# Documentation

Mathlib.Algebra.Star.Pointwise

# 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⋆∩ t)⋆ = s⋆ ∩ t⋆⋆ = s⋆ ∩ t⋆⋆ ∩ t⋆∩ t⋆⋆), if s t : Set α, then under suitable assumption on α, it is shown

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

The set (star s : Set α) is defined as {x | star x ∈ s}∈ s} in locale pointwise. In the usual case where star is involutive, it is equal to {star s | x ∈ s}∈ s}, see Set.image_star.

Equations
@[simp]
theorem Set.star_empty {α : Type u_1} [inst : Star α] :
@[simp]
theorem Set.star_univ {α : Type u_1} [inst : Star α] :
star Set.univ = Set.univ
@[simp]
theorem Set.nonempty_star {α : Type u_1} [inst : ] {s : Set α} :
theorem Set.Nonempty.star {α : Type u_1} [inst : ] {s : Set α} (h : ) :
@[simp]
theorem Set.mem_star {α : Type u_1} {s : Set α} {a : α} [inst : Star α] :
a star s star a s
theorem Set.star_mem_star {α : Type u_1} {s : Set α} {a : α} [inst : ] :
star a star s a s
@[simp]
theorem Set.star_preimage {α : Type u_1} {s : Set α} [inst : Star α] :
star ⁻¹' s = star s
@[simp]
theorem Set.image_star {α : Type u_1} {s : Set α} [inst : ] :
star '' s = star s
@[simp]
theorem Set.inter_star {α : Type u_1} {s : Set α} {t : Set α} [inst : Star α] :
star (s t) = star s star t
@[simp]
theorem Set.union_star {α : Type u_1} {s : Set α} {t : Set α} [inst : Star α] :
star (s t) = star s star t
@[simp]
theorem Set.interᵢ_star {α : Type u_2} {ι : Sort u_1} [inst : Star α] (s : ιSet α) :
star (Set.interᵢ fun i => s i) = Set.interᵢ fun i => star (s i)
@[simp]
theorem Set.unionᵢ_star {α : Type u_2} {ι : Sort u_1} [inst : Star α] (s : ιSet α) :
star (Set.unionᵢ fun i => s i) = Set.unionᵢ fun i => star (s i)
@[simp]
theorem Set.compl_star {α : Type u_1} {s : Set α} [inst : Star α] :
noncomputable instance Set.instInvolutiveStarSet {α : Type u_1} [inst : ] :
Equations
@[simp]
theorem Set.star_subset_star {α : Type u_1} [inst : ] {s : Set α} {t : Set α} :
star s star t s t
theorem Set.star_subset {α : Type u_1} [inst : ] {s : Set α} {t : Set α} :
star s t s star t
theorem Set.Finite.star {α : Type u_1} [inst : ] {s : Set α} (hs : ) :
theorem Set.star_singleton {β : Type u_1} [inst : ] (x : β) :
star {x} = {star x}
theorem Set.star_mul {α : Type u_1} [inst : ] [inst : ] (s : Set α) (t : Set α) :
star (s * t) = star t * star s
theorem Set.star_add {α : Type u_1} [inst : ] [inst : ] (s : Set α) (t : Set α) :
star (s + t) = star s + star t
@[simp]
instance Set.instTrivialStarSetStar {α : Type u_1} [inst : Star α] [inst : ] :
Equations
theorem Set.star_inv {α : Type u_1} [inst : ] [inst : ] (s : Set α) :
theorem Set.star_inv' {α : Type u_1} [inst : ] [inst : ] (s : Set α) :