mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²³ : Semifield R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : IsTopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : Field S
inst✝¹⁷ : StarRing S
inst✝¹⁶ : MetricSpace S
inst✝¹⁵ : IsTopologicalRing S
inst✝¹⁴ : ContinuousStar S
inst✝¹³ : NonUnitalRing A
inst✝¹² : StarRing A
inst✝¹¹ : Module S A
inst✝¹⁰ : IsScalarTower S A A
inst✝⁹ : SMulCommClass S A A
inst✝⁸ : Algebra R S
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus S A q
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
g : C(↑(σₙ R a), R)₀
s : S
hs :
  s ∈
    range
      ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
          (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
⊢ (algebraMap R S) (f s) = s

Tactic:
obtain ⟨r, rfl⟩ := hs

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²³ : Semifield R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : IsTopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : Field S
inst✝¹⁷ : StarRing S
inst✝¹⁶ : MetricSpace S
inst✝¹⁵ : IsTopologicalRing S
inst✝¹⁴ : ContinuousStar S
inst✝¹³ : NonUnitalRing A
inst✝¹² : StarRing A
inst✝¹¹ : Module S A
inst✝¹⁰ : IsScalarTower S A A
inst✝⁹ : SMulCommClass S A A
inst✝⁸ : Algebra R S
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus S A q
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
g : C(↑(σₙ R a), R)₀
r : ↑(σₙ S a)
⊢ (algebraMap R S)
      (f
        (({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
            (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
          r)) =
    ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
      r
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : p a ∧ ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ -f x) a = -cfc f a

pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ cfc (fun x ↦ -f x) a = -cfc f a

Tactic:
· obtain ⟨ha, hf⟩ := h
    rw [cfc_apply f a, ← map_neg, cfc_apply ..]
    congr

Goal state after:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ cfc (fun x ↦ -f x) a = -cfc f a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
ha : p a
⊢ cfcₙ 0 a = 0

Tactic:
exact cfcₙ_apply (0 : R → R) a ▸ map_zero (cfcₙHom ha)

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
⊢ cfcₙ f 0 = 0

Tactic:
by
  by_cases hf0 : f 0 = 0
  · nth_rw 2 [← cfcₙ_zero R 0]
    apply cfcₙ_congr
    simpa [CFC.quasispectrum_zero_eq]
  · exact cfcₙ_apply_of_not_map_zero _ hf0

Goal state after:
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
f : ↥A
g : ℝ[X]
⊢ (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (↑f).attachBound ∈ A

Tactic:
by
  rw [polynomial_comp_attachBound]
  apply SetLike.coe_mem

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.UnitizationL1

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 * y.ofLp.toProd.1‖ +
      ‖x.ofLp.toProd.1 • y.ofLp.toProd.2 + (y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2)‖ ≤
    ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.2‖ +
      (‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖)

Tactic:
add_assoc

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 * y.ofLp.toProd.1‖ +
      ‖x.ofLp.toProd.1 • y.ofLp.toProd.2 + (y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2)‖ ≤
    ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.1‖ +
      (‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.2‖ +
        (‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖))
Module: Mathlib.Topology.Algebra.Module.Spaces.WeakBilin

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
𝕝 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : CommSemiring 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : Module 𝕜 E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
inst✝ : ContinuousAdd 𝕜
⊢ ContinuousAdd (WeakBilin B)

Tactic:
refine ⟨continuous_induced_rng.2 ?_⟩

Goal state after:
α : Type u_1
𝕜 : Type u_2
𝕝 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : CommSemiring 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : Module 𝕜 E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
inst✝ : ContinuousAdd 𝕜
⊢ Continuous[instTopologicalSpaceProd, Pi.topologicalSpace] ((fun x y ↦ (B x) y) ∘ fun p ↦ p.1 + p.2)
Module: Mathlib.Topology.Algebra.Module.Spaces.WeakBilin

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
𝕝 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : CommSemiring 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : Module 𝕜 E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
inst✝ : ContinuousSMul 𝕜 𝕜
⊢ ContinuousSMul 𝕜 (WeakBilin B)

Tactic:
refine ⟨continuous_induced_rng.2 ?_⟩

Goal state after:
α : Type u_1
𝕜 : Type u_2
𝕝 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : CommSemiring 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : Module 𝕜 E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
inst✝ : ContinuousSMul 𝕜 𝕜
⊢ Continuous[instTopologicalSpaceProd, Pi.topologicalSpace] ((fun x y ↦ (B x) y) ∘ fun p ↦ p.1 • p.2)
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝] U
⊢ IsOpen[generateFrom (powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V})]
    {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
ofPred_and,

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝] U
⊢ IsOpen[generateFrom (powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V})]
    ({a | a ⊆ ⋃₀ u} ∩ {a | ∀ U ∈ u, (a ∩ U).Nonempty})
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
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, u).1
hu : (V, u).2.Finite
huB : (V, u).2 ⊆ B
right✝ : ∀ U ∈ (V, u).2, U ⊆ (V, u).1
⊢ IsOpen[TopologicalSpace.vietoris α] {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
ofPred_and,

Goal state after:
case refine_1
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, u).1
hu : (V, u).2.Finite
huB : (V, u).2 ⊆ B
right✝ : ∀ U ∈ (V, u).2, U ⊆ (V, u).1
⊢ IsOpen[TopologicalSpace.vietoris α] ({a | a ⊆ V} ∩ {a | ∀ U ∈ u, (a ∩ U).Nonempty})
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : IsScalarTower 𝕜 A A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ σ 𝕜
      (((↑(cfcHom ⋯)).comp
          ((↑(Homeomorph.compStarAlgEquiv' 𝕜 𝕜 (Homeomorph.setCongr ⋯))).comp ContinuousMapZero.toContinuousMapHom))
        f) =
    Set.range ⇑f

Tactic:
NonUnitalStarAlgHom.comp_apply

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : IsScalarTower 𝕜 A A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ σ 𝕜
      (↑(cfcHom ⋯)
        (((↑(Homeomorph.compStarAlgEquiv' 𝕜 𝕜 (Homeomorph.setCongr ⋯))).comp ContinuousMapZero.toContinuousMapHom) f)) =
    Set.range ⇑f
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case left
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
hU₁ : U ∈ 𝓤 α
hU₂ : IsClosed[instTopologicalSpaceProd] U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
x : α
hx : x ∈ ↑K
lx : Filter α := l ⊓ 𝓟 (UniformSpace.ball x U)
le_def : lx = l ⊓ 𝓟 (UniformSpace.ball x U)
⊢ x ∈ U.preimage ↑L

Tactic:
have hlx : lx.TotallyBounded := hl.mono inf_le_left

Goal state after:
case left
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
hU₁ : U ∈ 𝓤 α
hU₂ : IsClosed[instTopologicalSpaceProd] U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
x : α
hx : x ∈ ↑K
lx : Filter α := l ⊓ 𝓟 (UniformSpace.ball x U)
le_def : lx = l ⊓ 𝓟 (UniformSpace.ball x U)
hlx : lx.TotallyBounded
⊢ x ∈ U.preimage ↑L
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu : u.Finite
huB : u ⊆ B
⊢ IsOpen {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}

Tactic:
ofPred_and,

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu : u.Finite
huB : u ⊆ B
⊢ IsOpen ({a | ↑a ⊆ ⋃₀ u} ∩ {a | ∀ U ∈ u, (↑a ∩ U).Nonempty})
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁶ : NormedRing A
inst✝⁵ : CompleteSpace A
inst✝⁴ : SetLike SA A
inst✝³ : SubringClass SA A
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
h : IsPreconnected (σ 𝕜 ↑x)ᶜ
z : 𝕜
hz : z ∈ σ 𝕜 x \ σ 𝕜 ↑x
⊢ Bornology.IsBounded univ

Tactic:
obtain ⟨hz, hz'⟩ := mem_sdiff _ |>.mp hz

Goal state after:
𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁶ : NormedRing A
inst✝⁵ : CompleteSpace A
inst✝⁴ : SetLike SA A
inst✝³ : SubringClass SA A
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
h : IsPreconnected (σ 𝕜 ↑x)ᶜ
z : 𝕜
hz✝ : z ∈ σ 𝕜 x \ σ 𝕜 ↑x
hz : z ∈ σ 𝕜 x
hz' : z ∉ σ 𝕜 ↑x
⊢ Bornology.IsBounded univ
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜₁ : Type u_3
𝕜₂ : Type u_4
A : Type u_5
inst✝⁶ : NormedField 𝕜₁
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜₁ A
inst✝² : NormedAlgebra 𝕜₂ A
inst✝¹ : NormedAlgebra 𝕜₁ 𝕜₂
inst✝ : IsScalarTower 𝕜₁ 𝕜₂ A
f : 𝕜₂ → 𝕜₁
a : A
h : SpectrumRestricts a f
⊢ spectralRadius 𝕜₁ a = spectralRadius 𝕜₂ a

Tactic:
rw [spectralRadius, spectralRadius]
  have := algebraMap_isometry 𝕜₁ 𝕜₂ |>.nnnorm_map_of_map_zero (map_zero _)
  apply le_antisymm
  all_goals apply iSup₂_le fun x hx ↦ ?_
  · refine congr_arg ((↑) : ℝ≥0 → ℝ≥0∞) (this x) |>.symm.trans_le <| le_iSup₂ (α := ℝ≥0∞) _ ?_
    exact (spectrum.algebraMap_mem_iff _).mpr hx
  · have ⟨y, hy, hy'⟩ := h.algebraMap_image.symm ▸ hx
    subst hy'
    exact this y ▸ le_iSup₂ (α := ℝ≥0∞) y hy

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
n : ℕ
⊢ ↑‖(a⋆ * a) ^ n‖₊ ^ (1 / ↑n) = ((fun x ↦ x ^ 2) ∘ fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)) n

Tactic:
rw [Function.comp_apply, ← rpow_natCast, ← rpow_mul, mul_comm, rpow_mul, rpow_natCast, ←
      coe_pow, sq, ← nnnorm_star_mul_self, Commute.mul_pow (star_comm_self' a), star_pow]

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

Goal state before:
pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
hc : 0 < c
h : ∀ x ∈ σ ℝ≥0 a, f x < c
⊢ ‖cfc f a‖₊ < c

Tactic:
by
  obtain (_ | _) := subsingleton_or_nontrivial A
  · rw [Subsingleton.elim (cfc f a) 0]
    simpa
  · refine cfc_cases (‖·‖₊ < c) a f (by simpa) fun hf ha ↦ ?_
    simp only [← cfc_apply f a, (IsGreatest.nnnorm_cfc_nnreal f a hf ha |>.lt_iff)]
    rintro - ⟨x, hx, rfl⟩
    exact h x hx

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
⊢ {K | (↑K).Nonempty ∧ ↑K ⊆ s} ⊆ closure {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ s}

Tactic:
ofPred_and,

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
⊢ {a | (↑a).Nonempty} ∩ {a | ↑a ⊆ s} ⊆ closure {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ s}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
⊢ IsPreconnected {K | (↑K).Nonempty ∧ ↑K ⊆ s}

Tactic:
by
  refine (isPreconnected_nonempty_finite_subsets hs).subset_closure (by grind) ?_
  rw [ofPred_and, ofPred_and]
  simp_rw [Compacts.coe_nonempty, ← compl_singleton_eq]
  grw [← isClopen_singleton_bot.compl.isOpen.inter_closure, closure_finite_subsets,
    ← subset_closure]

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
U : Set (Compacts α)
hU :
  U ∈
    (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}}
this : IsPreconnected (U ∩ {K | (↑K).Finite})
⊢ U ⊆ U ∩ univ

Tactic:
inter_univ

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
U : Set (Compacts α)
hU :
  U ∈
    (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}}
this : IsPreconnected (U ∩ {K | (↑K).Finite})
⊢ U ⊆ U
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f g : α →C₀ β
h : f.toBCF = g.toBCF
⊢ f = g

Tactic:
by
  ext x
  simpa only using! DFunLike.congr_fun h x

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : LocallyConvexSpace 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
⊢ (𝓝 0).HasBasis (fun s ↦ s ∈ 𝓝 0 ∧ AbsConvex 𝕜 s) id

Tactic:
refine
    (LocallyConvexSpace.convex_basis_zero 𝕜 E).to_hasBasis (fun s hs => ?_) fun s hs =>
      ⟨s, ⟨hs.1, hs.2.2⟩, rfl.subset⟩

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : LocallyConvexSpace 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0 ∧ Convex 𝕜 s
⊢ ∃ i', (i' ∈ 𝓝 0 ∧ AbsConvex 𝕜 i') ∧ id i' ⊆ id s
Module: Mathlib.Analysis.Normed.Module.Dual

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : StrongDual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : ¬c = 0
eq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖
le : ∀ z ∈ s, ‖c⁻¹ • x' z‖ ≤ ‖c⁻¹‖ * ‖c‖
cancel : ‖c⁻¹‖ * ‖c‖ = 1
⊢ c⁻¹ • x' ∈ polar 𝕜 s

Tactic:
cancel

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : StrongDual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : ¬c = 0
eq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖
le : ∀ z ∈ s, ‖c⁻¹ • x' z‖ ≤ 1
cancel : ‖c⁻¹‖ * ‖c‖ = 1
⊢ c⁻¹ • x' ∈ polar 𝕜 s
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
⊢ a < (lineMap a b) r ↔ 0 < r

Tactic:
by
  rw [← lineMap_lt_lineMap_iff_of_lt' h, lineMap_apply_zero]

Goal state after:
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:
by
  rw [← lineMap_lt_lineMap_iff_of_lt' h, lineMap_apply_zero]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁸ : Field k
inst✝⁷ : LinearOrder k
inst✝⁶ : IsStrictOrderedRing k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r : k
hab : a < b
h₀ : 0 < r
h₁ : r < 1
⊢ slope f a ((lineMap a b) r) ≤ (lineMap (slope f ((lineMap a b) r) b) (slope f a ((lineMap a b) r))) r ↔
    slope f a ((lineMap a b) r) ≤ slope f ((lineMap a b) r) b

Tactic:
right_le_lineMap_iff_le h₁

Goal state after:
k : Type u_1
E : Type u_2
inst✝⁸ : Field k
inst✝⁷ : LinearOrder k
inst✝⁶ : IsStrictOrderedRing k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r : k
hab : a < b
h₀ : 0 < r
h₁ : r < 1
⊢ slope f a ((lineMap a b) r) ≤ slope f ((lineMap a b) r) b ↔ slope f a ((lineMap a b) r) ≤ slope f ((lineMap a b) r) b
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)
hI : IsClosed ↑I
⊢ I.IsMaximal ↔ ∃ x, idealOfSet 𝕜 {x}ᶜ = I

Tactic:
refine
    ⟨?_, fun h =>
      let ⟨x, hx⟩ := h
      hx ▸ idealOf_compl_singleton_isMaximal 𝕜 x⟩

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)
hI : IsClosed ↑I
⊢ I.IsMaximal → ∃ x, idealOfSet 𝕜 {x}ᶜ = I
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝³ : NormedCommRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
I : Ideal A
inst✝ : I.IsMaximal
a : A
ha : a ∈ I
⊢ ⋯.some = 0

Tactic:
simp_rw [Quotient.eq_zero_iff_mem.mpr ha, spectrum.zero_eq]

Goal state after:
A : Type u_1
inst✝³ : NormedCommRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
I : Ideal A
inst✝ : I.IsMaximal
a : A
ha : a ∈ I
⊢ ⋯.some = 0
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝ : CommCStarAlgebra A
⊢ ∀ {a : C(↑(characterSpace ℂ A), ℂ)},
    a ∈ (gelfandTransform ℂ A).range.carrier → star a ∈ (gelfandTransform ℂ A).range.carrier

Tactic:
by
        rintro - ⟨a, rfl⟩
        use star a
        ext1 φ
        dsimp
        simp only [map_star, RCLike.star_def]

Goal state after: