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

Goal state before:
pp: case inr
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 < c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ < c
h✝ : Nontrivial A
hf : ContinuousOn f (σ 𝕜 a)
ha : p a
⊢ ∀ x ∈ (fun x ↦ ‖f x‖) '' σ 𝕜 a, x < c

Tactic:
rintro - ⟨x, hx, rfl⟩

Goal state after:
case inr
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 < c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ < c
h✝ : Nontrivial A
hf : ContinuousOn f (σ 𝕜 a)
ha : p a
x : 𝕜
hx : x ∈ σ 𝕜 a
⊢ (fun x ↦ ‖f x‖) x < c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
x✝ : NonUnitalContinuousFunctionalCalculus 𝕜 A p := RCLike.nonUnitalContinuousFunctionalCalculus fun {x} ↦ hp₁
⊢ (inrNonUnitalStarAlgHom 𝕜 A).comp (cfcₙHom ha) = cfcₙAux ⋯ a ha

Tactic:
apply ContinuousMapZero.UniqueHom.eq_of_continuous_of_map_id _ _ _
    (Unitization.continuous_inr.comp <| cfcₙHom_continuous ha)
    (continuous_cfcₙAux hp₁ a ha)
    (by simp [cfcₙHom_id ha, cfcₙAux_id hp₁ a ha])

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
x✝ : NonUnitalContinuousFunctionalCalculus 𝕜 A p := ⋯
⊢ IsScalarTower 𝕜 (Unitization 𝕜 A) (Unitization 𝕜 A)

𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
x✝ : NonUnitalContinuousFunctionalCalculus 𝕜 A p := ⋯
⊢ SMulCommClass 𝕜 (Unitization 𝕜 A) (Unitization 𝕜 A)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t
⊢ ∀ x ∈ spectrum ℝ a, ‖↑t - x‖₊ ≤ t

Tactic:
simpa [spectralRadius, iSup₂_le_iff, ← spectrum.singleton_sub_eq] using h

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

Goal state before:
pp: case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
⊢ c ∈ upperBounds ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a)

Tactic:
rintro - ⟨x, hx, rfl⟩

Goal state after:
case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
x : 𝕜
hx : x ∈ σₙ 𝕜 a
⊢ (fun x ↦ ‖f x‖) x ≤ c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
⊢ ∀ x ∈ (fun x ↦ ‖f x‖) '' σₙ 𝕜 a, x < c

Tactic:
rintro - ⟨x, hx, rfl⟩

Goal state after:
case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
x : 𝕜
hx : x ∈ σₙ 𝕜 a
⊢ (fun x ↦ ‖f x‖) x < c
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
k : 𝕜
hk : k ∈ resolventSet 𝕜 a
H₁ :
  HasFDerivAt Ring.inverse (-((ContinuousLinearMap.mulLeftRight 𝕜 A) ↑(IsUnit.unit hk)⁻¹) ↑(IsUnit.unit hk)⁻¹)
    ((algebraMap 𝕜 A) k - a)
⊢ HasDerivAt (resolvent a) (-resolvent a k ^ 2) k

Tactic:
have H₂ : HasDerivAt (fun k => algebraMap 𝕜 A k - a) 1 k := by
    simpa using! (Algebra.linearMap 𝕜 A).hasDerivAt.sub_const a

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
k : 𝕜
hk : k ∈ resolventSet 𝕜 a
H₁ :
  HasFDerivAt Ring.inverse (-((ContinuousLinearMap.mulLeftRight 𝕜 A) ↑(IsUnit.unit hk)⁻¹) ↑(IsUnit.unit hk)⁻¹)
    ((algebraMap 𝕜 A) k - a)
H₂ : HasDerivAt (fun k ↦ (algebraMap 𝕜 A) k - a) 1 k
⊢ HasDerivAt (resolvent a) (-resolvent a k ^ 2) k
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case inr
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
h : ∀ x ∈ σ ℝ≥0 a, f x ≤ c
h✝ : Nontrivial A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
⊢ c ∈ upperBounds (f '' σ ℝ≥0 a)

Tactic:
rintro - ⟨x, hx, rfl⟩

Goal state after:
case inr
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
h : ∀ x ∈ σ ℝ≥0 a, f x ≤ c
h✝ : Nontrivial A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
x : ℝ≥0
hx : x ∈ σ ℝ≥0 a
⊢ f x ≤ c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case inr
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
h✝ : Nontrivial A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
⊢ ∀ x ∈ f '' σ ℝ≥0 a, x < c

Tactic:
rintro - ⟨x, hx, rfl⟩

Goal state after:
case inr
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
h✝ : Nontrivial A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
x : ℝ≥0
hx : x ∈ σ ℝ≥0 a
⊢ f x < c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x ≤ c
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : { toFun := (σₙ ℝ≥0 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : 0 ≤ a
⊢ c ∈ upperBounds (f '' σₙ ℝ≥0 a)

Tactic:
rintro - ⟨x, hx, rfl⟩

Goal state after:
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x ≤ c
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : { toFun := (σₙ ℝ≥0 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : 0 ≤ a
x : ℝ≥0
hx : x ∈ σₙ ℝ≥0 a
⊢ f x ≤ c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case refine_2
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x < c
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : { toFun := (σₙ ℝ≥0 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : 0 ≤ a
⊢ ∀ x ∈ f '' σₙ ℝ≥0 a, x < c

Tactic:
rintro - ⟨x, hx, rfl⟩

Goal state after:
case refine_2
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x < c
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : { toFun := (σₙ ℝ≥0 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : 0 ≤ a
x : ℝ≥0
hx : x ∈ σₙ ℝ≥0 a
⊢ f x < c
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr.refine_2
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✝ : RegularSpace α
U : Set α
hU : U ∈ {U | IsOpen[inst✝³] U}
K : Compacts α
x : α
hx₁ : x ∈ ↑K
hx₂ : x ∈ U
V W : Set α
hV : IsOpen[inst✝³] V
hW : IsOpen[inst✝³] W
hxV : {x} ⊆ V
hUW : Uᶜ ⊆ W
hVW : Disjoint V W
⊢ {K | (↑K ∩ V).Nonempty} ∈ 𝓝 K

Tactic:
(isOpen_inter_nonempty_of_isOpen hV).mem_nhds_iff

Goal state after:
case inr.refine_2
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✝ : RegularSpace α
U : Set α
hU : U ∈ {U | IsOpen[inst✝³] U}
K : Compacts α
x : α
hx₁ : x ∈ ↑K
hx₂ : x ∈ U
V W : Set α
hV : IsOpen[inst✝³] V
hW : IsOpen[inst✝³] W
hxV : {x} ⊆ V
hUW : Uᶜ ⊆ W
hVW : Disjoint V W
⊢ K ∈ {K | (↑K ∩ V).Nonempty}
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).Finite ∧ ↑K ⊆ s}

Tactic:
rw [← isEmbedding_coe.isPreconnected_image]
  convert vietoris.isPreconnected_nonempty_finite_subsets hs
  exact subset_antisymm (image_subset_iff.mpr .rfl) (fun t ht => ⟨⟨t, ht.2.1.isCompact⟩, ht, rfl⟩)

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

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

Tactic:
refine hB.compacts.isInducing isEmbedding_toCompacts.isInducing
    |>.isTopologicalBasis_of_exists_subset ?_ ?_ <;> simp_rw [forall_mem_image]

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

case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ ∀ ⦃x : Set (Set α)⦄,
    x ∈ {u | u.Finite ∧ u ⊆ B} →
      ∀ x_1 ∈ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty},
        ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u.Nonempty ∧ u ⊆ B},
          x_1 ∈ v ∧ v ⊆ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty}
Module: Mathlib.Analysis.LocallyConvex.Barrelled

Goal state before:
pp: ι : Sort u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : BarrelledSpace 𝕜 E
p : ι → Seminorm 𝕜 E
hp : ∀ (i : ι), Continuous[inst✝¹, _] ⇑(p i)
bdd : BddAbove (range p)
⊢ LowerSemicontinuous (⨆ i, ⇑(p i))

Tactic:
rw [Seminorm.bddAbove_range_iff] at bdd

Goal state after:
ι : Sort u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : BarrelledSpace 𝕜 E
p : ι → Seminorm 𝕜 E
hp : ∀ (i : ι), Continuous[inst✝¹, _] ⇑(p i)
bdd : ∀ (x : E), BddAbove (range fun i ↦ (p i) x)
⊢ LowerSemicontinuous (⨆ i, ⇑(p i))
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
hI' : I.IsMaximal
x : X
hx : setOfIdeal I = {x}ᶜ
⊢ idealOfSet 𝕜 {x}ᶜ = I

Tactic:
by
      simpa only [idealOfSet_ofIdeal_eq_closure, I.closure_eq_of_isClosed hI] using
        congr_arg (idealOfSet 𝕜) hx.symm

Goal state after:
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 := ⋯
c : Urysohns.CU P := ⋯
x : X
hx : x ∈ t
⊢ x ∉ c.U

Tactic:
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 : A
ha : IsSelfAdjoint a
⊢ -a⁻ ≤ a

Tactic:
simpa only [posPart_sub_negPart a, ← sub_eq_add_neg]
    using le_add_of_nonneg_left (a := -a⁻) (posPart_nonneg a)

Goal state after:
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝¹ : PseudoEMetricSpace β
inst✝ : Finite ↑𝔖
x✝ : Fintype ↑𝔖 := Fintype.ofFinite ↑𝔖
⊢ ⨅ i,
      ⨅ (_ : i > 0),
        comap
          (fun x ↦
            (fun s ↦ UniformFun.ofFun ((↑s).domRestrict ((toFun 𝔖) x.1)), fun s ↦
              UniformFun.ofFun ((↑s).domRestrict ((toFun 𝔖) x.2))))
          (𝓟 {p | edist p.1 p.2 < i}) =
    ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | edist p.1 p.2 < ε}

Tactic:
comap_principal,

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝¹ : PseudoEMetricSpace β
inst✝ : Finite ↑𝔖
x✝ : Fintype ↑𝔖 := Fintype.ofFinite ↑𝔖
⊢ ⨅ i,
      ⨅ (_ : i > 0),
        𝓟
          ((fun x ↦
              (fun s ↦ UniformFun.ofFun ((↑s).domRestrict ((toFun 𝔖) x.1)), fun s ↦
                UniformFun.ofFun ((↑s).domRestrict ((toFun 𝔖) x.2)))) ⁻¹'
            {p | edist p.1 p.2 < i}) =
    ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | edist p.1 p.2 < ε}
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:
rw [CFC.posPart_def, cfcₙ_eq_cfc, IsScalarTower.algebraMap_apply ℝ≥0 ℝ A]
  simp

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

Goal state before:
pp: case pos
A : Type u_1
inst✝¹³ : NonUnitalCStarAlgebra A
R : Type u_2
inst✝¹² : Semifield R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : Algebra R ℂ
inst✝³ : IsScalarTower R ℂ A
p : A → Prop
p' : Unitization ℂ A → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousFunctionalCalculus R (Unitization ℂ A) p'
inst✝ : ContinuousMapZero.UniqueHom R (Unitization ℂ A)
hp : ∀ {a : A}, p' ↑a ↔ p a
a : A
f : R → R
hf₀ : f 0 = 0
hf : ContinuousOn f (σₙ R a)
ha : p a
⊢ ↑(cfcₙ f a) = cfc f ↑a

Tactic:
rw [← cfcₙ_eq_cfc (quasispectrum_inr_eq R ℂ a ▸ hf)]

Goal state after:
case pos
A : Type u_1
inst✝¹³ : NonUnitalCStarAlgebra A
R : Type u_2
inst✝¹² : Semifield R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : Algebra R ℂ
inst✝³ : IsScalarTower R ℂ A
p : A → Prop
p' : Unitization ℂ A → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousFunctionalCalculus R (Unitization ℂ A) p'
inst✝ : ContinuousMapZero.UniqueHom R (Unitization ℂ A)
hp : ∀ {a : A}, p' ↑a ↔ p a
a : A
f : R → R
hf₀ : f 0 = 0
hf : ContinuousOn f (σₙ R a)
ha : p a
⊢ ↑(cfcₙ f a) = cfcₙ f ↑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 : Aˣ
ha : 0 ≤ ↑a
hab : ↑a ≤ ↑b
⊢ ↑b⁻¹ ≤ ↑a⁻¹

Tactic:
have hb := ha.trans hab

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : Aˣ
ha : 0 ≤ ↑a
hab : ↑a ≤ ↑b
hb : 0 ≤ ↑b
⊢ ↑b⁻¹ ≤ ↑a⁻¹
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 : Aˣ
ha : 1 ≤ ↑a
⊢ ↑a ^ (-1) ≤ 1

Tactic:
rpow_neg_one_eq_inv a (zero_le_one.trans ha)

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : Aˣ
ha : 1 ≤ ↑a
⊢ ↑a⁻¹ ≤ 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 : 0 ≤ a
hab : a ≤ b
this : ∀ (a b : Unitization ℂ A), 0 ≤ a → a ≤ b → ‖a‖ ≤ ‖b‖
⊢ ‖a‖ ≤ ‖b‖

Tactic:
have hb := ha.trans hab

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
ha : 0 ≤ a
hab : a ≤ b
this : ∀ (a b : Unitization ℂ A), 0 ≤ a → a ≤ b → ‖a‖ ≤ ‖b‖
hb : 0 ≤ b
⊢ ‖a‖ ≤ ‖b‖
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: m : Type u_3
n : Type u_4
α : Type u_5
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : SeminormedAddCommGroup α
A : Matrix m n α
⊢ ‖A‖ = ↑(Finset.univ.sup fun i ↦ Finset.univ.sup fun j ↦ ‖A i j‖₊)

Tactic:
simp_rw [Matrix.norm_def, Pi.norm_def, Pi.nnnorm_def]

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

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
hcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖
x : ℕ → A₁
hx : ∀ (n : ℕ), 0 ≤ x n ∧ ‖x n‖ = 1 ∧ 2 ^ (2 * n) < ‖f (x n)‖
⊢ False

Tactic:
simp only [forall_and] at hx

Goal state after:
A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
hcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖
x : ℕ → A₁
hx : (∀ (x_1 : ℕ), 0 ≤ x x_1) ∧ (∀ (x_1 : ℕ), ‖x x_1‖ = 1) ∧ ∀ (x_1 : ℕ), 2 ^ (2 * x_1) < ‖f (x x_1)‖
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.PositiveLinearMap

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
⊢ ∃ C, ∀ (a : A₁), ‖f a‖ ≤ ↑C * ‖a‖

Tactic:
by
  /- It suffices to only consider for positive `a`, by decomposing `a` into positive and negative
     parts of the real and imaginary parts. -/
  suffices h_nonneg : ∃ C : ℝ≥0, ∀ a, 0 ≤ a → ‖f a‖ ≤ C * ‖a‖ by
    obtain ⟨C, hmain⟩ := h_nonneg
    refine ⟨4 * C, fun x ↦ ?_⟩
    obtain ⟨y, hy_nonneg, hy_norm, hy⟩ := CStarAlgebra.exists_sum_four_nonneg x
    conv_lhs => rw [hy]
    simp only [map_sum, map_smul]
    apply norm_sum_le _ _ |>.trans
    simp only [norm_smul, norm_pow, norm_I, one_pow, one_mul]
    apply Finset.sum_le_sum (g := fun _ ↦ C * ‖x‖) (fun i _ ↦ ?_) |>.trans <| by simp [mul_assoc]
    apply hmain _ (hy_nonneg i) |>.trans
    gcongr
    exact hy_norm i
  -- Let's proceed by contradiction
  by_contra! hcontra
  -- Given `n : ℕ`, we can always choose a positive element of norm one with `2 ^ (2 * n) < ‖f x‖`
  have (n : ℕ) : ∃ x, 0 ≤ x ∧ ‖x‖ = 1 ∧ 2 ^ (2 * n) < ‖f x‖ := by
    obtain ⟨hx₁, hx₂⟩ := Classical.choose_spec (hcontra (2 ^ (2 * n)))
    set x := Classical.choose (hcontra (2 ^ (2 * n)))
    have hx := (eq_zero_or_norm_pos x).resolve_left (fun hx ↦ by simp_all)
    refine ⟨‖x‖⁻¹ • x, smul_nonneg (by positivity) hx₁, ?_, ?_⟩
    · simp [norm_smul, inv_mul_cancel₀ hx.ne']
    · simpa [norm_smul] using (lt_inv_mul_iff₀' hx).mpr hx₂
  -- Let `x n` be a sequence of nonnegative elements such that `‖x n‖ = 1` and `‖f (x n)‖ ≥ 4 ^ n`.
  choose x hx using this
  simp only [forall_and] at hx
  obtain ⟨hx_nonneg, hx_norm, hx⟩ := hx
  -- `∑ n, 2 ^ (-n) • x n` converges
  have x_summable : Summable fun n : ℕ => (2 : ℝ) ^ (-(n : ℤ)) • x n := by
    refine Summable.of_norm ?_
    have : (2 : ℝ)⁻¹ < 1 := by norm_num
    simp [norm_smul, hx_norm, ← inv_pow, this]
  -- There is some `n` such that `‖f (∑' m, 2 ^ (-m) • x m)‖ < 2 ^ n`
  obtain ⟨n, hn⟩ : ∃ n : ℕ, ‖f (∑' (n : ℕ), (2 : ℝ) ^ (-(n : ℤ)) • x n)‖ < (2 : ℝ) ^ n :=
    tendsto_pow_atTop_atTop_of_one_lt one_lt_two |>.eventually_gt_atTop _
      |>.exists
  -- But `2 ^ n ≤ ‖f (2 ^ (-n) • x n)‖ ≤ ‖f (∑' m, 2 ^ (-m) • x m)‖`, which is a contradiction.
  apply hn.not_ge
  trans ‖f ((2 : ℝ) ^ (-n : ℤ) • x n)‖
  · have := hx n |>.le
    rw [pow_mul', sq] at this
    simpa [norm_smul] using (le_inv_mul_iff₀ (show 0 < (2 : ℝ) ^ n by positivity)).mpr this
  · have (m : ℕ) : 0 ≤ ((2 : ℝ) ^ (-(m : ℤ)) • x m) := smul_nonneg (by positivity) (hx_nonneg m)
    refine CStarAlgebra.norm_le_norm_of_nonneg_of_le (f.map_nonneg (this n)) ?_
    gcongr
    exact x_summable.le_tsum n fun m _ ↦ this m

Goal state after: