The Eval(o, x=v) calling sequence substitutes the value $v$ for the parameter $x$ in the ordinal $o$, returning either an ordinal data structure, a nonnegative integer, or a polynomial with positive integer coefficients.

It is possible for $v$ to be a negative integer or a polynomial with some negative integer coefficients, provided that the result is a valid ordinal, which means it does not have any negative integer coefficients.

The resulting ordinal is simplified, namely, any coefficients that become zero are removed, and if only a single term with exponent $0$ is left after that, a nonnegative integer or a polynomial with positive integer coefficients is returned.

The Eval(o, l) calling sequence performs all the substitutions in l simultaneously.

This command can also be applied to a polynomial with positive integer coefficients representing a nonnegative integer ordinal.

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

$\left[\mathrm{`+`}\&comma;\mathrm{`.`}\&comma;\mathrm{`<`}\&comma;\mathrm{<=}\&comma;\mathrm{Add}\&comma;\mathrm{Base}\&comma;\mathrm{Dec}\&comma;\mathrm{Decompose}\&comma;\mathrm{Div}\&comma;\mathrm{Eval}\&comma;\mathrm{Factor}\&comma;\mathrm{Gcd}\&comma;\mathrm{Lcm}\&comma;\mathrm{LessThan}\&comma;\mathrm{Log}\&comma;\mathrm{Max}\&comma;\mathrm{Min}\&comma;\mathrm{Mult}\&comma;\mathrm{Ordinal}\&comma;\mathrm{Power}\&comma;\mathrm{Split}\&comma;\mathrm{Sub}\&comma;\mathrm{`^`}\&comma;\mathrm{degree}\&comma;\mathrm{lcoeff}\&comma;\log \&comma;\mathrm{lterm}\&comma;\mathrm{\omega }\&comma;\mathrm{quo}\&comma;\mathrm{rem}\&comma;\mathrm{tcoeff}\&comma;\mathrm{tdegree}\&comma;\mathrm{tterm}\right]$

$\mathrm{o1}≔\mathrm{Ordinal}\left(\left[\left[\mathrm{\omega }\&comma;x\right]\&comma;\left[2\&comma;3\right]\&comma;\left[1\&comma;y+1\right]\&comma;\left[0\&comma;4\right]\right]\right)$

$\mathrm{o1}≔{\mathbf{\omega }}^{\mathbf{\omega }}\cdot x\&plus;{\mathbf{\omega }}^2\cdot 3\&plus;\mathbf{\omega }\cdot \left(y+1\right)\&plus;4$

$\mathrm{Eval}\left(\mathrm{o1}\&comma;x=0\right)$

${\mathbf{\omega }}^2\cdot 3\&plus;\mathbf{\omega }\cdot \left(y+1\right)\&plus;4$

$\mathrm{Eval}\left(\mathrm{o1}\&comma;\left[x=x^2+1\&comma;y=4\right]\right)$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot \left(x^2+1\right)\&plus;{\mathbf{\omega }}^2\cdot 3\&plus;\mathbf{\omega }\cdot 5\&plus;4$

$\mathrm{o2}≔\mathrm{Ordinal}\left(\left[\left[2\&comma;x^2+x\right]\&comma;\left[1\&comma;x\right]\&comma;\left[0\&comma;4\right]\right]\right)$

$\mathrm{o2}≔{\mathbf{\omega }}^2\cdot \left(x^2+x\right)\&plus;\mathbf{\omega }\cdot x\&plus;4$

$\mathrm{Eval}\left(\mathrm{o2}\&comma;x=0\right)$

$4$

$\mathrm{Eval}\left(\mathrm{\omega }·x\&comma;x=0\right)$

$0$

$\mathrm{o3}≔\mathrm{Ordinal}\left(\left[\left[1\&comma;2x+2\right]\&comma;\left[0\&comma;3\right]\right]\right)$

$\mathrm{o3}≔\mathbf{\omega }\cdot \left(2x+2\right)\&plus;3$

$\mathrm{Eval}\left(\mathrm{o3}\&comma;x=-1\right)$

$3$

$\mathrm{Eval}\left(\mathrm{o3}\&comma;x=-2\right)$

Error, (in Ordinals:-Eval) invalid substitution; result is not a valid ordinal

$\mathrm{Eval}\left(\mathrm{o3}\&comma;x=x-1\right)$

$\mathbf{\omega }\cdot \left(2x\right)\&plus;3$

$\mathrm{Eval}\left(\mathrm{o3}\&comma;x=x-2\right)$

Error, (in Ordinals:-Eval) invalid substitution; result is not a valid ordinal

$\mathrm{Eval}\left(x^2+1\&comma;x=3\right)=\mathrm{eval}\left(x^2+1\&comma;x=3\right)$

$10=10$

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

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