mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
hb : Summable fun n ↦ (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
h₀ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = (a - ↑ₐ z) * b
h₁ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = b * (a - ↑ₐ z)
⊢ exp (a - ↑ₐ z) = 1 + (a - ↑ₐ z) * b

Tactic:
rw [exp_eq_tsum 𝕜]

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
hb : Summable fun n ↦ (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
h₀ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = (a - ↑ₐ z) * b
h₁ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = b * (a - ↑ₐ z)
⊢ (fun x ↦ ∑' (n : ℕ), (↑n.factorial)⁻¹ • x ^ n) (a - ↑ₐ z) = 1 + (a - ↑ₐ z) * b
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace α
f : Filter (Compacts α)
x✝ : Cauchy f
left✝ : f.NeBot
hf : ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∀ᶠ (K : Compacts α) (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
l : Filter α := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
hl : l.TotallyBounded
L : Compacts α := { carrier := {x | ClusterPt x l}, isCompact' := ⋯ }
⊢ ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsClosed[instTopologicalSpaceProd] i' →
      (fun y ↦ (y, L)) ⁻¹' Prod.map SetLike.coe SetLike.coe ⁻¹' hausdorffEntourage (id i') ∈ f

Tactic:
intro U

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace α
f : Filter (Compacts α)
x✝ : Cauchy f
left✝ : f.NeBot
hf : ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∀ᶠ (K : Compacts α) (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
l : Filter α := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
hl : l.TotallyBounded
L : Compacts α := { carrier := {x | ClusterPt x l}, isCompact' := ⋯ }
U : SetRel α α
⊢ U ∈ 𝓤 α ∧ IsClosed[instTopologicalSpaceProd] U →
    (fun y ↦ (y, L)) ⁻¹' Prod.map SetLike.coe SetLike.coe ⁻¹' hausdorffEntourage (id U) ∈ f
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set (Set α)
hs : IsPreconnected s
t : Set α
hts : t ∈ s
ht : IsPreconnected t
hts' : t ⊆ ⋃₀ s
⊢ IsPreconnected (⋃₀ s)

Tactic:
intro U

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s : Set (Set α)
hs : IsPreconnected s
t : Set α
hts : t ∈ s
ht : IsPreconnected t
hts' : t ⊆ ⋃₀ s
U : Set α
⊢ ∀ (v : Set α),
    IsOpen[inst✝] U →
      IsOpen[inst✝] v → ⋃₀ s ⊆ U ∪ v → (⋃₀ s ∩ U).Nonempty → (⋃₀ s ∩ v).Nonempty → (⋃₀ s ∩ (U ∩ v)).Nonempty
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
x✝ : CompleteSpace (NonemptyCompacts α)
f : Filter α
hf : Cauchy f
K : NonemptyCompacts α
hK : Filter.map (fun x ↦ {x}) f ≤ 𝓝 K
x : α
hx : x ∈ ↑K
⊢ ∀ (i' : Set α), x ∈ i' ∧ IsOpen[inst✝.toTopologicalSpace] i' → i' ∈ f

Tactic:
intro U

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
x✝ : CompleteSpace (NonemptyCompacts α)
f : Filter α
hf : Cauchy f
K : NonemptyCompacts α
hK : Filter.map (fun x ↦ {x}) f ≤ 𝓝 K
x : α
hx : x ∈ ↑K
U : Set α
⊢ x ∈ U ∧ IsOpen[inst✝.toTopologicalSpace] U → U ∈ f
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : PartialOrder A
inst✝⁷ : StarRing A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module ℝ A
inst✝³ : IsScalarTower ℝ A A
inst✝² : SMulCommClass ℝ A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
x : A
hx₁ : IsSelfAdjoint x
hx₂ : QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal
hb : IsSelfAdjoint (x * x) ∧ QuasispectrumRestricts (x * x) ⇑ContinuousMap.realToNNReal
h : Commute a (x * x)
hx : 0 ≤ x
⊢ QuasispectrumRestricts (a * (x * x)) ⇑ContinuousMap.realToNNReal

Tactic:
rw [← mul_assoc, quasispectrumRestricts_iff, quasispectrum.mul_comm,
    ← quasispectrumRestricts_iff, ← mul_assoc]

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : PartialOrder A
inst✝⁷ : StarRing A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module ℝ A
inst✝³ : IsScalarTower ℝ A A
inst✝² : SMulCommClass ℝ A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
x : A
hx₁ : IsSelfAdjoint x
hx₂ : QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal
hb : IsSelfAdjoint (x * x) ∧ QuasispectrumRestricts (x * x) ⇑ContinuousMap.realToNNReal
h : Commute a (x * x)
hx : 0 ≤ x
⊢ QuasispectrumRestricts (x * a * x) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃ i ∈ w, i
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            w.Finite →
              ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
                K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hw : ¬w.Finite
w' : Set (Set α)
h✝ : w' ⊆ w ∧ w'.Finite ∧ ↑K ⊆ ⋃ i ∈ w', i
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
apply this w' <;> grind [sUnion_eq_biUnion]

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

Goal state before:
pp: case refine_1
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 : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : MetricSpace A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMapZero.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts a ⇑f
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a)) R
x : ↑(σₙ S a)
⊢ dist
      (↑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
            (g₁.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
        x)
      (↑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
            (g₂.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
        x) ≤
    dist g₁ g₂

Tactic:
simpa [halg.dist_eq] using! ContinuousMap.dist_apply_le_dist _

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
k : 𝕜
hk : k ∈ resolventSet 𝕜 a
⊢ HasDerivAt (fun x ↦ resolvent x k) (resolvent a k ^ 2) a

Tactic:
by
  convert! hasFDerivAt_resolvent (𝕜 := A) hk |>.hasDerivAt
  simp [resolvent, pow_two]

Goal state after:
Module: Mathlib.Analysis.Convex.TotallyBounded

Goal state before:
pp: E : Type u_1
s : Set E
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : UniformSpace E
inst✝² : IsUniformAddGroup E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : ContinuousSMul ℝ E
hs : ∀ U ∈ nhds 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U
⊢ ∀ U ∈ nhds 0, ∃ t, t.Finite ∧ (convexHull ℝ) s ⊆ ⋃ y ∈ t, y +ᵥ U

Tactic:
intro U

Goal state after:
E : Type u_1
s : Set E
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : UniformSpace E
inst✝² : IsUniformAddGroup E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : ContinuousSMul ℝ E
hs : ∀ U ∈ nhds 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U
U : Set E
⊢ U ∈ nhds 0 → ∃ t, t.Finite ∧ (convexHull ℝ) s ⊆ ⋃ y ∈ t, y +ᵥ U
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
x✝ : F
⊢ x✝ ∈ B.polar ↑m ↔ x✝ ∈ ↑(⨅ x ∈ m, (B x).ker)

Tactic:
simp [polar_subMulAction]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.AbsConvexOpen

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommMonoid E
inst✝² : SeminormedRing 𝕜
inst✝¹ : SMul 𝕜 E
inst✝ : PartialOrder 𝕜
⊢ ∃ x, True

Tactic:
rw [Subtype.exists]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommMonoid E
inst✝² : SeminormedRing 𝕜
inst✝¹ : SMul 𝕜 E
inst✝ : PartialOrder 𝕜
⊢ ∃ a, ∃ (_ : 0 ∈ a ∧ IsOpen[inst✝⁴] a ∧ AbsConvex 𝕜 a), True
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: W : Type u_3
Q : Type u_4
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 W
p₁ p₂ : Q
h : p₁ ≠ p₂
c₁ c₂ : 𝕜
⊢ dist c₁ c₂ ≤ ↑(nndist p₁ p₂)⁻¹ * (dist c₁ c₂ * dist p₁ p₂)

Tactic:
NNReal.coe_inv,

Goal state after:
W : Type u_3
Q : Type u_4
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 W
p₁ p₂ : Q
h : p₁ ≠ p₂
c₁ c₂ : 𝕜
⊢ dist c₁ c₂ ≤ (↑(nndist p₁ p₂))⁻¹ * (dist c₁ c₂ * dist p₁ p₂)
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: W : Type u_3
Q : Type u_4
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 W
p₁ p₂ : Q
h : p₁ ≠ p₂
c₁ c₂ : 𝕜
⊢ dist c₁ c₂ ≤ (dist p₁ p₂)⁻¹ * (dist c₁ c₂ * dist p₁ p₂)

Tactic:
mul_left_comm,

Goal state after:
W : Type u_3
Q : Type u_4
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 W
p₁ p₂ : Q
h : p₁ ≠ p₂
c₁ c₂ : 𝕜
⊢ dist c₁ c₂ ≤ dist c₁ c₂ * ((dist p₁ p₂)⁻¹ * dist p₁ p₂)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi

Goal state before:
pp: case pos
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : (i : ι) → NonUnitalRing (A i)
inst✝⁹ : (i : ι) → Module S (A i)
inst✝⁸ : (i : ι) → Module R (A i)
inst✝⁷ : ∀ (i : ι), IsScalarTower R S (A i)
inst✝⁶ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R (A i) (A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMapZero.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, quasispectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
hf₀ : f 0 = 0
⊢ cfcₙ f a = fun i ↦ cfcₙ f (a i)

Tactic:
ext i

Goal state after:
case pos
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : (i : ι) → NonUnitalRing (A i)
inst✝⁹ : (i : ι) → Module S (A i)
inst✝⁸ : (i : ι) → Module R (A i)
inst✝⁷ : ∀ (i : ι), IsScalarTower R S (A i)
inst✝⁶ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R (A i) (A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMapZero.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, quasispectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
hf₀ : f 0 = 0
i : ι
⊢ cfcₙ f a i = cfcₙ f (a i)
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 u0 : Set X
⊢ IsClosed[inst✝¹] c →
    P c u0 →
      IsOpen[inst✝¹] u0 →
        c ⊆ u0 → ∃ v, IsOpen[inst✝¹] v ∧ c ⊆ v ∧ closure[inst✝¹] v ⊆ u0 ∧ P c v ∧ P (closure[inst✝¹] v) u0

Tactic:
cIsClosed

Goal state after:
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 u0 : Set X
cIsClosed : IsClosed[inst✝¹] c
⊢ P c u0 →
    IsOpen[inst✝¹] u0 →
      c ⊆ u0 → ∃ v, IsOpen[inst✝¹] v ∧ c ⊆ v ∧ closure[inst✝¹] v ⊆ u0 ∧ P c v ∧ P (closure[inst✝¹] v) u0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁻) a * cfcₙ (fun x ↦ x⁺) a = 0

pp: case neg
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : ¬IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁻) a * cfcₙ (fun x ↦ x⁺) a = 0

Tactic:
· rw [← cfcₙ_mul _ _, ← cfcₙ_zero ℝ a]
    refine cfcₙ_congr (fun x _ ↦ ?_)
    simp only [_root_.posPart_def, _root_.negPart_def]
    simpa using le_total 0 x

Goal state after:
case neg
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : ¬IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁻) a * cfcₙ (fun x ↦ x⁺) a = 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi

Goal state before:
pp: ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : (i : ι) → Ring (A i)
inst✝⁷ : (i : ι) → Algebra S (A i)
inst✝⁶ : (i : ι) → Algebra R (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R S (A i)
hinst : IsScalarTower R S ((i : ι) → A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : ContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), ContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMap.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, spectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
⊢ cfc f a = fun i ↦ cfc f (a i)

Tactic:
ext i

Goal state after:
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : (i : ι) → Ring (A i)
inst✝⁷ : (i : ι) → Algebra S (A i)
inst✝⁶ : (i : ι) → Algebra R (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R S (A i)
hinst : IsScalarTower R S ((i : ι) → A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : ContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), ContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMap.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, spectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
i : ι
⊢ cfc f a i = cfc f (a i)
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case h.refine_2
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 : Urysohns.CU P := { C := closure[inst✝¹] u, U := tᶜ, P_C_U := ⋯, closed_C := ⋯, open_U := ⋯, subset := ⋯, hP := ⋯ }
⊢ EqOn c.lim 1 t

Tactic:
intro x hx
    apply Urysohns.CU.lim_of_notMem_U
    exact notMem_compl_iff.mpr hx

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 : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : a = b - a⁻
hb : 0 ≤ b
ha : IsSelfAdjoint a
⊢ a⁺ = b

Tactic:
nth_rw 1 [← posPart_sub_negPart a] at hab

Goal state after:
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 : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : a⁺ - a⁻ = b - a⁻
hb : 0 ≤ b
ha : IsSelfAdjoint a
⊢ a⁺ = b
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
⊢ 1⁺ = 1

Tactic:
CFC.posPart_def,

Goal state after:
A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
⊢ cfcₙ (fun x ↦ x⁺) 1 = 1
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ
⊢ ((algebraMap ℝ A) r)⁺ = (algebraMap ℝ A) r⁺

Tactic:
CFC.posPart_def,

Goal state after:
A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ
⊢ cfcₙ (fun x ↦ x⁺) ((algebraMap ℝ A) r) = (algebraMap ℝ A) r⁺
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ≥0
⊢ ((algebraMap ℝ≥0 A) r)⁺ = (algebraMap ℝ≥0 A) r

Tactic:
CFC.posPart_def,

Goal state after:
A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ≥0
⊢ cfcₙ (fun x ↦ x⁺) ((algebraMap ℝ≥0 A) r) = (algebraMap ℝ≥0 A) r
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case mpr.add
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝¹ : PartialOrder A := spectralOrder A
x✝ p x y : A
hx✝ : x ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hy✝ : y ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hx : IsSelfAdjoint x ∧ QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal
hy : IsSelfAdjoint y ∧ QuasispectrumRestricts y ⇑ContinuousMap.realToNNReal
⊢ IsSelfAdjoint (x + y) ∧ QuasispectrumRestricts (x + y) ⇑ContinuousMap.realToNNReal

Tactic:
simp +singlePass only [← Unitization.isSelfAdjoint_inr (R := ℂ),
            quasispectrumRestricts_iff_spectrumRestricts_inr' ℂ] at hx hy ⊢

Goal state after:
case mpr.add
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝¹ : PartialOrder A := spectralOrder A
x✝ p x y : A
hx✝ : x ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hy✝ : y ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hx : IsSelfAdjoint ↑x ∧ SpectrumRestricts ↑x ⇑ContinuousMap.realToNNReal
hy : IsSelfAdjoint ↑y ∧ SpectrumRestricts ↑y ⇑ContinuousMap.realToNNReal
⊢ IsSelfAdjoint ↑(x + y) ∧ SpectrumRestricts ↑(x + y) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

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

Tactic:
by simp [sqrt]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«0».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨0, ⋯⟩)‖ ≤ ‖a‖

Tactic:
exact CStarAlgebra.norm_posPart_le _ |>.trans <| realPart.norm_le a

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ∀ (x : Fin 4),
    0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] x ∧
      ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] x‖ ≤ ‖a‖

Tactic:
intro i
    fin_cases i
    all_goals
      constructor
      · simp
        cfc_tac
    · exact CStarAlgebra.norm_posPart_le _ |>.trans <| realPart.norm_le a
    · exact CStarAlgebra.norm_posPart_le _ |>.trans <| imaginaryPart.norm_le a
    · exact CStarAlgebra.norm_negPart_le _ |>.trans <| realPart.norm_le a
    · exact CStarAlgebra.norm_negPart_le _ |>.trans <| imaginaryPart.norm_le a

Goal state after: