# Pointwise star operation on sets #

This file defines the star operation pointwise on sets and provides the basic API. Besides basic facts about about how the star operation acts on sets (e.g., (s ∩ t)⋆ = s⋆ ∩ t⋆), if s t : Set α, then under suitable assumption on α, it is shown

• (s + t)⋆ = s⋆ + t⋆
• (s * t)⋆ = t⋆ + s⋆
• (s⁻¹)⋆ = (s⋆)⁻¹
def Set.star {α : Type u_1} [Star α] :
Star (Set α)

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.

Equations
Instances For
@[simp]
theorem Set.star_empty {α : Type u_1} [Star α] :
@[simp]
theorem Set.star_univ {α : Type u_1} [Star α] :
star Set.univ = Set.univ
@[simp]
theorem Set.nonempty_star {α : Type u_1} [] {s : Set α} :
(star s).Nonempty s.Nonempty
theorem Set.Nonempty.star {α : Type u_1} [] {s : Set α} (h : s.Nonempty) :
(star s).Nonempty
@[simp]
theorem Set.mem_star {α : Type u_1} {s : Set α} {a : α} [Star α] :
a star s star a s
theorem Set.star_mem_star {α : Type u_1} {s : Set α} {a : α} [] :
star a star s a s
@[simp]
theorem Set.star_preimage {α : Type u_1} {s : Set α} [Star α] :
star ⁻¹' s = star s
@[simp]
theorem Set.image_star {α : Type u_1} {s : Set α} [] :
star '' s = star s
@[simp]
theorem Set.inter_star {α : Type u_1} {s : Set α} {t : Set α} [Star α] :
star (s t) = star s star t
@[simp]
theorem Set.union_star {α : Type u_1} {s : Set α} {t : Set α} [Star α] :
star (s t) = star s star t
@[simp]
theorem Set.iInter_star {α : Type u_1} {ι : Sort u_2} [Star α] (s : ιSet α) :
star (⋂ (i : ι), s i) = ⋂ (i : ι), star (s i)
@[simp]
theorem Set.iUnion_star {α : Type u_1} {ι : Sort u_2} [Star α] (s : ιSet α) :
star (⋃ (i : ι), s i) = ⋃ (i : ι), star (s i)
@[simp]
theorem Set.compl_star {α : Type u_1} {s : Set α} [Star α] :
= (star s)
instance Set.instInvolutiveStar {α : Type u_1} [] :
Equations
• Set.instInvolutiveStar =
@[simp]
theorem Set.star_subset_star {α : Type u_1} [] {s : Set α} {t : Set α} :
star s star t s t
theorem Set.star_subset {α : Type u_1} [] {s : Set α} {t : Set α} :
star s t s star t
theorem Set.Finite.star {α : Type u_1} [] {s : Set α} (hs : s.Finite) :
(star s).Finite
theorem Set.star_singleton {β : Type u_2} [] (x : β) :
star {x} = {star x}
theorem Set.star_mul {α : Type u_1} [Mul α] [] (s : Set α) (t : Set α) :
star (s * t) = star t * star s
theorem Set.star_add {α : Type u_1} [] [] (s : Set α) (t : Set α) :
star (s + t) = star s + star t
@[simp]
instance Set.instTrivialStar {α : Type u_1} [Star α] [] :
Equations
• =
theorem Set.star_inv {α : Type u_1} [] [] (s : Set α) :
theorem Set.star_inv' {α : Type u_1} [] [] (s : Set α) :
@[simp]
theorem StarMemClass.star_coe_eq {S : Type u_1} {α : Type u_2} [] [SetLike S α] [] (s : S) :
star s = s