On Mon, 15 Jul 2013 12:01:24 -0700 (PDT), Dan Christensen <Dan_Christensen@sympatico.ca> wrote:
>No doubt re-inventing wheel, but the explanations of double induction online seem quite confusing to me. I'm not sure how widely applicable the following may be, but you may find the following analogy to ordinary induction to be useful. > >With ordinary induction, we want to prove that for all x in N, P(x) where P is a unary predicate. > >With double induction, we want to prove that for all x, y in N, P(x,y) where P is a binary predicate. > >1. Base case: > >Ordinary induction: Prove P(1) > >Double induction: Prove P(1,1) > >2. Inductive step: > >Ordinary induction: For x in N, suppose P(x) and prove P(x+1) > >Double induction: For x, y in N, suppose P(x,y) and prove P(x+1,y) and P(x,y+1). > >Comments?
There's no need for a separate formalism, not that it hurts. Anything using "double induction" as above can be proved using ordinary induction:
Say you want to prove P(n,m) for all natural numbers n and m. This is the same as proving Q(N) for every natural number N, where Q(N) is
"P(n,m) holds for all natural numbers n, m with n + m <= N".
You can, I hope, verify for yourself that it's possible to prove Q(N) by induction if and only if it's possible to prove P(n,m) by double induction.