ordinal data structure, nonnegative integer, polynomial with positive integer coefficients, or $\mathrm{NULL}$.

The Dec(a) calling sequence decrements the ordinal number $a$, if possible. If $a=0$, then the return value is $\mathrm{NULL}$. Otherwise, if the trailing term is ${\mathbf{\omega }}^e\cdot c$, where $e$ is an ordinal and $c$ is a positive integer, then exactly one of the following happens:

If $e=0$ then $c$ is replaced by $c-1$.

If $e\ne 0$ and $c\ne 1$, then the trailing term is replaced by the sum of the two terms ${\mathbf{\omega }}^e\cdot \left(c-1\right)\+{\mathbf{\omega }}^{\mathrm{Dec}\left(e\right)}$.

Otherwise, if $e\ne 0$ and $c=1$, then the trailing exponent is replaced by $\mathrm{Dec}\left(e\right)$.

Note that in general $\mathrm{Dec}\left(a\right)$ is not the largest ordinal number smaller than $a$, because such an ordinal does not exist if $a$ is a limit ordinal, which means its trailing degree is nonzero.

If $a$ is a parametric ordinal number and $c-1$ is not a polynomial with nonnegative integer coefficients, an error is raised.

$\mathrm{with}\left(\mathrm{Ordinals}\right)\:$

$a≔\mathrm{Ordinal}\left(\left[\left[\mathrm{\omega }\,2\right]\,\left[2\,3\right]\,\left[0\,4\right]\right]\right)$

$a≔{\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+{\mathbf{\omega }}^2\cdot 3\+4$

$$\begin{gathered} \mathbf{while}a\ne 0\mathbf{do} \\ a≔\mathrm{Dec}\left(a\right); \\ \mathrm{print}\left(a\right) \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+{\mathbf{\omega }}^2\cdot 3\+3$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+{\mathbf{\omega }}^2\cdot 3\+2$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+{\mathbf{\omega }}^2\cdot 3\+1$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+{\mathbf{\omega }}^2\cdot 3$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+{\mathbf{\omega }}^2\cdot 2\+\mathbf{\omega }$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+{\mathbf{\omega }}^2\cdot 2\+1$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+{\mathbf{\omega }}^2\cdot 2$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+{\mathbf{\omega }}^2\+\mathbf{\omega }$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+{\mathbf{\omega }}^2\+1$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+{\mathbf{\omega }}^2$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+\mathbf{\omega }$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2\+1$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 2$

${\mathbf{\omega }}^{\mathbf{\omega }}\+\mathbf{\omega }$

${\mathbf{\omega }}^{\mathbf{\omega }}\+1$

${\mathbf{\omega }}^{\mathbf{\omega }}$

$\mathbf{\omega }$

$1$

$0$

$\mathrm{Dec}\left(5\right)$

$4$

$\mathrm{Dec}\left(x+3\right)$

$x+2$

$b≔\mathrm{Ordinal}\left(\left[\left[1\,3\right]\,\left[0\,x^2+x+2\right]\right]\right)$

$b≔\mathbf{\omega }\cdot 3\+\left(x^2+x+2\right)$

$\mathrm{Dec}\left(\right)$

$\mathbf{\omega }\cdot 3\+\left(x^2+x+1\right)$

$\mathrm{Dec}\left(\right)$

$\mathbf{\omega }\cdot 3\+\left(x^2+x\right)$

$\mathrm{Dec}\left(\right)$

Error, (in Ordinals:-Dec) cannot decrement, x^2+x

The Ordinals[Dec] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.