Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 233 — #241
7.5. Induction in Computer Science 233
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Figure 7.3
DET T 0 from Four Copies of DET T
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Figure 7.4
Trapezoid from Three Triangles
Properties of DETs were proved earlier by induction on the length of a side of the
triangle. Recognizing that the definition of DETs is recursive, we can instead prove
properties of DETs by structural induction.
(a) Prove by structural induction that a DET with one of its corner subtriangles
removed can be tiled with trapezoids built out of three subtriangles as in Figure 7.4.
(b) Explain why a DET with a triangle removed from the middle of one side can
also be tiled by trapezoids.
(c) In tiling a large square using L-shaped blocks as described in Section 5.1.5,
there was a tiling with any single subsquare removed. Part (b) indicates that trapezoidtilings
are possible for DETs with a non-corner subtriangle removed, so it’s natural
to make the mistaken guess that DETs have a corresponding property:
False Claim. A DET with any single subtriangle removed can be trapezoid-tiled.
We can try to prove the claim by structural induction as in part (a).
Bogus proof. The claim holds vacuously in the base case of a DET with a single
subtriangle.
Now let T 0 be a DET made of four copies of a DET T , and suppose we remove an
arbitrary subtriangle from T 0 .
The removed subtriangle must be a subtriangle of one of the copies of T . The
copies are the same, so for definiteness we assume the subtriangle was removed