File: Mathlib/Algebra/Star/Pointwise.lean

full_name: Set._aux_Mathlib_Algebra_Star_Pointwise___unexpand_Star_star_1
code: local postfix:max "⋆" => star
goal: Lean.PrettyPrinter.Unexpander

full_name: Set.«_aux_Mathlib_Algebra_Star_Pointwise___macroRules_Set_term_⋆_1»
code: local postfix:max "⋆" => star
goal: Lean.Macro

full_name: Set.star
code: /-- The set `(star s : Set α)` is defined as `{x | star x ∈ s}` in the locale `Pointwise`.
In the usual case where `star` is involutive, it is equal to `{star s | x ∈ s}`, see
`Set.image_star`. -/
protected def star [Star α] : Star (Set α) 
goal: Star (Set α)

full_name: Set.star_empty
code: @[simp]
theorem star_empty [Star α] : (∅ : Set α)⋆ = ∅ 
goal: star ∅ = ∅

full_name: Set.star_univ
code: @[simp]
theorem star_univ [Star α] : (univ : Set α)⋆ = univ 
goal: star Set.univ = Set.univ

full_name: Set.nonempty_star
code: @[simp]
theorem nonempty_star [InvolutiveStar α] {s : Set α} : s⋆.Nonempty ↔ s.Nonempty 
goal: (star s).Nonempty ↔ s.Nonempty

full_name: Set.Nonempty.star
code: theorem Nonempty.star [InvolutiveStar α] {s : Set α} (h : s.Nonempty) : s⋆.Nonempty 
goal: (star s).Nonempty

full_name: Set.mem_star
code: @[simp]
theorem mem_star [Star α] : a ∈ s⋆ ↔ a⋆ ∈ s 
goal: a ∈ star s ↔ star a ∈ s

full_name: Set.star_mem_star
code: theorem star_mem_star [InvolutiveStar α] : a⋆ ∈ s⋆ ↔ a ∈ s 
goal: star a ∈ star s ↔ a ∈ s

full_name: Set.star_preimage
code: @[simp]
theorem star_preimage [Star α] : Star.star ⁻¹' s = s⋆ 
goal: star ⁻¹' s = star s

full_name: Set.image_star
code: @[simp]
theorem image_star [InvolutiveStar α] : Star.star '' s = s⋆ 
goal: star '' s = star s

full_name: Set.inter_star
code: @[simp]
theorem inter_star [Star α] : (s ∩ t)⋆ = s⋆ ∩ t⋆ 
goal: star (s ∩ t) = star s ∩ star t

full_name: Set.union_star
code: @[simp]
theorem union_star [Star α] : (s ∪ t)⋆ = s⋆ ∪ t⋆ 
goal: star (s ∪ t) = star s ∪ star t

full_name: Set.iInter_star
code: @[simp]
theorem iInter_star {ι : Sort*} [Star α] (s : ι → Set α) : (⋂ i, s i)⋆ = ⋂ i, (s i)⋆ 
goal: star (⋂ i, s i) = ⋂ i, star (s i)

full_name: Set.iUnion_star
code: @[simp]
theorem iUnion_star {ι : Sort*} [Star α] (s : ι → Set α) : (⋃ i, s i)⋆ = ⋃ i, (s i)⋆ 
goal: star (⋃ i, s i) = ⋃ i, star (s i)

full_name: Set.compl_star
code: @[simp]
theorem compl_star [Star α] : sᶜ⋆ = s⋆ᶜ 
goal: star sᶜ = (star s)ᶜ

full_name: Set.instInvolutiveStar
code: @[simp]
instance [InvolutiveStar α] : InvolutiveStar (Set α) 
goal: InvolutiveStar (Set α)

full_name: Set.star_subset_star
code: @[simp]
theorem star_subset_star [InvolutiveStar α] {s t : Set α} : s⋆ ⊆ t⋆ ↔ s ⊆ t 
goal: star s ⊆ star t ↔ s ⊆ t

full_name: Set.star_subset
code: theorem star_subset [InvolutiveStar α] {s t : Set α} : s⋆ ⊆ t ↔ s ⊆ t⋆ 
goal: star s ⊆ t ↔ s ⊆ star t

full_name: Set.Finite.star
code: theorem Finite.star [InvolutiveStar α] {s : Set α} (hs : s.Finite) : s⋆.Finite 
goal: (star s).Finite

full_name: Set.star_singleton
code: theorem star_singleton {β : Type*} [InvolutiveStar β] (x : β) : ({x} : Set β)⋆ = {x⋆} 
goal: star {x} = {star x}

full_name: Set.star_mul
code: protected theorem star_mul [Mul α] [StarMul α] (s t : Set α) : (s * t)⋆ = t⋆ * s⋆ 
goal: star (s * t) = star t * star s

full_name: Set.star_add
code: protected theorem star_add [AddMonoid α] [StarAddMonoid α] (s t : Set α) : (s + t)⋆ = s⋆ + t⋆ 
goal: star (s + t) = star s + star t

full_name: Set.instTrivialStar
code: @[simp]
instance [Star α] [TrivialStar α] : TrivialStar (Set α) 
goal: TrivialStar (Set α)

full_name: Set.star_inv
code: protected theorem star_inv [Group α] [StarMul α] (s : Set α) : s⁻¹⋆ = s⋆⁻¹ 
goal: star s⁻¹ = (star s)⁻¹

full_name: Set.star_inv'
code: protected theorem star_inv' [DivisionSemiring α] [StarRing α] (s : Set α) : s⁻¹⋆ = s⋆⁻¹ 
goal: star s⁻¹ = (star s)⁻¹

full_name: StarMemClass.star_coe_eq
code: @[simp]
lemma StarMemClass.star_coe_eq {S α : Type*} [InvolutiveStar α] [SetLike S α]
    [StarMemClass S α] (s : S) : star (s : Set α) = s 
goal: star ↑s = ↑s