2 research outputs found
Formal study of plane Delaunay triangulation
This article presents the formal proof of correctness for a plane Delaunay
triangulation algorithm. It consists in repeating a sequence of edge flippings
from an initial triangulation until the Delaunay property is achieved. To
describe triangulations, we rely on a combinatorial hypermap specification
framework we have been developing for years. We embed hypermaps in the plane by
attaching coordinates to elements in a consistent way. We then describe what
are legal and illegal Delaunay edges and a flipping operation which we show
preserves hypermap, triangulation, and embedding invariants. To prove the
termination of the algorithm, we use a generic approach expressing that any
non-cyclic relation is well-founded when working on a finite set
Intuition in formal proof : a novel framework for combining mathematical tools
This doctoral thesis addresses one major difficulty in formal proof: removing obstructions
to intuition which hamper the proof endeavour. We investigate this in the context
of formally verifying geometric algorithms using the theorem prover Isabelle, by first
proving the Graham’s Scan algorithm for finding convex hulls, then using the challenges
we encountered as motivations for the design of a general, modular framework
for combining mathematical tools.
We introduce our integration framework — the Prover’s Palette, describing in detail
the guiding principles from software engineering and the key differentiator of our
approach — emphasising the role of the user. Two integrations are described, using
the framework to extend Eclipse Proof General so that the computer algebra systems
QEPCAD and Maple are directly available in an Isabelle proof context, capable of running
either fully automated or with user customisation. The versatility of the approach
is illustrated by showing a variety of ways that these tools can be used to streamline the
theorem proving process, enriching the user’s intuition rather than disrupting it. The
usefulness of our approach is then demonstrated through the formal verification of an
algorithm for computing Delaunay triangulations in the Prover’s Palette