Documentation

Mathlib.Data.Set.Accumulate

Accumulate #

The function Accumulate takes a set s and returns ⋃ y ≤ x, s y.

def Set.Accumulate {α : Type u_1} {β : Type u_2} [LE α] (s : αSet β) (x : α) :
Set β

Accumulate s is the union of s y for y ≤ x.

Equations
Instances For
    theorem Set.accumulate_def {α : Type u_1} {β : Type u_2} {s : αSet β} [LE α] {x : α} :
    Set.Accumulate s x = ⋃ (y : α), ⋃ (_ : y x), s y
    @[simp]
    theorem Set.mem_accumulate {α : Type u_1} {β : Type u_2} {s : αSet β} [LE α] {x : α} {z : β} :
    z Set.Accumulate s x yx, z s y
    theorem Set.subset_accumulate {α : Type u_1} {β : Type u_2} {s : αSet β} [Preorder α] {x : α} :
    theorem Set.accumulate_subset_iUnion {α : Type u_1} {β : Type u_2} {s : αSet β} [Preorder α] (x : α) :
    Set.Accumulate s x ⋃ (i : α), s i
    theorem Set.monotone_accumulate {α : Type u_1} {β : Type u_2} {s : αSet β} [Preorder α] :
    theorem Set.accumulate_subset_accumulate {α : Type u_1} {β : Type u_2} {s : αSet β} [Preorder α] {x : α} {y : α} (h : x y) :
    theorem Set.biUnion_accumulate {α : Type u_1} {β : Type u_2} {s : αSet β} [Preorder α] (x : α) :
    ⋃ (y : α), ⋃ (_ : y x), Set.Accumulate s y = ⋃ (y : α), ⋃ (_ : y x), s y
    theorem Set.iUnion_accumulate {α : Type u_1} {β : Type u_2} {s : αSet β} [Preorder α] :
    ⋃ (x : α), Set.Accumulate s x = ⋃ (x : α), s x