Documentation

Mathlib.SetTheory.Ordinal.CantorNormalForm

Cantor Normal Form #

The Cantor normal form of an ordinal is generally defined as its base ω expansion, with its non-zero exponents in decreasing order. Here, we more generally define a base b expansion Ordinal.CNF in this manner, which is well-behaved for any b ≥ 2.

Implementation notes #

We implement Ordinal.CNF as an association list, where keys are exponents and values are coefficients. This is because this structure intrinsically reflects two key properties of the Cantor normal form:

Todo #

Cantor normal form as a list #

@[irreducible]
def Ordinal.CNF.rec (b : Ordinal.{u_2}) {C : Ordinal.{u_2}Sort u_1} (H0 : C 0) (H : (o : Ordinal.{u_2}) → o 0C (o % b ^ log b o)C o) (o : Ordinal.{u_2}) :
C o

Inducts on the base b expansion of an ordinal.

Equations
Instances For
    @[deprecated Ordinal.CNF.rec (since := "2025-08-18")]
    def Ordinal.CNFRec (b : Ordinal.{u_2}) {C : Ordinal.{u_2}Sort u_1} (H0 : C 0) (H : (o : Ordinal.{u_2}) → o 0C (o % b ^ log b o)C o) (o : Ordinal.{u_2}) :
    C o

    Alias of Ordinal.CNF.rec.


    Inducts on the base b expansion of an ordinal.

    Equations
    Instances For
      @[simp]
      theorem Ordinal.CNF.rec_zero {C : Ordinal.{u_2}Sort u_1} (b : Ordinal.{u_2}) (H0 : C 0) (H : (o : Ordinal.{u_2}) → o 0C (o % b ^ log b o)C o) :
      CNF.rec b H0 H 0 = H0
      @[deprecated Ordinal.CNF.rec_zero (since := "2025-08-18")]
      theorem Ordinal.CNFRec_zero {C : Ordinal.{u_2}Sort u_1} (b : Ordinal.{u_2}) (H0 : C 0) (H : (o : Ordinal.{u_2}) → o 0C (o % b ^ log b o)C o) :
      CNF.rec b H0 H 0 = H0

      Alias of Ordinal.CNF.rec_zero.

      theorem Ordinal.CNF.rec_pos (b : Ordinal.{u_2}) {o : Ordinal.{u_2}} {C : Ordinal.{u_2}Sort u_1} (ho : o 0) (H0 : C 0) (H : (o : Ordinal.{u_2}) → o 0C (o % b ^ log b o)C o) :
      CNF.rec b H0 H o = H o ho (CNF.rec b H0 H (o % b ^ log b o))
      @[deprecated Ordinal.CNF.rec_pos (since := "2025-08-18")]
      theorem Ordinal.CNFRec_pos (b : Ordinal.{u_2}) {o : Ordinal.{u_2}} {C : Ordinal.{u_2}Sort u_1} (ho : o 0) (H0 : C 0) (H : (o : Ordinal.{u_2}) → o 0C (o % b ^ log b o)C o) :
      CNF.rec b H0 H o = H o ho (CNF.rec b H0 H (o % b ^ log b o))

      Alias of Ordinal.CNF.rec_pos.

      The Cantor normal form of an ordinal o is the list of coefficients and exponents in the base-b expansion of o.

      We special-case CNF 0 o = CNF 1 o = [(0, o)] for o ≠ 0.

      CNF b (b ^ u₁ * v₁ + b ^ u₂ * v₂) = [(u₁, v₁), (u₂, v₂)]

      Equations
      Instances For
        @[deprecated Ordinal.CNF.zero_right (since := "2025-08-18")]

        Alias of Ordinal.CNF.zero_right.

        theorem Ordinal.CNF.ne_zero {b o : Ordinal.{u_1}} (ho : o 0) :
        CNF b o = (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)

        Recursive definition for the Cantor normal form.

        @[deprecated Ordinal.CNF.ne_zero (since := "2025-08-18")]
        theorem Ordinal.CNF_ne_zero {b o : Ordinal.{u_1}} (ho : o 0) :
        CNF b o = (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)

        Alias of Ordinal.CNF.ne_zero.


        Recursive definition for the Cantor normal form.

        theorem Ordinal.CNF.zero_left {o : Ordinal.{u_1}} (ho : o 0) :
        CNF 0 o = [(0, o)]
        @[deprecated Ordinal.CNF.zero_left (since := "2025-08-18")]
        theorem Ordinal.zero_CNF {o : Ordinal.{u_1}} (ho : o 0) :
        CNF 0 o = [(0, o)]

        Alias of Ordinal.CNF.zero_left.

        theorem Ordinal.CNF.one_left {o : Ordinal.{u_1}} (ho : o 0) :
        CNF 1 o = [(0, o)]
        @[deprecated Ordinal.CNF.one_left (since := "2025-08-18")]
        theorem Ordinal.one_CNF {o : Ordinal.{u_1}} (ho : o 0) :
        CNF 1 o = [(0, o)]

        Alias of Ordinal.CNF.one_left.

        theorem Ordinal.CNF.of_le_one {b o : Ordinal.{u_1}} (hb : b 1) (ho : o 0) :
        CNF b o = [(0, o)]
        @[deprecated Ordinal.CNF.of_le_one (since := "2025-08-18")]
        theorem Ordinal.CNF_of_le_one {b o : Ordinal.{u_1}} (hb : b 1) (ho : o 0) :
        CNF b o = [(0, o)]

        Alias of Ordinal.CNF.of_le_one.

        theorem Ordinal.CNF.of_lt {b o : Ordinal.{u_1}} (ho : o 0) (hb : o < b) :
        CNF b o = [(0, o)]
        @[deprecated Ordinal.CNF.of_lt (since := "2025-08-18")]
        theorem Ordinal.CNF_of_lt {b o : Ordinal.{u_1}} (ho : o 0) (hb : o < b) :
        CNF b o = [(0, o)]

        Alias of Ordinal.CNF.of_lt.

        theorem Ordinal.CNF.foldr (b o : Ordinal.{u_1}) :
        List.foldr (fun (p : Ordinal.{u_1} × Ordinal.{u_1}) (r : Ordinal.{u_1}) => b ^ p.1 * p.2 + r) 0 (CNF b o) = o

        Evaluating the Cantor normal form of an ordinal returns the ordinal.

        @[deprecated Ordinal.CNF.foldr (since := "2025-08-18")]
        theorem Ordinal.CNF_foldr (b o : Ordinal.{u_1}) :
        List.foldr (fun (p : Ordinal.{u_1} × Ordinal.{u_1}) (r : Ordinal.{u_1}) => b ^ p.1 * p.2 + r) 0 (CNF b o) = o

        Alias of Ordinal.CNF.foldr.


        Evaluating the Cantor normal form of an ordinal returns the ordinal.

        theorem Ordinal.CNF.fst_le_log {b o : Ordinal.{u}} {x : Ordinal.{u} × Ordinal.{u}} :
        x CNF b ox.1 log b o

        Every exponent in the Cantor normal form CNF b o is less or equal to log b o.

        @[deprecated Ordinal.CNF.fst_le_log (since := "2025-08-18")]
        theorem Ordinal.CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal.{u} × Ordinal.{u}} :
        x CNF b ox.1 log b o

        Alias of Ordinal.CNF.fst_le_log.


        Every exponent in the Cantor normal form CNF b o is less or equal to log b o.

        theorem Ordinal.CNF.lt_snd {b o : Ordinal.{u}} {x : Ordinal.{u} × Ordinal.{u}} :
        x CNF b o0 < x.2

        Every coefficient in a Cantor normal form is positive.

        @[deprecated Ordinal.CNF.lt_snd (since := "2025-08-18")]
        theorem Ordinal.CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal.{u} × Ordinal.{u}} :
        x CNF b o0 < x.2

        Alias of Ordinal.CNF.lt_snd.


        Every coefficient in a Cantor normal form is positive.

        theorem Ordinal.CNF.snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal.{u} × Ordinal.{u}} :
        x CNF b ox.2 < b

        Every coefficient in the Cantor normal form CNF b o is less than b.

        @[deprecated Ordinal.CNF.snd_lt (since := "2025-08-18")]
        theorem Ordinal.CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal.{u} × Ordinal.{u}} :
        x CNF b ox.2 < b

        Alias of Ordinal.CNF.snd_lt.


        Every coefficient in the Cantor normal form CNF b o is less than b.

        theorem Ordinal.CNF.sorted (b o : Ordinal.{u_1}) :
        List.Sorted (fun (x1 x2 : Ordinal.{u_1}) => x1 > x2) (List.map Prod.fst (CNF b o))

        The exponents of the Cantor normal form are decreasing.

        @[deprecated Ordinal.CNF.sorted (since := "2025-08-18")]
        theorem Ordinal.CNF_sorted (b o : Ordinal.{u_1}) :
        List.Sorted (fun (x1 x2 : Ordinal.{u_1}) => x1 > x2) (List.map Prod.fst (CNF b o))

        Alias of Ordinal.CNF.sorted.


        The exponents of the Cantor normal form are decreasing.

        Cantor normal form as a finsupp #

        CNF.coeff b o is the finitely supported function returning the coefficient of b ^ e in the Cantor Normal Form (CNF) of o, for each e.

        Equations
        Instances For
          theorem Ordinal.CNF.coeff_of_mem_CNF {b o e c : Ordinal.{u_1}} (h : (e, c) CNF b o) :
          (coeff b o) e = c
          theorem Ordinal.CNF.coeff_of_not_mem_CNF {b o e : Ordinal.{u_1}} (h : eList.map Prod.fst (CNF b o)) :
          (coeff b o) e = 0