Prime Number: A natural number n is called prime iff n > 1 and its only positive divisors are 1 and n.
Claim: (\forall n \in \mathbb{N}) with n > 1 n is either prime or a product of primes.
Proof: We proceed by strong induction on n\in \mathbb{Z} \land n \geq 2. Define P(n) to be the variable proposition "n either prime or a product of primes"
Base Case: when n=2, the only possible divisor is 1 and 2. n is prime
Inductive Step: Let k \in \mathbb{N} with k \geq 2, assume P(i) holds for all i\in\mathbb{N} with 2 \leq i \leq k. let n=k+1
Case if k+1 is not prime: then let p, q \in \mathbb{N}, with 2\leq p \leq q \leq k, and k+1=pq. Fix p and q. By IH, p and q are either prime or a product of primes. Then (\exists a, b \in \mathbb{N})(p=ab) So we are good.
(break k+1 to k+1=a*b, then assume a=product_of_prime_or_prime and b=product_of_prime_or_prime, k+1 must equal to product of prime)
Claim: If Player I begins the game by choosing square (1, n) the top right corner, then Player 1 can always win the 2 x n game of Chomp. Proof: by strong induction. Let P(n) be the proposition.
Base Case: for 2 x 1 game of Chomp. Player 1 will win because...
Inductive Step
This is a strong induction because it require 2 x 1 is true to 2 x k.
Definition: A set X is called well-ordered iff every nonempty subset has a least element.
Example: \mathbb{N}, 2^n|n\in \mathbb{N}
Non-Example: \mathbb{Z}, [0, 1), \mathbb{R}
Claim: \mathbb{N} is well-ordered Proof: by SPMI for all n\in\mathbb{N}. Let P(n) be variable proposition "every subsets of N containing n has a least element" (read: assume the subset containing n has the property, prove subset containing all number has property)
Base: P(0) holds, cuz 0 is the least in N by definition (base case is crucial here)
IS:
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