mathlib3 documentation

set_theory.lists

A computable model of ZFA without infinity #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

In this file we define finite hereditary lists. This is useful for calculations in naive set theory.

We distinguish two kinds of ZFA lists:

Atoms. Directly correspond to an element of the original type.
Proper ZFA lists. Can be thought of (but aren't implemented) as a list of ZFA lists (not necessarily proper).

For example, lists ℕ contains stuff like 23, [], [37], [1, [[2], 3], 4].

Implementation note #

As we want to be able to append both atoms and proper ZFA lists to proper ZFA lists, it's handy that atoms and proper ZFA lists belong to the same type, even though atoms of α could be modelled as α directly. But we don't want to be able to append anything to atoms.

This calls for a two-steps definition of ZFA lists:

First, define ZFA prelists as atoms and proper ZFA prelists. Those proper ZFA prelists are defined by inductive appending of (not necessarily proper) ZFA lists.
Second, define ZFA lists by rubbing out the distinction between atoms and proper lists.

Main declarations #

lists' α ff: Atoms as ZFA prelists. Basically a copy of α.
lists' α tt: Proper ZFA prelists. Defined inductively from the empty ZFA prelist (lists'.nil) and from appending a ZFA prelist to a proper ZFA prelist (lists'.cons a l).
lists α: ZFA lists. Sum of the atoms and proper ZFA prelists.
finsets: ZFA sets. Defined as lists quotiented by lists.equiv, the extensional equivalence.

@[protected, instance]

def lists'.decidable_eq (α : Type u) [a : decidable_eq α] (ᾰ : bool) :

decidable_eq (lists' α ᾰ)

inductive lists' (α : Type u) :

bool → Type u

atom : Π {α : Type u}, α → lists' α bool.ff
nil : Π {α : Type u}, lists' α bool.tt
cons' : Π {α : Type u} {b : bool}, lists' α b → lists' α bool.tt → lists' α bool.tt

Prelists, helper type to define lists. lists' α ff are the "atoms", a copy of α. lists' α tt are the "proper" ZFA prelists, inductively defined from the empty ZFA prelist and from appending a ZFA prelist to a proper ZFA prelist. It is made so that you can't append anything to an atom while having only one appending function for appending both atoms and proper ZFC prelists to a proper ZFA prelist.

Instances for lists'

def lists (α : Type u_1) :

Type u_1

Hereditarily finite list, aka ZFA list. A ZFA list is either an "atom" (b = ff), corresponding to an element of α, or a "proper" ZFA list, inductively defined from the empty ZFA list and from appending a ZFA list to a proper ZFA list.

Equations

lists α = Σ (b : bool), lists' α b

Instances for lists

@[protected, instance]

def lists'.inhabited {α : Type u_1} [inhabited α] (b : bool) :

inhabited (lists' α b)

Equations

lists'.inhabited bool.tt = {default := lists'.nil α}
lists'.inhabited bool.ff = {default := lists'.atom inhabited.default}

def lists'.cons {α : Type u_1} :

lists α → lists' α bool.tt → lists' α bool.tt

Appending a ZFA list to a proper ZFA prelist.

Equations

lists'.cons ⟨b, a⟩ l = a.cons' l

@[simp]

def lists'.to_list {α : Type u_1} {b : bool} :

lists' α b → list (lists α)

Converts a ZFA prelist to a list of ZFA lists. Atoms are sent to [].

Equations

(a.cons' l).to_list = ⟨a_6, a⟩ :: l.to_list
lists'.nil.to_list = list.nil
(lists'.atom a).to_list = list.nil

@[simp]

theorem lists'.to_list_cons {α : Type u_1} (a : lists α) (l : lists' α bool.tt) :

(lists'.cons a l).to_list = a :: l.to_list

@[simp]

def lists'.of_list {α : Type u_1} :

list (lists α) → lists' α bool.tt

Converts a list of ZFA lists to a proper ZFA prelist.

Equations

lists'.of_list (a :: l) = lists'.cons a (lists'.of_list l)
lists'.of_list list.nil = lists'.nil

@[simp]

theorem lists'.to_of_list {α : Type u_1} (l : list (lists α)) :

(lists'.of_list l).to_list = l

@[simp]

theorem lists'.of_to_list {α : Type u_1} (l : lists' α bool.tt) :

lists'.of_list l.to_list = l

@[protected, instance]

def lists'.has_subset {α : Type u_1} :

has_subset (lists' α bool.tt)

Equations

lists'.has_subset = {subset := lists'.subset α}

@[protected, instance]

def lists'.has_mem {α : Type u_1} {b : bool} :

has_mem (lists α) (lists' α b)

ZFA prelist membership. A ZFA list is in a ZFA prelist if some element of this ZFA prelist is equivalent as a ZFA list to this ZFA list.

Equations

lists'.has_mem = {mem := λ (a : lists α) (l : lists' α b), ∃ (a' : lists α) (H : a' ∈ l.to_list), a.equiv a'}

theorem lists'.mem_def {α : Type u_1} {b : bool} {a : lists α} {l : lists' α b} :

a ∈ l ↔ ∃ (a' : lists α) (H : a' ∈ l.to_list), a.equiv a'

@[simp]

theorem lists'.mem_cons {α : Type u_1} {a y : lists α} {l : lists' α bool.tt} :

a ∈ lists'.cons y l ↔ a.equiv y ∨ a ∈ l

theorem lists'.cons_subset {α : Type u_1} {a : lists α} {l₁ l₂ : lists' α bool.tt} :

lists'.cons a l₁ ⊆ l₂ ↔ a ∈ l₂ ∧ l₁ ⊆ l₂

theorem lists'.of_list_subset {α : Type u_1} {l₁ l₂ : list (lists α)} (h : l₁ ⊆ l₂) :

lists'.of_list l₁ ⊆ lists'.of_list l₂

theorem lists'.subset_nil {α : Type u_1} {l : lists' α bool.tt} :

l ⊆ lists'.nil → l = lists'.nil

theorem lists'.mem_of_subset' {α : Type u_1} {a : lists α} {l₁ l₂ : lists' α bool.tt} (s : l₁ ⊆ l₂) (h : a ∈ l₁.to_list) :

a ∈ l₂

theorem lists'.subset_def {α : Type u_1} {l₁ l₂ : lists' α bool.tt} :

l₁ ⊆ l₂ ↔ ∀ (a : lists α), a ∈ l₁.to_list → a ∈ l₂

def lists.atom {α : Type u_1} (a : α) :

Sends a : α to the corresponding atom in lists α.

Equations

lists.atom a = ⟨bool.ff, lists'.atom a⟩

def lists.of' {α : Type u_1} (l : lists' α bool.tt) :

Converts a proper ZFA prelist to a ZFA list.

Equations

lists.of' l = ⟨bool.tt, l⟩

@[simp]

def lists.to_list {α : Type u_1} :

lists α → list (lists α)

Converts a ZFA list to a list of ZFA lists. Atoms are sent to [].

Equations

lists.to_list ⟨b, l⟩ = l.to_list

def lists.is_list {α : Type u_1} (l : lists α) :

Prop

Predicate stating that a ZFA list is proper.

Equations

l.is_list = ↥(l.fst)

def lists.of_list {α : Type u_1} (l : list (lists α)) :

Converts a list of ZFA lists to a ZFA list.

Equations

lists.of_list l = lists.of' (lists'.of_list l)

theorem lists.is_list_to_list {α : Type u_1} (l : list (lists α)) :

(lists.of_list l).is_list

theorem lists.to_of_list {α : Type u_1} (l : list (lists α)) :

(lists.of_list l).to_list = l

theorem lists.of_to_list {α : Type u_1} {l : lists α} :

l.is_list → lists.of_list l.to_list = l

@[protected, instance]

def lists.inhabited {α : Type u_1} :

inhabited (lists α)

Equations

lists.inhabited = {default := lists.of' lists'.nil}

@[protected, instance]

def lists.decidable_eq {α : Type u_1} [decidable_eq α] :

decidable_eq (lists α)

Equations

lists.decidable_eq = lists.decidable_eq._proof_1.mpr (λ (a b : Σ (b : bool), lists' α b), sigma.decidable_eq a b)

@[protected, instance]

def lists.has_sizeof {α : Type u_1} [has_sizeof α] :

has_sizeof (lists α)

Equations

lists.has_sizeof = lists.has_sizeof._proof_1.mpr (sigma.has_sizeof bool (λ (b : bool), lists' α b))

def lists.induction_mut {α : Type u_1} (C : lists α → Sort u_2) (D : lists' α bool.tt → Sort u_3) (C0 : Π (a : α), C (lists.atom a)) (C1 : Π (l : lists' α bool.tt), D l → C (lists.of' l)) (D0 : D lists'.nil) (D1 : Π (a : lists α) (l : lists' α bool.tt), C a → D l → D (lists'.cons a l)) :

pprod (Π (l : lists α), C l) (Π (l : lists' α bool.tt), D l)

A recursion principle for pairs of ZFA lists and proper ZFA prelists.

Equations

lists.induction_mut C D C0 C1 D0 D1 = ⟨λ (_x : lists α), lists.induction_mut._match_2 C D (λ (b : bool) (l : lists' α b), lists'.rec (λ (a : α), ⟨C0 a, punit.star⟩) ⟨C1 lists'.nil D0, D0⟩ (λ {b : bool} (a : lists' α b) (l : lists' α bool.tt) (IH₁ : pprod (C ⟨b, a⟩) (lists.induction_mut._match_1 D b a)) (IH₂ : pprod (C ⟨bool.tt, l⟩) (lists.induction_mut._match_1 D bool.tt l)), ⟨C1 (a.cons' l) (D1 ⟨b, a⟩ l IH₁.fst IH₂.snd), D1 ⟨b, a⟩ l IH₁.fst IH₂.snd⟩) l) _x, λ (l : lists' α bool.tt), ((λ (b : bool) (l : lists' α b), lists'.rec (λ (a : α), ⟨C0 a, punit.star⟩) ⟨C1 lists'.nil D0, D0⟩ (λ {b : bool} (a : lists' α b) (l : lists' α bool.tt) (IH₁ : pprod (C ⟨b, a⟩) (lists.induction_mut._match_1 D b a)) (IH₂ : pprod (C ⟨bool.tt, l⟩) (lists.induction_mut._match_1 D bool.tt l)), ⟨C1 (a.cons' l) (D1 ⟨b, a⟩ l IH₁.fst IH₂.snd), D1 ⟨b, a⟩ l IH₁.fst IH₂.snd⟩) l) bool.tt l).snd⟩
lists.induction_mut._match_2 C D this ⟨b, l⟩ = (this l).fst
lists.induction_mut._match_1 D bool.tt l = D l
lists.induction_mut._match_1 D bool.ff l = punit

def lists.mem {α : Type u_1} (a : lists α) :

lists α → Prop

Membership of ZFA list. A ZFA list belongs to a proper ZFA list if it belongs to the latter as a proper ZFA prelist. An atom has no members.

Equations

a.mem ⟨bool.tt, l⟩ = (a ∈ l)
a.mem ⟨bool.ff, l⟩ = false

@[protected, instance]

def lists.has_mem {α : Type u_1} :

has_mem (lists α) (lists α)

Equations

lists.has_mem = {mem := lists.mem α}

theorem lists.is_list_of_mem {α : Type u_1} {a l : lists α} :

a ∈ l → l.is_list

theorem lists.equiv_atom {α : Type u_1} {a : α} {l : lists α} :

(lists.atom a).equiv l ↔ lists.atom a = l

@[protected, instance]

def lists.setoid {α : Type u_1} :

setoid (lists α)

Equations

lists.setoid = {r := lists.equiv α, iseqv := _}

theorem lists.sizeof_pos {α : Type u_1} {b : bool} (l : lists' α b) :

theorem lists.lt_sizeof_cons' {α : Type u_1} {b : bool} (a : lists' α b) (l : lists' α bool.tt) :

sizeof ⟨b, a⟩ < sizeof (a.cons' l)

@[instance]

def lists.mem.decidable {α : Type u_1} [decidable_eq α] (a : lists α) (l : lists' α bool.tt) :

decidable (a ∈ l)

Equations

lists.mem.decidable a (b.cons' l₂) = decidable_of_iff' (a.equiv ⟨a_20, b⟩ ∨ a ∈ l₂) _
lists.mem.decidable a lists'.nil = decidable.is_false _

@[instance]

def lists.subset.decidable {α : Type u_1} [decidable_eq α] (l₁ l₂ : lists' α bool.tt) :

decidable (l₁ ⊆ l₂)

Equations

lists.subset.decidable (a.cons' l₁) l₂ = decidable_of_iff' (⟨b, a⟩ ∈ l₂ ∧ l₁ ⊆ l₂) _
lists.subset.decidable lists'.nil l₂ = decidable.is_true lists'.subset.nil

theorem lists'.mem_equiv_left {α : Type u_1} {l : lists' α bool.tt} {a a' : lists α} :

a.equiv a' → (a ∈ l ↔ a' ∈ l)

theorem lists'.mem_of_subset {α : Type u_1} {a : lists α} {l₁ l₂ : lists' α bool.tt} (s : l₁ ⊆ l₂) :

a ∈ l₁ → a ∈ l₂

def finsets (α : Type u_1) :

Type u_1

Equations

finsets α = quotient lists.setoid

Instances for finsets

@[protected, instance]

def finsets.has_emptyc {α : Type u_1} :

has_emptyc (finsets α)

Equations

finsets.has_emptyc = {emptyc := ⟦lists.of' lists'.nil⟧}

@[protected, instance]

def finsets.inhabited {α : Type u_1} :

inhabited (finsets α)

Equations

finsets.inhabited = {default := ∅}

@[protected, instance]

def finsets.decidable_eq {α : Type u_1} [decidable_eq α] :

decidable_eq (finsets α)

Equations

finsets.decidable_eq = finsets.decidable_eq._proof_1.mpr (λ (a b : quotient lists.setoid), quotient.decidable_eq a b)