Verification of the Redecoration Algorithm for Triangular Matrices

Abstract

Abstract. Triangular matrices with a dedicated type for the diagonal elements can be profitably represented by a nested datatype, i. e., a heterogeneous family of inductive datatypes. These families are fully supported since the version 8.1 of the Coq theorem proving environment, released in 2007. Redecoration of triangular matrices has a succinct implementation in this representation, thus giving the challenge of proving it correct. This has been achieved within Coq, using also induction with measures. An axiomatic approach allowed a verification in the Isabelle theorem prover, giving insights about the differences of both systems.

Terminal semantics for codata types in intensional Martin-Löf type theory

“…We make use of the notion of relative comonad, the dual to relative monads as introduced by [7]. One of our main examples, the codata type of infinite triangular matrices, is studied by [20]. Redecoration for both finite and infinite triangular matrices is used by [1] to exemplify the expressivity of the studied recursion schemes.…”

“…It is more sophisticated than the type of streams as one of its destructors is heterogeneous: Example 8. This codata type is studied in detail by [20]. We give a brief summary, but urge the reader to consult the given reference for an in-depth explanation.…”

“…Remark 15. A weak constructive comonad as defined by [20] to characterize the codata type Tri and redecoration on it, is precisely a comonad relative to the functor eq : Type → Setoid.…”

“…All our definitions and theorems are mechanized in the proof assistant Coq [11]. The formalization of infinite triangular matrices is taken from the work by [20], and only slightly adapted to compile with the version of Coq we use. The mechanization does not rely on any additional axioms.…”