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 : α → Set β) (x : α) : Set β

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

Equations

• Set.Accumulate s x = ⋃ (y : α), ⋃ (_ : y ≤ x), s y

theorem Set.accumulate_def {α : Type u_1} {β : Type u_2} {s : α → Set β} [LE α] {x : α} : Accumulate s x = ⋃ (y : α), ⋃ (_ : y ≤ x), s y

@[simp]

theorem Set.mem_accumulate {α : Type u_1} {β : Type u_2} {s : α → Set β} [LE α] {x : α} {z : β} : z ∈ Accumulate s x ↔ ∃ y ≤ x, z ∈ s y

theorem Set.subset_accumulate {α : Type u_1} {β : Type u_2} {s : α → Set β} [Preorder α] {x : α} : s x ⊆ Accumulate s x

theorem Set.accumulate_subset_iUnion {α : Type u_1} {β : Type u_2} {s : α → Set β} [LE α] (x : α) : Accumulate s x ⊆ ⋃ (i : α), s i

theorem Set.monotone_accumulate {α : Type u_1} {β : Type u_2} {s : α → Set β} [Preorder α] : Monotone (Accumulate s)

theorem Set.accumulate_subset_accumulate {α : Type u_1} {β : Type u_2} {s : α → Set β} [Preorder α] {x y : α} (h : x ≤ y) : Accumulate s x ⊆ Accumulate s y

theorem Set.biUnion_accumulate {α : Type u_1} {β : Type u_2} {s : α → Set β} [Preorder α] (x : α) : ⋃ (y : α), ⋃ (_ : y ≤ x), Accumulate s y = ⋃ (y : α), ⋃ (_ : y ≤ x), s y

theorem Set.iUnion_accumulate {α : Type u_1} {β : Type u_2} {s : α → Set β} [Preorder α] : ⋃ (x : α), Accumulate s x = ⋃ (x : α), s x

@[simp]

theorem Set.accumulate_bot {α : Type u_1} {β : Type u_2} [PartialOrder α] [OrderBot α] (s : α → Set β) : Accumulate s ⊥ = s ⊥

@[simp]

theorem Set.accumulate_zero_nat {β : Type u_2} (s : ℕ → Set β) : Accumulate s 0 = s 0

theorem Set.disjoint_accumulate {α : Type u_1} {β : Type u_2} {s : α → Set β} [Preorder α] (hs : Pairwise (Function.onFun Disjoint s)) {i j : α} (hij : i < j) : Disjoint (Accumulate s i) (s j)