Binary digit-sum inequality ($F(a+b)\ge F(a)+F(b)+\min (a,b)$)

For $F(a)={\sum }_{j=0}^{a-1}{s}_2(j)$ and nonnegative integers $a,b$, the displayed inequality follows by induction from $F(2r)=2F(r)+r$ and $F(2r+1)=F(r)+F(r+1)+r$. Splitting a vertex set of a hypercube graph into its two coordinate sections then proves the edge-isoperimetric theorem for binary initial segments.