mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ
this : Tendsto (fun x ↦ K * ⨍ (y : α) in a x, ‖f y - c‖ ∂μ) l (𝓝 0)
i : ι
hi : support (g i) ⊆ a i
h'i : ∀ (x : α), |g i x| ≤ K / μ.real (a i)
h''i : IntegrableOn f (a i) μ
hi_int : 0 < ∫ (y : α), g i y ∂μ
⊢ μ (a i) < ∞

Tactic:
rw [lt_top_iff_ne_top]

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ
this : Tendsto (fun x ↦ K * ⨍ (y : α) in a x, ‖f y - c‖ ∂μ) l (𝓝 0)
i : ι
hi : support (g i) ⊆ a i
h'i : ∀ (x : α), |g i x| ≤ K / μ.real (a i)
h''i : IntegrableOn f (a i) μ
hi_int : 0 < ∫ (y : α), g i y ∂μ
⊢ μ (a i) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : μ.HaveLebesgueDecomposition ν
⊢ r • μ = ↑r • (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν))

Tactic:
← haveLebesgueDecomposition_add μ ν,

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : μ.HaveLebesgueDecomposition ν
⊢ r • μ = ↑r • μ
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ₁ μ₂ ν : Measure α
inst✝¹ : μ₁.HaveLebesgueDecomposition ν
inst✝ : μ₂.HaveLebesgueDecomposition ν
⊢ μ₁ + μ₂ = μ₁.singularPart ν + ν.withDensity (μ₁.rnDeriv ν) + ν.withDensity (μ₂.rnDeriv ν) + μ₂.singularPart ν

Tactic:
rw [← haveLebesgueDecomposition_add μ₁ ν, add_assoc, add_comm (ν.withDensity (μ₂.rnDeriv ν)),
    ← haveLebesgueDecomposition_add μ₂ ν]

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 ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
x : α
hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
z : ℝ≥0∞
hz : z > 0
w : ℝ≥0
w_pos : 0 < ↑w
w_lt : ↑w < z
n : ℕ
hn : u n < w
a : Set α
ha : ρ a < ↑(u n) * μ a
⊢ 0 < μ a → μ a < ∞ → ρ a / μ a < z

Tactic:
μa_pos

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 ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
x : α
hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
z : ℝ≥0∞
hz : z > 0
w : ℝ≥0
w_pos : 0 < ↑w
w_lt : ↑w < z
n : ℕ
hn : u n < w
a : Set α
ha : ρ a < ↑(u n) * μ a
μa_pos : 0 < μ a
⊢ μ a < ∞ → ρ a / μ a < z
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) = g n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
hE₃ : ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
hε₂ : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν ≤ μ (A ∩ E)
hξε : (ξ + E.indicator fun x ↦ ↑ε) ∈ measurableLE ν μ
⊢ False

Tactic:
have : (∫⁻ a, ξ a + E.indicator (fun _ ↦ (ε : ℝ≥0∞)) a ∂ν) ≤ sSup (measurableLEEval ν μ) :=
        le_sSup ⟨ξ + E.indicator fun _ ↦ (ε : ℝ≥0∞), hξε, rfl⟩

Goal state after:
case refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) = g n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this✝ : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
hE₃ : ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
hε₂ : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν ≤ μ (A ∩ E)
hξε : (ξ + E.indicator fun x ↦ ↑ε) ∈ measurableLE ν μ
this : ∫⁻ (a : α), ξ a + E.indicator (fun x ↦ ↑ε) a ∂ν ≤ sSup (measurableLEEval ν μ)
⊢ False
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 f' : α → E
s : Set α
h : ∀ ⦃x : α⦄, x ∈ s → edist (f x) (f' x) = 0
⊢ eVariationOn f s = eVariationOn f' s

Tactic:
dsimp only [eVariationOn]

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f f' : α → E
s : Set α
h : ∀ ⦃x : α⦄, x ∈ s → edist (f x) (f' x) = 0
⊢ ⨆ p, ∑ i ∈ Finset.range p.1, edist (f (↑p.2 (i + 1))) (f (↑p.2 i)) =
    ⨆ p, ∑ i ∈ Finset.range p.1, edist (f' (↑p.2 (i + 1))) (f' (↑p.2 i))
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
h : ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0
⊢ eVariationOn f s = 0

Tactic:
dsimp only [eVariationOn]

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

Tactic:
have : Tendsto (fun j => ∑ i ∈ Finset.range n, edist (F j (u (i + 1))) (F j (u i))) p
      (𝓝 (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)))) := by
    apply tendsto_finsetSum
    exact fun i _ => Tendsto.edist (Ffs (u i.succ) (us i.succ)) (Ffs (u i) (us i))

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
h : μ ⟂ₘ ξ
s : Set α := h.nullSet
hs : MeasurableSet s
h_le_s : μ.restrict s + (ν - ξ).restrict s = μ.restrict s + ν.restrict s - ξ.restrict s
⊢ μ.restrict sᶜ + (ν.restrict sᶜ - ξ.restrict sᶜ) = μ.restrict sᶜ + ν.restrict sᶜ - ξ.restrict sᶜ

Tactic:
h.restrict_compl_nullSet

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
h : μ ⟂ₘ ξ
s : Set α := h.nullSet
hs : MeasurableSet s
h_le_s : μ.restrict s + (ν - ξ).restrict s = μ.restrict s + ν.restrict s - ξ.restrict s
⊢ μ.restrict sᶜ + (ν.restrict sᶜ - 0) = μ.restrict sᶜ + ν.restrict sᶜ - 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 : 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)
hw : Monotone w
Npos : 0 < N
⊢ ∑ i ∈ Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i)) + edist (f (w (N + 1))) (f (w (N - 1))) +
      ∑ i ∈ Finset.Ico (N + 1) (n + 1), edist (f (w (i + 1))) (f (w i)) ≤
    ∑ i ∈ Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i)) +
        ∑ i ∈ Finset.Ico (N - 1) (N + 1), edist (f (w (i + 1))) (f (w i)) +
      ∑ i ∈ Finset.Ico (N + 1) (n + 1), edist (f (w (i + 1))) (f (w i))

Tactic:
refine add_le_add (add_le_add le_rfl ?_) le_rfl

Goal state after:
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)
hw : Monotone w
Npos : 0 < N
⊢ edist (f (w (N + 1))) (f (w (N - 1))) ≤ ∑ i ∈ Finset.Ico (N - 1) (N + 1), edist (f (w (i + 1))) (f (w i))
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝³ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
hs : MeasurableSet s
this : IsLocallyFiniteMeasure (μ.restrict s)
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.indicator 1 x))

Tactic:
filter_upwards [ae_tendsto_rnDeriv v (μ.restrict s), rnDeriv_restrict_self μ hs]

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
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
hs : MeasurableSet s
this : IsLocallyFiniteMeasure (μ.restrict s)
⊢ ∀ (a : α),
    Tendsto (fun a ↦ (μ.restrict s) a / μ a) (v.filterAt a) (𝓝 ((μ.restrict s).rnDeriv μ a)) →
      (μ.restrict s).rnDeriv μ a = s.indicator 1 a →
        Tendsto (fun a ↦ μ (s ∩ a) / μ a) (v.filterAt a) (𝓝 (s.indicator 1 a))
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr.inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
a b : α
hab : a ≤ b
n : ℕ
u : ℕ → α
hmono : StrictMonoOn u (Iic n)
hi : ∀ i ∈ Iic n, u i ∈ {a, b}
hn : 2 ≤ n
⊢ ∑ i ∈ Finset.range ⟨n, ⟨u, ⋯⟩⟩.fst, edist (f (↑⟨n, ⟨u, ⋯⟩⟩.snd (i + 1))) (f (↑⟨n, ⟨u, ⋯⟩⟩.snd i)) ≤ edist (f a) (f b)

Tactic:
· have := hmono (by simp) (by grind) zero_lt_one
      have := hmono (by grind) (by grind) one_lt_two
      grind

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
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 := ⋯ })
⊢ P (cfc f a)

Tactic:
by
  by_cases h : p a ∧ ContinuousOn f (spectrum R a)
  · rw [cfc_apply f a h.1 h.2]
    exact haf h.2 h.1
  · simp only [not_and_or] at h
    obtain (h | h) := h
    · rwa [cfc_apply_of_not_predicate _ h]
    · rwa [cfc_apply_of_not_continuousOn _ h]

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

Goal state before:
pp: case monomial
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
ha : p a
hf : ContinuousOn f (spectrum R a)
n : ℕ
r : R
a✝ : cfc (fun x ↦ Polynomial.eval (f x) (Polynomial.C r * X ^ n)) a = (aeval (cfc f a)) (Polynomial.C r * X ^ n)
⊢ cfc (fun x ↦ Polynomial.eval (f x) (Polynomial.C r * X ^ (n + 1))) a =
    (aeval (cfc f a)) (Polynomial.C r * X ^ (n + 1))

Tactic:
simp only [eval_mul, eval_C, eval_X_pow, map_mul, aeval_C, map_pow, aeval_X]

Goal state after:
case monomial
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
ha : p a
hf : ContinuousOn f (spectrum R a)
n : ℕ
r : R
a✝ : cfc (fun x ↦ Polynomial.eval (f x) (Polynomial.C r * X ^ n)) a = (aeval (cfc f a)) (Polynomial.C r * X ^ n)
⊢ cfc (fun x ↦ r * f x ^ (n + 1)) a = (algebraMap R A) r * cfc f a ^ (n + 1)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: case neg
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ¬(ContinuousOn g (σₙ R a) ∧ g 0 = 0)
⊢ cfcₙ g a = cfcₙ (fun x ↦ (algebraMap R S) (g (f x))) a

Tactic:
simp only [not_and_or] at hg

Goal state after:
case neg
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ¬ContinuousOn g (σₙ R a) ∨ ¬g 0 = 0
⊢ cfcₙ g a = cfcₙ (fun x ↦ (algebraMap R S) (g (f x))) a
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
a : A
h : ∀ x ∈ spectrum R a, x ≤ 1
ha : p a
⊢ a ≤ 1

Tactic:
by
  simpa using le_algebraMap_of_spectrum_le h

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommRing R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalRing R
inst✝⁹ : ContinuousStar R
inst✝⁸ : ContinuousSqrt R
inst✝⁷ : StarOrderedRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
f : R → R
r : R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ cfc f a ≤ (algebraMap R A) r ↔ ∀ x ∈ spectrum R a, f x ≤ r

Tactic:
rw [← cfc_const r a, cfc_le_iff ..]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommRing R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalRing R
inst✝⁹ : ContinuousStar R
inst✝⁸ : ContinuousSqrt R
inst✝⁷ : StarOrderedRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
f : R → R
r : R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ (algebraMap R A) r ≤ cfc f a ↔ ∀ x ∈ spectrum R a, r ≤ f x

Tactic:
rw [← cfc_const r a, cfc_le_iff ..]

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

Goal state before:
pp: case succ.succ
R : Type u_1
inst✝ : CommRing R
k : ℕ
ih : ∀ (n ν : ℕ), k ≤ ν → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n (ν + 1))) = 0
n ν : ℕ
⊢ k + 1 ≤ ν → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k + 1] (bernsteinPolynomial R n (ν + 1))) = 0

Tactic:
simp only [derivative_succ, Function.comp_apply,
        Function.iterate_succ, Polynomial.iterate_derivative_sub,
        Polynomial.iterate_derivative_natCast_mul, Polynomial.eval_mul, Polynomial.eval_natCast,
        Polynomial.eval_sub]

Goal state after:
case succ.succ
R : Type u_1
inst✝ : CommRing R
k : ℕ
ih : ∀ (n ν : ℕ), k ≤ ν → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n (ν + 1))) = 0
n ν : ℕ
⊢ k + 1 ≤ ν →
    ↑n *
        (Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R (n - 1) ν)) -
          Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R (n - 1) (ν + 1)))) =
      0
Module: Mathlib.Topology.ContinuousMap.Weierstrass

Goal state before:
pp: case inl
a b : ℝ
h : a < b
⊢ (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤

Tactic:
let W : C(Set.Icc a b, ℝ) →ₐ[ℝ] C(I, ℝ) :=
      compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm

Goal state after:
case inl
a b : ℝ
h : a < b
W : C(↑(Set.Icc a b), ℝ) →ₐ[ℝ] C(↑I, ℝ) := compRightAlgHom ℝ ℝ ↑(iccHomeoI a b h).symm
⊢ (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤
Module: Mathlib.Analysis.SpecialFunctions.Bernstein

Goal state before:
pp: n : ℕ
x : ↑I
⊢ ∑ k, (bernstein n ↑k) x = 1

Tactic:
have := bernsteinPolynomial.sum ℝ n

Goal state after:
n : ℕ
x : ↑I
this : ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial ℝ n ν = 1
⊢ ∑ k, (bernstein n ↑k) x = 1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : Zero X
A : Type u_2
inst✝² : NonUnitalRing A
inst✝¹ : StarRing A
inst✝ : Module ℝ A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ.realContinuousMapZeroOfNNReal ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f) = φ f

Tactic:
simp only [realContinuousMapZeroOfNNReal_apply]

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : Zero X
A : Type u_2
inst✝² : NonUnitalRing A
inst✝¹ : StarRing A
inst✝ : Module ℝ A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal -
      φ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal =
    φ f
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
f : C(X, 𝕜)
hfA : f ∈ ↑A.toSubalgebra
hf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂
F : C(X, 𝕜) := f - const X (f x₂)
⊢ F ∈ A

Tactic:
refine A.sub_mem hfA (@Eq.subst _ (· ∈ A) _ _ ?_ <| A.smul_mem A.one_mem <| f x₂)

Goal state after:
𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
f : C(X, 𝕜)
hfA : f ∈ ↑A.toSubalgebra
hf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂
F : C(X, 𝕜) := f - const X (f x₂)
⊢ f x₂ • 1 = const X (f x₂)
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
f : C(X, 𝕜)
hfA : f ∈ ↑A.toSubalgebra
hf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂
F : C(X, 𝕜) := f - const X (f x₂)
⊢ f x₂ • 1 = const X (f x₂)

Tactic:
ext1

Goal state after:
𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
f : C(X, 𝕜)
hfA : f ∈ ↑A.toSubalgebra
hf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂
F : C(X, 𝕜) := f - const X (f x₂)
a✝ : X
⊢ (f x₂ • 1) a✝ = (const X (f x₂)) a✝