityMeasure.ne_zero μ) hf

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ.HaveLebesgueDecomposition ν
⊢ Measurable (Classical.choose ⋯).2 ∧
    (if h : μ.HaveLebesgueDecomposition ν then (Classical.choose ⋯).1 else 0) ⟂ₘ ν ∧
      μ =
        (if h : μ.HaveLebesgueDecomposition ν then (Classical.choose ⋯).1 else 0) + ν.withDensity (Classical.choose ⋯).2

Tactic:
dif_pos h

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ.HaveLebesgueDecomposition ν
⊢ Measurable (Classical.choose ⋯).2 ∧
    (Classical.choose ⋯).1 ⟂ₘ ν ∧ μ = (Classical.choose ⋯).1 + ν.withDensity (Classical.choose ⋯).2
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : μ s ≠ ∞
hl : μ.HaveLebesgueDecomposition ν
⊢ ∫⁻ (x : α) in toMeasurable μ s, μ.rnDeriv ν x ∂ν < ∞

Tactic:
rw [← withDensity_apply _ (measurableSet_toMeasurable _ _)]

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : μ s ≠ ∞
hl : μ.HaveLebesgueDecomposition ν
⊢ (ν.withDensity (μ.rnDeriv ν)) (toMeasurable μ s) < ∞
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
⊢ ∀ᵐ (x : α) ∂ν, μ.rnDeriv ν x < ∞

Tactic:
suffices ∀ n, ∀ᵐ x ∂ν, x ∈ spanningSets μ n → μ.rnDeriv ν x < ∞ by
    filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _)

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
⊢ ∀ (n : ℕ), ∀ᵐ (x : α) ∂ν, x ∈ spanningSets μ n → μ.rnDeriv ν x < ∞
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
⊢ ν Sᶜ + 0 ≤ 0

Tactic:
hS₃,

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
⊢ 0 + 0 ≤ 0
Module: Mathlib.MeasureTheory.Covering.DensityTheorem

Goal state before:
pp: case pos
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := ⋯
H : closedBall y r ⊆ closedBall x (R / 4)
⊢ ∃ r_1,
    closedBall y r ⊆ closedBall x r_1 ∧
      μ (closedBall x (3 * r_1)) ≤ ↑(scalingConstantOf μ (max (4 * K + 3) 3)) * μ (closedBall y r)

Tactic:
rcases le_or_gt r R with (hr | hr)

Goal state after:
case pos.inl
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ (max (4 * K + 3) 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ ∃ r_1,
    closedBall y r ⊆ closedBall x r_1 ∧
      μ (closedBall x (3 * r_1)) ≤ ↑(scalingConstantOf μ (max (4 * K + 3) 3)) * μ (closedBall y r)

case pos.inr
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ (max (4 * K + 3) 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : R < r
⊢ ∃ r_1,
    closedBall y r ⊆ closedBall x r_1 ∧
      μ (closedBall x (3 * r_1)) ≤ ↑(scalingConstantOf μ (max (4 * K + 3) 3)) * μ (closedBall y r)
Module: Mathlib.MeasureTheory.Covering.DensityTheorem

Goal state before:
pp: case pos.inl.right
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := ⋯
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ μ (closedBall x (3 * ((K + 1) * r))) ≤ ↑(scalingConstantOf μ (max (4 * K + 3) 3)) * μ (closedBall y r)

Tactic:
· have I1 : closedBall x (3 * ((K + 1) * r)) ⊆ closedBall y ((4 * K + 3) * r) := by
        apply closedBall_subset_closedBall'
        linarith
      have I2 : closedBall y ((4 * K + 3) * r) ⊆ closedBall y (max (4 * K + 3) 3 * r) := by
        gcongr
        exact le_max_left ..
      apply (measure_mono (I1.trans I2)).trans
      exact measure_mul_le_scalingConstantOf_mul _
        ⟨zero_lt_three.trans_le (le_max_right _ _), le_rfl⟩ hr

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

Goal state before:
pp: case refine_3
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
C : ℝ≥0
h : ∀ (x : α), ∃ᶠ (r : ℝ) in 𝓝[>] 0, μ (closedBall x (3 * r)) ≤ ↑C * μ (closedBall x r)
s : Set α
f : α → Set (Set α)
fsubset :
  ∀ x ∈ s,
    f x ⊆
      {a |
        IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a ∧
          (interior a).Nonempty ∧ ∃ r, a ⊆ closedBall x r ∧ μ (closedBall x (3 * r)) ≤ ↑C * μ a}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
t : Set (ℝ × α × Set α) :=
  {p |
    p.2.2 ⊆ closedBall p.2.1 p.1 ∧
      μ (closedBall p.2.1 (3 * p.1)) ≤ ↑C * μ p.2.2 ∧
        (interior p.2.2).Nonempty ∧
          IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] p.2.2 ∧ p.2.2 ∈ f p.2.1 ∧ p.2.1 ∈ s}
A : ∀ x ∈ s, ∀ ε > 0, ∃ p ∈ t, p.1 ≤ ε ∧ p.2.1 = x
t' : Set (ℝ × α × Set α)
t't : t' ⊆ t
left✝ : t'.Countable
t'_disj : t'.PairwiseDisjoint fun p ↦ p.2.2
μt' : μ (s \ ⋃ a ∈ t', a.2.2) = 0
q : ℝ × α × Set α
hq : q ∈ t'
⊢ ((fun p ↦ p.2) q).2 ∈ f ((fun p ↦ p.2) q).1

Tactic:
exact (t't hq).2.2.2.2.1

Goal state after:
Module: Mathlib.Order.Monotone.Extension

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : Disjoint (Iic x) s
hy : ¬Disjoint (Iic y) s
⊢ a ≤ sSup (f '' (Iic y ∩ s))

Tactic:
rcases not_disjoint_iff_nonempty_inter.1 hy with ⟨z, hz⟩

Goal state after:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : Disjoint (Iic x) s
hy : ¬Disjoint (Iic y) s
z : α
hz : z ∈ Iic y ∩ s
⊢ a ≤ sSup (f '' (Iic y ∩ s))
Module: Mathlib.Order.Monotone.Extension

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : Disjoint (Iic x) s
hy : ¬Disjoint (Iic y) s
⊢ a ≤ sSup (f '' (Iic y ∩ s))

pp: case pos
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : ¬Disjoint (Iic x) s
hy : Disjoint (Iic y) s
⊢ sSup (f '' (Iic x ∩ s)) ≤ a

pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : ¬Disjoint (Iic x) s
hy : ¬Disjoint (Iic y) s
⊢ sSup (f '' (Iic x ∩ s)) ≤ sSup (f '' (Iic y ∩ s))

Tactic:
· rcases not_disjoint_iff_nonempty_inter.1 hy with ⟨z, hz⟩
      exact le_csSup_of_le (hu' _) (mem_image_of_mem _ hz) (ha <| mem_image_of_mem _ hz.2)

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : ¬Disjoint (Iic x) s
hy : Disjoint (Iic y) s
⊢ sSup (f '' (Iic x ∩ s)) ≤ a

case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : ¬Disjoint (Iic x) s
hy : ¬Disjoint (Iic y) s
⊢ sSup (f '' (Iic x ∩ s)) ≤ sSup (f '' (Iic y ∩ s))
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν
⊢ (fun a ↦ max (f a) (g a)) ∈ measurableLE μ ν

Tactic:
by
  refine ⟨Measurable.max hf.1 hg.1, fun A hA ↦ ?_⟩
  have h₁ := hA.inter (measurableSet_le hf.1 hg.1)
  have h₂ := hA.inter (measurableSet_lt hg.1 hf.1)
  rw [setLIntegral_max hf.1 hg.1]
  refine (add_le_add (hg.2 _ h₁) (hf.2 _ h₂)).trans_eq ?_
  simp only [← not_le, ← compl_ofPred, ← sdiff_eq]
  exact measure_inter_add_sdiff _ (measurableSet_le hf.1 hg.1)

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
⊢ μ.withDensity (v.limRatioMeas hρ) = ρ

Tactic:
by
  ext1 s hs
  refine le_antisymm ?_ ?_
  · have : Tendsto (fun t : ℝ≥0 =>
        ((t : ℝ≥0∞) ^ 2 * ρ s : ℝ≥0∞)) (𝓝[>] 1) (𝓝 ((1 : ℝ≥0∞) ^ 2 * ρ s)) := by
      refine ENNReal.Tendsto.mul ?_ ?_ tendsto_const_nhds ?_
      · exact ENNReal.Tendsto.pow (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds)
      · simp
      · simp only [one_pow, Ne, or_true, ENNReal.one_ne_top, not_false_iff]
    simp only [one_pow, one_mul] at this
    refine ge_of_tendsto this ?_
    filter_upwards [self_mem_nhdsWithin] with _ ht
    exact v.withDensity_le_mul hρ hs ht
  · have :
      Tendsto (fun t : ℝ≥0 => (t : ℝ≥0∞) * μ.withDensity (v.limRatioMeas hρ) s) (𝓝[>] 1)
        (𝓝 ((1 : ℝ≥0∞) * μ.withDensity (v.limRatioMeas hρ) s)) := by
      refine ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds) ?_
      simp only [true_or, Ne, not_false_iff, one_ne_zero]
    simp only [one_mul] at this
    refine ge_of_tendsto this ?_
    filter_upwards [self_mem_nhdsWithin] with _ ht
    exact v.le_mul_withDensity hρ hs ht

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

Goal state before:
pp: case left
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
I : ℕ → α
hI : Monotone I
n i✝ : ℕ
a✝¹ : 0 < i✝
a✝ : i✝ < n
⊢ I 0 ≤ I i✝

Tactic:
apply hI; lia

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

Goal state before:
pp: case right
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
I : ℕ → α
hI : Monotone I
n i✝ : ℕ
a✝¹ : 0 < i✝
a✝ : i✝ < n
⊢ I i✝ ≤ I n

Tactic:
apply hI; lia

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

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : MonotoneOn φ t
x y : β
hx : x ∈ t
hy : y ∈ t
⊢ eVariationOn (f ∘ φ) (t ∩ Icc x y) = eVariationOn f (φ '' t ∩ Icc (φ x) (φ y))

Tactic:
rcases le_total x y with (h | h)
  · convert! comp_eq_of_monotoneOn f φ (hφ.mono Set.inter_subset_left)
    apply le_antisymm
    · rintro _ ⟨⟨u, us, rfl⟩, vφx, vφy⟩
      rcases le_total x u with (xu | ux)
      · rcases le_total u y with (uy | yu)
        · exact ⟨u, ⟨us, ⟨xu, uy⟩⟩, rfl⟩
        · rw [le_antisymm vφy (hφ hy us yu)]
          exact ⟨y, ⟨hy, ⟨h, le_rfl⟩⟩, rfl⟩
      · rw [← le_antisymm vφx (hφ us hx ux)]
        exact ⟨x, ⟨hx, ⟨le_rfl, h⟩⟩, rfl⟩
    · rintro _ ⟨u, ⟨⟨hu, xu, uy⟩, rfl⟩⟩
      exact ⟨⟨u, hu, rfl⟩, ⟨hφ hx hu xu, hφ hu hy uy⟩⟩
  · rw [eVariationOn.subsingleton, eVariationOn.subsingleton]
    exacts [(Set.subsingleton_Icc_of_ge (hφ hy hx h)).anti Set.inter_subset_right,
      (Set.subsingleton_Icc_of_ge h).anti Set.inter_subset_right]

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

Goal state before:
pp: a b : ℝ
⊢ eVariationOn id (Set.Icc a b) = ENNReal.ofReal (b - a)

Tactic:
simpa using eVariationOn_id (s := univ) (by simp) (by simp)

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

Goal state before:
pp: 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
hf : ContinuousOn f✝ (σₙ R a✝)
hf0 : f✝ 0 = 0
hg : ContinuousOn g (σₙ R a✝)
hg0 : g 0 = 0
ha : p a✝
f : R → R
a : A
x✝² : ContinuousOn f (σₙ R a)
x✝¹ : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
x✝ : p a
⊢ star ((cfcₙHom x✝) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := x✝¹ }) *
      (cfcₙHom x✝) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := x✝¹ } =
    (cfcₙHom x✝) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := x✝¹ } *
      star ((cfcₙHom x✝) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := x✝¹ })

Tactic:
← cfcₙ_apply f a,

Goal state after:
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
hf : ContinuousOn f✝ (σₙ R a✝)
hf0 : f✝ 0 = 0
hg : ContinuousOn g (σₙ R a✝)
hg0 : g 0 = 0
ha : p a✝
f : R → R
a : A
x✝² : ContinuousOn f (σₙ R a)
x✝¹ : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
x✝ : p a
⊢ star (cfcₙ f a) * cfcₙ f a = cfcₙ f a * star (cfcₙ f a)
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
⊢ (pderiv true) x = 1

Tactic:
rw [pderiv_X]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
⊢ Pi.single true 1 true = 1
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
⊢ (pderiv true) y = 0

Tactic:
rw [pderiv_X]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
⊢ Pi.single true 1 false = 0