Abstract
We mechanise a proof of Green’s theorem in Isabelle/HOL. We use a novel proof that avoids the ubiquitous line integral cancellation argument. This eliminates the need to formalise orientations and region boundaries explicitly with respect to the outwards-pointing normal vector. Instead we appeal to a homological argument about equivalences between paths. Contributions include mechanised theories of line integrals and partial derivatives, as well as the first mechanisation of Green’s theorem.
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Notes
The gauge integral [15] is a generalisation of the well-known Riemann integral.
Formally, this observation follows immediately from theorem
.This is not exactly true, since the instantiations
and
are obtained using a pair of orthonormal unit vectors
and
. If
and
are to be assigned to
and
, then the instantiations of
in
and
can be seen as the x-axis and y-axis Green statements.Rutter [13] explains curve speeds and velocities.
bitbucket.org/MohammadAbdulaziz/isabellegeometry/
.
and
are obtained using a pair of orthonormal unit vectors
and
. If
and
are to be assigned to
and
, then the instantiations of
in
and
can be seen as the x-axis and y-axis Green statements.References
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Acknowledgements
We thank Johannes Hoelzl, Fabian Immler and Manuel Eberl for their help with different aspects of the Analysis Library and for the useful discussions we had with them. We also note that the first author is supported by the DFG Koselleck Grant NI 491/16-1 and that the second author is supported by the ERC Advanced Grant ALEXANDRIA (Project 742178).
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Abdulaziz, M., Paulson, L.C. An Isabelle/HOL Formalisation of Green’s Theorem. J Autom Reasoning 63, 763–786 (2019). https://doi.org/10.1007/s10817-018-9495-z
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DOI: https://doi.org/10.1007/s10817-018-9495-z