mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : LinearOrder k
inst✝⁵ : IsStrictOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b : E
r : k
h : a < b
⊢ (lineMap a b) r < a ↔ r < 0

Tactic:
← lineMap_lt_lineMap_iff_of_lt' h,

Goal state after:
k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : LinearOrder k
inst✝⁵ : IsStrictOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b : E
r : k
h : a < b
⊢ (lineMap a b) r < a ↔ (lineMap a b) r < (lineMap a b) 0
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : LinearOrder k
inst✝⁵ : IsStrictOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b : E
r : k
h : a < b
⊢ b < (lineMap a b) r ↔ 1 < r

Tactic:
← lineMap_lt_lineMap_iff_of_lt' h,

Goal state after:
k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : LinearOrder k
inst✝⁵ : IsStrictOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b : E
r : k
h : a < b
⊢ b < (lineMap a b) r ↔ (lineMap a b) 1 < (lineMap a b) r
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : LinearOrder k
inst✝⁵ : IsStrictOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b : E
r : k
h : a < b
⊢ (lineMap a b) r < b ↔ r < 1

Tactic:
← lineMap_lt_lineMap_iff_of_lt' h,

Goal state after:
k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : LinearOrder k
inst✝⁵ : IsStrictOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b : E
r : k
h : a < b
⊢ (lineMap a b) r < b ↔ (lineMap a b) r < (lineMap a b) 1
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
g' : C(X, ℝ≥0)
hI' : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I
hgt' : ∀ x ∈ t, 0 < g' x
c : ℝ≥0
hc : c > 0
hgc' : t ⊆ {y | c ≤ g' y}
⊢ ∃ g, (↑(algebraMapCLM ℝ≥0 𝕜)).comp g ∈ I ∧ (∀ (x : X), g x ≤ 1) ∧ Set.EqOn (⇑g) 1 t

Tactic:
obtain ⟨g, hg, hgc⟩ := exists_mul_le_one_eqOn_ge g' hc

Goal state after:
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
g' : C(X, ℝ≥0)
hI' : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I
hgt' : ∀ x ∈ t, 0 < g' x
c : ℝ≥0
hc : c > 0
hgc' : t ⊆ {y | c ≤ g' y}
g : C(X, ℝ≥0)
hg : ∀ (x : X), (g * g') x ≤ 1
hgc : Set.EqOn (⇑(g * g')) 1 {x | c ≤ g' x}
⊢ ∃ g, (↑(algebraMapCLM ℝ≥0 𝕜)).comp g ∈ I ∧ (∀ (x : X), g x ≤ 1) ∧ Set.EqOn (⇑g) 1 t
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
⊢ ∀ {c u : Set X},
    IsClosed[inst✝¹] c →
      P c u →
        IsOpen[inst✝¹] u →
          c ⊆ u → ∃ v, IsOpen[inst✝¹] v ∧ c ⊆ v ∧ closure[inst✝¹] v ⊆ u ∧ P c v ∧ P (closure[inst✝¹] v) u

Tactic:
intro c u0 cIsClosed Pc u0IsOpen csubu0
      obtain ⟨u1, hu1⟩ := SeparatedNhds.of_isClosed_isCompact_closure_compl_isClosed cIsClosed
        (IsCompact.of_isClosed_subset hscp isClosed_closure
        (closure_mono (compl_subset_compl.mpr Pc)))
        (isClosed_compl_iff.mpr u0IsOpen) (LE.le.disjoint_compl_right csubu0)
      simp_rw [← subset_compl_iff_disjoint_right, compl_subset_comm (s := u0)] at hu1
      obtain ⟨v1, hu1, hv1, hcu1, hv1u, hu1v1⟩ := hu1
      refine ⟨u1, hu1, hcu1, ?_, Pc, (Pc.trans hcu1).trans subset_closure⟩
      exact closure_minimal hu1v1 hv1.isClosed_compl |>.trans hv1u

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ
hr : r ≤ 0
a : A
⊢ (r • a)⁻ = -r • a⁺

Tactic:
by
  conv_lhs => rw [← neg_neg r, neg_smul, negPart_neg, posPart_smul_of_nonneg (by simpa)]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : a ≤ 0
⊢ a = -a⁻

Tactic:
rw [← neg_nonneg] at ha

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ -a
⊢ a = -a⁻
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : ∀ x ∈ spectrum ℝ a, 0 ≤ x
r : ℝ
hr : 0 ≤ r
⊢ ∀ x ∈ spectrum ℝ (r • a), 0 ≤ x

Tactic:
nontriviality A

Goal state after:
A : Type u_2
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : ∀ x ∈ spectrum ℝ a, 0 ≤ x
r : ℝ
hr : 0 ≤ r
a✝ : Nontrivial A
⊢ ∀ x ∈ spectrum ℝ (r • a), 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case zero
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ∀ x ∈ spectrum ℝ 0, 0 ≤ x

Tactic:
nontriviality A

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
a✝ : Nontrivial A
⊢ ∀ x ∈ spectrum ℝ 0, 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
⊢ ∀ x ∈ spectrum ℝ ↑0, 0 ≤ x

Tactic:
nontriviality A

Goal state after:
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a✝ : Nontrivial A
⊢ ∀ x ∈ spectrum ℝ ↑0, 0 ≤ x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
y : ℝ
hy : y ≠ 0
ha : 0 ≤ a
⊢ IsUnit (a ^ y) ↔ IsUnit a

Tactic:
nontriviality A

Goal state after:
A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
y : ℝ
hy : y ≠ 0
ha : 0 ≤ a
a✝ : Nontrivial A
⊢ IsUnit (a ^ y) ↔ IsUnit a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : 0 ≤ a
⊢ 1 * 1 = a ↔ a = 1

Tactic:
mul_one,

Goal state after:
A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : 0 ≤ a
⊢ 1 = a ↔ a = 1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
ha : IsSelfAdjoint a
hb : IsSelfAdjoint b
⊢ ↑a ≤ ↑b ↔ a ≤ b

Tactic:
by
  -- TODO: prove the more general result for star monomorphisms and use it here.
  rw [← sub_nonneg, ← sub_nonneg (a := b), StarOrderedRing.nonneg_iff_spectrum_nonneg (R := ℝ) _,
    ← inr_sub ℂ b a, ← Unitization.quasispectrum_eq_spectrum_inr' ℝ ℂ]
  exact StarOrderedRing.nonneg_iff_quasispectrum_nonneg _ |>.symm

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ IsStrictlyPositive a⁻¹ʳ ↔ IsStrictlyPositive a

Tactic:
nontriviality A

Goal state after:
A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
a✝ : Nontrivial A
⊢ IsStrictlyPositive a⁻¹ʳ ↔ IsStrictlyPositive a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : a ≤ b
h : IsStrictlyPositive a
⊢ IsUnit b

Tactic:
nontriviality A

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : a ≤ b
h : IsStrictlyPositive a
a✝ : Nontrivial A
⊢ IsUnit b
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: A : Type u_2
inst✝¹⁰ : NormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedAlgebra ℝ A
inst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : ContinuousStar A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : CompleteSpace A
s : Set ℝ≥0
x✝¹ : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
f : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a : A
x✝ : (f, a) ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s}
ha : 0 ≤ (f, a).2
has : s ∈ 𝓝ˢ (spectrum ℝ≥0 (f, a).2)
hs : IsCompact (spectrum ℝ≥0 a)
k : Set ℝ≥0
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (spectrum ℝ≥0 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (spectrum ℝ≥0 x)}
⊢ ∃ t,
    IsOpen[instTopologicalSpaceProd] t ∧
      (f, a) ∈ t ∧
        ContinuousOn (fun fa ↦ cfc ((toFun {s}) fa.1) fa.2)
          ({f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)} ∩ t)

Tactic:
refine ⟨Set.univ ×ˢ {x | k ∈ 𝓝ˢ (spectrum ℝ≥0 x)}, isOpen_univ.prod this, by simpa, ?_⟩

Goal state after:
A : Type u_2
inst✝¹⁰ : NormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedAlgebra ℝ A
inst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : ContinuousStar A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : CompleteSpace A
s : Set ℝ≥0
x✝¹ : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
f : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a : A
x✝ : (f, a) ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s}
ha : 0 ≤ (f, a).2
has : s ∈ 𝓝ˢ (spectrum ℝ≥0 (f, a).2)
hs : IsCompact (spectrum ℝ≥0 a)
k : Set ℝ≥0
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (spectrum ℝ≥0 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (spectrum ℝ≥0 x)}
⊢ ContinuousOn (fun fa ↦ cfc ((toFun {s}) fa.1) fa.2)
    ({f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)} ∩
      Set.univ ×ˢ {x | k ∈ 𝓝ˢ (spectrum ℝ≥0 x)})
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
⊢ cfcₙ (f ∘ g) a = cfcₙ f (cfcₙ g a)

Tactic:
rw [cfcₙ_comp f g a ?_ (by simp [f, tsub_self]) ?_ (by simp [g]) ha₁]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
⊢ ContinuousOn f (g '' σₙ ℝ≥0 a)

A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
⊢ ContinuousOn g (σₙ ℝ≥0 a)
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ ∀ᶠ (x : A) in pure 1, ‖x‖ ≤ 1

Tactic:
nontriviality A

Goal state after:
A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a✝ : Nontrivial A
⊢ ∀ᶠ (x : A) in pure 1, ‖x‖ ≤ 1
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hy : y ∈ Set.Icc x 1
c : ℝ≥0
h : ‖star z * (1 - x) * z‖₊ ≤ c ^ 2
hy₀ : y ∈ Set.Icc 0 1
hy' : 1 - y ∈ Set.Icc 0 1
⊢ (1 - y) ^ 2 ≤ 1 - x

Tactic:
trans (1 - y)

Goal state after:
A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hy : y ∈ Set.Icc x 1
c : ℝ≥0
h : ‖star z * (1 - x) * z‖₊ ≤ c ^ 2
hy₀ : y ∈ Set.Icc 0 1
hy' : 1 - y ∈ Set.Icc 0 1
⊢ (1 - y) ^ 2 ≤ 1 - y

A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hy : y ∈ Set.Icc x 1
c : ℝ≥0
h : ‖star z * (1 - x) * z‖₊ ≤ c ^ 2
hy₀ : y ∈ Set.Icc 0 1
hy' : 1 - y ∈ Set.Icc 0 1
⊢ 1 - y ≤ 1 - x
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: case inr
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
hN : ∀ (x : n → α), ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N * ‖x‖₊
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
⊢ ∑ j, ‖A i j‖₊ ≤ N

Tactic:
let x : n → α := fun j => unitOf (A i j)

Goal state after:
case inr
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
hN : ∀ (x : n → α), ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N * ‖x‖₊
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
x : n → α := fun j ↦ unitOf (A i j)
⊢ ∑ j, ‖A i j‖₊ ≤ N
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: case inr
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
x : n → α := fun j ↦ unitOf (A i j)
hN : ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N * 1
hxn : ‖x‖₊ = 1
⊢ ∑ j, ‖A i j‖₊ ≤ N

Tactic:
mul_one,

Goal state after:
case inr
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
x : n → α := fun j ↦ unitOf (A i j)
hN : ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N
hxn : ‖x‖₊ = 1
⊢ ∑ j, ‖A i j‖₊ ≤ N
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMap

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : TopologicalSpace Y
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompactSpace Y
f : X → Y → E
g : C(Y, E)
f_ae_cont : ∀ᵐ (x : X) ∂μ, Continuous (f x)
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖f x y‖ ≤ bound x
⊢ HasFiniteIntegral (fun x ↦ mkD (f x) g) μ

Tactic:
by
  refine hasFiniteIntegral_of_bound _ bound bound_int ?_
  filter_upwards [bound_ge, f_ae_cont] with x bound_ge_x cont_x
  simpa only [mkD_apply_of_continuous cont_x] using bound_ge_x

Goal state after:
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁷ : MeasurableSpace X
μ : Measure X
inst✝⁶ : TopologicalSpace Y
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : CompactSpace Y
inst✝³ : Zero Y
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : SecondCountableTopologyEither X C(Y, E)
f : X → Y → E
g : C(Y, E)₀
f_cont : Continuous (Function.uncurry f)
f_zero : ∀ᵐ (x : X) ∂μ, f x 0 = 0
⊢ AEStronglyMeasurable (fun x ↦ mkD (f x) g) μ

Tactic:
← ContinuousMapZero.isEmbedding_toContinuousMap.aestronglyMeasurable_comp_iff

Goal state after:
X : Type u_1
Y : Type u_2
inst✝⁷ : MeasurableSpace X
μ : Measure X
inst✝⁶ : TopologicalSpace Y
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : CompactSpace Y
inst✝³ : Zero Y
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : SecondCountableTopologyEither X C(Y, E)
f : X → Y → E
g : C(Y, E)₀
f_cont : Continuous (Function.uncurry f)
f_zero : ∀ᵐ (x : X) ∂μ, f x 0 = 0
⊢ AEStronglyMeasurable (fun x ↦ ↑(mkD (f x) g)) μ
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁷ : MeasurableSpace X
μ : Measure X
inst✝⁶ : TopologicalSpace Y
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : CompactSpace Y
inst✝³ : Zero Y
s : Set X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : SecondCountableTopologyEither X C(Y, E)
hs : MeasurableSet s
f : X → Y → E
g : C(Y, E)₀
f_cont : ContinuousOn (Function.uncurry f) (s ×ˢ univ)
f_zero : ∀ᵐ (x : X) ∂μ.restrict s, f x 0 = 0
⊢ AEStronglyMeasurable (fun x ↦ mkD (f x) g) (μ.restrict s)

Tactic:
← ContinuousMapZero.isEmbedding_toContinuousMap.aestronglyMeasurable_comp_iff

Goal state after:
X : Type u_1
Y : Type u_2
inst✝⁷ : MeasurableSpace X
μ : Measure X
inst✝⁶ : TopologicalSpace Y
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : CompactSpace Y
inst✝³ : Zero Y
s : Set X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : SecondCountableTopologyEither X C(Y, E)
hs : MeasurableSet s
f : X → Y → E
g : C(Y, E)₀
f_cont : ContinuousOn (Function.uncurry f) (s ×ˢ univ)
f_zero : ∀ᵐ (x : X) ∂μ.restrict s, f x 0 = 0
⊢ AEStronglyMeasurable (fun x ↦ ↑(mkD (f x) g)) (μ.restrict s)
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁷ : MeasurableSpace X
μ : Measure X
inst✝⁶ : TopologicalSpace Y
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
t : Set Y
inst✝⁴ : CompactSpace ↑t
inst✝³ : Zero ↑t
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : SecondCountableTopologyEither X C(↑t, E)
f : X → Y → E
g : C(↑t, E)₀
f_cont : ContinuousOn (Function.uncurry f) (univ ×ˢ t)
f_zero : ∀ᵐ (x : X) ∂μ, f x ↑0 = 0
⊢ AEStronglyMeasurable (fun x ↦ mkD (t.domRestrict (f x)) g) μ

Tactic:
← ContinuousMapZero.isEmbedding_toContinuousMap.aestronglyMeasurable_comp_iff

Goal state after:
X : Type u_1
Y : Type u_2
inst✝⁷ : MeasurableSpace X
μ : Measure X
inst✝⁶ : TopologicalSpace Y
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
t : Set Y
inst✝⁴ : CompactSpace ↑t
inst✝³ : Zero ↑t
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : SecondCountableTopologyEither X C(↑t, E)
f : X → Y → E
g : C(↑t, E)₀
f_cont : ContinuousOn (Function.uncurry f) (univ ×ˢ t)
f_zero : ∀ᵐ (x : X) ∂μ, f x ↑0 = 0
⊢ AEStronglyMeasurable (fun x ↦ ↑(mkD (t.domRestrict (f x)) g)) μ
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁷ : MeasurableSpace X
μ : Measure X
inst✝⁶ : TopologicalSpace Y
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
s : Set X
t : Set Y
inst✝⁴ : CompactSpace ↑t
inst✝³ : Zero ↑t
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : SecondCountableTopologyEither X C(↑t, E)
hs : MeasurableSet s
f : X → Y → E
g : C(↑t, E)₀
f_cont : ContinuousOn (Function.uncurry f) (s ×ˢ t)
f_zero : ∀ᵐ (x : X) ∂μ.restrict s, f x ↑0 = 0
⊢ AEStronglyMeasurable (fun x ↦ mkD (t.domRestrict (f x)) g) (μ.restrict s)

Tactic:
← ContinuousMapZero.isEmbedding_toContinuousMap.aestronglyMeasurable_comp_iff

Goal state after:
X : Type u_1
Y : Type u_2
inst✝⁷ : MeasurableSpace X
μ : Measure X
inst✝⁶ : TopologicalSpace Y
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
s : Set X
t : Set Y
inst✝⁴ : CompactSpace ↑t
inst✝³ : Zero ↑t
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : SecondCountableTopologyEither X C(↑t, E)
hs : MeasurableSet s
f : X → Y → E
g : C(↑t, E)₀
f_cont : ContinuousOn (Function.uncurry f) (s ×ˢ t)
f_zero : ∀ᵐ (x : X) ∂μ.restrict s, f x ↑0 = 0
⊢ AEStronglyMeasurable (fun x ↦ ↑(mkD (t.domRestrict (f x)) g)) (μ.restrict s)
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: n : Type u_2
A : Type u_5
inst✝ : Star A
f : CStarMatrix n n A
i j : n
⊢ star f i j = star (f j i)

Tactic:
by
  rw [star_eq_conjTranspose, conjTranspose_apply]

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

Goal state before:
pp: case hf₀
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ :
  ∀ (z : ↑(quasispectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
z : ↑(quasispectrum 𝕜 a)
⊢ ∫ (x : X), f x ↑0 ∂μ = 0

Tactic:
· exact integral_eq_zero_of_ae hf₂

Goal state after: