Safety and conservativity of definitions in HOL and Isabelle/HOL

Kunčar, Ondřej; Popescu, Andrei

doi:10.1145/3158112

Cited by 16 publications

(13 citation statements)

References 48 publications

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“…They gave extended checks to avoid these circularities and proved that these checks ensure consistency of theories that extend the initial HOL theory (comprising the standard HOL axioms) with overloaded constant definitions and type definitions. In the end [KP18], they strengthened the result to show that such definitional extensions are in fact meta-safe over the initial HOL theory.…”

Section: Logical Foundationsmentioning

confidence: 85%

From LCF to Isabelle/HOL

Paulson

Nipkow

Wenzel³

2019

Form. Asp. Comput.

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Interactive theorem provers have developed dramatically over the past four decades, from primitive beginnings to today's powerful systems. Here, we focus on Isabelle/HOL and its distinctive strengths. They include automatic proof search, borrowing techniques from the world of first order theorem proving, but also the automatic search for counterexamples. They include a highly readable structured language of proofs and a unique interactive development environment for editing live proof documents. Everything rests on the foundation conceived by Robin Milner for Edinburgh LCF: a proof kernel, using abstract types to ensure soundness and eliminate the need to store proofs. Compared with the research prototypes of the 1970s, Isabelle is a practical and versatile tool. It is used by system designers, mathematicians and many others.Isabelle is a leading interactive theorem prover. It is generic, supporting a number of different formal calculi, but by far the most important of these is its instantiation to higher-order logic: Isabelle/HOL. Already during the 1990s, Isabelle/HOL was being applied with great success to the task of verifying cryptographic protocols [Pau98]. Turning to mathematics, it played a critical role in Hales's Flyspeck project, which verified his proof of the Kepler conjecture [HAB + 17]. It is the basis for the seL4 project, under which an entire operating system kernel was verified, proving full functional correctness [KAE + 10]. It was adopted by researchers outside the verification milieu for specifying and verifying algorithms for replicated datatypes that provide "eventual consistency" [GKMB17]. Numerous other projects are underway around the world. Like other proof assistants, Isabelle is not directly concerned with program verification, i.e. with verifying code written in a programming language, but it can be used as a back end to prove verification conditions. Isabelle can also be used as a verified programming environment, where mathematical functions can be proved correct and then automatically translated to executable code in one of several different programming languages. The translation process itself is currently unverified, but even this is likely to change in the near future.This essay focuses on Isabelle/HOL and therefore has little to say about techniques common to most systems. For example, simplification by rewriting-coupled with recursive simplification to handle conditional rewrite rules-was already realised in both the Boyer/Moore theorem prover [BM79] and Edinburgh LCF by the end of the 1970s. Recursive datatype and function definitions, as well as inductive definitions, were commonplace by around 1995. Linear arithmetic decision procedures were also widely available by then.Our title echoes Mike Gordon's paper "From LCF to HOL: a Short History" [Gor00]. Like Mike, we begin with LCF, the source of the most fundamental ideas. But we pass over this material quickly in order to focus on Isabelle. There is no way to surpass Mike's account of the early years. He starts in 1969, wi...

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Section: Logical Foundationsmentioning

confidence: 85%

From LCF to Isabelle/HOL

Paulson

Nipkow

Wenzel³

2019

Form. Asp. Comput.

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“…We follow his approach of reflecting type classes in the logic but cannot remove them completely because of our intentional implementation bias (see above). Kunčar and Popescu [15,16,17,18] focus on the subtleties of definition principles for HOL with overloading and prove that under certain conditions, type and constant definitions preserve consistency.Åman Pohjola et al [1] formalize [15,18].…”

Section: Related Workmentioning

confidence: 99%

Isabelle’s Metalogic: Formalization and Proof Checker

Nipkow

Roßkopf

2021

Automated Deduction – CADE 28

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Isabelle is a generic theorem prover with a fragment of higher-order logic as a metalogic for defining object logics. Isabelle also provides proof terms. We formalize this metalogic and the language of proof terms in Isabelle/HOL, define an executable (but inefficient) proof term checker and prove its correctness w.r.t. the metalogic. We integrate the proof checker with Isabelle and run it on a range of logics and theories to check the correctness of all the proofs in those theories.

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“…In another paper, they use an unfolding that stays in the logic of HOL [11] by relativising defined types in formulas to a predicate on the defined type's host type. The proof-theoretic conservativity result holds for any definitional theory unfolded into initial HOL, and motivates a dual model-theoretic conservativity result where any model of initial HOL can be extended to a model of a given definitional theory.…”

Section: Related Workmentioning

confidence: 99%

Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

Gengelbach

Pohjola

Weber

2021

Electron. Proc. Theor. Comput. Sci.

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Definitions of new symbols merely abbreviate expressions in logical frameworks, and no new facts (regarding previously defined symbols) should hold because of a new definition. In Isabelle/HOL, definable symbols are types and constants. The latter may be ad-hoc overloaded, i. e. have different definitions for non-overlapping types. We prove that symbols that are independent of a new definition may keep their interpretation in a model extension. This work revises our earlier notion of modeltheoretic conservative extension and generalises an earlier model construction. We obtain consistency of theories of definitions in higher-order logic (HOL) with ad-hoc overloading as a corollary. Our results are mechanised in the HOL4 theorem prover. Const c ty 1 # Const d ty 2 def = c = d ∨ ty 1 # ty 2. A user may introduce (non-built-in) types and constants by theory extension, as described in Section 2.3. For types and constants we generally say symbols. Built-ins We abbreviate Bool for Tyapp «bool» [ ] x → y for Tyapp «fun» [x; y] Equal ty for Const «=» (ty → ty → Bool) s === t for Comb (Comb (Equal (typeof s)) s) t Any type with any of these type constructors at the top level is built-in, as is the constant Equal. These are the only symbols which are not user-defined. A formula is a term of type Bool.

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Safety and conservativity of definitions in HOL and Isabelle/HOL

Cited by 16 publications

References 48 publications

From LCF to Isabelle/HOL

From LCF to Isabelle/HOL

Isabelle’s Metalogic: Formalization and Proof Checker

Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

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