# Documentation

Mathlib.Order.Disjointed

# Consecutive differences of sets #

This file defines the way to make a sequence of elements into a sequence of disjoint elements with the same partial sups.

For a sequence f : ℕ → α, this new sequence will be f 0, f 1 \ f 0, f 2 \ (f 0 ⊔ f 1). It is actually unique, as disjointed_unique shows.

## Main declarations #

• disjointed f: The sequence f 0, f 1 \ f 0, f 2 \ (f 0 ⊔ f 1), ....
• partialSups_disjointed: disjointed f has the same partial sups as f.
• disjoint_disjointed: The elements of disjointed f are pairwise disjoint.
• disjointed_unique: disjointed f is the only pairwise disjoint sequence having the same partial sups as f.
• iSup_disjointed: disjointed f has the same supremum as f. Limiting case of partialSups_disjointed.

We also provide set notation variants of some lemmas.

## TODO #

Find a useful statement of disjointedRec_succ.

One could generalize disjointed to any locally finite bot preorder domain, in place of ℕ. Related to the TODO in the module docstring of Mathlib.Order.PartialSups.

def disjointed {α : Type u_1} (f : α) :
α

If f : ℕ → α is a sequence of elements, then disjointed f is the sequence formed by subtracting each element from the nexts. This is the unique disjoint sequence whose partial sups are the same as the original sequence.

Instances For
@[simp]
theorem disjointed_zero {α : Type u_1} (f : α) :
= f 0
theorem disjointed_succ {α : Type u_1} (f : α) (n : ) :
disjointed f (n + 1) = f (n + 1) \ ↑() n
theorem disjointed_le_id {α : Type u_1} :
disjointed id
theorem disjointed_le {α : Type u_1} (f : α) :
f
theorem disjoint_disjointed {α : Type u_1} (f : α) :
Pairwise (Disjoint on )
def disjointedRec {α : Type u_1} {f : α} {p : αSort u_3} (hdiff : t : α⦄ → i : ⦄ → p tp (t \ f i)) ⦃n : :
p (f n)p ()

An induction principle for disjointed. To define/prove something on disjointed f n, it's enough to define/prove it for f n and being able to extend through diffs.

Instances For
@[simp]
theorem disjointedRec_zero {α : Type u_1} {f : α} {p : αSort u_3} (hdiff : t : α⦄ → i : ⦄ → p tp (t \ f i)) (h₀ : p (f 0)) :
disjointedRec hdiff h₀ = h₀
theorem Monotone.disjointed_eq {α : Type u_1} {f : α} (hf : ) (n : ) :
disjointed f (n + 1) = f (n + 1) \ f n
@[simp]
theorem partialSups_disjointed {α : Type u_1} (f : α) :
theorem disjointed_unique {α : Type u_1} {f : α} {d : α} (hdisj : Pairwise (Disjoint on d)) (hsups : ) :
d =

disjointed f is the unique sequence that is pairwise disjoint and has the same partial sups as f.

theorem iSup_disjointed {α : Type u_1} (f : α) :
⨆ (n : ), = ⨆ (n : ), f n
theorem disjointed_eq_inf_compl {α : Type u_1} (f : α) (n : ) :
= f n ⨅ (i : ) (_ : i < n), (f i)

### Set notation variants of lemmas #

theorem disjointed_subset {α : Type u_1} (f : Set α) (n : ) :
f n
theorem iUnion_disjointed {α : Type u_1} {f : Set α} :
⋃ (n : ), = ⋃ (n : ), f n
theorem disjointed_eq_inter_compl {α : Type u_1} (f : Set α) (n : ) :
= f n ⋂ (i : ) (_ : i < n), (f i)
theorem preimage_find_eq_disjointed {α : Type u_1} (s : Set α) (H : ∀ (x : α), n, x s n) [(x : α) → (n : ) → Decidable (x s n)] (n : ) :
(fun x => Nat.find (H x)) ⁻¹' {n} =