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→ α, this new sequence will be f 0, f 1 \ f 0, f 2 \ (f 0 ⊔ f 1)⊔ 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)⊔ 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.
supᵢ_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} [inst : GeneralizedBooleanAlgebra α] (f : ℕ → α) :

ℕ → α

If f : ℕ → α→ 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.

Equations

disjointed f x = match x with | 0 => f 0 | Nat.succ n => f (n + 1) \ ↑(partialSups f) n

@[simp]

theorem disjointed_zero {α : Type u_1} [inst : GeneralizedBooleanAlgebra α] (f : ℕ → α) :

disjointed f 0 = f 0

theorem disjointed_succ {α : Type u_1} [inst : GeneralizedBooleanAlgebra α] (f : ℕ → α) (n : ℕ) :

disjointed f (n + 1) = f (n + 1) \ ↑(partialSups f) n

theorem disjointed_le_id {α : Type u_1} [inst : GeneralizedBooleanAlgebra α] :

disjointed ≤ id

theorem disjointed_le {α : Type u_1} [inst : GeneralizedBooleanAlgebra α] (f : ℕ → α) :

disjointed f ≤ f

theorem disjoint_disjointed {α : Type u_1} [inst : GeneralizedBooleanAlgebra α] (f : ℕ → α) :

Pairwise (Disjoint on disjointed f)

def disjointedRec {α : Type u_1} [inst : GeneralizedBooleanAlgebra α] {f : ℕ → α} {p : α → Sort u_2} (hdiff : ⦃t : α⦄ → ⦃i : ℕ⦄ → p t → p (t \ f i)) ⦃n : ℕ⦄ :

p (f n) → p (disjointed f n)

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.

Equations

One or more equations did not get rendered due to their size.

@[simp]

theorem disjointedRec_zero {α : Type u_2} [inst : GeneralizedBooleanAlgebra α] {f : ℕ → α} {p : α → Sort u_1} (hdiff : ⦃t : α⦄ → ⦃i : ℕ⦄ → p t → p (t \ f i)) (h₀ : p (f 0)) :

disjointedRec hdiff h₀ = h₀

theorem Monotone.disjointed_eq {α : Type u_1} [inst : GeneralizedBooleanAlgebra α] {f : ℕ → α} (hf : Monotone f) (n : ℕ) :

disjointed f (n + 1) = f (n + 1) \ f n

@[simp]

theorem partialSups_disjointed {α : Type u_1} [inst : GeneralizedBooleanAlgebra α] (f : ℕ → α) :

partialSups (disjointed f) = partialSups f

theorem disjointed_unique {α : Type u_1} [inst : GeneralizedBooleanAlgebra α] {f : ℕ → α} {d : ℕ → α} (hdisj : Pairwise (Disjoint on d)) (hsups : partialSups d = partialSups f) :

d = disjointed f

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

theorem supᵢ_disjointed {α : Type u_1} [inst : CompleteBooleanAlgebra α] (f : ℕ → α) :

(⨆ n, disjointed f n) = ⨆ n, f n

theorem disjointed_eq_inf_compl {α : Type u_1} [inst : CompleteBooleanAlgebra α] (f : ℕ → α) (n : ℕ) :

disjointed f n = f n ⊓ ⨅ i, ⨅ h, f iᶜ

Set notation variants of lemmas #

theorem disjointed_subset {α : Type u_1} (f : ℕ → Set α) (n : ℕ) :

disjointed f n ⊆ f n

theorem unionᵢ_disjointed {α : Type u_1} {f : ℕ → Set α} :

(Set.unionᵢ fun n => disjointed f n) = Set.unionᵢ fun n => f n

theorem disjointed_eq_inter_compl {α : Type u_1} (f : ℕ → Set α) (n : ℕ) :

disjointed f n = f n ∩ Set.interᵢ fun i => Set.interᵢ fun h => f iᶜ

theorem preimage_find_eq_disjointed {α : Type u_1} (s : ℕ → Set α) (H : ∀ (x : α), ∃ n, x ∈ s n) [inst : (x : α) → (n : ℕ) → Decidable (x ∈ s n)] (n : ℕ) :

(fun x => Nat.find (H x)) ⁻¹' {n} = disjointed s n