238 research outputs found
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ISABELLE - THE NEXT 700 THEOREM PROVERS
Isabelle is a generic theorem prover, designed for interactive reasoning in a
variety of formal theories. At present it provides useful proof procedures for
Constructive Type Theory, various first-order logics, Zermelo-Fraenkel set
theory, and higher-order logic. This survey of Isabelle serves as an
introduction to the literature. It explains why generic theorem proving is
beneficial. It gives a thorough history of Isabelle, beginning with its origins
in the LCF system. It presents an account of how logics are represented,
illustrated using classical logic. The approach is compared with the Edinburgh
Logical Framework. Several of the Isabelle object-logics are presented
The foundation of a generic theorem prover
Isabelle is an interactive theorem prover that supports a variety of logics.
It represents rules as propositions (not as functions) and builds proofs by
combining rules. These operations constitute a meta-logic (or `logical
framework') in which the object-logics are formalized. Isabelle is now based on
higher-order logic -- a precise and well-understood foundation. Examples
illustrate use of this meta-logic to formalize logics and proofs. Axioms for
first-order logic are shown sound and complete. Backwards proof is formalized
by meta-reasoning about object-level entailment. Higher-order logic has several
practical advantages over other meta-logics. Many proof techniques are known,
such as Huet's higher-order unification procedure
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A MACHINE-ASSISTED PROOF OF GĂDEL'S INCOMPLETENESS THEOREMS FOR THE THEORY OF HEREDITARILY FINITE SETS
A formalisation of G\"odel's incompleteness theorems using the Isabelle proof
assistant is described. This is apparently the first mechanical verification of
the second incompleteness theorem. The work closely follows {\'S}wierczkowski
(2003), who gave a detailed proof using hereditarily finite set theory. The
adoption of this theory is generally beneficial, but it poses certain technical
issues that do not arise for Peano arithmetic. The formalisation itself should
be useful to logicians, particularly concerning the second incompleteness
theorem, where existing proofs are lacking in detail.This is the author accepted manuscript. The final version is available from Cambridge University Press via https://doi.org/10.1017/S175502031400011
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Proving termination of normalization functions for conditional expressions
Boyer and Moore have discussed a recursive function that puts conditional
expressions into normal form [1]. It is difficult to prove that this function
terminates on all inputs. Three termination proofs are compared: (1) using a
measure function, (2) in domain theory using LCF, (3) showing that its
recursion relation, defined by the pattern of recursive calls, is well-founded.
The last two proofs are essentially the same though conducted in markedly
different logical frameworks. An obviously total variant of the normalize
function is presented as the `computational meaning' of those two proofs. A
related function makes nested recursive calls. The three termination proofs
become more complex: termination and correctness must be proved simultaneously.
The recursion relation approach seems flexible enough to handle subtle
termination proofs where previously domain theory seemed essential
NATURAL DEDUCTION AS HIGHER-ORDER RESOLUTION
An interactive theorem prover, Isabelle, is under development. In LCF, each
inference rule is represented by one function for forwards proof and another (a
tactic) for backwards proof. In Isabelle, each inference rule is represented by
a Horn clause. Resolution gives both forwards and backwards proof, supporting a
large class of logics. Isabelle has been used to prove theorems in
Martin-L\"of's Constructive Type Theory. Quantifiers pose several difficulties:
substitution, bound variables, Skolemization. Isabelle's representation of
logical syntax is the typed lambda-calculus, requiring higher- order
unification. It may have potential for logic programming. Depth-first
subgoaling along inference rules constitutes a higher-order Prolog
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A Mechanised Proof of GĂśdelâs Incompleteness Theorems Using Nominal Isabelle
An Isabelle/HOL formalisation of G\"odel's two incompleteness theorems is
presented. The work follows \'Swierczkowski's detailed proof of the theorems
using hereditarily finite (HF) set theory. Avoiding the usual arithmetical
encodings of syntax eliminates the necessity to formalise elementary number
theory within an embedded logical calculus. The Isabelle formalisation uses two
separate treatments of variable binding: the nominal package is shown to scale
to a development of this complexity, while de Bruijn indices turn out to be
ideal for coding syntax. Critical details of the Isabelle proof are described,
in particular gaps and errors found in the literature.Jesse Alama drew my attention to Swierczkowski, the source material for this ´
project. Christian Urban assisted with nominal aspects of some of the proofs, even
writing code. Brian Huffman provided the core formalisation of type hf. Dana Scott
offered advice and drew my attention to Kirby. Matt Kaufmann and the referees
made many insightful comments.This is the author accepted manuscript. The final version is available from Springer at http://link.springer.com/article/10.1007%2Fs10817-015-9322-
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ACKERMANNâS FUNCTION IN ITERATIVE FORM: A PROOF ASSISTANT EXPERIMENT
Ackermann's function can be expressed using an iterative algorithm, which
essentially takes the form of a term rewriting system. Although the termination
of this algorithm is far from obvious, its equivalence to the traditional
recursive formulation--and therefore its totality--has a simple proof in
Isabelle/HOL. This is a small example of formalising mathematics using a proof
assistant, with a focus on the treatment of difficult recursions.ERC Advanced Grant ALEXANDRIA (Project GA 742178
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A modular, efficient formalisation of real algebraic numbers
This paper presents a construction of the real algebraic numbers with executable arithmetic operations in Isabelle/HOL. Instead of verified resultants, arithmetic operations on real algebraic numbers are based on a decision procedure to decide the sign of a bivariate polynomial (with rational coefficients) at a real algebraic point. The modular design allows the safe use of fast external code. This work can be the basis for decision procedures that rely on real algebraic numbers.The CSC Cambridge International Scholarship is generously funding Wenda Liâs Ph.D. course.This is the author accepted manuscript. The final version is available from the Association for Computing Machinery via http://dx.doi.org/10.1145/2854065.285407
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An Isabelle/HOL Formalisation of Greenâs Theorem
We mechanise a proof of Greenâs theorem in Isabelle/HOL. We use a novel proof that avoids the ubiquitous line integral cancellation argument. This eliminates the need to formalise orientations and region boundaries explicitly with respect to the outwards-pointing normal vector. Instead we appeal to a homological argument about equivalences between paths. Contributions include mechanised theories of line integrals and partial derivatives, as well as the ďŹrst mechanisation of Greenâs theorem
Algebraically Closed Fields in Isabelle/HOL
A fundamental theorem states that every field admits an algebraically closed extension. Despite its central importance, this theorem has never before been formalised in a proof assistant. We fill this gap by documenting its formalisation in Isabelle/HOL, describing the difficulties that impeded this development and their solutions.ERC Advanced Grant ALEXANDRIA (Project GA 742178
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