# Isabelle (proof assistant)

Original author(s) Isabelle/jEdit running on macOS Lawrence Paulson University of Cambridge and Technical University of Munich et al. 1986[1] Isabelle2021 / February 2021; 1 year ago Standard ML and Scala Linux, Windows, macOS Mathematics BSD license isabelle.in.tum.de

The Isabelle[a] automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring — yet supporting — explicit proof objects.

Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle Archive of Formal Proofs (Isabelle AFP)[2]

Isabelle was named by Lawrence Paulson after Gérard Huet's daughter.[3]

The Isabelle theorem prover is free software, released under the revised BSD license.

## Features

Isabelle is generic: it provides a meta-logic (a weak type theory), which is used to encode object logics like first-order logic (FOL), higher-order logic (HOL) or Zermelo–Fraenkel set theory (ZFC). The most widely used object logic is Isabelle/HOL, although significant set theory developments were completed in Isabelle/ZF. Isabelle's main proof method is a higher-order version of resolution, based on higher-order unification.

Though interactive, Isabelle features efficient automatic reasoning tools, such as a term rewriting engine and a tableaux prover, various decision procedures, and, through the Sledgehammer proof-automation interface, external satisfiability modulo theories (SMT) solvers (including CVC4) and resolution-based automated theorem provers (ATPs), including E and SPASS (the Metis[b] proof method reconstructs resolution proofs generated by these ATPs).[4] It also features two model finders (counterexample generators): Nitpick[5] and Nunchaku.[6]

Isabelle features locales which are modules that structure large proofs. A locale fixes types, constants, and assumptions within a specified scope[5] so that they do not have to be repeated for every lemma.

Isar ("intelligible semi-automated reasoning") is Isabelle's formal proof language. It is inspired by the Mizar system.[5]

## Example proof

Isabelle allows proofs to be written in two different styles, the procedural and the declarative. Procedural proofs specify a series of tactics (theorem proving functions/procedures) to apply; while reflecting the procedure that a human mathematician might apply to proving a result, they are typically hard to read as they do not describe the outcome of these steps. Declarative proofs (supported by Isabelle's proof language, Isar), on the other hand, specify the actual mathematical operations to be performed, and are therefore more easily read and checked by humans.

The procedural style has been deprecated in recent versions of Isabelle.[citation needed]

For example, a declarative proof by contradiction in Isar that the square root of two is not rational can be written as follows.

```theorem sqrt2_not_rational:
"sqrt 2 ∉ ${\displaystyle \mathbb {Q} }$"
proof
let ?x = "sqrt 2"
assume "?x ∈ ${\displaystyle \mathbb {Q} }$"
then obtain m n :: nat where
sqrt_rat: "¦?x¦ = m / n" and lowest_terms: "coprime m n"
by (rule Rats_abs_nat_div_natE)
hence "m^2 = ?x^2 * n^2" by (auto simp add: power2_eq_square)
hence eq: "m^2 = 2 * n^2" using of_nat_eq_iff power2_eq_square by fastforce
hence "2 dvd m^2" by simp
hence "2 dvd m" by simp
have "2 dvd n" proof -
from ‹2 dvd m› obtain k where "m = 2 * k" ..
with eq have "2 * n^2 = 2^2 * k^2" by simp
hence "2 dvd n^2" by simp
thus "2 dvd n" by simp
qed
with ‹2 dvd m› have "2 dvd gcd m n" by (rule gcd_greatest)
with lowest_terms have "2 dvd 1" by simp
thus False using odd_one by blast
qed
```

## Applications

Isabelle has been used to aid formal methods for the specification, development and verification of software and hardware systems.

Isabelle has been used to formalize numerous theorems from mathematics and computer science, like Gödel's completeness theorem, Gödel's theorem about the consistency of the axiom of choice, the prime number theorem, correctness of security protocols, and properties of programming language semantics. Many of the formal proofs are, as mentioned, maintained in the Archive of Formal Proofs, which contains (as of 2019) at least 500 articles with over 2 million lines of proof in total.[7]

• In 2009, the L4.verified project at NICTA produced the first formal proof of functional correctness of a general-purpose operating system kernel:[8] the seL4 (secure embedded L4) microkernel. The proof is constructed and checked in Isabelle/HOL and comprises over 200,000 lines of proof script to verify 7,500 lines of C. The verification covers code, design, and implementation, and the main theorem states that the C code correctly implements the formal specification of the kernel. The proof uncovered 144 bugs in an early version of the C code of the seL4 kernel, and about 150 issues in each of design and specification.

Larry Paulson keeps a list of research projects that use Isabelle.[10]

## Alternatives

Several languages and systems provide similar functionality:

## References

1. ^ Paulson, L. C. (1986). "Natural deduction as higher-order resolution". The Journal of Logic Programming. 3 (3): 237–258. arXiv:cs/9301104. doi:10.1016/0743-1066(86)90015-4. S2CID 27085090.
2. ^ Eberl, Manuel; Klein, Gerwin; Nipkow, Tobias; Paulson, Larry; Thiemann, René. "Archive of Formal Proofs". Retrieved 1 May 2021.
3. ^ Gordon, Mike (1994-11-16). "1.2 History". Isabelle and HOL. Cambridge AR Research (The Automated Reasoning Group). Retrieved 2016-04-28.
4. ^ Jasmin Christian Blanchette, Lukas Bulwahn, Tobias Nipkow, "Automatic Proof and Disproof in Isabelle/HOL", in: Cesare Tinelli, Viorica Sofronie-Stokkermans (eds.), International Symposium on Frontiers of Combining Systems – FroCoS 2011, Springer, 2011.
5. ^ a b c Jasmin Christian Blanchette, Mathias Fleury, Peter Lammich & Christoph Weidenbach, "A Verified SAT Solver Framework with Learn, Forget, Restart, and Incrementality", Journal of Automated Reasoning 61:333–365 (2018).
6. ^ Andrew Reynolds, Jasmin Christian Blanchette, Simon Cruanes, Cesare Tinelli, "Model Finding for Recursive Functions in SMT", in: Nicola Olivetti, Ashish Tiwari (eds.), 8th International Joint Conference on Automated Reasoning, Springer, 2016.
7. ^ Eberl, Manuel; Klein, Gerwin; Nipkow, Tobias; Paulson, Larry; Thiemann, René. "Archive of Formal Proofs". Retrieved 22 October 2019.
8. ^ Klein, Gerwin; Elphinstone, Kevin; Heiser, Gernot; Andronick, June; Cock, David; Derrin, Philip; Elkaduwe, Dhammika; Engelhardt, Kai; Kolanski, Rafal; Norrish, Michael; Sewell, Thomas; Tuch, Harvey; Winwood, Simon (October 2009). "seL4: Formal verification of an OS kernel" (PDF). 22nd ACM Symposium on Operating System Principles. Big Sky, Montana, US. pp. 207–200.
9. ^ Strniša, Rok; Parkinson, Matthew (2011-02-07). "Lightweight Java". Archive of Formal Proofs (Feb 2011 ed.). ISSN 2150-914X. Retrieved 2019-11-25.
10. ^