mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Bernstein

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
f : C(↑I, E)
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E
U : Set E
hU₀ : U ∈ 𝓝 0
hUc : Convex ℝ U
C : ℝ
hC : ∀ (x y : ↑I), gauge U (f x - f y) ≤ C
hC₀ : 0 ≤ C
δ : ℝ
hδ₀ : δ > 0
hδ : ∀ (x y : ↑I), dist x y < δ → gauge U (f x - f y) < 1 / 2
nhds_zero : Tendsto (fun n ↦ C / δ ^ 2 / ↑n) atTop (𝓝 0)
n : ℕ
nh : C / δ ^ 2 / ↑n < 1 / 2
hn₀ : n ≠ 0
x : ↑I
S : Finset (Fin (n + 1)) := {k | dist k/ₙ x < δ}
⊢ ∑ k, gauge U ((bernstein n ↑k) x • (f k/ₙ - f x)) = ∑ k, (bernstein n ↑k) x * gauge U (f k/ₙ - f x)

Tactic:
by
      simp only [gauge_smul_of_nonneg, bernstein_nonneg, smul_eq_mul]

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : L.SeparatesPointsStrongly
⊢ closure L = ⊤

Tactic:
eq_top_iff

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : L.SeparatesPointsStrongly
⊢ ⊤ ⊆ closure L
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y : X), g x y ∈ L
w₁ : ∀ (x y : X), (g x y) x = f x
w₂ : ∀ (x y : X), (g x y) y = f y
U : X → X → Set X := fun x y ↦ {z | f z - ε < (g x y) z}
x y : X
⊢ y ∈ U x y

Tactic:
Set.mem_ofPred_eq,

Goal state after:
case refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y : X), g x y ∈ L
w₁ : ∀ (x y : X), (g x y) x = f x
w₂ : ∀ (x y : X), (g x y) y = f y
U : X → X → Set X := fun x y ↦ {z | f z - ε < (g x y) z}
x y : X
⊢ f y - ε < (g x y) y
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f g : C(X, ℝ)
⊢ φ (f * g).toNNReal - φ (-(f * g)).toNNReal = (φ f.toNNReal - φ (-f).toNNReal) * (φ g.toNNReal - φ (-g).toNNReal)

Tactic:
by
    have := congr(φ $(f.toNNReal_mul_add_neg_mul_add_mul_neg_eq g))
    simp only [map_add, map_mul, sub_mul, mul_sub] at this ⊢
    rw [← sub_eq_zero] at this ⊢
    convert! this using 1
    abel

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

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
⊢ φ (r • f).toNNReal - φ (-(r • f)).toNNReal = (MonoidHom.id ℝ) r • (φ f.toNNReal - φ (-f).toNNReal)

Tactic:
simp only [MonoidHom.id_apply]
    by_cases! hr : 0 ≤ r
    · lift r to ℝ≥0 using hr
      simp only [← smul_def, toNNReal_smul, map_smul, toNNReal_neg_smul, smul_sub]
    · rw [← neg_pos] at hr
      rw [← neg_smul]
      nth_rw 1 [← neg_neg r]
      nth_rw 3 [← neg_neg r]
      lift -r to ℝ≥0 using hr.le with r
      simp only [neg_smul, ← smul_def, toNNReal_neg_smul, map_smul, toNNReal_smul, smul_sub,
        sub_neg_eq_add]
      rw [sub_eq_add_neg, add_comm]

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
ι : Type u_5
inst✝ : RegularSpace β
x₀ : α
l : Filter ι
hf : UpperHemicontinuousAt f x₀
hf_closed : IsClosed[inst✝¹] (f x₀)
x : ι → α
hx : Tendsto x l (𝓝 x₀)
y : ι → β
hy : ∃ᶠ (n : ι) in l, y n ∈ f (x n)
y₀ : β
hy₀ : Tendsto y l (𝓝 y₀)
this : y₀ ∉ f x₀
s : Set β
hs : s ∈ 𝓝ˢ (f x₀)
t : Set β
ht : t ∈ 𝓝 y₀
hst : Disjoint s t
⊢ False

Tactic:
suffices ∃ᶠ n in l, y n ∈ s by
    apply this
    filter_upwards [hy₀ ht] with n hn hyn
    exact hst.notMem_of_mem_left hyn hn

Goal state after:
α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
ι : Type u_5
inst✝ : RegularSpace β
x₀ : α
l : Filter ι
hf : UpperHemicontinuousAt f x₀
hf_closed : IsClosed[inst✝¹] (f x₀)
x : ι → α
hx : Tendsto x l (𝓝 x₀)
y : ι → β
hy : ∃ᶠ (n : ι) in l, y n ∈ f (x n)
y₀ : β
hy₀ : Tendsto y l (𝓝 y₀)
this : y₀ ∉ f x₀
s : Set β
hs : s ∈ 𝓝ˢ (f x₀)
t : Set β
ht : t ∈ 𝓝 y₀
hst : Disjoint s t
⊢ ∃ᶠ (n : ι) in l, y n ∈ s
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 p ↦ {s | s ⊆ p.1 ∧ ∀ U ∈ p.2, (s ∩ U).Nonempty}) ''
      {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1})

Tactic:
refine isTopologicalBasis.isTopologicalBasis_of_exists_subset ?_ ?_

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ ∀
    u ∈
      (fun p ↦ {s | s ⊆ p.1 ∧ ∀ U ∈ p.2, (s ∩ U).Nonempty}) ''
        {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1},
    IsOpen[TopologicalSpace.vietoris α] u

case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ ∀ u ∈ (fun u ↦ {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}) '' {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U},
    ∀ x ∈ u,
      ∃
        v ∈
          (fun p ↦ {s | s ⊆ p.1 ∧ ∀ U ∈ p.2, (s ∩ U).Nonempty}) ''
            {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1},
        x ∈ v ∧ v ⊆ u
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
k : 𝕜
h : ‖a‖ * ‖1‖ < ‖k‖
⊢ k ∈ {r | IsUnit (↑ₐ r - a)}

Tactic:
Set.mem_ofPred_eq,

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
k : 𝕜
h : ‖a‖ * ‖1‖ < ‖k‖
⊢ IsUnit (↑ₐ k - a)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case pos
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
h : ‖a‖₊ = 0
⊢ ‖a‖₊ ^ n.succ * r ^ (n + 1) ≤ 1

Tactic:
simp [h, pow_succ']

Goal state after:
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
x✝ : CompactSpace (Closeds α)
ι : Type u_1
F : ι → Set α
hF₁ : ∀ (i : ι), IsClosed[inst✝.toTopologicalSpace] (F i)
hF₂ : ⋂ i, F i = ∅
this : {⊥}ᶜ ∩ ⋂ i, {C | ↑C ⊆ F i} = ∅ → ∃ u, {⊥}ᶜ ∩ ⋂ i ∈ u, {C | ↑C ⊆ F i} = ∅
⊢ ∃ u, ⋂ i ∈ u, F i = ∅

Tactic:
simp_rw [← Set.disjoint_iff_inter_eq_empty, Set.disjoint_compl_left_iff_subset,
    ← Set.ofPred_forall, ← Set.subset_iInter_iff, hF₂, Set.subset_empty_iff, coe_eq_empty,
    Set.ofPred_eq_eq_singleton] at this

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
x✝ : CompactSpace (Closeds α)
ι : Type u_1
F : ι → Set α
hF₁ : ∀ (i : ι), IsClosed[inst✝.toTopologicalSpace] (F i)
hF₂ : ⋂ i, F i = ∅
this : {⊥} ⊆ {⊥} → ∃ u, {x | ↑x ⊆ ⋂ i ∈ u, F i} ⊆ {⊥}
⊢ ∃ u, ⋂ i ∈ u, F i = ∅
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case neg
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (L \ ⋃₀ u).Nonempty
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q

Tactic:
replace hsu L (h : L ∈ s) : (K \ ⋃₀ u ∩ L).Nonempty := (hsu L h).mono <| by grind

Goal state after:
case neg
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q
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 := ⋯
hl : l.TotallyBounded
L : Compacts α := ⋯
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
⊢ ↑K ⊆ U.preimage ↑L

Tactic:
intro x hx
    set lx := l ⊓ 𝓟 (UniformSpace.ball x U) with le_def
    have hlx : lx.TotallyBounded := hl.mono inf_le_left
    have : lx.NeBot := by
      rw [le_def, Filter.lift'_inf_principal_eq, Filter.lift'_neBot_iff fun _ _ h =>
        Set.inter_subset_inter_left _ <| Set.biUnion_subset_biUnion_left h]
      intro s hs
      obtain ⟨K', ⟨h₁, -⟩, h₂⟩ := Filter.nonempty_of_mem <| Filter.inter_mem hK hs
      obtain ⟨y, hy, hxy⟩ := h₁ hx
      exact ⟨y, Set.mem_iUnion₂_of_mem h₂ hy, hxy⟩
    obtain ⟨y, hy⟩ := hlx.exists_clusterPt
    have hy₁ : ClusterPt y l := .of_inf_left hy
    have hy₂ : ClusterPt y (𝓟 (UniformSpace.ball x U)) := .of_inf_right hy
    rw [← mem_closure_iff_clusterPt, (UniformSpace.isClosed_ball x hU₂).closure_eq] at hy₂
    exact ⟨y, hy₁, hy₂⟩

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

Goal state before:
pp: case e'_4
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : Nontrivial A
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σ 𝕜 a)
ha : p a
x : 𝕜
hx' : x ∈ σ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ‖{ toFun := (σ 𝕜 a).domRestrict f, continuous_toFun := ⋯ }‖ = (fun x ↦ ‖f x‖) x

Tactic:
apply le_antisymm

Goal state after:
case e'_4.a
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : Nontrivial A
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σ 𝕜 a)
ha : p a
x : 𝕜
hx' : x ∈ σ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ‖{ toFun := (σ 𝕜 a).domRestrict f, continuous_toFun := ⋯ }‖ ≤ (fun x ↦ ‖f x‖) x

case a
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : Nontrivial A
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σ 𝕜 a)
ha : p a
x : 𝕜
hx' : x ∈ σ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ (fun x ↦ ‖f x‖) x ≤ ‖{ toFun := (σ 𝕜 a).domRestrict f, continuous_toFun := ⋯ }‖
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁵ : NormedRing A
inst✝⁴ : CompleteSpace A
inst✝³ : SetLike SA A
inst✝² : SubringClass SA A
inst✝¹ : NormedField 𝕜
inst✝ : NormedAlgebra 𝕜 A
instSMulMem : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
this : CompleteSpace ↥S
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (σ 𝕜 x \ σ 𝕜 ↑x)

Tactic:
← (spectrum.isClosed (𝕜 := 𝕜) x).closure_eq,

Goal state after:
𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁵ : NormedRing A
inst✝⁴ : CompleteSpace A
inst✝³ : SetLike SA A
inst✝² : SubringClass SA A
inst✝¹ : NormedField 𝕜
inst✝ : NormedAlgebra 𝕜 A
instSMulMem : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
this : CompleteSpace ↥S
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (σ 𝕜 x) \ σ 𝕜 ↑x)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : Nontrivial A
a : A
ha : p a
⊢ IsGreatest ((fun x ↦ ‖x‖₊) '' σ 𝕜 a) ‖a‖₊

Tactic:
simpa only [cfc_id 𝕜 a] using! IsGreatest.nnnorm_cfc (id : 𝕜 → 𝕜) a

Goal state after:
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
this : ∀ (x : 𝕜₁), ‖(algebraMap 𝕜₁ 𝕜₂) x‖₊ = ‖x‖₊
⊢ ⨆ k ∈ spectrum 𝕜₁ a, ↑‖k‖₊ = ⨆ k ∈ spectrum 𝕜₂ a, ↑‖k‖₊

Tactic:
apply le_antisymm

Goal state after:
case a
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
this : ∀ (x : 𝕜₁), ‖(algebraMap 𝕜₁ 𝕜₂) x‖₊ = ‖x‖₊
⊢ ⨆ k ∈ spectrum 𝕜₁ a, ↑‖k‖₊ ≤ ⨆ k ∈ spectrum 𝕜₂ a, ↑‖k‖₊

case a
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
this : ∀ (x : 𝕜₁), ‖(algebraMap 𝕜₁ 𝕜₂) x‖₊ = ‖x‖₊
⊢ ⨆ k ∈ spectrum 𝕜₂ a, ↑‖k‖₊ ≤ ⨆ k ∈ spectrum 𝕜₁ a, ↑‖k‖₊
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : MetricSpace A
inst✝² : IsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts a ⇑f
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C(↑(σ R a), R)
x : ↑(σ R a)
x' : ↑(σ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ dist (g₁ x) (g₂ x) =
    dist (g₁ (Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x)))
      (g₂ (Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x)))

Tactic:
congr!

Goal state after:
case e'_3
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : MetricSpace A
inst✝² : IsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts a ⇑f
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C(↑(σ R a), R)
x : ↑(σ R a)
x' : ↑(σ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ x = Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x)

case e'_4
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : MetricSpace A
inst✝² : IsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts a ⇑f
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C(↑(σ R a), R)
x : ↑(σ R a)
x' : ↑(σ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ x = Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x)
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
⊢ Unitization.inr ∘ ⇑(cfcₙHom ha) = ⇑(cfcₙAux ⋯ a ha)

Tactic:
congrm (⇑$(inrNonUnitalStarAlgHom_comp_cfcₙHom_eq_cfcₙAux hp₁ a ha))

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

Goal state before:
pp: case e'_4
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
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ‖{ toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }‖ = (fun x ↦ ‖f x‖) x

Tactic:
apply le_antisymm

Goal state after:
case e'_4.a
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
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ‖{ toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }‖ ≤ (fun x ↦ ‖f x‖) x

case a
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
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ (fun x ↦ ‖f x‖) x ≤ ‖{ toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝¹¹ : TopologicalSpace A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : PartialOrder A
inst✝⁸ : StarRing A
inst✝⁷ : StarOrderedRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : IsSemitopologicalRing A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : SMulCommClass ℝ A A
inst✝² : T2Space A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ → ℝ
a : A
hf_nonneg : ∀ x ∈ σₙ ℝ a, 0 ≤ f x
ha : 0 ≤ a
x : ℝ
hx : x ∈ σₙ ℝ a
⊢ f x = ↑(f ↑x.toNNReal).toNNReal

Tactic:
rw [x.coe_toNNReal (quasispectrum_nonneg_of_nonneg _ ha _ hx),
    (f x).coe_toNNReal (hf_nonneg x hx)]

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

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝⁵ : NormedField 𝕜
E : Type u_2
inst✝⁴ : NormedRing E
inst✝³ : StarRing E
inst✝² : CStarRing E
inst✝¹ : NormedAlgebra 𝕜 E
inst✝ : CompleteSpace E
u : ↥(unitary E)
a✝ : Nontrivial E
k : 𝕜
hk : k⁻¹ ∈ σ 𝕜 ↑(toUnits u)⁻¹
hnk : k ≠ 0
⊢ 1 ≤ ‖k‖

Tactic:
have : ‖k‖⁻¹ ≤ ‖(↑(Unitary.toUnits u)⁻¹ : E)‖ := by
      simpa only [norm_inv] using norm_le_norm_of_mem hk

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝⁵ : NormedField 𝕜
E : Type u_2
inst✝⁴ : NormedRing E
inst✝³ : StarRing E
inst✝² : CStarRing E
inst✝¹ : NormedAlgebra 𝕜 E
inst✝ : CompleteSpace E
u : ↥(unitary E)
a✝ : Nontrivial E
k : 𝕜
hk : k⁻¹ ∈ σ 𝕜 ↑(toUnits u)⁻¹
hnk : k ≠ 0
this : ‖k‖⁻¹ ≤ ‖↑(toUnits u)⁻¹‖
⊢ 1 ≤ ‖k‖
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
⊢ (spectralRadius ℂ (a * a⋆)).toReal = ‖a‖ ^ 2

Tactic:
← norm_star a,

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
⊢ (spectralRadius ℂ (a * a⋆)).toReal = ‖a⋆‖ ^ 2
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ σ ℂ a
⊢ (ofReal ∘ re) z ∈ σ ℂ a

Tactic:
simpa only [(ha.mem_spectrum_eq_re hz).symm, Function.comp_apply] using hz

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✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
⊢ ContinuousSup (Compacts α)

Tactic:
by
  constructor
  simp_rw [isEmbedding_coe.continuous_iff, Function.comp_def, coe_sup]
  fun_prop

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

Goal state before:
pp: case lower
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ {z | z.im < 0}
hz' : z ∈ σ ℂ a
⊢ False

Tactic:
Set.mem_ofPred_eq,

Goal state after:
case lower
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z.im < 0
hz' : z ∈ σ ℂ a
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: case upper
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ {z | 0 < z.im}
hz' : z ∈ σ ℂ a
⊢ False

Tactic:
Set.mem_ofPred_eq,

Goal state after:
case upper
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : 0 < z.im
hz' : z ∈ σ ℂ a
⊢ False
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✝ : CompactSpace α
⊢ IsCompact univ

Tactic:
by simpa using isCompact_subsets_of_isCompact isCompact_univ

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

Goal state before:
pp: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
⊢ Continuous expUnitary

Tactic:
simp only [continuous_induced_rng, Function.comp_def, selfAdjoint.expUnitary_coe]

Goal state after:
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
⊢ Continuous fun x ↦ exp (I • ↑x)