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Topic: Proposed Schema for Double Induction
Replies: 4   Last Post: Jul 18, 2013 12:00 PM

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Dan Christensen

Posts: 8,219
Registered: 7/9/08
Proposed Schema for Double Induction
Posted: Jul 18, 2013 12:33 AM
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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)


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


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