IntegerPart[z + n] == IntegerPart[z] + n - UnitStep[z + n] + UnitStep[z] /; Element[n, Integers] && !Element[z, Integers]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["IntegerPart", "[", RowBox[List["z", "+", "n"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["IntegerPart", "[", "z", "]"]], "+", "n", "-", RowBox[List["UnitStep", "[", RowBox[List["z", "+", "n"]], "]"]], "+", RowBox[List["UnitStep", "[", "z", "]"]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["z", ",", "Integers"]], "]"]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> int </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> int </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mrow> <mi> &#952; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#952; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mtext> &#8203; </mtext> <mrow> <mi> z </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> int </ci> <apply> <plus /> <ci> n </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <ci> int </ci> <ci> z </ci> </apply> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> &#952; </ci> <apply> <plus /> <ci> n </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> &#952; </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <notin /> <ci> z </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["IntegerPart", "[", RowBox[List["z_", "+", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["IntegerPart", "[", "z", "]"]], "+", "n", "-", RowBox[List["UnitStep", "[", RowBox[List["z", "+", "n"]], "]"]], "+", RowBox[List["UnitStep", "[", "z", "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["!", RowBox[List["z", "\[Element]", "Integers"]]]]]]]]]]]]

2001-10-29