mathlib-initiative/mathlib-tactics huggingface dataset:
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
M : (s \ ⋃ a ∈ u, B a) ∩ ball x (R x) ⊆ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)
⊢ μ ((s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) ≤ ε

Tactic:
have : Countable v := (u_count.mono vu).to_subtype

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
M : (s \ ⋃ a ∈ u, B a) ∩ ball x (R x) ⊆ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)
this : Countable ↑v
⊢ μ ((s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) ≤ ε
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)
A : Set α
⊢ MeasurableSet A → (ν.withDensity f) A = ((ν.withDensity f).restrict (S ∩ T)) A

Tactic:
hA

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)
A : Set α
hA : MeasurableSet A
⊢ (ν.withDensity f) A = ((ν.withDensity f).restrict (S ∩ T)) A
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝ : NormedAddCommGroup E
x : α
f : α → E
hf : LocallyIntegrable f μ
⊢ ∀ᶠ (a : Set α) in v.filterAt x, IntegrableOn f a μ

Tactic:
rcases hf x with ⟨w, w_nhds, hw⟩
  filter_upwards [v.eventually_filterAt_subset_of_nhds w_nhds] with a ha
  exact hw.mono_set ha

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

Goal state before:
pp: x y : ℝ
hxy : x < y
⊢ Icc x y ∈ (vitaliFamily volume 1).setsAt x

Tactic:
Icc_eq_closedBall

Goal state after:
x y : ℝ
hxy : x < y
⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ (vitaliFamily volume 1).setsAt x
Module: Mathlib.MeasureTheory.Covering.OneDim

Goal state before:
pp: x : ℝ
⊢ Tendsto (fun y ↦ Icc x y) (𝓝[>] x) ((vitaliFamily volume 1).filterAt x)

Tactic:
by
  refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
  · filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy
  · intro ε εpos
    filter_upwards [Icc_mem_nhdsGT <| show x < x + ε by linarith] with y hy
    rw [closedBall_eq_Icc]
    exact Icc_subset_Icc (by linarith) hy.2

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

Goal state before:
pp: x y : ℝ
hxy : x < y
⊢ Icc x y ∈ (vitaliFamily volume 1).setsAt y

Tactic:
Icc_eq_closedBall

Goal state after:
x y : ℝ
hxy : x < y
⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ (vitaliFamily volume 1).setsAt y
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case zero.right
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν
A : Set α
⊢ MeasurableSet A → ∫⁻ (x : α) in A, (fun x ↦ ⨆ k, ⨆ (_ : k ≤ 0), f k x) x ∂μ ≤ ν A

Tactic:
hA

Goal state after:
case zero.right
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
⊢ ∫⁻ (x : α) in A, (fun x ↦ ⨆ k, ⨆ (_ : k ≤ 0), f k x) x ∂μ ≤ ν A
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⟩)

Goal state after:
case inl
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
h : LocallyBoundedVariationOn f ∅
⊢ ∃ p q,
    MonotoneOn p ∅ ∧ MonotoneOn q ∅ ∧ f = p - q ∧ ∀ x ∈ ∅, ∀ y ∈ ∅, p y - p x + (q y - q x) = variationOnFromTo f ∅ x y

case inr
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
h : LocallyBoundedVariationOn f s
c : α
cs : c ∈ 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
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
h : LocallyBoundedVariationOn f ∅
x : α
hx : x ∈ ∅
y : α
hy : y ∈ ∅
⊢ f y - f x + (0 y - 0 x) = variationOnFromTo f ∅ x y

Tactic:
by simp at hx

Goal state after:
Module: Mathlib.Analysis.BoundedVariation

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
f : ℝ → V
a b : ℝ
h : BoundedVariationOn f (uIcc a b)
h₁ : ∀ᵐ (x : ℝ), x ≠ min a b
h₂ : ∀ᵐ (x : ℝ), x ≠ max a b
x : ℝ
hx₁ : x ∈ uIcc a b → DifferentiableWithinAt ℝ f (uIcc a b) x
hx₂ : x ≠ min a b
hx₃ : x ≠ max a b
⊢ x ∈ uIcc a b → DifferentiableAt ℝ f x

Tactic:
hx₄

Goal state after:
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
f : ℝ → V
a b : ℝ
h : BoundedVariationOn f (uIcc a b)
h₁ : ∀ᵐ (x : ℝ), x ≠ min a b
h₂ : ∀ᵐ (x : ℝ), x ≠ max a b
x : ℝ
hx₁ : x ∈ uIcc a b → DifferentiableWithinAt ℝ f (uIcc a b) x
hx₂ : x ≠ min a b
hx₃ : x ≠ max a b
hx₄ : x ∈ uIcc a b
⊢ DifferentiableAt ℝ f x
Module: Mathlib.Analysis.BoundedVariation

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
f : ℝ → V
h : LocallyBoundedVariationOn f univ
x : ℝ
hx : x ∈ univ → DifferentiableWithinAt ℝ f univ x
⊢ DifferentiableAt ℝ f x

Tactic:
differentiableWithinAt_univ

Goal state after:
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
f : ℝ → V
h : LocallyBoundedVariationOn f univ
x : ℝ
hx : x ∈ univ → DifferentiableAt ℝ f x
⊢ DifferentiableAt ℝ f x
Module: Mathlib.Topology.ContinuousMap.StarOrdered

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : Zero α
R : Type u_2
inst✝⁸ : TopologicalSpace R
inst✝⁷ : CommSemiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : StarRing R
inst✝³ : StarOrderedRing R
inst✝² : IsTopologicalSemiring R
inst✝¹ : ContinuousStar R
inst✝ : StarOrderedRing C(α, R)
f g : C(α, R)₀
⊢ (∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p) → f ≤ g

Tactic:
· rintro ⟨p, hp, rfl⟩
      induction hp using AddSubmonoid.closure_induction generalizing f with
      | mem s s_mem =>
        obtain ⟨s, rfl⟩ := s_mem
        exact fun x ↦ le_add_of_nonneg_right (star_mul_self_nonneg (s x))
      | zero => simp
      | add g₁ g₂ _ _ h₁ h₂ => calc
          f ≤ f + g₁ := h₁ f
          _ ≤ (f + g₁) + g₂ := h₂ (f + g₁)
          _ = f + (g₁ + g₂) := add_assoc _ _ _

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

Goal state before:
pp: 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

pp: 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

Tactic:
· exact (vst i).elim id (fun h ↦ (hv hi.2).antisymm (ht.2 h) ▸ hs.1)

Goal state after:
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.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr
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
h✝ : Nonempty β
⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t

Tactic:
let ψ := φ.invFunOn t

Goal state after:
case inr
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
h✝ : Nonempty β
ψ : α → β := Function.invFunOn φ t
⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr
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 : AntitoneOn φ t
h✝ : Nonempty β
⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t

Tactic:
let ψ := φ.invFunOn t

Goal state after:
case inr
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 : AntitoneOn φ t
h✝ : Nonempty β
ψ : α → β := Function.invFunOn φ t
⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (spectrum R a)
ha : p a
⊢ cfc (∑ i ∈ s, f i) a = ∑ i ∈ s, cfc (f i) a

pp: case neg
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (spectrum R a)
ha : ¬p a
⊢ cfc (∑ i ∈ s, f i) a = ∑ i ∈ s, cfc (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) (spectrum 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 [cfc_apply_pi _ a ha (fun ⟨i, hi⟩ => hf i hi), ← map_sum, cfc_apply _ a ha hf']
    congr 1
    ext
    simp

Goal state after:
case neg
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (spectrum R a)
ha : ¬p a
⊢ cfc (∑ i ∈ s, f i) a = ∑ i ∈ s, cfc (f i) 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 : α
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s ∩ Iio x] x) (𝓝 0)

Tactic:
← Iio_inter_Ici,

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
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ (Iio x ∩ Ici y))) (𝓝[s ∩ Iio x] x) (𝓝 0)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr.refine_1
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 ∩ Iic x) x
h' : (𝓝[s ∩ Iio x] x).NeBot
y : α
hy : y < x
⊢ (𝓝[s ∩ Icc y x ∩ Iio x] x).NeBot

Tactic:
rwa [show s ∩ Icc y x ∩ Iio x = (s ∩ Iio x) ∩ Ici y by grind, nhdsWithin_inter_of_mem']

Goal state after:
case inr.refine_1
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 ∩ Iic x) x
h' : (𝓝[s ∩ Iio x] x).NeBot
y : α
hy : y < x
⊢ Ici y ∈ 𝓝[s ∩ Iio x] x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_3
A : Type u_4
inst✝⁴ : Semifield R
inst✝³ : Ring A
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousInv₀ R
inst✝ : Algebra R A
a : Aˣ
⊢ spectrum R ↑a ⊆ {0}ᶜ

Tactic:
simpa only [Set.subset_compl_singleton_iff] using spectrum.zero_notMem R a.isUnit

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

Goal state before:
pp: R : Type u_3
A : Type u_4
inst✝⁵ : Semifield R
inst✝⁴ : Ring A
inst✝³ : TopologicalSpace R
inst✝² : ContinuousInv₀ R
inst✝¹ : Algebra R A
inst✝ : ContinuousMul R
a : Aˣ
n : ℤ
⊢ spectrum R ↑a ⊆ {0}ᶜ

Tactic:
simpa only [Set.subset_compl_singleton_iff] using spectrum.zero_notMem R a.isUnit

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

Goal state before:
pp: case neg
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 : R → R
a : A
ha : p a
f_cont : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ f a = (cfcₙHom ha) (mkD ((σₙ R a).domRestrict f) 0)

Tactic:
rw [cfcₙ_apply_of_not_continuousOn a f_cont, mkD_of_not_continuousOn f_cont, map_zero]

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

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

Tactic:
by
  simpa using algebraMap_le_cfc f 1 a h

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

Goal state before:
pp: case neg.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)
ha : ¬p a
⊢ cfcₙ f a = cfcₙ g a

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

pp: case neg.inr.inr
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)
h0 : ¬g 0 = 0
⊢ cfcₙ f a = cfcₙ g a

Tactic:
· simp [cfcₙ_apply_of_not_predicate a ha]

Goal state after:
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

case neg.inr.inr
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)
h0 : ¬g 0 = 0
⊢ cfcₙ f a = cfcₙ g a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inr.inr
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)
h0 : ¬g 0 = 0
⊢ cfcₙ f a = cfcₙ g a

Tactic:
cfcₙ_apply_of_not_map_zero a h0,

Goal state after:
case neg.inr.inr
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)
h0 : ¬g 0 = 0
⊢ cfcₙ f a = 0
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 g : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : p a
⊢ cfc f a ≤ cfc g a ↔ ∀ x ∈ spectrum R a, f x ≤ g x

Tactic:
rw [cfc_apply f a, cfc_apply g a, cfcHom_le_iff (show p a from ha), ContinuousMap.le_def]

Goal state after:
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 g : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : p a
⊢ (∀ (a_1 : ↑(spectrum R a)),
      { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } a_1 ≤
        { toFun := (spectrum R a).domRestrict g, continuous_toFun := ⋯ } a_1) ↔
    ∀ x ∈ spectrum R a, f x ≤ g x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg
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
⊢ cfcₙ (fun x ↦ f x * g x) a = cfcₙ f a * cfcₙ g a

Tactic:
· simp [cfcₙ_apply_of_not_predicate a ha]

Goal state after: