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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommRing R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalRing R
inst✝⁹ : ContinuousStar R
inst✝⁸ : ContinuousSqrt R
inst✝⁷ : StarOrderedRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
r : R
a : A
ha : p a
⊢ a ≤ (algebraMap R A) r ↔ ∀ x ∈ spectrum R a, x ≤ r

Tactic:
← cfc_id R a

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommRing R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalRing R
inst✝⁹ : ContinuousStar R
inst✝⁸ : ContinuousSqrt R
inst✝⁷ : StarOrderedRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
r : R
a : A
ha : p a
⊢ cfc id a ≤ (algebraMap R A) r ↔ ∀ x ∈ spectrum R a, x ≤ r
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommRing R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalRing R
inst✝⁹ : ContinuousStar R
inst✝⁸ : ContinuousSqrt R
inst✝⁷ : StarOrderedRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
r : R
a : A
ha : p a
⊢ (algebraMap R A) r ≤ a ↔ ∀ x ∈ spectrum R a, r ≤ x

Tactic:
← cfc_id R a

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommRing R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalRing R
inst✝⁹ : ContinuousStar R
inst✝⁸ : ContinuousSqrt R
inst✝⁷ : StarOrderedRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
r : R
a : A
ha : p a
⊢ (algebraMap R A) r ≤ cfc id a ↔ ∀ x ∈ spectrum R a, r ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inr.inl
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
a : A
hf : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ (fun x ↦ star (f x)) a = star 0

Tactic:
cfcₙ_apply_of_not_continuousOn,

Goal state after:
case neg.inr.inl
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
a : A
hf : ¬ContinuousOn f (σₙ R a)
⊢ 0 = star 0

case neg.inr.inl.hf
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
a : A
hf : ¬ContinuousOn f (σₙ R a)
⊢ ¬ContinuousOn (fun x ↦ star (f x)) (σₙ R a)
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
h : ν = 0
⊢ Polynomial.eval 0 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = 1

pp: case neg
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
h : ¬ν = 0
⊢ Polynomial.eval 0 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = 0

Tactic:
· subst h; simp

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
h : ¬ν = 0
⊢ Polynomial.eval 0 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = 0
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
h : ν = n
⊢ Polynomial.eval 1 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = 1

pp: case neg
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
h : ¬ν = n
⊢ Polynomial.eval 1 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = 0

Tactic:
· subst h; simp

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
h : ¬ν = n
⊢ Polynomial.eval 1 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = 0
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case refine_3
n k : ℕ
h : k ≤ n
p : ℚ[X]
m : p ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
x y : ℚ[X]
hx✝ : x ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
hy✝ : y ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
hx : eval 1 ((⇑derivative)^[n - k] x) = 0
⊢ eval 1 ((⇑derivative)^[n - k] y) = 0 → eval 1 ((⇑derivative)^[n - k] (x + y)) = 0

Tactic:
hy

Goal state after:
case refine_3
n k : ℕ
h : k ≤ n
p : ℚ[X]
m : p ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
x y : ℚ[X]
hx✝ : x ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
hy✝ : y ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
hx : eval 1 ((⇑derivative)^[n - k] x) = 0
hy : eval 1 ((⇑derivative)^[n - k] y) = 0
⊢ eval 1 ((⇑derivative)^[n - k] (x + y)) = 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
hf : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ (fun x ↦ -f x) a = -0

Tactic:
cfcₙ_apply_of_not_continuousOn,

Goal state after:
case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
hf : ¬ContinuousOn f (σₙ R a)
⊢ 0 = -0

case neg.inr.inl.hf
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
hf : ¬ContinuousOn f (σₙ R a)
⊢ ¬ContinuousOn (fun x ↦ -f x) (σₙ R a)
Module: Mathlib.Analysis.SpecialFunctions.Bernstein

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module ℝ E
inst✝ : ContinuousSMul ℝ E
n : ℕ
f : C(↑I, E)
x : ↑I
⊢ (bernsteinApproximation n f) x = ∑ k, (bernstein n ↑k) x • f k/ₙ

Tactic:
by
  simp [bernsteinApproximation]

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

Goal state before:
pp: case neg
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
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : ¬Nonempty X
⊢ ∃ x, dist x f < ε ∧ x ∈ L

Tactic:
exact ⟨nA.some, (dist_lt_iff pos).mpr fun x => False.elim (nX ⟨x⟩), nA.choose_spec⟩

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : NonUnitalNonAssocSemiring A
inst✝¹ : TopologicalSpace A
inst✝ : Module 𝕜 A
x✝¹ x✝ : A
⊢ 0 (x✝¹ * x✝) = 0 x✝¹ * 0 x✝

Tactic:
by exact (zero_mul (0 : 𝕜)).symm

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
inst✝ : CompactSpace ↑s
⊢ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)) =
    closure ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) ∩ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))

Tactic:
convert! (Set.univ_inter _).symm

Goal state after:
case e'_3
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
inst✝ : CompactSpace ↑s
⊢ closure ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) = Set.univ
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
F : Set α
h : IsClosed[inst✝] F
⊢ IsClosed[TopologicalSpace.vietoris α] {a | a ⊆ Fᶜ}ᶜ

Tactic:
exact h.isOpen_compl.powerset_vietoris.isClosed_compl

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

Goal state before:
pp: case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h✝ : s.Nonempty
v : Set (Set (Set α))
hv : v ⊆ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {U | IsOpen[inst✝] U}
u : Set (Set α)
hu₁ : u ⊆ {U | IsOpen[inst✝] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U | IsOpen[inst✝] U}
ht₁ : (powerset '' u).Finite ∧ v.Finite
ht₂ : powerset '' u ∪ v ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hs : s ∈ ⋂₀ (powerset '' u ∪ v)
⊢ ∃ 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 (powerset '' u ∪ v)

Tactic:
obtain ⟨v, hv₁, hv₂, rfl⟩ := exists_subset_image_finite_and.mp ⟨v, hv, ht₁.2, rfl⟩

Goal state after:
case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h✝ : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U | IsOpen[inst✝] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U | IsOpen[inst✝] U}
v : Set (Set α)
hv₁ : v ⊆ {U | IsOpen[inst✝] U}
hv₂ : v.Finite
hv : (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {U | IsOpen[inst✝] U}
ht₁ : (powerset '' u).Finite ∧ ((fun V ↦ {s | (s ∩ V).Nonempty}) '' v).Finite
ht₂ :
  powerset '' u ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hs : s ∈ ⋂₀ (powerset '' u ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' v)
⊢ ∃ 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 (powerset '' u ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' v)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU : U ∈ 𝓤 α
⊢ ∃ t, (↑t, s) ∈ hausdorffEntourage U

Tactic:
by
  obtain ⟨t, ht₁, ht₂⟩ := hs _ (symm_le_uniformity hU)
  lift t to Finset α using ht₁
  classical
  refine ⟨{x ∈ t | ∃ y ∈ s, (x, y) ∈ U}, ?_⟩
  rw [Finset.coe_filter]
  refine ⟨fun _ h => h.2, fun x hx => ?_⟩
  obtain ⟨y, hy, hxy⟩ := Set.mem_iUnion₂.mp (ht₂ hx)
  exact ⟨y, ⟨hy, x, hx, hxy⟩, hxy⟩

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

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
⊢ Filter.comap (fun x ↦ ({x.1}, {x.2})) (𝓤 (Set α)) = 𝓤 α

Tactic:
by
    change Filter.comap _ (Filter.lift' _ _) = _
    simp_rw [Filter.comap_lift'_eq, Function.comp_def, Set.preimage,
      singleton_mem_hausdorffEntourage]
    exact Filter.lift'_id

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

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
⊢ UniformContinuous closure[inst✝.toTopologicalSpace]

Tactic:
UniformContinuous,

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
⊢ Filter.Tendsto (fun x ↦ (closure[inst✝.toTopologicalSpace] x.1, closure[inst✝.toTopologicalSpace] x.2)) (𝓤 (Set α))
    (𝓤 (Set α))
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hf_compact : ∀ x ∈ s, IsCompact (f x)
x₀ : α
hx₀s : x₀ ∈ s
u : Set β
hu : IsOpen[inst✝.toTopologicalSpace] u
hx₀u : f x₀ ⊆ u
W : Set (β × β)
hW : W ∈ uniformity β
left✝ : IsOpen[instTopologicalSpaceProd] W
hWu : ∀ x ∈ f x₀, ball x W ⊆ u
V : Set (β × β)
hV : V ∈ uniformity β
hVsym : SetRel.IsSymm V
hVcomp : SetRel.comp V V ⊆ W
U : SetRel β β
hUsub : id U ⊆ V
hUmem : U ∈ uniformity β
hUopen : IsOpen[instTopologicalSpaceProd] U
hHU : hausdorffEntourage U ∈ uniformity (Set β)
N : ι
hN : ∀ x ∈ s, (f x, F N x) ∈ hausdorffEntourage U
hFN_image : F N x₀ ⊆ U.image (f x₀)
hF : ∀ (n : ι), ∀ x ∈ s, UpperHemicontinuousWithinAt (F n) s x
hFN_uhc : ∀ᶠ (x : α) in 𝓝[s] x₀, F N x ⊆ U.image (f x₀)
x : α
hFNx : F N x ⊆ U.image (f x₀)
hx_s : x ∈ s
y : β
⊢ y ∈ f x → y ∈ u

Tactic:
hy

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hf_compact : ∀ x ∈ s, IsCompact (f x)
x₀ : α
hx₀s : x₀ ∈ s
u : Set β
hu : IsOpen[inst✝.toTopologicalSpace] u
hx₀u : f x₀ ⊆ u
W : Set (β × β)
hW : W ∈ uniformity β
left✝ : IsOpen[instTopologicalSpaceProd] W
hWu : ∀ x ∈ f x₀, ball x W ⊆ u
V : Set (β × β)
hV : V ∈ uniformity β
hVsym : SetRel.IsSymm V
hVcomp : SetRel.comp V V ⊆ W
U : SetRel β β
hUsub : id U ⊆ V
hUmem : U ∈ uniformity β
hUopen : IsOpen[instTopologicalSpaceProd] U
hHU : hausdorffEntourage U ∈ uniformity (Set β)
N : ι
hN : ∀ x ∈ s, (f x, F N x) ∈ hausdorffEntourage U
hFN_image : F N x₀ ⊆ U.image (f x₀)
hF : ∀ (n : ι), ∀ x ∈ s, UpperHemicontinuousWithinAt (F n) s x
hFN_uhc : ∀ᶠ (x : α) in 𝓝[s] x₀, F N x ⊆ U.image (f x₀)
x : α
hFNx : F N x ⊆ U.image (f x₀)
hx_s : x ∈ s
y : β
hy : y ∈ f x
⊢ y ∈ u
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝¹ : UniformSpace α
inst✝ : T0Space α
⊢ IsClosed (SetLike.coe ⁻¹' Set.range (SetLike.coe ∘ fun x ↦ {x}))

Tactic:
exact UniformSpace.hausdorff.isClosedEmbedding_singleton.isClosed_range.preimage
      uniformContinuous_coe.continuous

Goal state after:
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
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ p (ψ f)

Tactic:
exact hp₁.mp <| coe_ψ _ ▸ cfcHom_predicate (R := 𝕜) (hp₁.mpr ha) _

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

Goal state before:
pp: case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
              K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, (↑K ∩ W).Nonempty
x : α
hx : x ∈ ↑K
⊢ x ∈ ⋃₀ {W | W ∈ w ∧ (↑K ∩ W).Nonempty}

Tactic:
obtain ⟨W, hWw, hxW⟩ := mem_sUnion.mp (hKV hx)

Goal state after:
case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
              K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, (↑K ∩ W).Nonempty
x : α
hx : x ∈ ↑K
W : Set α
hWw : W ∈ w
hxW : x ∈ W
⊢ x ∈ ⋃₀ {W | W ∈ w ∧ (↑K ∩ W).Nonempty}
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

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

Tactic:
quasispectrumRestricts_iff,

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : PartialOrder A
inst✝⁷ : StarRing A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module ℝ A
inst✝³ : IsScalarTower ℝ A A
inst✝² : SMulCommClass ℝ A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
x : A
hx₁ : IsSelfAdjoint x
hx₂ : QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal
hb : IsSelfAdjoint (x * x) ∧ QuasispectrumRestricts (x * x) ⇑ContinuousMap.realToNNReal
h : Commute a (x * x)
hx : 0 ≤ x
⊢ Set.RightInvOn (⇑ContinuousMap.realToNNReal) (⇑(algebraMap NNReal ℝ)) (σₙ ℝ (a * x * x)) ∧
    Function.LeftInverse ⇑ContinuousMap.realToNNReal ⇑(algebraMap NNReal ℝ)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : T2Space A
f : ℂ → ℂ
a : A
hf_real : ∀ x ∈ σ ℂ a, star (f x) = f x
ha : IsSelfAdjoint a
⊢ cfc f a = cfc (fun x ↦ (f ↑x).re) a

Tactic:
cfc_real_eq_complex ..

Goal state after:
A : Type u_1
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : T2Space A
f : ℂ → ℂ
a : A
hf_real : ∀ x ∈ σ ℂ a, star (f x) = f x
ha : IsSelfAdjoint a
⊢ cfc f a = cfc (fun x ↦ ↑(f ↑x.re).re) a
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_4
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_5
inst✝² : AddCommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : Module 𝕜 E
a : E
y : StrongDual 𝕜 E
⊢ y ∈ polar 𝕜 {a} ↔ ‖y a‖ ≤ 1

Tactic:
by
  simp only [polar_singleton, mem_ofPred_eq]

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
x✝ : LocallyCompactSpace (NonemptyCompacts α)
x : α
U : Set α
hU : ↑{x} ⊆ interior U
K : Set (NonemptyCompacts α)
hK : IsCompact K
hxK : K ∈ 𝓝 {x}
hKU : K ⊆ {K | ↑K ⊆ interior U}
y : α
⊢ y ∈ (fun x ↦ {x}) ⁻¹' K → y ∈ ⋃ L ∈ K, ↑L

Tactic:
hy

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
x✝ : LocallyCompactSpace (NonemptyCompacts α)
x : α
U : Set α
hU : ↑{x} ⊆ interior U
K : Set (NonemptyCompacts α)
hK : IsCompact K
hxK : K ∈ 𝓝 {x}
hKU : K ⊆ {K | ↑K ⊆ interior U}
y : α
hy : y ∈ (fun x ↦ {x}) ⁻¹' K
⊢ y ∈ ⋃ L ∈ K, ↑L
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
h : LocallyConnectedSpace (NonemptyCompacts α)
x : α
U : Set α
x✝ : x ∈ U ∧ IsOpen[inst✝] U
hx : x ∈ U
hU : IsOpen[inst✝] U
V : Set (NonemptyCompacts α)
hV₁ : IsOpen V
hV₂ : IsPreconnected V
hxV : {x} ∈ V
hKV : V ⊆ {K | ↑K ⊆ U}
y : α
⊢ y ∈ (fun x ↦ {x}) ⁻¹' V → y ∈ ⋃ L ∈ V, ↑L

Tactic:
hy

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
h : LocallyConnectedSpace (NonemptyCompacts α)
x : α
U : Set α
x✝ : x ∈ U ∧ IsOpen[inst✝] U
hx : x ∈ U
hU : IsOpen[inst✝] U
V : Set (NonemptyCompacts α)
hV₁ : IsOpen V
hV₂ : IsPreconnected V
hxV : {x} ∈ V
hKV : V ⊆ {K | ↑K ⊆ U}
y : α
hy : y ∈ (fun x ↦ {x}) ⁻¹' V
⊢ y ∈ ⋃ L ∈ V, ↑L
Module: Mathlib.Analysis.Normed.Operator.Completeness

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
Fₗ : Type u_5
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup Fₗ
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜 Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
f✝ g : E →SL[σ₁₂] F
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f : E' → F
s : Set (E' →SL[σ₁₂] F)
hs : Bornology.IsBounded s
hf : f ∈ closure[Pi.topologicalSpace] (DFunLike.coe '' s)
⊢ E' →SL[σ₁₂] F

Tactic:
by
  -- `f` is a linear map due to `linearMapOfMemClosureRangeCoe`
  refine (linearMapOfMemClosureRangeCoe f ?_).mkContinuousOfExistsBound ?_
  · refine closure_mono (image_subset_iff.2 fun g _ => ?_) hf
    exact ⟨g, rfl⟩
  · -- We need to show that `f` has bounded norm. Choose `C` such that `‖g‖ ≤ C` for all `g ∈ s`.
    rcases isBounded_iff_forall_norm_le.1 hs with ⟨C, hC⟩
    -- Then `‖g x‖ ≤ C * ‖x‖` for all `g ∈ s`, `x : E`, hence `‖f x‖ ≤ C * ‖x‖` for all `x`.
    have : ∀ x, IsClosed { g : E' → F | ‖g x‖ ≤ C * ‖x‖ } := fun x =>
      isClosed_Iic.preimage (@continuous_apply E' (fun _ => F) _ x).norm
    refine ⟨C, fun x => (this x).closure_subset_iff.2 (image_subset_iff.2 fun g hg => ?_) hf⟩
    exact g.le_of_opNorm_le (hC _ hg) _

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝⁸ : RingHomIsometric σ₁₂
inst✝⁷ : AddCommGroup E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜₁ E
inst✝ : BaireSpace E
p : Seminorm 𝕜₁ E
hp : LowerSemicontinuous ⇑p
h₁ : ∀ (n : ℕ), IsClosed[inst✝³] (p.closedBall 0 ↑n)
h₂ : ⋃ n, p.closedBall 0 ↑n = univ
n : ℕ
x : E
hxn : {y | p y ≤ ↑n} ∈ map (fun x_1 ↦ x + x_1) (𝓝 0)
hxn' : p x ≤ ↑n
y : E
⊢ y ∈ (fun x_1 ↦ x + x_1) ⁻¹' {y | p y ≤ ↑n} → p y ≤ ↑n + ↑n

Tactic:
hy

Goal state after:
α : Type u_1
ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝⁸ : RingHomIsometric σ₁₂
inst✝⁷ : AddCommGroup E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜₁ E
inst✝ : BaireSpace E
p : Seminorm 𝕜₁ E
hp : LowerSemicontinuous ⇑p
h₁ : ∀ (n : ℕ), IsClosed[inst✝³] (p.closedBall 0 ↑n)
h₂ : ⋃ n, p.closedBall 0 ↑n = univ
n : ℕ
x : E
hxn : {y | p y ≤ ↑n} ∈ map (fun x_1 ↦ x + x_1) (𝓝 0)
hxn' : p x ≤ ↑n
y : E
hy : y ∈ (fun x_1 ↦ x + x_1) ⁻¹' {y | p y ≤ ↑n}
⊢ p y ≤ ↑n + ↑n