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- Johannes Hölzl, Armin Heller
- ITP
- 2011

Currently published HOL formalizations of measure theory concentrate on the Lebesgue integral and they are restricted to realvalued measures. We lift this restriction by introducing the extended real numbers. We define the Borel σ-algebra for an arbitrary type forming a topological space. Then, we introduce measure spaces with extended real numbers as… (More)

- Fabian Immler, Johannes Hölzl
- ITP
- 2012

Many ordinary differential equations (ODEs) do not have a closed solution, therefore approximating them is an important problem in numerical analysis. This work formalizes a method to approximate solutions of ODEs in Isabelle/HOL. We formalize initial value problems (IVPs) of ODEs and prove the existence of a unique solution, i.e. the Picard-Lindelöf… (More)

- Jeremy Avigad, Johannes Hölzl, Luke Serafin
- Journal of Automated Reasoning
- 2017

We describe a proof of the Central Limit Theorem that has been formally verified in the Isabelle proof assistant. Our formalization builds upon and extends Isabelle’s libraries for analysis and measure-theoretic probability. The proof of the theorem uses characteristic functions, which are a kind of Fourier transform, to demonstrate that, under suitable… (More)

Sparse matrix formats are typically implemented with low-level imperative programs. The optimized nature of these implementations hides the structural organization of the sparse format and complicates its verification. We define a variable-free functional language (LL) in which even advanced formats can be expressed naturally, as a pipeline-style… (More)

- Johannes Hölzl, Tobias Nipkow
- QFM
- 2012

Probabilistic model checkers like PRISM only check probabilistic systems of a fixed size. To guarantee the desired properties for an arbitrary size, mathematical analysis is necessary. We show for two case studies how this can be done in the interactive proof assistant Isabelle/HOL. The first case study is a detailed description of how we verified… (More)

- Johannes Hölzl
- ITP
- 2016

We extended Isabelle/HOL with a pair of definitional commands for datatypes and codatatypes. They support mutual and nested (co)recursion through well-behaved type constructors, including mixed recursion–corecursion, and are complemented by syntaxes for introducing primitive (co)recursive functions and by a general proof method for reasoning coinductively.… (More)

- Johannes Hölzl, Fabian Immler, Brian Huffman
- ITP
- 2013

The theory of analysis in Isabelle/HOL derives from earlier formalizations that were limited to specific concrete types: R, C and R. Isabelle’s new analysis theory unifies and generalizes these earlier efforts. The improvements are centered on two primary contributions: a generic theory of limits based on filters, and a new hierarchy of type classes that… (More)

- Fabian Immler, Johannes Hölzl
- Archive of Formal Proofs
- 2012

- Andreas Lochbihler, Johannes Hölzl
- ITP
- 2014