mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2.inl
α : Type u_1
inst✝ : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ : t ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hs : ∅ ∈ ⋂₀ t
⊢ ∃ x ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U},
    ∅ ∈ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ∧ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ t

pp: case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ : t ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
s : Set α
hs : s ∈ ⋂₀ t
h✝ : s.Nonempty
⊢ ∃ x ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U},
    s ∈ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ∧ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ t

Tactic:
·

Goal state after:
case refine_2.inl
α : Type u_1
inst✝ : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ : t ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hs : ∅ ∈ ⋂₀ t
⊢ ∃ x ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U},
    ∅ ∈ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ∧ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ t
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
U V : SetRel α α
h : U ⊆ V
⊢ hausdorffEntourage U ⊆ hausdorffEntourage V

Tactic:
unfold hausdorffEntourage

Goal state after:
α : Type u_1
U V : SetRel α α
h : U ⊆ V
⊢ {x | x.1 ⊆ U.preimage x.2 ∧ x.2 ⊆ U.image x.1} ⊆ {x | x.1 ⊆ V.preimage x.2 ∧ x.2 ⊆ V.image x.1}
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case a
α : Type u_1
U V : SetRel α α
⊢ hausdorffEntourage (U ○ V) ⊆ hausdorffEntourage U ○ hausdorffEntourage V

pp: case a
α : Type u_1
U V : SetRel α α
⊢ hausdorffEntourage U ○ hausdorffEntourage V ⊆ hausdorffEntourage (U ○ V)

Tactic:
·

Goal state after:
case a
α : Type u_1
U V : SetRel α α
⊢ hausdorffEntourage (U ○ V) ⊆ hausdorffEntourage U ○ hausdorffEntourage V
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case a.refine_1
α : Type u_1
U V : SetRel α α
s t : Set α
hst : s ⊆ (U ○ V).preimage t
hts : t ⊆ (U ○ V).image s
⊢ s ⊆ U.preimage (U.image s ∩ V.preimage t)

pp: case a.refine_2
α : Type u_1
U V : SetRel α α
s t : Set α
hst : s ⊆ (U ○ V).preimage t
hts : t ⊆ (U ○ V).image s
⊢ t ⊆ V.image (U.image s ∩ V.preimage t)

Tactic:
·

Goal state after:
case a.refine_1
α : Type u_1
U V : SetRel α α
s t : Set α
hst : s ⊆ (U ○ V).preimage t
hts : t ⊆ (U ○ V).image s
⊢ s ⊆ U.preimage (U.image s ∩ V.preimage t)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case a.refine_2
α : Type u_1
U V : SetRel α α
s t : Set α
hst : s ⊆ (U ○ V).preimage t
hts : t ⊆ (U ○ V).image s
⊢ t ⊆ V.image (U.image s ∩ V.preimage t)

Tactic:
·

Goal state after:
case a.refine_2
α : Type u_1
U V : SetRel α α
s t : Set α
hst : s ⊆ (U ○ V).preimage t
hts : t ⊆ (U ○ V).image s
⊢ t ⊆ V.image (U.image s ∩ V.preimage t)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case a
α : Type u_1
U V : SetRel α α
⊢ hausdorffEntourage U ○ hausdorffEntourage V ⊆ hausdorffEntourage (U ○ V)

Tactic:
·

Goal state after:
case a
α : Type u_1
U V : SetRel α α
⊢ hausdorffEntourage U ○ hausdorffEntourage V ⊆ hausdorffEntourage (U ○ V)
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
⊢ ∀ ⦃x : Set α × Set (Set α)⦄,
    x ∈ {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1} →
      IsOpen[TopologicalSpace.vietoris α] {s | s ⊆ x.1 ∧ ∀ U ∈ x.2, (s ∩ U).Nonempty}

pp: 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 ∈ u, IsOpen[inst✝] U} →
      ∀ x_1 ∈ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty},
        ∃
          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_1 ∈ v ∧ v ⊆ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty}

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ ∀ ⦃x : Set α × Set (Set α)⦄,
    x ∈ {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1} →
      IsOpen[TopologicalSpace.vietoris α] {s | s ⊆ x.1 ∧ ∀ U ∈ x.2, (s ∩ U).Nonempty}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: 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 ∈ u, IsOpen[inst✝] U} →
      ∀ x_1 ∈ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty},
        ∃
          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_1 ∈ v ∧ v ⊆ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty}

Tactic:
·

Goal state after:
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 ∈ u, IsOpen[inst✝] U} →
      ∀ x_1 ∈ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty},
        ∃
          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_1 ∈ v ∧ v ⊆ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty}
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case left
α : Type u_1
inst✝ : UniformSpace α
⊢ IsClosed[(UniformSpace.hausdorff α).toTopologicalSpace] {∅}

pp: case right
α : Type u_1
inst✝ : UniformSpace α
⊢ IsOpen[(UniformSpace.hausdorff α).toTopologicalSpace] {∅}

Tactic:
·

Goal state after:
case left
α : Type u_1
inst✝ : UniformSpace α
⊢ IsClosed[(UniformSpace.hausdorff α).toTopologicalSpace] {∅}
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case left
α : Type u_1
inst✝ : UniformSpace α
⊢ IsClosed[(UniformSpace.hausdorff α).toTopologicalSpace] {∅}

Tactic:
← Set.powerset_empty

Goal state after:
case left
α : Type u_1
inst✝ : UniformSpace α
⊢ IsClosed[(UniformSpace.hausdorff α).toTopologicalSpace] (𝒫 ∅)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case right
α : Type u_1
inst✝ : UniformSpace α
⊢ IsOpen[(UniformSpace.hausdorff α).toTopologicalSpace] {∅}

Tactic:
·

Goal state after:
case right
α : Type u_1
inst✝ : UniformSpace α
⊢ IsOpen[(UniformSpace.hausdorff α).toTopologicalSpace] {∅}
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case zero
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
a : A
⊢ spectralRadius 𝕜 a ^ 0 ≤ spectralRadius 𝕜 (a ^ 0)

pp: case succ
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
a : A
n✝ : ℕ
⊢ spectralRadius 𝕜 a ^ (n✝ + 1) ≤ spectralRadius 𝕜 (a ^ (n✝ + 1))

Tactic:
·

Goal state after:
case zero
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
a : A
⊢ spectralRadius 𝕜 a ^ 0 ≤ spectralRadius 𝕜 (a ^ 0)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case succ
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
a : A
n✝ : ℕ
⊢ spectralRadius 𝕜 a ^ (n✝ + 1) ≤ spectralRadius 𝕜 (a ^ (n✝ + 1))

Tactic:
·

Goal state after:
case succ
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
a : A
n✝ : ℕ
⊢ spectralRadius 𝕜 a ^ (n✝ + 1) ≤ spectralRadius 𝕜 (a ^ (n✝ + 1))
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case left
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ : (s, t) ∈ hausdorffEntourage V
hst : s ⊆ V.preimage t
hts : t ⊆ V.image s
⊢ closure[inst✝.toTopologicalSpace] s ⊆ SetRel.preimage U (closure[inst✝.toTopologicalSpace] t)

pp: case right
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ : (s, t) ∈ hausdorffEntourage V
hst : s ⊆ V.preimage t
hts : t ⊆ V.image s
⊢ closure[inst✝.toTopologicalSpace] t ⊆ SetRel.image U (closure[inst✝.toTopologicalSpace] s)

Tactic:
·

Goal state after:
case left
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ : (s, t) ∈ hausdorffEntourage V
hst : s ⊆ V.preimage t
hts : t ⊆ V.image s
⊢ closure[inst✝.toTopologicalSpace] s ⊆ SetRel.preimage U (closure[inst✝.toTopologicalSpace] t)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case right
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ : (s, t) ∈ hausdorffEntourage V
hst : s ⊆ V.preimage t
hts : t ⊆ V.image s
⊢ closure[inst✝.toTopologicalSpace] t ⊆ SetRel.image U (closure[inst✝.toTopologicalSpace] s)

Tactic:
·

Goal state after:
case right
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ : (s, t) ∈ hausdorffEntourage V
hst : s ⊆ V.preimage t
hts : t ⊆ V.image s
⊢ closure[inst✝.toTopologicalSpace] t ⊆ SetRel.image U (closure[inst✝.toTopologicalSpace] s)
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
⊢ closure[TopologicalSpace.vietoris α] {t | t.Finite ∧ t ⊆ s} ⊆ 𝒫 closure[inst✝] s

pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
s K : Set α
hKs : K ∈ 𝒫 closure[inst✝] s
⊢ K ∈ closure[TopologicalSpace.vietoris α] {t | t.Finite ∧ t ⊆ s}

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
⊢ closure[TopologicalSpace.vietoris α] {t | t.Finite ∧ t ⊆ s} ⊆ 𝒫 closure[inst✝] s
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
s K : Set α
hKs : K ∈ 𝒫 closure[inst✝] s
⊢ K ∈ closure[TopologicalSpace.vietoris α] {t | t.Finite ∧ t ⊆ s}

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
s K : Set α
hKs : K ∈ 𝒫 closure[inst✝] s
⊢ K ∈ closure[TopologicalSpace.vietoris α] {t | t.Finite ∧ t ⊆ s}
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case e'_3
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
n : ℕ
k : 𝕜
hk : k ∈ σ a
pow_mem : k ^ (n + 1) ∈ σ (a ^ (n + 1))
nnnorm_pow_le : ↑(‖k‖₊ ^ (n + 1)) ≤ ↑‖a ^ (n + 1)‖₊ * ↑‖1‖₊
hn : 0 < ↑(n + 1)
⊢ ↑‖k‖₊ = (↑‖k‖₊ ^ (n + 1)) ^ (1 / ↑(n + 1))

pp: case e'_4
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
n : ℕ
k : 𝕜
hk : k ∈ σ a
pow_mem : k ^ (n + 1) ∈ σ (a ^ (n + 1))
nnnorm_pow_le : ↑(‖k‖₊ ^ (n + 1)) ≤ ↑‖a ^ (n + 1)‖₊ * ↑‖1‖₊
hn : 0 < ↑(n + 1)
⊢ ↑‖a ^ (n + 1)‖₊ ^ (1 / (↑n + 1)) * ↑‖1‖₊ ^ (1 / (↑n + 1)) = (↑‖a ^ (n + 1)‖₊ * ↑‖1‖₊) ^ (1 / ↑(n + 1))

Tactic:
·

Goal state after:
case e'_3
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
n : ℕ
k : 𝕜
hk : k ∈ σ a
pow_mem : k ^ (n + 1) ∈ σ (a ^ (n + 1))
nnnorm_pow_le : ↑(‖k‖₊ ^ (n + 1)) ≤ ↑‖a ^ (n + 1)‖₊ * ↑‖1‖₊
hn : 0 < ↑(n + 1)
⊢ ↑‖k‖₊ = (↑‖k‖₊ ^ (n + 1)) ^ (1 / ↑(n + 1))
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case left
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ :
  (s, t) ∈
    (fun x ↦ (closure[inst✝.toTopologicalSpace] x.1, closure[inst✝.toTopologicalSpace] x.2)) ⁻¹' hausdorffEntourage V
hst : closure[inst✝.toTopologicalSpace] s ⊆ V.preimage (closure[inst✝.toTopologicalSpace] t)
hts : closure[inst✝.toTopologicalSpace] t ⊆ V.image (closure[inst✝.toTopologicalSpace] s)
⊢ s ⊆ SetRel.preimage U t

pp: case right
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ :
  (s, t) ∈
    (fun x ↦ (closure[inst✝.toTopologicalSpace] x.1, closure[inst✝.toTopologicalSpace] x.2)) ⁻¹' hausdorffEntourage V
hst : closure[inst✝.toTopologicalSpace] s ⊆ V.preimage (closure[inst✝.toTopologicalSpace] t)
hts : closure[inst✝.toTopologicalSpace] t ⊆ V.image (closure[inst✝.toTopologicalSpace] s)
⊢ t ⊆ SetRel.image U s

Tactic:
·

Goal state after:
case left
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ :
  (s, t) ∈
    (fun x ↦ (closure[inst✝.toTopologicalSpace] x.1, closure[inst✝.toTopologicalSpace] x.2)) ⁻¹' hausdorffEntourage V
hst : closure[inst✝.toTopologicalSpace] s ⊆ V.preimage (closure[inst✝.toTopologicalSpace] t)
hts : closure[inst✝.toTopologicalSpace] t ⊆ V.image (closure[inst✝.toTopologicalSpace] s)
⊢ s ⊆ SetRel.preimage U t
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case pos
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
ε : ℝ≥0∞
hε : ε < 1
h : ε = 0
⊢ ε * spectralRadius 𝕜 a ≤ liminf (fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)) atTop

pp: case neg
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
ε : ℝ≥0∞
hε : ε < 1
h : ¬ε = 0
⊢ ε * spectralRadius 𝕜 a ≤ liminf (fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)) atTop

Tactic:
·

Goal state after:
case pos
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
ε : ℝ≥0∞
hε : ε < 1
h : ε = 0
⊢ ε * spectralRadius 𝕜 a ≤ liminf (fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)) atTop
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case right
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ :
  (s, t) ∈
    (fun x ↦ (closure[inst✝.toTopologicalSpace] x.1, closure[inst✝.toTopologicalSpace] x.2)) ⁻¹' hausdorffEntourage V
hst : closure[inst✝.toTopologicalSpace] s ⊆ V.preimage (closure[inst✝.toTopologicalSpace] t)
hts : closure[inst✝.toTopologicalSpace] t ⊆ V.image (closure[inst✝.toTopologicalSpace] s)
⊢ t ⊆ SetRel.image U s

Tactic:
·

Goal state after:
case right
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ :
  (s, t) ∈
    (fun x ↦ (closure[inst✝.toTopologicalSpace] x.1, closure[inst✝.toTopologicalSpace] x.2)) ⁻¹' hausdorffEntourage V
hst : closure[inst✝.toTopologicalSpace] s ⊆ V.preimage (closure[inst✝.toTopologicalSpace] t)
hts : closure[inst✝.toTopologicalSpace] t ⊆ V.image (closure[inst✝.toTopologicalSpace] s)
⊢ t ⊆ SetRel.image U s
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
ε : ℝ≥0∞
hε : 1 < ε⁻¹
h : ¬ε = 0
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ↑‖1‖₊ ^ (1 / ↑b) ≤ ε⁻¹
n : ℕ
⊢ N ≤ n + N + 1

Tactic:
by lia

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

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
x✝ :
  (∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)) ∧
    ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
h₁ : ∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)
h₂ : ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
U : Set β
hU : U ∈ {U | IsOpen[inst✝] U}
⊢ IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)

pp: case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
x✝ :
  (∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)) ∧
    ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
h₁ : ∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)
h₂ : ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
U : Set β
hU : U ∈ {V | IsOpen[inst✝] V}
⊢ IsOpen[inst✝¹] (f ⁻¹' (fun V ↦ {s | (s ∩ V).Nonempty}) U)

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
x✝ :
  (∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)) ∧
    ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
h₁ : ∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)
h₂ : ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
U : Set β
hU : U ∈ {U | IsOpen[inst✝] U}
⊢ IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
x✝ :
  (∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)) ∧
    ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
h₁ : ∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)
h₂ : ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
U : Set β
hU : U ∈ {V | IsOpen[inst✝] V}
⊢ IsOpen[inst✝¹] (f ⁻¹' (fun V ↦ {s | (s ∩ V).Nonempty}) U)

Tactic:
·

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
x✝ :
  (∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)) ∧
    ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
h₁ : ∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)
h₂ : ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
U : Set β
hU : U ∈ {V | IsOpen[inst✝] V}
⊢ IsOpen[inst✝¹] (f ⁻¹' (fun V ↦ {s | (s ∩ V).Nonempty}) U)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case zero
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
⊢ ‖a ^ 0‖₊ * r ^ 0 ≤ max 1 ‖1‖₊

pp: case succ
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
⊢ ‖a ^ (n + 1)‖₊ * r ^ (n + 1) ≤ max 1 ‖1‖₊

Tactic:
·

Goal state after:
case zero
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
⊢ ‖a ^ 0‖₊ * r ^ 0 ≤ max 1 ‖1‖₊
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case succ
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
⊢ ‖a ^ (n + 1)‖₊ * r ^ (n + 1) ≤ max 1 ‖1‖₊

Tactic:
·

Goal state after:
case succ
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
⊢ ‖a ^ (n + 1)‖₊ * r ^ (n + 1) ≤ max 1 ‖1‖₊
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

pp: case neg
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:
·

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

Goal state before:
pp: case neg
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:
·

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

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
hs : ∅ ∈ (range fun x ↦ {x})ᶜ
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 ∅

pp: case inr
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
s : Set α
hs : s ∈ (range fun x ↦ {x})ᶜ
h : s.Nonempty
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 s

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
hs : ∅ ∈ (range fun x ↦ {x})ᶜ
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 ∅
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
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ‖a‖₊ = 0
⊢ ‖y • a‖ < 1

pp: case neg
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
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
⊢ ‖y • a‖ < 1

Tactic:
·

Goal state after:
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
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ‖a‖₊ = 0
⊢ ‖y • a‖ < 1
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr.inl
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
x : α
hs : {x} ∈ (range fun x ↦ {x})ᶜ
h : {x}.Nonempty
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 {x}

pp: case inr.inr
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
s : Set α
hs : s ∈ (range fun x ↦ {x})ᶜ
h : s.Nonempty
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≠ y
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 s

Tactic:
·

Goal state after:
case inr.inl
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
x : α
hs : {x} ∈ (range fun x ↦ {x})ᶜ
h : {x}.Nonempty
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 {x}
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case neg
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
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
⊢ ‖y • a‖ < 1

Tactic:
·

Goal state after:
case neg
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
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
⊢ ‖y • a‖ < 1