General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

26 CHAPTER 3. ELEMENTARY DISCRETE MATH
1. the distance between Si and S s(i) is smaller than the height of Si,
2. Si is much higher than wide, so it is unstable, and
3. Si and S s(i) have the same weight.
If S0 is pushed towards S1 so that it falls, then all dominos will fall.
The Domino Induction
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©: Michael Kohlhase 37
Proof: We prove the assertion by induction over i with the property P that “Si falls in the
direction of S s(i)”.
P.1 We have to consider two cases
P.1.1 base case: i is zero:
P.1.1.1 We have assumed that “S0 is pushed towards S1, so that it falls”
P.1.2 step case: i = s(j) for some unary natural number j:
P.1.2.1 We assume that P holds for Sj, i.e. Sj falls in the direction of S s(j) = Si.
P.1.2.2 But we know that Sj has the same weight as Si, which is unstable,
P.1.2.3 so Si falls into the direction opposite to Sj, i.e. towards S s(i) (we have a linear
sequence of dominos)
P.2 We have considered all the cases, so we have proven that P holds for all unary natural
numbers i. (by induction)
P.3 Now, the assertion follows trivially, since if “Si falls in the direction of S s(i)”, then in
particular “Si falls”.
©: Michael Kohlhase 38
If we look closely at the proof above, we see another recurring pattern. To get the proof to go
through, we had to use a property P that is a little stronger than what we need for the assertion
alone. In effect, the additional clause “... in the direction ...” in property P is used to make the
step condition go through: we we can use the stronger inductive hypothesis in the proof of step
case, which is simpler.
Often the key idea in an induction proof is to find a suitable strengthening of the assertion to
get the step case to go through.
3.3 Defining Operations on Natural Numbers
The next thing we want to do is to define operations on unary natural numbers, i.e. ways to
do something with numbers. Without really committing what “operations” are, we build on the
intuition that they take (unary natural) numbers as input and return numbers. The important
thing in this is not what operations are but how we define them.