Mechanizing the metatheory of sledgehammer

This paper presents an Isabelle/HOL formalization of recent research in automated reasoning: efficient encodings of sorts in unsorted first-order logic, as implemented in Isabelle’s Sledgehammer proof tool. The formalization provides the general-purpose machinery to reason about formulas and models, emulating the theory of institutions. Quantifiers are represented using a nominal-like approach designed for interpreting syntax in semantic domains

Housing markets and independence in old age: expanding the opportunities

This report highlights the benefits of specialised private retirement accommodation and recommends a number of simple policy changes at no cost to the public purse to help increase its supply and address the challenges of housing an ageing population

A Decision Support Tool For Thrift Savings Plan Investors

The Thrift Savings Plan is-the defined contribution retirement plan for federal government employees. It is one of the largest retirement plans in the United States. The plan offers five different investment options: two funds concentrate on government securities and corporate bonds, two funds span the United States stock market, and one fund focuses on international stocks. These funds give investors the opportunity to diversify among a wide range of securities. This thesis examines the funds offered by the plan and creates a portfolio selection tool that uses investor inputs. The tool uses three mathematical models: optimization, based on Markowitz\u27s (1952) Modern Portfolio Theory; simulation, based on the combination of optimization results and investor profiles; and selection, based on the simulation results and investor preferences. Results are presented for four investors. Other investors can use these results to gain insight, but the greatest benefit is derived from individual implementation. The tool requires only a few user inputs, and it can be operated without any external assistance. The research has further found that one investment option should not be part of any portfolio, and some portfolios can be risky to individuals for differing reasons

Foundational, compositional (co)datatypes for higher-order logic: category theory applied to theorem proving

Interactive theorem provers based on higher-order logic (HOL) traditionally follow the definitional approach, reducing high-level specifications to logical primitives. This also applies to the support for datatype definitions. However, the internal datatype construction used in HOL4, HOL Light, and Isabelle/HOL is fundamentally noncompositional, limiting its efficiency and flexibility, and it does not cater for codatatypes. We present a fully modular framework for constructing (co)datatypes in HOL, with support for mixed mutual and nested (co)recursion. Mixed (co)recursion enables type definitions involving both datatypes and codatatypes, such as the type of finitely branching trees of possibly infinite depth. Our framework draws heavily from category theory. The key notion is that of a bounded natural functor—an enriched type constructor satisfying specific properties preserved by interesting categorical operations. Our ideas are implemented as a definitional package in Isabelle, addressing a frequent request from users

Witnessing (co)datatypes

Datatypes and codatatypes are useful for specifying and reasoning about (possibly infinite) computational processes. The Isabelle/HOL proof assistant has recently been extended with a definitional package that supports both. We describe a complete procedure for deriving nonemptiness witnesses in the general mutually recursive, nested case—nonemptiness being a proviso for introducing types in higher-order logic

Unified classical logic completeness: a coinductive pearl

Codatatypes are absent from many programming languages and proof assistants. We make a case for their importance by revisiting a classic result: the completeness theorem for first-order logic established through a Gentzen system. The core of the proof establishes an abstract property of possibly infinite derivation trees, independently of the concrete syntax or inference rules. This separation of concerns simplifies the presentation. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems as well as various flavors of first order logic. The corresponding Isabelle/HOL formalization demonstrates the recently introduced support for codatatypes and the Haskell code generator

Soundness and completeness proofs by coinductive methods

We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL’s recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL’s new coinductive specification language such as nesting through non-free types and mixed recursion–corecursion

SAT-Inspired Higher-Order Eliminations

We generalize several propositional preprocessing techniques to higher-orderlogic, building on existing first-order generalizations. These techniqueseliminate literals, clauses, or predicate symbols from the problem, with theaim of making it more amenable to automatic proof search. We also introduce anew technique, which we call quasipure literal elimination, that strictlysubsumes pure literal elimination. The new techniques are implemented in theZipperposition theorem prover. Our evaluation shows that they sometimes helpprove problems originating from the TPTP library and Isabelle formalizations.<br

Cardinals in Isabelle/HOL

We report on a formalization of ordinals and cardinals in Isabelle/HOL. A main challenge we faced was the inability of higher-order logic to represent ordinals canonically, as transitive sets (as done in set theory). We resolved this into a “decentralized” representation identifying ordinals with well-orders, with all concepts and results proved to be invariant under order isomorphism. We also discuss several applications of this general theory in formal developments

Foundational extensible corecursion: a proof assistant perspective

This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under “friendly” operations, including constructors. Friendly corecursive functions can be registered as such, thereby increasing the corecursor’s expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic