
Proposed Schema for Double Induction
Posted:
Jul 18, 2013 12:33 AM


THE PROPOSED SCHEMA
To prove for all x, y in N, we have P(x,y) where P is a binary predicate:
Step 1: Prove by ordinary induction that for all x in N, P(1,x).
Step 2: Prove for all x, y in N, if P(x,y) then P(x+1,y)
EXAMPLE
Given the associativity of natural number addition, prove its commutativity.
Here, we have P(x,y) <> x+y = y+x
Step 1: Prove that for all x in N, P(1,x), i.e. 1+x=x+1
Prove P(1,1), i.e. 1+1 = 1+1 (reflexivity) Suppose P(1,x), i.e. 1+x = x+1. Prove P(1,x+1), i.e. 1+(x+1) = x+1+1. x+1+1 = 1+x+1 = 1+(x+1)
Step 2: Show for all x, y in N, if P(x,y) then P(x+1,y), i.e. x+y = y+x > x+1+y = y+(x+1)
Suppose P(x,y), i.e. x+y = y+x. Show P(x+1,y), i.e. x+1+y = y+(x+1) y+(x+1) = y+x+1 = x+y+1 = x+(y+1) = x+(1+y) = x+1+y
Comments?
Dan Download my DC Proof 2.0 software at http://www.dcproof.com

