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

Goal state before:
pp: 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)
y : ℝ
hy : y < x
⊢ (↑hf.stieltjesFunction y - f x) / (y - x) ≤ (f y - f x) / (y - x)

Tactic:
← sub_neg

Goal state after:
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)
y : ℝ
hy : y - x < 0
⊢ (↑hf.stieltjesFunction y - f x) / (y - x) ≤ (f y - f x) / (y - x)
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: 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)
y : ℝ
hy : y < x
⊢ (f y - f x) / (y - x) ≤ (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)

Tactic:
← sub_neg

Goal state after:
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)
y : ℝ
hy : y - x < 0
⊢ (f y - f x) / (y - x) ≤ (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: 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
x y z : α
xy : x ≤ y
yz : y ≤ z
xs : x ∈ s
ys : y ∈ s
zs : z ∈ s
⊢ variationOnFromTo f s x z = variationOnFromTo f s x y + variationOnFromTo f s y z

Tactic:
variationOnFromTo.eq_of_le f s xy,

Goal state after:
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
x y z : α
xy : x ≤ y
yz : y ≤ z
xs : x ∈ s
ys : y ∈ s
zs : z ∈ s
⊢ variationOnFromTo f s x z = (eVariationOn f (s ∩ Icc x y)).toReal + variationOnFromTo f s y z
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: 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
x y z : α
xy : x ≤ y
yz : y ≤ z
xs : x ∈ s
ys : y ∈ s
zs : z ∈ s
⊢ (eVariationOn f (s ∩ Icc x z)).toReal = (eVariationOn f (s ∩ Icc x y) + eVariationOn f (s ∩ Icc y z)).toReal

Tactic:
eVariationOn.Icc_add_Icc f xy yz ys

Goal state after:
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
x y z : α
xy : x ≤ y
yz : y ≤ z
xs : x ∈ s
ys : y ∈ s
zs : z ∈ s
⊢ (eVariationOn f (s ∩ Icc x z)).toReal = (eVariationOn f (s ∩ Icc x z)).toReal
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
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
⊢ StrictMonoOn u (Iic 0)

Tactic:
by grind [StrictMonoOn]

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
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
n m : ℕ
v : ℕ → α
v_mono : StrictMonoOn v (Iic m)
v_mem : ∀ i ∈ Iic m, v i ∈ s
hv : ↑⟨m, ⟨v, ⋯⟩⟩.snd ⟨m, ⟨v, ⋯⟩⟩.fst = u n
h'v :
  ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x)) =
    ∑ i ∈ Finset.range ⟨m, ⟨v, ⋯⟩⟩.fst, edist (f (↑⟨m, ⟨v, ⋯⟩⟩.snd (i + 1))) (f (↑⟨m, ⟨v, ⋯⟩⟩.snd i))
hn : u n < u (n + 1)
⊢ StrictMonoOn (fun i ↦ if i ≤ m then v i else u (n + 1)) (Iic (m + 1)) ∧
    ∀ i ≤ m + 1, (fun i ↦ if i ≤ m then v i else u (n + 1)) i ∈ s

Tactic:
by grind [StrictMonoOn]

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
hst : t ⊆ s
⊢ eVariationOn f t ≤ eVariationOn f s

Tactic:
by
  apply iSup_le _
  rintro ⟨n, ⟨u, hu, ut⟩⟩
  exact sum_le hu fun i => hst (ut i)

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ ν₁ + ν₂ = ν₁.singularPart μ + ν₂.singularPart μ + μ.withDensity (ν₁.rnDeriv μ + ν₂.rnDeriv μ)

Tactic:
withDensity_add_left (measurable_rnDeriv _ _),

Goal state after:
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ ν₁ + ν₂ = ν₁.singularPart μ + ν₂.singularPart μ + (μ.withDensity (ν₁.rnDeriv μ) + μ.withDensity (ν₂.rnDeriv μ))
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr.inl
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
i : ℕ
hi : i + 1 = N
⊢ w i ≤ w (i + 1)

pp: case inr.inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
i : ℕ
hi : N < i + 1
⊢ w i ≤ w (i + 1)

Tactic:
· have A : i < N := hi ▸ i.lt_succ_self
      have := Nat.find_min exists_N A
      grind

Goal state after:
case inr.inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
i : ℕ
hi : N < i + 1
⊢ w i ≤ w (i + 1)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case neg.hab
α : Type u_1
inst✝¹ : LinearOrder α
E✝ : Type u_2
inst✝ : PseudoEMetricSpace E✝
f : α → E✝
s : Set α
E : ℕ → α
hE : Monotone E
n : ℕ
ih :
  (∀ (i : ℕ), 0 < i → i < n → E i ∈ s) →
    ∑ i ∈ Finset.range n, eVariationOn f (s ∩ Icc (E i) (E (i + 1))) = eVariationOn f (s ∩ Icc (E 0) (E n))
hn : ∀ (i : ℕ), 0 < i → i < n + 1 → E i ∈ s
hn₀ : ¬n = 0
⊢ E 0 ≤ E n

Tactic:
apply hE; lia

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

Goal state before:
pp: case neg.hbc
α : Type u_1
inst✝¹ : LinearOrder α
E✝ : Type u_2
inst✝ : PseudoEMetricSpace E✝
f : α → E✝
s : Set α
E : ℕ → α
hE : Monotone E
n : ℕ
ih :
  (∀ (i : ℕ), 0 < i → i < n → E i ∈ s) →
    ∑ i ∈ Finset.range n, eVariationOn f (s ∩ Icc (E i) (E (i + 1))) = eVariationOn f (s ∩ Icc (E 0) (E n))
hn : ∀ (i : ℕ), 0 < i → i < n + 1 → E i ∈ s
hn₀ : ¬n = 0
⊢ E n ≤ E (n + 1)

Tactic:
apply hE; lia

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

Goal state before:
pp: case hf
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} := ⋯
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
n : ℕ
c : E
hi✝ : c ∈ t
⊢ Measurable fun y ↦ ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ

Tactic:
refine (h'f.sub ?_).enorm

Goal state after:
case hf
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
⊢ StronglyMeasurable ((A.set n).indicator fun x ↦ c)
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
a : α
l : E
h : (𝓝[s ∩ Iio a] a).NeBot
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s ∩ Iic a
this : u n ≤ a
hn : u n < a
⊢ ∀ i ≤ n, u i ∈ s ∩ Iio a

Tactic:
by grind [StrictMonoOn]

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
a : α
l : E
h : (𝓝[s ∩ Iio a] a).NeBot
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
u : ℕ → α
n : ℕ
u_mono : StrictMonoOn u (Iic (n + 1))
u_mem : ∀ i ∈ Iic (n + 1), u i ∈ s ∩ Iic a
this✝ : u (n + 1) ≤ a
hn : u (n + 1) = a
this :
  Tendsto (fun y ↦ eVariationOn f (s ∩ Iio a) + edist (f a) (f y)) (𝓝[s ∩ Iio a] a)
    (𝓝 (eVariationOn f (s ∩ Iio a) + edist (f a) l))
⊢ u n < a

Tactic:
by grind [StrictMonoOn]

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
a : α
l : E
h : (𝓝[s ∩ Iio a] a).NeBot
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
⊢ (s ∩ Iio a).Nonempty

Tactic:
contrapose! h

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
a : α
l : E
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
h : s ∩ Iio a = ∅
⊢ 𝓝[s ∩ Iio a] 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
a : α
l : E
h : (𝓝[s ∩ Iio a] a).NeBot
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
b : α
hb : b ∈ s ∩ Iio a
⊢ StrictMonoOn (fun i ↦ b) (Iic 0) ∧ ∀ i ∈ Iic 0, (fun i ↦ b) i ∈ s ∩ Iio a

Tactic:
by grind [StrictMonoOn]

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
a : α
l : E
h : (𝓝[s ∩ Iio a] a).NeBot
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
b : α
hb : b ∈ s ∩ Iio a
this✝ : Nonempty ((n : ℕ) × { u // StrictMonoOn u (Iic n) ∧ ∀ i ∈ Iic n, u i ∈ s ∩ Iio a })
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s ∩ Iio a
this :
  Tendsto (fun y ↦ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f y)) (𝓝[s ∩ Iio a] a)
    (𝓝 (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) l))
⊢ u n < a

Tactic:
by grind [StrictMonoOn]

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 : LocallyIntegrable f μ
u : ℕ → Set α
u_open : ∀ (n : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (u n)
u_univ : ⋃ n, u n = univ
hu : ∀ (n : ℕ), IntegrableOn f (u n) μ
this :
  ∀ (n : ℕ),
    ∀ᵐ (x : α) ∂μ,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u n).indicator f y - (u n).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
x : α
hx :
  ∀ (i : ℕ),
    Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u i).indicator f y - (u i).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
⊢ Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)

Tactic:
obtain ⟨n, hn⟩ : ∃ n, x ∈ u n := by simpa only [← u_univ, mem_iUnion] using mem_univ x

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 : LocallyIntegrable f μ
u : ℕ → Set α
u_open : ∀ (n : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (u n)
u_univ : ⋃ n, u n = univ
hu : ∀ (n : ℕ), IntegrableOn f (u n) μ
this :
  ∀ (n : ℕ),
    ∀ᵐ (x : α) ∂μ,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u n).indicator f y - (u n).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
x : α
hx :
  ∀ (i : ℕ),
    Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u i).indicator f y - (u i).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
n : ℕ
hn : x ∈ u n
⊢ Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 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
ε : ℝ≥0∞
s : Set α
h : ε < eVariationOn f s
n : ℕ
u : ℕ → α
hu : ε < ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x))
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
A : ε < eVariationOn f (s ∩ Icc (u 0) (u n))
this✝ : u n ≤ u 0
this : (s ∩ Icc (u 0) (u n)).Subsingleton
⊢ False

Tactic:
simp [this] at A

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

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

pp: case neg.inr
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))
hg : ¬ContinuousOn g (spectrum R a)
⊢ cfc f a = cfc g a

Tactic:
· simp [cfc_apply_of_not_predicate a ha]

Goal state after:
case neg.inr
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))
hg : ¬ContinuousOn g (spectrum R a)
⊢ cfc f a = cfc g a
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
L : Filter α
hL : ∀ y ∈ s, s ∩ Ici y ∈ L
x₀ : α
hx₀ : x₀ ∈ s
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ < ε
H : ∃ᶠ (x : α) in L, ε ≤ eVariationOn f (s ∩ Ici x)
y : α → α
y_mem : ∀ y_1 ∈ s, y y_1 ∈ s ∩ Ici y_1
le_y : ∀ y_1 ∈ s, δ ≤ eVariationOn f (s ∩ Icc y_1 (y y_1))
v : ℕ → α := fun n ↦ y^[n] x₀
⊢ False

Tactic:
have I n : v n ∈ s := by
    induction n with
    | zero => simpa [v] using hx₀
    | succ n ih =>
      simp only [Function.iterate_succ', Function.comp_apply, v]
      exact (y_mem _ ih).1

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
L : Filter α
hL : ∀ y ∈ s, s ∩ Ici y ∈ L
x₀ : α
hx₀ : x₀ ∈ s
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ < ε
H : ∃ᶠ (x : α) in L, ε ≤ eVariationOn f (s ∩ Ici x)
y : α → α
y_mem : ∀ y_1 ∈ s, y y_1 ∈ s ∩ Ici y_1
le_y : ∀ y_1 ∈ s, δ ≤ eVariationOn f (s ∩ Icc y_1 (y y_1))
v : ℕ → α := fun n ↦ y^[n] x₀
I : ∀ (n : ℕ), v n ∈ s
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
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
f g : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : ¬p a
⊢ cfc (fun x ↦ f x * g x) a = cfc f a * cfc g a

Tactic:
· simp [cfc_apply_of_not_predicate a ha]

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

Goal state before:
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
a : A
f g : R → R
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : ¬p a
⊢ cfc (fun x ↦ f x + g x) a = cfc f a + cfc g a

Tactic:
· simp [cfc_apply_of_not_predicate a ha]

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

Goal state before:
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:
· simp [cfc_apply_of_not_predicate a ha]

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

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

Tactic:
· simp [cfc_apply_of_not_predicate a ha]

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

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

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

Tactic:
· simp [cfc_apply_of_not_predicate a ha]

Goal state after:
case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)
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
r : R
f : R → R
h₁ : ContinuousOn f (spectrum R ((algebraMap R A) r))
⊢ cfc f ((algebraMap R A) r) = (algebraMap R A) (f r)

Tactic:
cfc_apply f (algebraMap R A r) (cfc_predicate_algebraMap r),

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
r : R
f : R → R
h₁ : ContinuousOn f (spectrum R ((algebraMap R A) r))
⊢ (cfcHom ⋯) { toFun := (spectrum R ((algebraMap R A) r)).domRestrict f, continuous_toFun := ⋯ } =
    (algebraMap R A) (f r)