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Mathlib.SetTheory.Ordinal.Notation

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 :

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.

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    def instDecidableEqONote.decEq (x✝ x✝¹ : ONote) :
    Decidable (x✝ = x✝¹)
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      Notation for 0

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      @[simp]
      theorem ONote.zero_def :

      Notation for 1

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      Notation for ω

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        noncomputable def ONote.repr :

        The ordinal denoted by a notation

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          @[simp]
          theorem ONote.repr_zero :
          repr 0 = 0

          Print an ordinal notation

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            def ONote.repr' (prec : ) :

            Print an ordinal notation

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              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

              Convert a Nat into an ordinal

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                @[simp]
                theorem ONote.ofNat_zero :
                0 = 0
                @[simp]
                theorem ONote.ofNat_succ (n : ) :
                n.succ = oadd 0 n.succPNat 0
                @[instance 100]
                instance ONote.nat (n : ) :
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                @[simp]
                theorem ONote.ofNat_one :
                1 = 1
                @[simp]
                theorem ONote.repr_ofNat (n : ) :
                (↑n).repr = n
                @[simp]
                theorem ONote.repr_one :
                repr 1 = 1
                theorem ONote.oadd_pos (e : ONote) (n : ℕ+) (a : ONote) :
                0 < e.oadd n a

                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.

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                  theorem ONote.eq_of_cmp_eq {o₁ o₂ : ONote} :
                  o₁.cmp o₂ = Ordering.eqo₁ = o₂
                  inductive ONote.NFBelow :

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

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                    class ONote.NF (o : ONote) :

                    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.

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                      theorem ONote.NFBelow.oadd {e : ONote} {n : ℕ+} {a : ONote} {b : Ordinal.{0}} :
                      e.NFa.NFBelow e.repre.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).NFe.NF
                      theorem ONote.NFBelow.snd {e : ONote} {n : ℕ+} {a : ONote} {b : Ordinal.{0}} (h : (e.oadd n a).NFBelow b) :
                      theorem ONote.NF.snd' {e : ONote} {n : ℕ+} {a : ONote} :
                      (e.oadd n a).NFa.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.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.nfBelow_ofNat (n : ) :
                      (↑n).NFBelow 1
                      instance ONote.nf_ofNat (n : ) :
                      (↑n).NF
                      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) :
                      theorem ONote.NF.of_dvd_omega0 {e : ONote} {n : ℕ+} {a : ONote} (h : (e.oadd n a).NF) :

                      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.

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

                        Auxiliary definition for add

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                          def ONote.add :
                          ONoteONoteONote

                          Addition of ordinal notations (correct only for normal input)

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                            @[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 :
                            ONoteONoteONote

                            Subtraction of ordinal notations (correct only for normal input)

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                              theorem ONote.add_nfBelow {b : Ordinal.{0}} {o₁ o₂ : ONote} :
                              o₁.NFBelow bo₂.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 bo₂.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 :
                              ONoteONoteONote

                              Multiplication of ordinal notations (correct only for normal input)

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                                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

                                Calculate division and remainder of o mod ω:

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

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                                  Calculate division and remainder of o mod ω:

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

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                                    def ONote.scale (x : ONote) :

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

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

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

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

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

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

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                                            def ONote.opow (o₁ o₂ : ONote) :

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

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                                              theorem ONote.opow_def (o₁ o₂ : ONote) :
                                              o₁ ^ o₂ = o₂.opowAux2 o₁.split
                                              theorem ONote.split_eq_scale_split' {o o' : ONote} {m : } [o.NF] :
                                              o.split' = (o', m)o.split = (scale 1 o', m)
                                              theorem ONote.nf_repr_split' {o o' : ONote} {m : } [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] :
                                              theorem ONote.nf_repr_split {o o' : ONote} {m : } [o.NF] (h : o.split = (o', m)) :
                                              o'.NF o.repr = o'.repr + m
                                              theorem ONote.split_dvd {o o' : ONote} {m : } [o.NF] (h : o.split = (o', m)) :
                                              theorem ONote.split_add_lt {o e : ONote} {n : ℕ+} {a : ONote} {m : } [o.NF] (h : o.split = (e.oadd n a, m)) :
                                              @[simp]
                                              theorem ONote.mulNat_eq_mul (n : ) (o : ONote) :
                                              o.mulNat n = o * n
                                              instance ONote.nf_mulNat (o : ONote) [o.NF] (n : ) :
                                              (o.mulNat n).NF
                                              @[irreducible]
                                              instance ONote.nf_opowAux (e a0 a : ONote) [e.NF] [a0.NF] [a.NF] (k m : ) :
                                              (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 : ) :
                                              (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+) :
                                              theorem ONote.repr_opow_aux₂ {a0 a' : ONote} [N0 : a0.NF] [Na' : a'.NF] (m : ) (d : Ordinal.omega0 a'.repr) (e0 : a0.repr 0) (h : a'.repr + m < Ordinal.omega0 ^ a0.repr) (n : ℕ+) (k : ) :
                                              have R := (opowAux 0 a0 (a0.oadd n a' * m) k m).repr; (k 0R < (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

                                              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
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                                                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
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                                                  theorem ONote.fundamentalSequenceProp_inr (o : ONote) (f : ONote) :
                                                  o.FundamentalSequenceProp (Sum.inr f) Order.IsSuccLimit o.repr (∀ (i : ), f i < f (i + 1) f i < o (o.NF(f i).NF)) a < o.repr, ∃ (i : ), a < (f i).repr
                                                  @[irreducible]

                                                  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
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                                                    theorem ONote.fastGrowing_def {o : ONote} {x : Option ONote (ONote)} (e : o.fundamentalSequence = x) :
                                                    o.fastGrowing = match (motive := (x : Option ONote (ONote)) → o.FundamentalSequenceProp x) x, with | Sum.inl none, x => Nat.succ | Sum.inl (some a), x => fun (i : ) => a.fastGrowing^[i] i | Sum.inr f, x => fun (i : ) => (f i).fastGrowing i
                                                    theorem ONote.fastGrowing_limit (o : ONote) {f : ONote} (h : o.fundamentalSequence = Sum.inr f) :
                                                    o.fastGrowing = fun (i : ) => (f i).fastGrowing i
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                                                    theorem ONote.fastGrowing_one :
                                                    fastGrowing 1 = fun (n : ) => 2 * n
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                                                    theorem ONote.fastGrowing_two :
                                                    fastGrowing 2 = fun (n : ) => 2 ^ 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.

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                                                      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.

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

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

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                                                          noncomputable def NONote.repr (o : NONote) :

                                                          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.

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

                                                            Convert a natural number to an ordinal notation

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                                                              Compare ordinal notations

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

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

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

                                                                  The oadd pseudo-constructor for NONote

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

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                                                                      Addition of ordinal notations

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

                                                                      Subtraction of ordinal notations

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

                                                                      Multiplication of ordinal notations

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

                                                                      Exponentiation of ordinal notations

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