awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (goal)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 0 1) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
h : Nontrivial A
p_pos : 0 < (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
⊢ HasFiniteIntegral bound (μ.restrict (Ioi 0))
have rw : HasFiniteIntegral (ε := ℝ) (fun (t : ℝ) ↦ f t maxr) (μ.restrict (Ioi 0))
Tactic: rw [hasFiniteIntegral_norm_iff]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (hypothesis)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 0 1) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
h : Nontrivial A
p_pos : 0 < (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
⊢ IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a ∂μ
have hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
Tactic: have hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0)) := by
    rw [hasFiniteIntegral_norm_iff]
    exact (hμ maxr maxr_nonneg).1.2
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (goal)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 0 1) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
h : Nontrivial A
p_pos : 0 < (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
⊢ ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p : ℝ).rpowIntegrand₀₁ x 0 = 0
have h : ∀ (a : ℝ), Membership.mem (γ := Set ℝ) (Ioi 0) a → (↑p : ℝ).rpowIntegrand₀₁ a 0 = 0
Tactic: filter_upwards [ae_restrict_mem measurableSet_Ioi]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (hypothesis)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 0 1) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
h : Nontrivial A
p_pos : 0 < (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
⊢ IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a ∂μ
have hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p : ℝ).rpowIntegrand₀₁ x 0 = 0
Tactic: have hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), rpowIntegrand₀₁ p x 0 = 0 := by
    filter_upwards [ae_restrict_mem measurableSet_Ioi]
    simp
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (goal)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 0 1) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
h : Nontrivial A
p_pos : 0 < (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p : ℝ).rpowIntegrand₀₁ x 0 = 0
⊢ IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a ∂μ
have refine : IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a) (Ioi 0) μ
Tactic: refine ⟨?integrable, ?integral⟩
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (goal)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 0 1) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
h : Nontrivial A
p_pos : 0 < (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p : ℝ).rpowIntegrand₀₁ x 0 = 0
refine :
  IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a) (Ioi 0) μ
⊢ IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a ∂μ
have refine₁ : a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a ∂μ
Tactic: refine ⟨?integrable, ?integral⟩
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (goal)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 0 1) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
h : Nontrivial A
p_pos : 0 < (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p : ℝ).rpowIntegrand₀₁ x 0 = 0
⊢ a ^ p = cfcₙ (p := IsSelfAdjoint) (fun (r : ℝ) ↦ ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t r ∂μ) a
have rw : cfcₙ (p := IsSelfAdjoint) (fun (x : ℝ) ↦ x ^ (↑p : ℝ)) a =
  cfcₙ (p := IsSelfAdjoint) (fun (r : ℝ) ↦ ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t r ∂μ) a
Tactic: rw [nnrpow_eq_cfcₙ_real _ _]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (goal)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 0 1) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
h : Nontrivial A
p_pos : 0 < (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p : ℝ).rpowIntegrand₀₁ x 0 = 0
⊢ a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a ∂μ
have a₁ : a ^ p = cfcₙ (p := IsSelfAdjoint) (fun (r : ℝ) ↦ ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t r ∂μ) a
Tactic: calc
    a ^ p = cfcₙ (fun r => ∫ t in Ioi 0, rpowIntegrand₀₁ p t r ∂μ) a := by
      rw [nnrpow_eq_cfcₙ_real _ _]
      exact cfcₙ_congr fun r _ ↦ (hμ r (by grind)).2
    _ = _ := cfcₙ_setIntegral measurableSet_Ioi _ bound a hf hmapzero hbound
                hbound_finite_integral ha.isSelfAdjoint
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (goal)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 0 1) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
h : Nontrivial A
p_pos : 0 < (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p : ℝ).rpowIntegrand₀₁ x 0 = 0
⊢ a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₀₁ t) a ∂μ
have case : a ^ p = cfcₙ (p := IsSelfAdjoint) (fun (r : ℝ) ↦ ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₀₁ t r ∂μ) a
Tactic: case integral => calc
    a ^ p = cfcₙ (fun r => ∫ t in Ioi 0, rpowIntegrand₀₁ p t r ∂μ) a := by
      rw [nnrpow_eq_cfcₙ_real _ _]
      exact cfcₙ_congr fun r _ ↦ (hμ r (by grind)).2
    _ = _ := cfcₙ_setIntegral measurableSet_Ioi _ bound a hf hmapzero hbound
                hbound_finite_integral ha.isSelfAdjoint
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (goal)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 1 2) p
⊢ ∃ (μ : Measure ℝ),
    ∀ (a : A),
      Membership.mem (γ := Set A) (Ici 0) a →
        IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a) (Ioi 0)
            μ ∧
          a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a ∂μ
have obtain : ∀ (μ : Measure ℝ)
  (hμ :
    ∀ (x : ℝ),
      Membership.mem (γ := Set ℝ) (Ici 0) x →
        IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧
          x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₁₂ t x ∂μ),
  ∃ (μ : Measure ℝ),
    ∀ (a : A),
      Membership.mem (γ := Set A) (Ici 0) a →
        IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a) (Ioi 0)
            μ ∧
          a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a ∂μ
Tactic: obtain ⟨μ, hμ⟩ := exists_measure_rpow_eq_integral_rpowIntegrand₁₂ hp
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (goal)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 1 2) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₁₂ t x ∂μ
⊢ ∃ (μ : Measure ℝ),
    ∀ (a : A),
      Membership.mem (γ := Set A) (Ici 0) a →
        IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a) (Ioi 0)
            μ ∧
          a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a ∂μ
have refine : ∀ (a : A) (ha : 0 ≤ a),
  IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a ∂μ
Tactic: refine ⟨μ, fun a (ha : 0 ≤ a) => ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (hypothesis)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 1 2) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
⊢ IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a ∂μ
have hpcoe : Membership.mem (γ := Set ℝ) (Ioo 1 2) (↑p : ℝ)
Tactic: have hpcoe : (p : ℝ) ∈ Ioo 1 2 := by exact_mod_cast hp
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (hypothesis)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 1 2) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : Membership.mem (γ := Set ℝ) (Ioo 1 2) (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
⊢ IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a ∂μ
have maxr_nonneg : 0 ≤ maxr
Tactic: have maxr_nonneg : 0 ≤ maxr :=
    le_csSup_of_le (b := 0) (IsCompact.bddAbove (quasispectrum.isCompact _)) (by simp) le_rfl
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (hypothesis)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 1 2) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : Membership.mem (γ := Set ℝ) (Ioo 1 2) (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
⊢ IntegrableOn.{0, u_1} (ε := A) (fun (t : ℝ) ↦ cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ (p := IsSelfAdjoint) ((↑p : ℝ).rpowIntegrand₁₂ t) a ∂μ
have hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
Tactic: have hf : ContinuousOn (Function.uncurry f) (Ioi (0 : ℝ) ×ˢ quasispectrum ℝ a) :=
    continuousOn_rpowIntegrand₁₂_uncurry hpcoe.1 (quasispectrum ℝ a) (by grind)
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (goal)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 1 2) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : Membership.mem (γ := Set ℝ) (Ioo 1 2) (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
⊢ ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
have h : ∀ (t : ℝ) (ht : Membership.mem (γ := Set ℝ) (Ioi 0) t), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
Tactic: filter_upwards [ae_restrict_mem measurableSet_Ioi] with t ht
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation (hypothesis)
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : SMulCommClass ℝ A A
inst_4 : IsScalarTower ℝ A A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_9 : CompleteSpace A
p : ℝ≥0
hp : Membership.mem (γ := Set ℝ≥0) (Ioo 1 2) p
μ : Measure ℝ
hμ :
  ∀ (x : ℝ),
    Membership.mem (γ := Set ℝ) (Ici 0) x →
      IntegrableOn (ε := ℝ) (fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧
        x ^ (↑p : ℝ) = ∫ (t : ℝ) in Ioi 0, (↑p : ℝ).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : Membership.mem (γ := Set ℝ) (Ioo 1 2) (↑p : ℝ)
f : (t x : ℝ) → ℝ := fun (t : ℝ) ↦ (↑p : ℝ).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : (t : ℝ) → ℝ := fun (t : ℝ) ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (SProd.sprod (α := Set ℝ) (Ioi 0) (quasispectrum ℝ a))
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Ioi 0) t
z : ℝ
hz : z ∈ quasispectrum ℝ a
⊢ ‖f t z‖ ≤ bound t
have hz' : 0 ≤ z
Tactic: have hz' : 0 ≤ z := by grind