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 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] :

The set (star s : Set α) is defined as {x | star x ∈ s} in the scope 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 univ = univ

@[simp]

theorem Set.nonempty_star {α : Type u_1} [InvolutiveStar α] {s : Set α} :

(star s).Nonempty ↔ s.Nonempty

theorem Set.Nonempty.star {α : Type u_1} [InvolutiveStar α] {s : Set α} (h : s.Nonempty) :

(Star.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} [InvolutiveStar 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} [InvolutiveStar α] :

star '' s = star s

@[simp]

theorem Set.inter_star {α : Type u_1} {s t : Set α} [Star α] :

star (s ∩ t) = star s ∩ star t

@[simp]

theorem Set.union_star {α : Type u_1} {s 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 : ι → Set 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 : ι → Set 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 sᶜ = (star s)ᶜ

instance Set.instInvolutiveStar {α : Type u_1} [InvolutiveStar α] :

InvolutiveStar (Set α)

Equations

@[simp]

theorem Set.star_subset_star {α : Type u_1} [InvolutiveStar α] {s t : Set α} :

star s ⊆ star t ↔ s ⊆ t

theorem Set.star_subset {α : Type u_1} [InvolutiveStar α] {s t : Set α} :

star s ⊆ t ↔ s ⊆ star t

theorem Set.Finite.star {α : Type u_1} [InvolutiveStar α] {s : Set α} (hs : s.Finite) :

(Star.star s).Finite

theorem Set.star_singleton {β : Type u_2} [InvolutiveStar β] (x : β) :

star {x} = {star x}

theorem Set.star_mul {α : Type u_1} [Mul α] [StarMul α] (s t : Set α) :

star (s * t) = star t * star s

theorem Set.star_add {α : Type u_1} [AddMonoid α] [StarAddMonoid α] (s t : Set α) :

star (s + t) = star s + star t

@[simp]

instance Set.instTrivialStar {α : Type u_1} [Star α] [TrivialStar α] :

TrivialStar (Set α)

theorem Set.star_inv {α : Type u_1} [Group α] [StarMul α] (s : Set α) :

star s⁻¹ = (star s)⁻¹

theorem Set.star_inv' {α : Type u_1} [GroupWithZero α] [StarMul α] (s : Set α) :

star s⁻¹ = (star s)⁻¹

@[simp]

theorem StarMemClass.star_coe_eq {S : Type u_1} {α : Type u_2} [InvolutiveStar α] [SetLike S α] [StarMemClass S α] (s : S) :

star ↑s = ↑s