The command IsStronglyNormalized(rc, R) returns true if and only if rc is strongly normalized.

This command is part of the RegularChains[ChainTools] package, so it can be used in the form IsStronglyNormalized(..) only after executing the command with(RegularChains[ChainTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][IsStronglyNormalized](..).

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

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

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,z\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{pz}≔\left(z-1\right)\left(z-2\right)$

$\mathrm{pz}≔\left(z-1\right)\left(z-2\right)$

$\mathrm{py}≔y^2\+z$

$\mathrm{py}≔y^2+z$

$\mathrm{px1}≔x\+y\+z$

$\mathrm{px1}≔x+y+z$

$\mathrm{px2}≔zx\+y\+z$

$\mathrm{px2}≔zx+y+z$

$\mathrm{rc}≔\mathrm{Empty}\left(R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{rc1}≔\mathrm{Chain}\left(\left[\mathrm{pz}\,\mathrm{py}\,\mathrm{px1}\right]\,\mathrm{rc}\,R\right)$

$\mathrm{rc1}≔\mathrm{regular\_chain}$

$\mathrm{IsStronglyNormalized}\left(\mathrm{rc1}\,R\right)$

$\mathrm{true}$

$\mathrm{rc2}≔\mathrm{Chain}\left(\left[\mathrm{pz}\,\mathrm{py}\,\mathrm{px2}\right]\,\mathrm{rc}\,R\right)$

$\mathrm{rc2}≔\mathrm{regular\_chain}$

$\mathrm{IsStronglyNormalized}\left(\mathrm{rc2}\,R\right)$

$\mathrm{false}$