### Abstract

Lenses provide an abstract interface for manipulating data types
through spatially-separated views. They are defined abstractly in
terms of two functions,

BSD License*get*, the return a value from the source type, and*put*that updates the value. We mechanise the underlying theory of lenses, in terms of an algebraic hierarchy of lenses, including well-behaved and very well-behaved lenses, each lens class being characterised by a set of lens laws. We also mechanise a lens algebra in Isabelle that enables their composition and comparison, so as to allow construction of complex lenses. This is accompanied by a large library of algebraic laws. Moreover we also show how the lens classes can be applied by instantiating them with a number of Isabelle data types.### Change history

[2020-03-02] Added partial bijective and symmetric lenses. Improved alphabet command generating additional lenses and results. Several additional lens relations, including observational equivalence. Additional theorems throughout. Adaptations for Isabelle 2020. (revision 44e2e5c) [2021-01-27] Addition of new theorems throughout, particularly for prisms. New "chantype" command allows the definition of an algebraic datatype with generated prisms. New "dataspace" command allows the definition of a local-based state space, including lenses and prisms. Addition of various examples for the above. (revision 89cf045a)