User blog:Rgetar/Converting generalized Veblen function to Madore function

Definitions of eo, cofbeo, leo, lest see here.

asttr means "array shift to the right"

asttr(X) is array such as

eo(asttr(X); X1) = eo(X; 1 + X1) if cofbeo(X1) = 0 (that is X1 is ordinal)

eo(asttr(X); X1) = eo(X; X1) if cofbeo(X1) ≠ 0 (that is X1 is not ordinal)

ate means "array to exponent"

ate(0) = 0

ate(⟨X1⟩α1, ⟨X2⟩α2, ⟨X3⟩α3, ...) = Ωate(X1)α1 + Ωate(X2)α2 + Ωate(X3)α3

ata means "array to argument"

ata(1, α) = α

ata(1 + β, α) = Ωβ(1 + α)

ata(X) = Ωate(asttr(X))(1 + α) if cofbeo(X) > 1

vtm means "Veblen to Madore"

Let α - standard form of ordinal expressed using generalized Veblen function φ(X). Then vtm(α) is α expressed using Madore function ψ(α).

vtm(0) = 0

vtm(1) = 1

vtm(α, β) = vtm(α) + vtm(β)

vtm(ωα) = ωvtm(α)

If cofbeo(X) ≠ 0 and leo(X) = φ(X1) + β then

vtm(φ(X)) = ψ(ata(X1) + ata(lest(X; β)))