mathlib-initiative/mathlib-tactics huggingface dataset:
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
u v : Set (Set (Set α))
hu : u ⊆ powerset '' {U | IsOpen[inst✝] U}
hv : v ⊆ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {U | IsOpen[inst✝] U}
ht₁ : u.Finite ∧ v.Finite
ht₂ : u ∪ v ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hs : s ∈ ⋂₀ (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 (u ∪ v)

Tactic:
obtain ⟨u, hu₁, hu₂, rfl⟩ := exists_subset_image_finite_and.mp ⟨u, hu, ht₁.1, rfl⟩

Goal state after:
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)
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
u : Set (Set α)
⊢ u ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U} →
    ∀ x ∈ {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (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 ∈ v ∧ v ⊆ {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
⟨hu₁, hu₂⟩

Goal state after:
case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝] U
⊢ ∀ x ∈ {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (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 ∈ v ∧ v ⊆ {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
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 : Finset α
ht₂ : s ⊆ ⋃ y ∈ ↑t, {x | (x, y) ∈ Prod.swap ⁻¹' U}
⊢ ∃ t, (↑t, s) ∈ hausdorffEntourage U

Tactic:
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.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‖⁻¹ * ‖k‖
a✝ : Nontrivial A
hk : k ≠ 0
ku : Aˣ := (Units.map ↑↑ₐ) (Units.mk0 k hk)
hku : ‖-a‖ < ‖↑ku⁻¹‖⁻¹
⊢ IsUnit (k • 1 - a)

Tactic:
simpa [ku, sub_eq_add_neg, Algebra.algebraMap_eq_smul_one] using (ku.add (-a) hku).isUnit

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

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
s : Set α
⊢ 𝓝 (closure[inst✝.toTopologicalSpace] s) = 𝓝 s

Tactic:
isUniformInducing_closure.isInducing.nhds_eq_comap,

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
s : Set α
⊢ Filter.comap closure[inst✝.toTopologicalSpace]
      (𝓝 (closure[inst✝.toTopologicalSpace] (closure[inst✝.toTopologicalSpace] s))) =
    Filter.comap closure[inst✝.toTopologicalSpace] (𝓝 (closure[inst✝.toTopologicalSpace] s))
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

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✝ : HasSummableGeomSeries A
a : A
⊢ (↑‖a‖₊)⁻¹ ≤ radius fun n ↦ ContinuousMultilinearMap.mkPiRing 𝕜 (Fin n) (a ^ n)

Tactic:
by
      refine le_of_forall_nnreal_lt fun r hr =>
        le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => ?_
      rw [← norm_toNNReal, norm_mkPiRing, norm_toNNReal]
      rcases n with - | n
      · simp
      · grw [nnnorm_pow_le' a n.succ_pos, ← le_max_left]
        by_cases h : ‖a‖₊ = 0
        · simp [h, pow_succ']
        · rw [← coe_inv h, coe_lt_coe, NNReal.lt_inv_iff_mul_lt h] at hr
          simpa only [← mul_pow, mul_comm] using! pow_le_one' hr.le n.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:
← Set.disjoint_iff_inter_eq_empty,

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 : Disjoint {⊥}ᶜ (⋂ i, {C | ↑C ⊆ F i}) → ∃ u, Disjoint {⊥}ᶜ (⋂ i ∈ u, {C | ↑C ⊆ F i})
⊢ ∃ u, ⋂ i ∈ u, F i = ∅
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

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

Tactic:
funext n

Goal state after:
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
n : ℕ
⊢ ↑‖(a⋆ * a) ^ n‖₊ ^ (1 / ↑n) = ((fun x ↦ x ^ 2) ∘ fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)) n
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
heq : (fun n ↦ ↑‖(a⋆ * a) ^ n‖₊ ^ (1 / ↑n)) = (fun x ↦ x ^ 2) ∘ fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)
h₂ : Tendsto (fun n ↦ ↑‖(a⋆ * a) ^ n‖₊ ^ (1 / ↑n)) atTop (𝓝 (spectralRadius ℂ a ^ 2))
⊢ ↑‖a‖₊ ^ 2 = spectralRadius ℂ (a⋆ * a)

Tactic:
(IsSelfAdjoint.star_mul_self a).spectralRadius_eq_nnnorm,

Goal state after:
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
heq : (fun n ↦ ↑‖(a⋆ * a) ^ n‖₊ ^ (1 / ↑n)) = (fun x ↦ x ^ 2) ∘ fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)
h₂ : Tendsto (fun n ↦ ↑‖(a⋆ * a) ^ n‖₊ ^ (1 / ↑n)) atTop (𝓝 (spectralRadius ℂ a ^ 2))
⊢ ↑‖a‖₊ ^ 2 = ↑‖a⋆ * a‖₊
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✝ : T2Space α
K₁ K₂ : Compacts α
h : K₁ ≠ K₂
x : α
hx₁ : x ∈ K₁
hx₂ : x ∉ K₂
U V : Set α
hU : IsOpen[inst✝³] U
hV : IsOpen[inst✝³] V
hU' : ↑K₂ ⊆ U
hV' : x ∈ V
hUV : Disjoint U V
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ K₁ ∈ u ∧ K₂ ∈ v ∧ Disjoint u v

Tactic:
exact ⟨_, _, isOpen_inter_nonempty_of_isOpen hV, isOpen_subsets_of_isOpen hU, ⟨x, hx₁, hV'⟩,
      hU', by grind [Set.Nonempty]⟩

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

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f : α →ᵇ β
hf : ∀ U ∈ 𝓝 f, (U ∩ range toBCF).Nonempty
ε : ℝ
hε : ε > 0
g : α →C₀ β
hg : g.toBCF ∈ ball f (ε / 2)
x : α
hx : dist (g x) 0 < ε / 2
⊢ dist g.toBCF f + ε / 2 ≤ ε

Tactic:
grw [mem_ball.1 hg, add_halves ε]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : LocallyConvexSpace 𝕜 E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ZeroLEOneClass 𝕜
⊢ (𝓝 0).HasBasis (fun s ↦ s ∈ 𝓝 0 ∧ IsClosed[inst✝⁴] s ∧ AbsConvex 𝕜 s) id

Tactic:
refine (nhds_basis_opens 0).to_hasBasis ?_
    fun s ⟨hs_nhds, _, _⟩ ↦ ⟨interior s,
      by simp [interior_subset, mem_interior_iff_mem_nhds.mpr hs_nhds]⟩
  intro s ⟨hs_zero, hs_open⟩
  obtain ⟨W, hW_open, hW_zero, hW_add⟩ :=
    exists_open_nhds_zero_add_subset (hs_open.mem_nhds hs_zero)
  obtain ⟨V, ⟨hV_zero, hV_open, hV_abs⟩, hVW⟩ :=
    (nhds_hasBasis_absConvex_open 𝕜 E).mem_iff.mp (hW_open.mem_nhds hW_zero)
  exact ⟨closure V,
    ⟨Filter.mem_of_superset (hV_open.mem_nhds hV_zero) subset_closure, isClosed_closure,
     hV_abs.closure⟩,
    (closure_subset_add_self_of_mem_nhds_zero (hV_open.mem_nhds hV_zero)).trans
      ((add_subset_add hVW hVW).trans hW_add)⟩

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

Goal state before:
pp: case g
α : Type u_1
inst✝ : TopologicalSpace α
h : ∀ (s : Set (NonemptyCompacts α)), IsClopen s → s = ∅ ∨ s = univ
s : Set α
hs : IsClopen s
⊢ {K | ↑K ⊆ s} = univ → s = univ

Tactic:
· simp only [Set.eq_univ_iff_forall]
    exact fun h x => Set.singleton_subset_iff.mp (h {x})

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
⊢ ConnectedSpace (NonemptyCompacts α) ↔ ConnectedSpace α

Tactic:
simp [connectedSpace_iff]

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

Goal state before:
pp: ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
t₂ : Finset ι → TopologicalSpace E := fun s ↦ ⨅ i, induced (⇑(f ↑i)) t𝕜
t₁_group : IsTopologicalAddGroup E
t₂_group : ∀ (s : Finset ι), IsTopologicalAddGroup E
t₁_smul : ContinuousSMul 𝕜 E
t₂_smul : ∀ (s : Finset ι), ContinuousSMul 𝕜 E
⊢ ((normSeminorm 𝕜 𝕜).comp φ).ball 0 1 ∈ ⨅ i, 𝓝 0 ↔ ∃ s, Continuous[t₂ s, _] ⇑((normSeminorm 𝕜 𝕜).comp φ)

Tactic:
conv in Continuous _ =>
    rw [letI := t₂ s; Seminorm.continuous_iff one_pos, nhds_iInf, iInf_subtype]

Goal state after:
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
t₂ : Finset ι → TopologicalSpace E := fun s ↦ ⨅ i, induced (⇑(f ↑i)) t𝕜
t₁_group : IsTopologicalAddGroup E
t₂_group : ∀ (s : Finset ι), IsTopologicalAddGroup E
t₁_smul : ContinuousSMul 𝕜 E
t₂_smul : ∀ (s : Finset ι), ContinuousSMul 𝕜 E
⊢ ((normSeminorm 𝕜 𝕜).comp φ).ball 0 1 ∈ ⨅ i, 𝓝 0 ↔ ∃ s, ((normSeminorm 𝕜 𝕜).comp φ).ball 0 1 ∈ ⨅ i, ⨅ (h : i ∈ s), 𝓝 0
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
x : X
hx : x ∈ s
⊢ { toFun := fun x ↦ 1 - f x, continuous_toFun := ⋯ } x = 1 x

Tactic:
by simpa using hfs hx

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

Goal state before:
pp: case refine_3.refine_1
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)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
s₁ s₂ : Set X
g₁ : C(X, ℝ≥0)
hI₁ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₁ ∈ I
hgt₁ : ∀ x ∈ s₁, 0 < g₁ x
g₂ : C(X, ℝ≥0)
hI₂ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₂ ∈ I
hgt₂ : ∀ x ∈ s₂, 0 < g₂ x
⊢ (↑(algebraMapCLM ℝ≥0 𝕜)).comp (g₁ + g₂) ∈ I

Tactic:
convert! I.add_mem hI₁ hI₂

Goal state after:
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
s₁ s₂ : Set X
g₁ : C(X, ℝ≥0)
hI₁ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₁ ∈ I
hgt₁ : ∀ x ∈ s₁, 0 < g₁ x
g₂ : C(X, ℝ≥0)
hI₂ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₂ ∈ I
hgt₂ : ∀ x ∈ s₂, 0 < g₂ x
⊢ (↑(algebraMapCLM ℝ≥0 𝕜)).comp (g₁ + g₂) = (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₁ + (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₂
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝ : CommCStarAlgebra A
a : A
this : spectralRadius ℂ ((gelfandTransform ℂ A) (star a) * (gelfandTransform ℂ A) a) = spectralRadius ℂ (star a * a)
⊢ ‖(gelfandTransform ℂ A) a‖ = ‖a‖

Tactic:
(IsSelfAdjoint.star_mul_self a).spectralRadius_eq_nnnorm,

Goal state after:
A : Type u_1
inst✝ : CommCStarAlgebra A
a : A
this : spectralRadius ℂ ((gelfandTransform ℂ A) (star a) * (gelfandTransform ℂ A) a) = ↑‖star a * a‖₊
⊢ ‖(gelfandTransform ℂ A) a‖ = ‖a‖
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 b : A
hab : a = b - a⁻
hb : 0 ≤ b
⊢ a⁺ = b

Tactic:
by
  have ha := hab.symm ▸ hb.isSelfAdjoint.sub (negPart_nonneg a).isSelfAdjoint
  nth_rw 1 [← posPart_sub_negPart a] at hab
  simpa using hab

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✝¹ : PseudoEMetricSpace γ
inst✝ : PseudoEMetricSpace β
this : 𝓤 (α →ᵤ β) = comap (fun x ↦ edist x.1 x.2) (𝓝 0)
⊢ 𝓤 (α →ᵤ β) = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | edist p.1 p.2 < ε}

Tactic:
simp [this, ENNReal.nhds_zero_basis.comap _ |>.eq_biInf, Set.Iio]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ β
C : ℝ
hC : 0 ≤ C
⊢ (edist f g).toReal ≤ C ↔ ∀ (x : α), (edist (toFun f x) (toFun g x)).toReal ≤ C

Tactic:
← ENNReal.le_ofReal_iff_toReal_le (edist_ne_top _ _) hC,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ β
C : ℝ
hC : 0 ≤ C
⊢ edist f g ≤ ENNReal.ofReal C ↔ ∀ (x : α), edist (toFun f x) (toFun g x) ≤ ENNReal.ofReal C
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
i : Fin 4
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] i ∧
    ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] i‖ ≤ ‖a‖

Tactic:
fin_cases i

Goal state after:
case h.left.«0»
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨0, ⋯⟩) ∧
    ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨0, ⋯⟩)‖ ≤ ‖a‖

case h.left.«1»
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨1, ⋯⟩) ∧
    ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨1, ⋯⟩)‖ ≤ ‖a‖

case h.left.«2»
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨2, ⋯⟩) ∧
    ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨2, ⋯⟩)‖ ≤ ‖a‖

case h.left.«3»
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨3, ⋯⟩) ∧
    ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨3, ⋯⟩)‖ ≤ ‖a‖
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : StarRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : 0 ≤ a
⊢ sqrt a = 0 ↔ a = 0

Tactic:
sqrt_eq_iff a _,

Goal state after:
A : Type u_1
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : StarRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : 0 ≤ a
⊢ 0 * 0 = a ↔ a = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : 0 ≤ a
⊢ sqrt a = 1 ↔ a = 1

Tactic:
sqrt_eq_iff a _,

Goal state after:
A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : 0 ≤ a
⊢ 1 * 1 = a ↔ a = 1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : IsTopologicalRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
f g : ℝ≥0 → ℝ≥0
a : A
ha_spec : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hf : ContinuousOn f (spectrum ℝ≥0 a)
hg : ContinuousOn g (spectrum ℝ≥0 a)
ha : 0 ≤ a
hf' : ContinuousOn (fun x ↦ ↑(f x.toNNReal)) (spectrum ℝ a)
⊢ cfc f a ≤ cfc g a ↔ ∀ x ∈ spectrum ℝ≥0 a, f x ≤ g x

Tactic:
have hg' := hg.ofReal_map_toNNReal <| ha_spec.image ▸ Set.mapsTo_image ..

Goal state after:
A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : IsTopologicalRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
f g : ℝ≥0 → ℝ≥0
a : A
ha_spec : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hf : ContinuousOn f (spectrum ℝ≥0 a)
hg : ContinuousOn g (spectrum ℝ≥0 a)
ha : 0 ≤ a
hf' : ContinuousOn (fun x ↦ ↑(f x.toNNReal)) (spectrum ℝ a)
hg' : ContinuousOn (fun x ↦ ↑(g x.toNNReal)) (spectrum ℝ a)
⊢ cfc f a ≤ cfc g a ↔ ∀ x ∈ spectrum ℝ≥0 a, f x ≤ g x
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: case inl
m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
M : CStarMatrix m n A
i : m
j : n
this : ‖M i j‖ * ‖M i j‖ ≤ ‖M‖ * ‖M i j‖
h : M i j = 0
⊢ ‖M i j‖ ≤ ‖M‖

Tactic:
simp [h, norm_def]

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

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : RCLike 𝕜
inst✝¹¹ : MeasurableSpace X
μ : Measure X
inst✝¹⁰ : NonUnitalNormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedSpace 𝕜 A
inst✝⁷ : IsScalarTower 𝕜 A A
inst✝⁶ : SMulCommClass 𝕜 A A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : CompleteSpace A
inst✝³ : NormedSpace ℝ A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(quasispectrum 𝕜 a), 𝕜)
hf : ContinuousOn (uncurry f) (univ ×ˢ quasispectrum 𝕜 a)
f_zero : ∀ᵐ (x : X) ∂μ, f x 0 = 0
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ z ∈ quasispectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound μ
ha : p a
this : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
⊢ cfcₙ (fun r ↦ ∫ (x : X), f x r ∂μ) a = ∫ (x : X), cfcₙ (f x) a ∂μ

Tactic:
refine cfcₙ_integral' _ _ this f_zero ⟨?_, ?_⟩ ha

Goal state after:
case refine_1
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : RCLike 𝕜
inst✝¹¹ : MeasurableSpace X
μ : Measure X
inst✝¹⁰ : NonUnitalNormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedSpace 𝕜 A
inst✝⁷ : IsScalarTower 𝕜 A A
inst✝⁶ : SMulCommClass 𝕜 A A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : CompleteSpace A
inst✝³ : NormedSpace ℝ A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(quasispectrum 𝕜 a), 𝕜)
hf : ContinuousOn (uncurry f) (univ ×ˢ quasispectrum 𝕜 a)
f_zero : ∀ᵐ (x : X) ∂μ, f x 0 = 0
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ z ∈ quasispectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound μ
ha : p a
this : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
⊢ AEStronglyMeasurable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ

case refine_2
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : RCLike 𝕜
inst✝¹¹ : MeasurableSpace X
μ : Measure X
inst✝¹⁰ : NonUnitalNormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedSpace 𝕜 A
inst✝⁷ : IsScalarTower 𝕜 A A
inst✝⁶ : SMulCommClass 𝕜 A A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : CompleteSpace A
inst✝³ : NormedSpace ℝ A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(quasispectrum 𝕜 a), 𝕜)
hf : ContinuousOn (uncurry f) (univ ×ˢ quasispectrum 𝕜 a)
f_zero : ∀ᵐ (x : X) ∂μ, f x 0 = 0
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ z ∈ quasispectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound μ
ha : p a
this : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
⊢ HasFiniteIntegral (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
Module: Mathlib.Topology.OpenPartialHomeomorph.Composition

Goal state before:
pp: case e_a
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Set X
e' : OpenPartialHomeomorph X Y
hs : IsOpen[inst✝¹] s
ht : IsOpen[inst✝] (e'.target ∩ ↑e'.symm ⁻¹' s)
⊢ e'.target ∩ ↑e'.symm ⁻¹' s = interior (e'.target ∩ ↑e'.symm ⁻¹' s)

Tactic:
interior_eq_iff_isOpen.mpr ht

Goal state after:
case e_a
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Set X
e' : OpenPartialHomeomorph X Y
hs : IsOpen[inst✝¹] s
ht : IsOpen[inst✝] (e'.target ∩ ↑e'.symm ⁻¹' s)
⊢ e'.target ∩ ↑e'.symm ⁻¹' s = e'.target ∩ ↑e'.symm ⁻¹' s
Module: Mathlib.Topology.IsLocalHomeomorph

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s : Set X
x : ↑s
e : OpenPartialHomeomorph X Y
hx : ↑x ∈ e.source
h : IsLocalHomeomorphOn (↑e) s
inst✝ : DiscreteTopology ↑(↑e '' s)
⊢ IsOpen[instTopologicalSpaceSubtype] {x}

Tactic:
have ⟨U, hU, eq⟩ := isOpen_discrete {(⟨_, _, x.2, rfl⟩ : e '' s)}

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s : Set X
x : ↑s
e : OpenPartialHomeomorph X Y
hx : ↑x ∈ e.source
h : IsLocalHomeomorphOn (↑e) s
inst✝ : DiscreteTopology ↑(↑e '' s)
U : Set Y
hU : IsOpen[inst✝¹] U
eq : Subtype.val ⁻¹' U = {⟨↑e ↑x, ⋯⟩}
⊢ IsOpen[instTopologicalSpaceSubtype] {x}