User blog comment:B1mb0w/Strong D Function/@comment-1605058-20150702183610/@comment-1605058-20150808193152

What is meant here is that the term is built from other functions using repeated applications of substitutions. First, I'm going to handwave that it is possible to create, given m-ary functions $$f,g$$ and using primitive recursion, a m+1-ary function $$h$$ such that $$h(0,x_1,...,x_m)=f(x_1,...,x_m),h(i,x_1,...,x_m)=g(x_1,...,x_m)$$ for $$i>0$$. This assures us that we can work by cases.

Now this is how you can define $$D(x_1,x_2)$$: using work by cases, as above, define it to be $$x_2+1$$ for $$x_1=0$$, $$D(x_1-1,D(x_1-1,x_1))$$ if $$x_2=0$$, and otherwise $$D(x_1-1,D(x_1,x_2-1))$$. Similarly you define $$D(x_1,x_2,x_3)$$.