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

Theorem 36 is a very useful fact to know, it tells us something about the form of unary natural
numbers, which lets us streamline induction proofs and bring them more into the form you may
know from school: to show that some property P holds for every natural number, we analyze an
arbitrary number n by its form in two cases, either it is zero (the base case), or it is a successor of
another number (the step case). In the first case we prove the base condition and in the latter, we
prove the step condition and use the induction axiom to conclude that all natural numbers have
property P . We will show the form of this proof in the domino-induction below.
The Domino Theorem
Theorem 37 Let S0, S1, . . . be a linear sequence of dominos, such that for any unary natural
number i we know that
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
• • • • • •
c○: Michael Kohlhase 36
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”.
c○: Michael Kohlhase 37
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
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