IntegerPart[x] == x + 1 - Mod[x, 1] /; Element[x, Reals] && x < 0 && !Element[x, Integers]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["IntegerPart", "[", "x", "]"]], "\[Equal]", RowBox[List["x", "+", "1", "-", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["x", "\[Element]", "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> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> <mo> - </mo> <semantics> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </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> <ci> x </ci> </apply> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> <apply> <notin /> <ci> x </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["IntegerPart", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["x", "+", "1", "-", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]], "&&", RowBox[List["!", RowBox[List["x", "\[Element]", "Integers"]]]]]]]]]]]]

2007-05-02