mathlib-initiative/mathlib-tactics huggingface dataset:
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
this : ∃ᶠ (n : ι) in l, y n ∈ s
n : ι
hn : n ∈ y ⁻¹' t
hyn : y n ∈ s
⊢ False

Tactic:
exact hst.notMem_of_mem_left hyn hn

Goal state after:
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

Tactic:
subset_closure (s := s),

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)
⊢ closure[inst✝.toTopologicalSpace] s ⊆ SetRel.preimage U t
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
⊢ Continuous[instTopologicalSpaceProd, TopologicalSpace.vietoris α] fun x ↦ x.1 ∪ x.2

Tactic:
simp_rw [continuous_iff, powerset, preimage_ofPred_eq, union_subset_iff, ofPred_and]
  exact ⟨
    fun U hU => .inter
      (hU.powerset_vietoris.preimage continuous_fst)
      (hU.powerset_vietoris.preimage continuous_snd),
    fun F hF => .inter
      (hF.powerset_vietoris.preimage continuous_fst)
      (hF.powerset_vietoris.preimage continuous_snd)⟩

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

Goal state before:
pp: 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✝¹ : 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 := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
U : SetRel α α
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage (Prod.swap ⁻¹' V)
t : Set α
ht₁ : t.Finite
ht₂ : ↑K ⊆ V.preimage t
K' : Compacts α
left✝ : (↑K, ↑K').1 ⊆ SetRel.preimage (Prod.swap ⁻¹' V) (↑K, ↑K').2
hK' : ↑K' ⊆ V.preimage ↑K
⊢ ↑K' ⊆ V.preimage (V.preimage t)

Tactic:
← ht₂,

Goal state after:
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✝¹ : 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 := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
U : SetRel α α
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage (Prod.swap ⁻¹' V)
t : Set α
ht₁ : t.Finite
ht₂ : ↑K ⊆ V.preimage t
K' : Compacts α
left✝ : (↑K, ↑K').1 ⊆ SetRel.preimage (Prod.swap ⁻¹' V) (↑K, ↑K').2
hK' : ↑K' ⊆ V.preimage ↑K
⊢ ↑K' ⊆ V.preimage ↑K
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: 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 := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
hl : l.TotallyBounded
L : Compacts α := { carrier := {x | ClusterPt x l}, isCompact' := ⋯ }
⊢ f ≤ Filter.comap (fun y ↦ (y, L)) (𝓤 (Compacts α))

Tactic:
uniformity_hasBasis_closed.uniformity_compacts.comap _ |>.ge_iff

Goal state after:
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 := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
hl : l.TotallyBounded
L : Compacts α := { carrier := {x | ClusterPt x l}, isCompact' := ⋯ }
⊢ ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsClosed[instTopologicalSpaceProd] i' →
      (fun y ↦ (y, L)) ⁻¹' Prod.map SetLike.coe SetLike.coe ⁻¹' hausdorffEntourage (id i') ∈ f
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: 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 := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
hl : l.TotallyBounded
L : Compacts α := { carrier := {x | ClusterPt x l}, isCompact' := ⋯ }
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
⊢ Prod.map SetLike.coe SetLike.coe (K, L) ∈ hausdorffEntourage (id U)

Tactic:
Prod.map,

Goal state after:
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 := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
hl : l.TotallyBounded
L : Compacts α := { carrier := {x | ClusterPt x l}, isCompact' := ⋯ }
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, ↑L) ∈ hausdorffEntourage (id U)
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
⊢ p₁ ↑0

Tactic:
Unitization.inr_zero 𝕜

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
⊢ p₁ 0
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
ha : IsSelfAdjoint a
x : ℂ
hx : x ∈ σ ℂ a
hf_real : ∀ x ∈ σ ℂ a, ↑(f x).re = f x
⊢ f x = ↑(f ↑x.re).re

Tactic:
RCLike.ofReal_eq_complex_ofReal

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
ha : IsSelfAdjoint a
x : ℂ
hx : x ∈ σ ℂ a
hf_real : ∀ x ∈ σ ℂ a, ↑(f x).re = f x
⊢ f x = ↑(f ↑x.re).re
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : T2Space A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
f : ℂ → ℂ
a : A
ha : IsSelfAdjoint a
x : ℂ
hx : x ∈ σₙ ℂ a
hf_real : ∀ x ∈ σₙ ℂ a, ↑(f x).re = f x
⊢ f x = ↑(f ↑x.re).re

Tactic:
RCLike.ofReal_eq_complex_ofReal

Goal state after:
A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : T2Space A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
f : ℂ → ℂ
a : A
ha : IsSelfAdjoint a
x : ℂ
hx : x ∈ σₙ ℂ a
hf_real : ∀ x ∈ σₙ ℂ a, ↑(f x).re = f x
⊢ f x = ↑(f ↑x.re).re
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

Goal state before:
pp: case e_u
A : Type u_2
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
r_pos : 0 < r
r_lt : ↑r < (spectralRadius ℂ a)⁻¹
p : FormalMultilinearSeries ℂ ℂ A := fun n ↦ ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)
h : ↑r ≤ p.radius
n : ℕ
⊢ 1 / ↑‖a ^ n‖.toNNReal ^ (1 / ↑n) = 1 / ↑(‖a ^ n‖.toNNReal ^ (1 / ↑n))

Tactic:
norm_toNNReal,

Goal state after:
case e_u
A : Type u_2
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
r_pos : 0 < r
r_lt : ↑r < (spectralRadius ℂ a)⁻¹
p : FormalMultilinearSeries ℂ ℂ A := fun n ↦ ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)
h : ↑r ≤ p.radius
n : ℕ
⊢ 1 / ↑‖a ^ n‖₊ ^ (1 / ↑n) = 1 / ↑(‖a ^ n‖₊ ^ (1 / ↑n))
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

Goal state before:
pp: A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
⊢ resolventSet ℂ a = Set.univ

Tactic:
spectrum,

Goal state after:
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : (resolventSet ℂ a)ᶜ = ∅
⊢ resolventSet ℂ a = Set.univ
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
x✝¹ : E
x✝ : x✝¹ ∈ s
⊢ ‖(B x✝¹) 0‖ ≤ 1

Tactic:
by
  simp only [map_zero, norm_zero, zero_le_one]

Goal state after:
Module: Mathlib.Analysis.Convex.TotallyBounded

Goal state before:
pp: E : Type u_1
s : Set E
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : UniformSpace E
inst✝² : IsUniformAddGroup E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : ContinuousSMul ℝ E
hs : ∀ U ∈ nhds 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U
U : Set E
hU : U ∈ nhds 0
⊢ ∃ t, t.Finite ∧ (convexHull ℝ) s ⊆ ⋃ y ∈ t, y +ᵥ U

Tactic:
obtain ⟨W, hW₁, hW₂⟩ := exists_nhds_zero_half hU

Goal state after:
E : Type u_1
s : Set E
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : UniformSpace E
inst✝² : IsUniformAddGroup E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : ContinuousSMul ℝ E
hs : ∀ U ∈ nhds 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U
U : Set E
hU : U ∈ nhds 0
W : Set E
hW₁ : W ∈ nhds 0
hW₂ : ∀ v ∈ W, ∀ w ∈ W, v + w ∈ U
⊢ ∃ t, t.Finite ∧ (convexHull ℝ) s ⊆ ⋃ y ∈ t, y +ᵥ U
Module: Mathlib.Analysis.Convex.TotallyBounded

Goal state before:
pp: case right
E : Type u_1
s : Set E
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : UniformSpace E
inst✝² : IsUniformAddGroup E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : ContinuousSMul ℝ E
hs : ∀ U ∈ nhds 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U
U : Set E
hU : U ∈ nhds 0
W : Set E
hW₁ : W ∈ nhds 0
hW₂ : ∀ v ∈ W, ∀ w ∈ W, v + w ∈ U
V : Set E
hV₁ : V ∈ nhds 0
hV₂ : Convex ℝ V
hV₃ : V ⊆ W
t : Set E
htf : t.Finite
t' : Set E
htf' : t'.Finite
hts : s ⊆ t + V
hts' : (convexHull ℝ) t ⊆ t' + V
⊢ t' + (W + W) ⊆ t' + U

Tactic:
add_subset_iff.mpr hW₂

Goal state after:
case right
E : Type u_1
s : Set E
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : UniformSpace E
inst✝² : IsUniformAddGroup E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : ContinuousSMul ℝ E
hs : ∀ U ∈ nhds 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U
U : Set E
hU : U ∈ nhds 0
W : Set E
hW₁ : W ∈ nhds 0
hW₂ : ∀ v ∈ W, ∀ w ∈ W, v + w ∈ U
V : Set E
hV₁ : V ∈ nhds 0
hV₂ : Convex ℝ V
hV₃ : V ⊆ W
t : Set E
htf : t.Finite
t' : Set E
htf' : t'.Finite
hts : s ⊆ t + V
hts' : (convexHull ℝ) t ⊆ t' + V
⊢ t' + U ⊆ t' + U
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: case e'_3
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
x : ℝ
u : ↥(unitary A) := ⟨NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a), ⋯⟩
⊢ NormedSpace.exp (x • star a) * NormedSpace.exp (-(x • a)) *
      (NormedSpace.exp (x • a) * NormedSpace.exp (-(x • star a))) =
    1

Tactic:
let _ := invertibleExp (𝔸 := A)

Goal state after:
case e'_3
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝¹ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
x : ℝ
u : ↥(unitary A) := ⟨NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a), ⋯⟩
x✝ : (x : A) → Invertible (NormedSpace.exp x) := invertibleExp
⊢ NormedSpace.exp (x • star a) * NormedSpace.exp (-(x • a)) *
      (NormedSpace.exp (x • a) * NormedSpace.exp (-(x • star a))) =
    1
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
⊢ closure {K | (↑K).Finite ∧ ↑K ⊆ s} = {K | ↑K ⊆ closure[inst✝] s}

Tactic:
simpa [isOpenEmbedding_toCompacts.isOpenMap.preimage_closure_eq_closure_preimage
    continuous_toCompacts] using congr(toCompacts ⁻¹' $(Compacts.closure_finite_subsets s))

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : PartialOrder 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : LocallyConvexSpace 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ZeroLEOneClass 𝕜
inst✝ : FirstCountableTopology E
u : ℕ → Set E
hu_basis : (𝓝 0).HasAntitoneBasis u
hu_zero : ∀ (n : ℕ), 0 ∈ interior (u n)
v : ℕ → Set E := fun n ↦ (absConvexHull 𝕜) (interior (u n))
hv_open : ∀ (n : ℕ), IsOpen[inst✝⁵] (v n)
hv_nhds : ∀ (n : ℕ), v n ∈ 𝓝 0
hv_basis : (𝓝 0).HasAntitoneBasis v
⊢ ∃ x,
    (𝓝 0).HasAntitoneBasis x ∧
      ∀ (n : ℕ),
        IsOpen[inst✝⁵] (x n) ∧ AbsConvex 𝕜 (x n) ∧ x (n + 1) + x (n + 1) ⊆ x n ∧ closure[inst✝⁵] (x (n + 1)) ⊆ x n

Tactic:
obtain ⟨φ, -, hφ_add, hφ_basis⟩ := hv_basis.subbasis_with_rel
    (r := fun i j ↦ v j + v j ⊆ v i) fun m ↦ by
      obtain ⟨W, hW_open, hW_zero, hW_add⟩ := exists_open_nhds_zero_add_subset (hv_nhds m)
      obtain ⟨N, hN⟩ := hv_basis.mem_iff.mp (hW_open.mem_nhds hW_zero)
      filter_upwards [Filter.eventually_ge_atTop N] with M hM
      exact (add_subset_add ((hv_basis.antitone hM).trans hN)
        ((hv_basis.antitone hM).trans hN)).trans hW_add

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : PartialOrder 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : LocallyConvexSpace 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ZeroLEOneClass 𝕜
inst✝ : FirstCountableTopology E
u : ℕ → Set E
hu_basis : (𝓝 0).HasAntitoneBasis u
hu_zero : ∀ (n : ℕ), 0 ∈ interior (u n)
v : ℕ → Set E := fun n ↦ (absConvexHull 𝕜) (interior (u n))
hv_open : ∀ (n : ℕ), IsOpen[inst✝⁵] (v n)
hv_nhds : ∀ (n : ℕ), v n ∈ 𝓝 0
hv_basis : (𝓝 0).HasAntitoneBasis v
φ : ℕ → ℕ
hφ_add : ∀ ⦃m n : ℕ⦄, m < n → v (φ n) + v (φ n) ⊆ v (φ m)
hφ_basis : (𝓝 0).HasAntitoneBasis (v ∘ φ)
⊢ ∃ x,
    (𝓝 0).HasAntitoneBasis x ∧
      ∀ (n : ℕ),
        IsOpen[inst✝⁵] (x n) ∧ AbsConvex 𝕜 (x n) ∧ x (n + 1) + x (n + 1) ⊆ x n ∧ closure[inst✝⁵] (x (n + 1)) ⊆ x n
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
⊢ IsPreconnected {K | (↑K).Finite ∧ ↑K ⊆ s}

Tactic:
by
  rw [← isEmbedding_toCompacts.isPreconnected_image]
  convert Compacts.isPreconnected_nonempty_finite_subsets hs
  exact subset_antisymm
    (image_subset_iff.mpr fun K hK => ⟨K.nonempty, hK⟩)
    (fun K hK => ⟨⟨K, hK.1⟩, hK.2, rfl⟩)

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Dual

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : StrongDual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : ¬c = 0
eq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖
le : ∀ z ∈ s, ‖c⁻¹ • x' z‖ ≤ ‖c⁻¹‖ * ‖c‖
⊢ ‖c⁻¹‖ * ‖c‖ = 1

Tactic:
simp only [c_zero, norm_eq_zero, Ne, not_false_iff, inv_mul_cancel₀, norm_inv]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Completeness

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f₀ : E' →SL[σ₁₂] F
r : ℝ
f : E' →SL[σ₁₂] F
hf : ⇑f ∈ closure[Pi.topologicalSpace] (DFunLike.coe '' closedBall f₀ r)
⊢ f ∈ closedBall f₀ r

Tactic:
by
  have hr : 0 ≤ r := nonempty_closedBall.1 (closure_nonempty_iff.1 ⟨_, hf⟩).of_image
  refine mem_closedBall_iff_norm.2 (opNorm_le_bound _ hr fun x => ?_)
  have : IsClosed { g : E' → F | ‖g x - f₀ x‖ ≤ r * ‖x‖ } :=
    isClosed_Iic.preimage ((@continuous_apply E' (fun _ => F) _ x).sub continuous_const).norm
  refine this.closure_subset_iff.2 (image_subset_iff.2 fun g hg => ?_) hf
  exact (g - f₀).le_of_opNorm_le (mem_closedBall_iff_norm.1 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 : x ∈ interior {y | p y ≤ ↑n}
hxn' : p x ≤ ↑n
⊢ {y | p y ≤ ↑n + ↑n} ∈ 𝓝 0

Tactic:
mem_interior_iff_mem_nhds,

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} ∈ 𝓝 x
hxn' : p x ≤ ↑n
⊢ {y | p y ≤ ↑n + ↑n} ∈ 𝓝 0
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor V P
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
p₁ p₂ : P
c : 𝕜
⊢ dist ((homothety p₁ c) p₂) p₂ = ‖1 - c‖ * dist p₁ p₂

Tactic:
by
  rw [homothety_eq_lineMap, dist_lineMap_right]

Goal state after:
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
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
x : X
hx : x ∈ mᶜ
B : ∀ (n : ℕ), (f n) x = 0
⊢ g x = 0 x

Tactic:
simp [g, B]

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
hgmc : EqOn g 0 mᶜ
I : ∀ (n : ℕ) (x : X), u n * (f n) x ≤ u n
S : ∀ (x : X), Summable fun n ↦ u n * (f n) x
⊢ HasCompactSupport ⇑{ toFun := g, continuous_toFun := ⋯ }

Tactic:
exact HasCompactSupport.of_support_subset_isCompact m_comp
      (Function.support_subset_iff'.mpr hgmc)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ -a
x✝ : IsSelfAdjoint a
x : ℝ
hx : x ∈ quasispectrum ℝ a
⊢ id x = -x⁻

Tactic:
rw [Unitization.quasispectrum_eq_spectrum_inr ℝ, ← neg_neg x, ← Set.mem_neg,
    spectrum.neg_eq, ← Unitization.inr_neg, ← Unitization.quasispectrum_eq_spectrum_inr ℝ] at hx

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ -a
x✝ : IsSelfAdjoint a
x : ℝ
hx : -x ∈ quasispectrum ℝ (-a)
⊢ id x = -x⁻
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
x : ℝ≥0
⊢ a ^ (1 / 2 * x) = a ^ (x / 2)

Tactic:
one_div_mul_eq_div 2 x

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
x : ℝ≥0
⊢ a ^ (x / 2) = a ^ (x / 2)