# Isabelle (proof assistant)

(Redirected from Isabelle (theorem prover))
Original author(s) Isabelle/jEdit running on macOS. Lawrence Paulson University of Cambridge and Technical University of Munich et al. 1986[1] 2017 / October 2017 Standard ML and Scala Linux, Windows, Mac OS X Mathematics BSD license isabelle.in.tum.de

The Isabelle theorem prover is an interactive theorem prover, a Higher Order Logic (HOL) theorem prover. It is an LCF-style theorem prover (written in Standard ML), so it is based on a small logical core to ease logical correctness. 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). Isabelle's main proof method is a higher-order version of resolution, based on higher-order unification. Though interactive, Isabelle also features efficient automatic reasoning tools, such as a term rewriting engine and a tableaux prover, as well as various decision procedures.

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

Isabelle was named by Lawrence Paulson after Gérard Huet's daughter.[2]

## Example proof

Isabelle allows proofs to be written in two different styles, the procedural and the declarative. Procedural proofs specify a series of tactics or 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. The Archive of Formal Proofs also recommends the declarative style[3].

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 (real 2) ∉ ℚ"
proof
let ?x = "sqrt (real 2)"
assume "?x ∈ ℚ"
then obtain m n :: nat where
sqrt_rat: "¦?x¦ = real m / real n" and lowest_terms: "coprime m n"
by (rule Rats_abs_nat_div_natE)
hence "real (m^2) = ?x^2 * real (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.

• The use of Isabelle by Hewlett-Packard in the design of the HP 9000 line of servers' Runway bus led to the discovery of a number of bugs uncaught by previous testing and simulation.[4]
• In 2009, the L4.verified project at NICTA produced the first formal proof of functional correctness of a general-purpose operating system kernel:[5] 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.

## Alternatives

The following programs provide similar functionality to Isabelle:

## References

1. ^ Paulson, L. C. (1986). "Natural deduction as higher-order resolution". The Journal of Logic Programming. 3 (3): 237. doi:10.1016/0743-1066(86)90015-4.
2. ^ Gordon, Mike (1994-11-16). "1.2 History". Isabelle and HOL. Cambridge AR Research (The Automated Reasoning Group). Retrieved 2016-04-28.
3. ^
4. ^ Philip Wadler's "An Angry Half-Dozen" (1998) attributes this result to: Albert J. Camilleri. "A hybrid approach to verifying liveness in a symmetric multiprocessor". 10th International Conference on Theorem Proving in Higher-Order Logics, Elsa Gunter and Amy Felty, editors, Murray Hill, New Jersey, August 1997. Lecture Notes in Computer Science (LNCS) Vol. 1275, Springer Verlag, 1997
5. ^ 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.