Ordinal notation

Constructive ordinal arithmetic for ordinals below ε₀.

We define a type ONote, with constructors 0 : ONote and ONote.oadd e n a representing ω ^ e * n + a. We say that o is in Cantor normal form - ONote.NF o - if either o = 0 or o = ω ^ e * n + a with a < ω ^ e and a in Cantor normal form.

The type NONote is the type of ordinals below ε₀ in Cantor normal form. Various operations (addition, subtraction, multiplication, exponentiation) are defined on ONote and NONote.

inductive ONote : Type

Recursive definition of an ordinal notation. zero denotes the ordinal 0, and oadd e n a is intended to refer to ω ^ e * n + a. For this to be a valid Cantor normal form, we must have the exponents decrease to the right, but we can't state this condition until we've defined repr, so we make it a separate definition NF.

• zero : ONote
• oadd : ONote → ℕ+ → ONote → ONote

instance instDecidableEqONote : DecidableEq ONote

instance ONote.instZero : Zero ONote

Notation for 0

@[simp]

theorem ONote.zero_def : zero = 0

instance ONote.instInhabited : Inhabited ONote

instance ONote.instOne : One ONote

Notation for 1

def ONote.omega : ONote

Notation for ω

noncomputable def ONote.repr : ONote → Ordinal.{0}

The ordinal denoted by a notation

@[simp]

theorem ONote.repr_zero : repr 0 = 0

def ONote.toString : ONote → String

Print an ordinal notation

def ONote.repr' (prec : ℕ) : ONote → Std.Format

Print an ordinal notation

instance ONote.instToString : ToString ONote

instance ONote.instRepr : Repr ONote

instance ONote.instPreorder : Preorder ONote

theorem ONote.lt_def {x y : ONote} : x < y ↔ x.repr < y.repr

theorem ONote.le_def {x y : ONote} : x ≤ y ↔ x.repr ≤ y.repr

instance ONote.instWellFoundedRelation : WellFoundedRelation ONote

def ONote.ofNat : ℕ → ONote

Convert a Nat into an ordinal

@[simp]

theorem ONote.ofNat_zero : ↑0 = 0

@[simp]

theorem ONote.ofNat_succ (n : ℕ) : ↑n.succ = oadd 0 n.succPNat 0

@[instance 100]

instance ONote.nat (n : ℕ) : OfNat ONote n

@[simp]

theorem ONote.ofNat_one : ↑1 = 1

@[simp]

theorem ONote.repr_ofNat (n : ℕ) : (↑n).repr = ↑n

@[simp]

theorem ONote.repr_one : repr 1 = ↑1

theorem ONote.omega0_le_oadd (e : ONote) (n : ℕ+) (a : ONote) : Ordinal.omega0 ^ e.repr ≤ (e.oadd n a).repr

theorem ONote.oadd_pos (e : ONote) (n : ℕ+) (a : ONote) : 0 < e.oadd n a

def ONote.cmp : ONote → ONote → Ordering

Comparison of ordinal notations:

ω ^ e₁ * n₁ + a₁ is less than ω ^ e₂ * n₂ + a₂ when either e₁ < e₂, or e₁ = e₂ and n₁ < n₂, or e₁ = e₂, n₁ = n₂, and a₁ < a₂.

theorem ONote.eq_of_cmp_eq {o₁ o₂ : ONote} : o₁.cmp o₂ = Ordering.eq → o₁ = o₂

theorem ONote.zero_lt_one : 0 < 1

inductive ONote.NFBelow : ONote → Ordinal.{0} → Prop

NFBelow o b says that o is a normal form ordinal notation satisfying repr o < ω ^ b.

• zero {b : Ordinal.{0}} : NFBelow 0 b
• oadd' {e : ONote} {n : ℕ+} {a : ONote} {eb b : Ordinal.{0}} : e.NFBelow eb → a.NFBelow e.repr → e.repr < b → (e.oadd n a).NFBelow b

class ONote.NF (o : ONote) : Prop

A normal form ordinal notation has the form

ω ^ a₁ * n₁ + ω ^ a₂ * n₂ + ⋯ + ω ^ aₖ * nₖ

where a₁ > a₂ > ⋯ > aₖ and all the aᵢ are also in normal form.

We will essentially only be interested in normal form ordinal notations, but to avoid complicating the algorithms, we define everything over general ordinal notations and only prove correctness with normal form as an invariant.

• out : Exists o.NFBelow

instance ONote.NF.zero : NF 0

theorem ONote.NFBelow.oadd {e : ONote} {n : ℕ+} {a : ONote} {b : Ordinal.{0}} : e.NF → a.NFBelow e.repr → e.repr < b → (e.oadd n a).NFBelow b

theorem ONote.NFBelow.fst {e : ONote} {n : ℕ+} {a : ONote} {b : Ordinal.{0}} (h : (e.oadd n a).NFBelow b) : e.NF

theorem ONote.NF.fst {e : ONote} {n : ℕ+} {a : ONote} : (e.oadd n a).NF → e.NF

theorem ONote.NFBelow.snd {e : ONote} {n : ℕ+} {a : ONote} {b : Ordinal.{0}} (h : (e.oadd n a).NFBelow b) : a.NFBelow e.repr

theorem ONote.NF.snd' {e : ONote} {n : ℕ+} {a : ONote} : (e.oadd n a).NF → a.NFBelow e.repr

theorem ONote.NF.snd {e : ONote} {n : ℕ+} {a : ONote} (h : (e.oadd n a).NF) : a.NF

theorem ONote.NF.oadd {e a : ONote} (h₁ : e.NF) (n : ℕ+) (h₂ : a.NFBelow e.repr) : (e.oadd n a).NF

instance ONote.NF.oadd_zero (e : ONote) (n : ℕ+) [h : e.NF] : (e.oadd n 0).NF

theorem ONote.NFBelow.lt {e : ONote} {n : ℕ+} {a : ONote} {b : Ordinal.{0}} (h : (e.oadd n a).NFBelow b) : e.repr < b

theorem ONote.NFBelow_zero {o : ONote} : o.NFBelow 0 ↔ o = 0

theorem ONote.NF.zero_of_zero {e : ONote} {n : ℕ+} {a : ONote} (h : (e.oadd n a).NF) (e0 : e = 0) : a = 0

theorem ONote.NFBelow.repr_lt {o : ONote} {b : Ordinal.{0}} (h : o.NFBelow b) : o.repr < Ordinal.omega0 ^ b

theorem ONote.NFBelow.mono {o : ONote} {b₁ b₂ : Ordinal.{0}} (bb : b₁ ≤ b₂) (h : o.NFBelow b₁) : o.NFBelow b₂

theorem ONote.NF.below_of_lt {e : ONote} {n : ℕ+} {a : ONote} {b : Ordinal.{0}} (H : e.repr < b) : (e.oadd n a).NF → (e.oadd n a).NFBelow b

theorem ONote.NF.below_of_lt' {o : ONote} {b : Ordinal.{0}} : o.repr < Ordinal.omega0 ^ b → o.NF → o.NFBelow b

theorem ONote.nfBelow_ofNat (n : ℕ) : (↑n).NFBelow 1

instance ONote.nf_ofNat (n : ℕ) : (↑n).NF

instance ONote.nf_one : NF 1

theorem ONote.oadd_lt_oadd_1 {e₁ : ONote} {n₁ : ℕ+} {o₁ e₂ : ONote} {n₂ : ℕ+} {o₂ : ONote} (h₁ : (e₁.oadd n₁ o₁).NF) (h : e₁ < e₂) : e₁.oadd n₁ o₁ < e₂.oadd n₂ o₂

theorem ONote.oadd_lt_oadd_2 {e o₁ o₂ : ONote} {n₁ n₂ : ℕ+} (h₁ : (e.oadd n₁ o₁).NF) (h : ↑n₁ < ↑n₂) : e.oadd n₁ o₁ < e.oadd n₂ o₂

theorem ONote.oadd_lt_oadd_3 {e : ONote} {n : ℕ+} {a₁ a₂ : ONote} (h : a₁ < a₂) : e.oadd n a₁ < e.oadd n a₂

theorem ONote.cmp_compares (a b : ONote) [a.NF] [b.NF] : (a.cmp b).Compares a b

theorem ONote.repr_inj {a b : ONote} [a.NF] [b.NF] : a.repr = b.repr ↔ a = b

theorem ONote.NF.of_dvd_omega0_opow {b : Ordinal.{0}} {e : ONote} {n : ℕ+} {a : ONote} (h : (e.oadd n a).NF) (d : Ordinal.omega0 ^ b ∣ (e.oadd n a).repr) : b ≤ e.repr ∧ Ordinal.omega0 ^ b ∣ a.repr

theorem ONote.NF.of_dvd_omega0 {e : ONote} {n : ℕ+} {a : ONote} (h : (e.oadd n a).NF) : Ordinal.omega0 ∣ (e.oadd n a).repr → e.repr ≠ 0 ∧ Ordinal.omega0 ∣ a.repr

def ONote.TopBelow (b : ONote) : ONote → Prop

TopBelow b o asserts that the largest exponent in o, if it exists, is less than b. This is an auxiliary definition for decidability of NF.

instance ONote.decidableTopBelow : DecidableRel TopBelow

theorem ONote.nfBelow_iff_topBelow {b : ONote} [b.NF] {o : ONote} : o.NFBelow b.repr ↔ o.NF ∧ b.TopBelow o

instance ONote.decidableNF : DecidablePred NF

def ONote.addAux (e : ONote) (n : ℕ+) (o : ONote) : ONote

Auxiliary definition for add

def ONote.add : ONote → ONote → ONote

Addition of ordinal notations (correct only for normal input)

instance ONote.instAdd : Add ONote

@[simp]

theorem ONote.zero_add (o : ONote) : 0 + o = o

theorem ONote.oadd_add (e : ONote) (n : ℕ+) (a o : ONote) : e.oadd n a + o = e.addAux n (a + o)

def ONote.sub : ONote → ONote → ONote

Subtraction of ordinal notations (correct only for normal input)

instance ONote.instSub : Sub ONote

theorem ONote.add_nfBelow {b : Ordinal.{0}} {o₁ o₂ : ONote} : o₁.NFBelow b → o₂.NFBelow b → (o₁ + o₂).NFBelow b

instance ONote.add_nf (o₁ o₂ : ONote) [o₁.NF] [o₂.NF] : (o₁ + o₂).NF

@[simp]

theorem ONote.repr_add (o₁ o₂ : ONote) [o₁.NF] [o₂.NF] : (o₁ + o₂).repr = o₁.repr + o₂.repr

theorem ONote.sub_nfBelow {o₁ o₂ : ONote} {b : Ordinal.{0}} : o₁.NFBelow b → o₂.NF → (o₁ - o₂).NFBelow b

instance ONote.sub_nf (o₁ o₂ : ONote) [o₁.NF] [o₂.NF] : (o₁ - o₂).NF

@[simp]

theorem ONote.repr_sub (o₁ o₂ : ONote) [o₁.NF] [o₂.NF] : (o₁ - o₂).repr = o₁.repr - o₂.repr

def ONote.mul : ONote → ONote → ONote

Multiplication of ordinal notations (correct only for normal input)

instance ONote.instMul : Mul ONote

instance ONote.instMulZeroClass : MulZeroClass ONote

theorem ONote.oadd_mul (e₁ : ONote) (n₁ : ℕ+) (a₁ e₂ : ONote) (n₂ : ℕ+) (a₂ : ONote) : e₁.oadd n₁ a₁ * e₂.oadd n₂ a₂ = if e₂ = 0 then e₁.oadd (n₁ * n₂) a₁ else (e₁ + e₂).oadd n₂ (e₁.oadd n₁ a₁ * a₂)

theorem ONote.oadd_mul_nfBelow {e₁ : ONote} {n₁ : ℕ+} {a₁ : ONote} {b₁ : Ordinal.{0}} (h₁ : (e₁.oadd n₁ a₁).NFBelow b₁) {o₂ : ONote} {b₂ : Ordinal.{0}} : o₂.NFBelow b₂ → (e₁.oadd n₁ a₁ * o₂).NFBelow (e₁.repr + b₂)

instance ONote.mul_nf (o₁ o₂ : ONote) [o₁.NF] [o₂.NF] : (o₁ * o₂).NF

@[simp]

theorem ONote.repr_mul (o₁ o₂ : ONote) [o₁.NF] [o₂.NF] : (o₁ * o₂).repr = o₁.repr * o₂.repr

def ONote.split' : ONote → ONote × ℕ

Calculate division and remainder of o mod ω:

split' o = (a, n) means o = ω * a + n.

def ONote.split : ONote → ONote × ℕ

Calculate division and remainder of o mod ω:

split o = (a, n) means o = a + n, where ω ∣ a.

def ONote.scale (x : ONote) : ONote → ONote

scale x o is the ordinal notation for ω ^ x * o.

def ONote.mulNat : ONote → ℕ → ONote

mulNat o n is the ordinal notation for o * n.

def ONote.opowAux (e a0 a : ONote) : ℕ → ℕ → ONote

Auxiliary definition to compute the ordinal notation for the ordinal exponentiation in opow

def ONote.opowAux2 (o₂ : ONote) (o₁ : ONote × ℕ) : ONote

Auxiliary definition to compute the ordinal notation for the ordinal exponentiation in opow

def ONote.opow (o₁ o₂ : ONote) : ONote

opow o₁ o₂ calculates the ordinal notation for the ordinal exponential o₁ ^ o₂.

instance ONote.instPow : Pow ONote ONote

theorem ONote.opow_def (o₁ o₂ : ONote) : o₁ ^ o₂ = o₂.opowAux2 o₁.split

theorem ONote.split_eq_scale_split' {o o' : ONote} {m : ℕ} [o.NF] : o.split' = (o', m) → o.split = (scale 1 o', m)

theorem ONote.nf_repr_split' {o o' : ONote} {m : ℕ} [o.NF] : o.split' = (o', m) → o'.NF ∧ o.repr = Ordinal.omega0 * o'.repr + ↑m

theorem ONote.scale_eq_mul (x : ONote) [x.NF] (o : ONote) [o.NF] : x.scale o = x.oadd 1 0 * o

instance ONote.nf_scale (x : ONote) [x.NF] (o : ONote) [o.NF] : (x.scale o).NF

@[simp]

theorem ONote.repr_scale (x : ONote) [x.NF] (o : ONote) [o.NF] : (x.scale o).repr = Ordinal.omega0 ^ x.repr * o.repr

theorem ONote.nf_repr_split {o o' : ONote} {m : ℕ} [o.NF] (h : o.split = (o', m)) : o'.NF ∧ o.repr = o'.repr + ↑m

theorem ONote.split_dvd {o o' : ONote} {m : ℕ} [o.NF] (h : o.split = (o', m)) : Ordinal.omega0 ∣ o'.repr

theorem ONote.split_add_lt {o e : ONote} {n : ℕ+} {a : ONote} {m : ℕ} [o.NF] (h : o.split = (e.oadd n a, m)) : a.repr + ↑m < Ordinal.omega0 ^ e.repr

@[simp]

theorem ONote.mulNat_eq_mul (n : ℕ) (o : ONote) : o.mulNat n = o * ↑n

instance ONote.nf_mulNat (o : ONote) [o.NF] (n : ℕ) : (o.mulNat n).NF

@[irreducible]

instance ONote.nf_opowAux (e a0 a : ONote) [e.NF] [a0.NF] [a.NF] (k m : ℕ) : (e.opowAux a0 a k m).NF

instance ONote.nf_opow (o₁ o₂ : ONote) [o₁.NF] [o₂.NF] : (o₁ ^ o₂).NF

theorem ONote.scale_opowAux (e a0 a : ONote) [e.NF] [a0.NF] [a.NF] (k m : ℕ) : (e.opowAux a0 a k m).repr = Ordinal.omega0 ^ e.repr * (opowAux 0 a0 a k m).repr

theorem ONote.repr_opow_aux₁ {e a : ONote} [Ne : e.NF] [Na : a.NF] {a' : Ordinal.{0}} (e0 : e.repr ≠ 0) (h : a' < Ordinal.omega0 ^ e.repr) (aa : a.repr = a') (n : ℕ+) : (Ordinal.omega0 ^ e.repr * ↑↑n + a') ^ Ordinal.omega0 = (Ordinal.omega0 ^ e.repr) ^ Ordinal.omega0

theorem ONote.repr_opow_aux₂ {a0 a' : ONote} [N0 : a0.NF] [Na' : a'.NF] (m : ℕ) (d : Ordinal.omega0 ∣ a'.repr) (e0 : a0.repr ≠ 0) (h : a'.repr + ↑m < Ordinal.omega0 ^ a0.repr) (n : ℕ+) (k : ℕ) : have R := (opowAux 0 a0 (a0.oadd n a' * ↑m) k m).repr; (k ≠ 0 → R < (Ordinal.omega0 ^ a0.repr) ^ Order.succ ↑k) ∧ (Ordinal.omega0 ^ a0.repr) ^ ↑k * (Ordinal.omega0 ^ a0.repr * ↑↑n + a'.repr) + R = (Ordinal.omega0 ^ a0.repr * ↑↑n + a'.repr + ↑m) ^ Order.succ ↑k

theorem ONote.repr_opow (o₁ o₂ : ONote) [o₁.NF] [o₂.NF] : (o₁ ^ o₂).repr = o₁.repr ^ o₂.repr

def ONote.fundamentalSequence : ONote → Option ONote ⊕ (ℕ → ONote)

Given an ordinal, returns:

• inl none for 0
• inl (some a) for a + 1
• inr f for a limit ordinal a, where f i is a sequence converging to a

def ONote.FundamentalSequenceProp (o : ONote) : Option ONote ⊕ (ℕ → ONote) → Prop

The property satisfied by fundamentalSequence o:

• inl none means o = 0
• inl (some a) means o = succ a
• inr f means o is a limit ordinal and f is a strictly increasing sequence which converges to o

theorem ONote.fundamentalSequenceProp_inl_none (o : ONote) : o.FundamentalSequenceProp (Sum.inl none) ↔ o = 0

theorem ONote.fundamentalSequenceProp_inl_some (o a : ONote) : o.FundamentalSequenceProp (Sum.inl (some a)) ↔ o.repr = Order.succ a.repr ∧ (o.NF → a.NF)

theorem ONote.fundamentalSequenceProp_inr (o : ONote) (f : ℕ → ONote) : o.FundamentalSequenceProp (Sum.inr f) ↔ Order.IsSuccLimit o.repr ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < o ∧ (o.NF → (f i).NF)) ∧ ∀ a < o.repr, ∃ (i : ℕ), a < (f i).repr

theorem ONote.fundamentalSequence_has_prop (o : ONote) : o.FundamentalSequenceProp o.fundamentalSequence

@[irreducible]

def ONote.fastGrowing : ONote → ℕ → ℕ

The fast growing hierarchy for ordinal notations < ε₀. This is a sequence of functions ℕ → ℕ indexed by ordinals, with the definition:

• f_0(n) = n + 1
• f_(α + 1)(n) = f_α^[n](n)
• f_α(n) = f_(α[n])(n) where α is a limit ordinal and α[i] is the fundamental sequence converging to α

theorem ONote.fastGrowing_def {o : ONote} {x : Option ONote ⊕ (ℕ → ONote)} (e : o.fundamentalSequence = x) : o.fastGrowing = match (motive := (x : Option ONote ⊕ (ℕ → ONote)) → o.FundamentalSequenceProp x → ℕ → ℕ) x, ⋯ with | Sum.inl none, x => Nat.succ | Sum.inl (some a), x => fun (i : ℕ) => a.fastGrowing^[i] i | Sum.inr f, x => fun (i : ℕ) => (f i).fastGrowing i

theorem ONote.fastGrowing_zero' (o : ONote) (h : o.fundamentalSequence = Sum.inl none) : o.fastGrowing = Nat.succ

theorem ONote.fastGrowing_succ (o : ONote) {a : ONote} (h : o.fundamentalSequence = Sum.inl (some a)) : o.fastGrowing = fun (i : ℕ) => a.fastGrowing^[i] i

theorem ONote.fastGrowing_limit (o : ONote) {f : ℕ → ONote} (h : o.fundamentalSequence = Sum.inr f) : o.fastGrowing = fun (i : ℕ) => (f i).fastGrowing i

@[simp]

theorem ONote.fastGrowing_zero : fastGrowing 0 = Nat.succ

@[simp]

theorem ONote.fastGrowing_one : fastGrowing 1 = fun (n : ℕ) => 2 * n

@[simp]

theorem ONote.fastGrowing_two : fastGrowing 2 = fun (n : ℕ) => 2 ^ n * n

def ONote.fastGrowingε₀ (i : ℕ) : ℕ

We can extend the fast growing hierarchy one more step to ε₀ itself, using ω ^ (ω ^ (⋯ ^ ω)) as the fundamental sequence converging to ε₀ (which is not an ONote). Extending the fast growing hierarchy beyond this requires a definition of fundamental sequence for larger ordinals.

theorem ONote.fastGrowingε₀_zero : fastGrowingε₀ 0 = 1

theorem ONote.fastGrowingε₀_one : fastGrowingε₀ 1 = 2

theorem ONote.fastGrowingε₀_two : fastGrowingε₀ 2 = 2048

def NONote : Type

The type of normal ordinal notations.

It would have been nicer to define this right in the inductive type, but NF o requires repr which requires ONote, so all these things would have to be defined at once, which messes up the VM representation.

instance instDecidableEqNONote : DecidableEq NONote

instance NONote.NF (o : NONote) : (↑o).NF

def NONote.mk (o : ONote) [h : o.NF] : NONote

Construct a NONote from an ordinal notation (and infer normality)

noncomputable def NONote.repr (o : NONote) : Ordinal.{0}

The ordinal represented by an ordinal notation.

This function is noncomputable because ordinal arithmetic is noncomputable. In computational applications NONote can be used exclusively without reference to Ordinal, but this function allows for correctness results to be stated.

instance NONote.instToString : ToString NONote

instance NONote.instRepr : Repr NONote

instance NONote.instPreorder : Preorder NONote

instance NONote.instZero : Zero NONote

instance NONote.instInhabited : Inhabited NONote

theorem NONote.lt_wf : WellFounded fun (x1 x2 : NONote) => x1 < x2

instance NONote.instWellFoundedLT : WellFoundedLT NONote

instance NONote.instWellFoundedRelation : WellFoundedRelation NONote

def NONote.ofNat (n : ℕ) : NONote

Convert a natural number to an ordinal notation

def NONote.cmp (a b : NONote) : Ordering

Compare ordinal notations

theorem NONote.cmp_compares (a b : NONote) : (a.cmp b).Compares a b

instance NONote.instLinearOrder : LinearOrder NONote

def NONote.below (a b : NONote) : Prop

Asserts that repr a < ω ^ repr b. Used in NONote.recOn.

def NONote.oadd (e : NONote) (n : ℕ+) (a : NONote) (h : a.below e) : NONote

The oadd pseudo-constructor for NONote

def NONote.recOn {C : NONote → Sort u_1} (o : NONote) (H0 : C 0) (H1 : (e : NONote) → (n : ℕ+) → (a : NONote) → (h : a.below e) → C e → C a → C (e.oadd n a h)) : C o

This is a recursor-like theorem for NONote suggesting an inductive definition, which can't actually be defined this way due to conflicting dependencies.

instance NONote.instAdd : Add NONote

Addition of ordinal notations

theorem NONote.repr_add (a b : NONote) : (a + b).repr = a.repr + b.repr

instance NONote.instSub : Sub NONote

Subtraction of ordinal notations

theorem NONote.repr_sub (a b : NONote) : (a - b).repr = a.repr - b.repr

instance NONote.instMul : Mul NONote

Multiplication of ordinal notations

theorem NONote.repr_mul (a b : NONote) : (a * b).repr = a.repr * b.repr

def NONote.opow (x y : NONote) : NONote

Exponentiation of ordinal notations

theorem NONote.repr_opow (a b : NONote) : (a.opow b).repr = a.repr ^ b.repr