mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Hahn

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

Tactic:
have : n ≤ m + 1 := le_of_lt (Nat.succ_le_succ hnm)

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ∞
hν : ∀ (s : Set α), ν s ≠ ∞
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
n m : ℕ
hnm : n ≤ m
this : n ≤ m + 1
⊢ f n (m + 1) = f n m ∩ e (m + 1)
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
ρ ν : Measure α
inst✝ : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ν a
ε : ℝ≥0
εpos : 0 < ε
x✝ : ν s < ∞
U : Set α
sU : s ⊆ U
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
νU : ν U ≤ ν s + ↑ε
f : α → Set (Set α) := ⋯
x : α
hx : x ∈ s
this : ∃ᶠ (x_1 : Set α) in v.filterAt x, ρ x_1 ≤ ν x_1 ∧ x_1 ∈ v.setsAt x ∧ x_1 ⊆ U
⊢ ∀ (x_1 : Set α), ρ x_1 ≤ ν x_1 ∧ x_1 ∈ v.setsAt x ∧ x_1 ⊆ U → x_1 ∈ f x

Tactic:
rintro a ⟨ρa, _, aU⟩

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
ρ ν : Measure α
inst✝ : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ν a
ε : ℝ≥0
εpos : 0 < ε
x✝ : ν s < ∞
U : Set α
sU : s ⊆ U
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
νU : ν U ≤ ν s + ↑ε
f : α → Set (Set α) := fun x ↦ {a | ρ a ≤ ν a ∧ a ⊆ U}
x : α
hx : x ∈ s
this : ∃ᶠ (x_1 : Set α) in v.filterAt x, ρ x_1 ≤ ν x_1 ∧ x_1 ∈ v.setsAt x ∧ x_1 ⊆ U
a : Set α
ρa : ρ a ≤ ν a
left✝ : a ∈ v.setsAt x
aU : a ⊆ U
⊢ a ∈ f 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
hνinter : ν (S ∩ T)ᶜ = 0
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
heq' : ∀ (A : Set α), MeasurableSet A → (ν.withDensity f) A = ((ν.withDensity f).restrict (S ∩ T)) A
A : Set α
hA : MeasurableSet A
hνrn : (ν.withDensity (μ.rnDeriv ν)) (A ∩ (S ∩ T)ᶜ) = 0
⊢ ((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)) A + 0 = (ν.withDensity (μ.rnDeriv ν)) A

Tactic:
← hνrn,

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
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
heq' : ∀ (A : Set α), MeasurableSet A → (ν.withDensity f) A = ((ν.withDensity f).restrict (S ∩ T)) A
A : Set α
hA : MeasurableSet A
hνrn : (ν.withDensity (μ.rnDeriv ν)) (A ∩ (S ∩ T)ᶜ) = 0
⊢ ((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)) A + (ν.withDensity (μ.rnDeriv ν)) (A ∩ (S ∩ T)ᶜ) =
    (ν.withDensity (μ.rnDeriv ν)) 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
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
H : ∀ (m n : ℕ), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
A :
  (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆
    toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
⊢ μ (toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ 0

Tactic:
calc
      μ (toMeasurable μ sᶜ ∪
        ⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
          μ (toMeasurable μ sᶜ) +
            μ (⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :=
        measure_union_le _ _
      _ = μ (⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := by
        have : μ sᶜ = 0 := v.ae_tendsto_div hρ; rw [measure_toMeasurable, this, zero_add]
      _ ≤ ∑' (m) (n), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :=
        ((measure_iUnion_le _).trans (ENNReal.tsum_le_tsum fun m => measure_iUnion_le _))
      _ = 0 := by simp only [H, tsum_zero]

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

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
f : α → E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
hf : BoundedVariationOn f univ
a x : α
hx : ContinuousWithinAt f (Ici x) x
this : variationOnFromTo f univ a = fun y ↦ variationOnFromTo f univ a x + variationOnFromTo f univ x y
⊢ Tendsto (fun y ↦ (eVariationOn f (univ ∩ Icc x y)).toReal) (𝓝[≥] x) (𝓝 (eVariationOn f (univ ∩ {x})).toReal)

Tactic:
eVariationOn.subsingleton _ (by grind [Set.Subsingleton])

Goal state after:
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
f : α → E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
hf : BoundedVariationOn f univ
a x : α
hx : ContinuousWithinAt f (Ici x) x
this : variationOnFromTo f univ a = fun y ↦ variationOnFromTo f univ a x + variationOnFromTo f univ x y
⊢ Tendsto (fun y ↦ (eVariationOn f (univ ∩ Icc x y)).toReal) (𝓝[≥] x) (𝓝 (ENNReal.toReal 0))
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 := μ.withDensity (v.limRatioMeas hρ)
f : α → ℝ≥0∞ := v.limRatioMeas hρ
f_meas : Measurable f
⊢ ν (s ∩ f ⁻¹' {0}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})

Tactic:
apply zero_le.trans'

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
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
⊢ ν (s ∩ f ⁻¹' {0}) ≤ 0
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
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
m : ℕ
v : ℕ → α
hv : Monotone v
vt : ∀ (i : ℕ), v i ∈ t
w : ℕ → α := fun i ↦ if i ≤ n then u i else v (i - (n + 1))
⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + ∑ i ∈ Finset.range m, edist (f (v (i + 1))) (f (v i)) =
    ∑ i ∈ Finset.range n, edist (f (w (i + 1))) (f (w i)) +
      ∑ i ∈ Finset.range m, edist (f (w (n + 1 + i + 1))) (f (w (n + 1 + i)))

Tactic:
dsimp only [w]
      congr 1
      · grind [Finset.sum_congr]
      · grind

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
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
m : ℕ
v : ℕ → α
hv : Monotone v
vt : ∀ (i : ℕ), v i ∈ t
w : ℕ → α := fun i ↦ if i ≤ n then u i else v (i - (n + 1))
⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + ∑ i ∈ Finset.range m, edist (f (v (i + 1))) (f (v i)) =
    ∑ i ∈ Finset.range n, edist (f (w (i + 1))) (f (w i)) +
      ∑ i ∈ Finset.range m, edist (f (w (n + 1 + i + 1))) (f (w (n + 1 + i)))

Tactic:
by
      dsimp only [w]
      congr 1
      · grind [Finset.sum_congr]
      · grind

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
a b c : α
hab : a ≤ b
hbc : b ≤ c
hb : b ∈ s
A : IsGreatest (s ∩ Icc a b) b
B : IsLeast (s ∩ Icc b c) b
⊢ eVariationOn f (s ∩ (Icc a b ∪ Icc b c)) = eVariationOn f (s ∩ Icc a c)

Tactic:
Icc_union_Icc_eq_Icc hab hbc

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b c : α
hab : a ≤ b
hbc : b ≤ c
hb : b ∈ s
A : IsGreatest (s ∩ Icc a b) b
B : IsLeast (s ∩ Icc b c) b
⊢ eVariationOn f (s ∩ Icc a c) = eVariationOn f (s ∩ Icc a c)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case succ.ht
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
u : ℕ → α
hu : Monotone u
n : ℕ
ih : eVariationOn f (u '' Iic n) = ∑ i ∈ Finset.range n, edist (f (u i)) (f (u (i + 1)))
⊢ IsLeast {u n, u (n + 1)} (u n)

Tactic:
simp [IsLeast, hu n.le_succ]

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
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} :=
  μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
⊢ ∀ (i : ℕ),
    ∀ i_1 ∈ t,
      ∀ᵐ (x : α) ∂μ,
        Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set i).indicator (fun x ↦ i_1) y‖ₑ ∂μ) / μ a) (v.filterAt x)
          (𝓝 ‖f x - (A.set i).indicator (fun x ↦ i_1) x‖ₑ)

Tactic:
intro n c _

Goal state after:
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 : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} :=
  μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
n : ℕ
c : E
hi✝ : c ∈ t
⊢ ∀ᵐ (x : α) ∂μ,
    Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
      (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
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
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (spectrum R a)) (ha : p a),
    P ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
h : p a ∧ ContinuousOn f (spectrum R a)
⊢ P ((cfcHom ⋯) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })

Tactic:
exact haf h.2 h.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✝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
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : p a
h : (cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } = cfc g a
⊢ Set.EqOn f g (spectrum R a)

Tactic:
cfc_apply g a

Goal state after:
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
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : p a
h :
  (cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } =
    (cfcHom ha) { toFun := (spectrum R a).domRestrict g, continuous_toFun := ⋯ }
⊢ Set.EqOn f g (spectrum R a)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: 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
hf : BoundedVariationOn f s
x y : α
⊢ eVariationOn f (s ∩ Ioc x y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ ⇑ofDual ⁻¹' Ioc x y)

Tactic:
← preimage_inter,

Goal state after:
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
hf : BoundedVariationOn f s
x y : α
⊢ eVariationOn f (s ∩ Ioc x y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' (s ∩ Ioc x y))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: 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
h_spec : spectrum R a ⊆ {0}
ha : p a
⊢ a = 0

Tactic:
simpa using eq_algebraMap_of_spectrum_subset_singleton a 0 h_spec

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

Goal state before:
pp: 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 ∩ Ioi x] x) (𝓝 l)
hx : x ∈ s
y : α
⊢ eVariationOn f (s ∩ Icc x y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ ⇑ofDual ⁻¹' Icc x y)

Tactic:
← preimage_inter,

Goal state after:
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 ∩ Ioi x] x) (𝓝 l)
hx : x ∈ s
y : α
⊢ eVariationOn f (s ∩ Icc x y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' (s ∩ Icc x y))
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: 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 : LocallyBoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Ioi x] x) (𝓝 l)
hx : x ∈ s
y : α
⊢ eVariationOn f (s ∩ Icc x y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ ⇑ofDual ⁻¹' Icc x y)

Tactic:
← preimage_inter,

Goal state after:
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 : LocallyBoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Ioi x] x) (𝓝 l)
hx : x ∈ s
y : α
⊢ eVariationOn f (s ∩ Icc x y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' (s ∩ Icc x y))
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: 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
hf : BoundedVariationOn f s
x : α
h : ContinuousWithinAt f (s ∩ Ici x) x
y : α
⊢ eVariationOn f (s ∩ Icc x y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ ⇑ofDual ⁻¹' Icc x y)

Tactic:
← preimage_inter,

Goal state after:
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
hf : BoundedVariationOn f s
x : α
h : ContinuousWithinAt f (s ∩ Ici x) x
y : α
⊢ eVariationOn f (s ∩ Icc x y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' (s ∩ Icc x y))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case mpr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommSemiring R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring 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
a : A
ha : p a
f : C(↑(spectrum R a), R)
⊢ 0 ≤ f → 0 ≤ (cfcHom ha) f

Tactic:
simpa using (cfcHom_mono ha (f := 0) (g := f) ·)

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
⊢ (cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } ≤ cfc g a

Tactic:
cfc_apply g a

Goal state after:
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
⊢ (cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } ≤
    (cfcHom ha) { toFun := (spectrum R a).domRestrict g, continuous_toFun := ⋯ }
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
y : α
⊢ eVariationOn f (s ∩ Iic y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ ⇑ofDual ⁻¹' Iic y)

Tactic:
← preimage_inter,

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
y : α
⊢ eVariationOn f (s ∩ Iic y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' (s ∩ Iic y))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
⊢ cfcₙ (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (f i) a

Tactic:
have hsum : s.sum f = fun z => ∑ i ∈ s, f i z := by ext; simp
    have hf' : ContinuousOn (∑ i : s, f i) (σₙ R a) := by
      rw [sum_coe_sort s, hsum]
      exact continuousOn_finsetSum s fun i hi => hf i hi
    rw [← sum_coe_sort s, ← sum_coe_sort s]
    rw [cfcₙ_apply_pi _ a ha (fun ⟨i, hi⟩ => hf i hi), ← map_sum, cfcₙ_apply _ a hf']
    congr 1
    ext
    simp

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ ¬eval (↑(n - (ν - 1))) (ascPochhammer R ν) = 0

Tactic:
simp only [← ascPochhammer_eval_cast]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ ¬↑(eval (n - (ν - 1)) (ascPochhammer ℕ ν)) = 0