Find a function $f(x,y) : \mathbb{N}^2 \to \mathbb{N}^2$ such that:
$1 + f(x+1,y) - f(x,y) = a$
$1 + f(x, y+1) - f(x,y) = b$
$k + f(x-k, y) - f(x,y) = c \space \forall k \leq x$
$k + f(x,y-k) - f(x,y) = d \space \forall k \leq y$
$k + f(x-k, y+k) - f(x,y) = e \space \forall k \leq x$
where $a,b,c,d,e$ are some constants.
$f(x,y) = x+y$ works for the first 4 conditions,but I've been trying to find one that will satisfy the fifth as well or prove no such function can exist and I'm stuck.
Any help is appreciated.