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# Provably Awesome. [![](img/badge.svg)](http://awesome.re)
This is a curated list of links and resources related to mathematical proofs
about the properties of computer programs.
#### Table of Contents
* [Languages](#languages) - Languages with good ability to use formal type
safety
* [Proof Assistants](#proof-assistants) - Interactive theorem provers used to
prove properties of programs.
* [Projects](#projects) - Projects involving provably correct code.
* [Books](#books) - Textbooks commonly referred to
* [Courses](#courses) - Online courses (youtube, university courses)
* [More Links](#more) - Video presentations about formal proof of code topics
## Languages
* [Idris](https://www.idris-lang.org/) : Idris is a general purpose pure
functional programming language with dependent types. Dependent types allow
types to be predicated on values, meaning that some aspects of a program’s
behaviour can be specified precisely in the type. It is compiled, with eager
evaluation. Its features are influenced by Haskell and ML.
* [Idris docs](http://docs.idris-lang.org/en/latest/)
* [Idris tutorial](http://docs.idris-lang.org/en/latest/tutorial/index.html#tutorial-index)
* [Theorem proving with Idris tutorial](http://docs.idris-lang.org/en/latest/proofs/index.html)
* [Agda](http://wiki.portal.chalmers.se/agda/pmwiki.php) : Dependently
typed functional programming language. It has inductive families, i.e., data
types which depend on values, such as the type of vectors of a given length.
It also has parametrised modules, mixfix operators, Unicode characters, and an
interactive Emacs interface which can assist the programmer in writing the
program.
* [Agda Github](https://github.com/agda/agda)
* [Agda User Manual](http://agda.readthedocs.io/en/v2.5.2/)
* [UR/Web](http://www.impredicative.com/ur/) : Ur/Web is Ur plus a special
standard library and associated rules for parsing and optimization. Ur/Web
supports construction of dynamic web applications backed by SQL databases.
The signature of the standard library is such that well-typed Ur/Web programs
"don't go wrong" in a very broad sense. Not only do they not crash during
particular page generations, but they also may not:
* Suffer from any kinds of code-injection attacks
* Return invalid HTML
* Contain dead intra-application links
* Have mismatches between HTML forms and the fields expected by their handlers
* Include client-side code that makes incorrect assumptions about the
"AJAX"-style services that the remote web server provides
* Attempt invalid SQL queries
* Use improper marshaling or unmarshaling in communication with SQL databases
or between browsers and web servers
* [Haskell](https://www.haskell.org/) : An advanced, purely functional
programming language.
* [Liquid Haskell](https://ucsd-progsys.github.io/liquidhaskell-blog/) :
LiquidHaskell (LH) refines Haskell's types with logical predicates that let
you enforce critical properties at compile time. (Guarantee functions are total,
keep pointers within bounds, avoid infinite loops, enforce correctness
properties, prove laws by writing code)
* [Elm](http://elm-lang.org/) : A type-safe functional programming language for
declaratively creating web browser-based graphical user interfaces.
Implemented in Haskell. It generates JavaScript with great performance and
*no runtime exceptions*.
## Proof Assistants
* [Coq](https://coq.inria.fr/) : Coq is a formal proof management system. It
provides a formal language to write mathematical definitions, executable
algorithms and theorems together with an environment for semi-interactive
development of machine-checked proofs.
[[current stable version](https://coq.inria.fr/download)]
[[reference manual](https://coq.inria.fr/distrib/current/refman)]
* https://github.com/Ptival/PeaCoq
* https://math-comp.github.io/mcb/
* [Isabelle](https://isabelle.in.tum.de/) : Isabelle is a generic proof assistant.
It allows mathematical formulas to be expressed in a formal language and
provides tools for proving those formulas in a logical calculus.
[[overview](https://isabelle.in.tum.de/overview.html)]
* [HOL](https://hol-theorem-prover.org/) : The HOL interactive theorem prover is
a proof assistant for higher-order logic: a programming environment in which
theorems can be proved and proof tools implemented. Built-in decision
procedures and theorem provers can automatically establish many simple
theorems (users may have to prove the hard theorems themselves!) An oracle
mechanism gives access to external programs such as SMT and BDD engines.
HOL is particularly suitable as a platform for implementing combinations of
deduction, execution and property checking.
[[Other HOLS](https://hol-theorem-prover.org/other-hols.html)]
* [LEAN](https://leanprover.github.io/) : Lean is a new open source theorem
prover being developed at Microsoft Research, and its standard library at
Carnegie Mellon University. Lean aims to bridge the gap between interactive and
automated theorem proving, by situating automated tools and methods in a framework
that supports user interaction and the construction of fully specified axiomatic
proofs. The goal is to support both mathematical reasoning and reasoning about
complex systems, and to verify claims in both domains.
[Online version](https://leanprover.github.io/live/latest/)
* [K Framework](http://www.kframework.org/index.php/Main_Page) : The K framework
is a rewrite-based executable semantic framework in which programming languages,
type systems and formal analysis tools can be defined using configurations,
computations and rules. Configurations organize the state in units called cells,
which are labeled and can be nested. Computations carry computational meaning as
special nested list structures sequentializing computational tasks, such as fragments
of program. Computations extend the original language abstract syntax. K (rewrite)
rules make it explicit which parts of the term they read-only, write-only, read-write,
or do not care about. This makes K suitable for defining truly concurrent languages
even in the presence of sharing. Computations are like any other terms in a rewriting
environment: they can be matched, moved from one place to another, modified, or
deleted. This makes K suitable for defining control-intensive features such as
abrupt termination, exceptions or call/cc.
* [K Tutorial](http://www.kframework.org/index.php/K_Tutorial) by [Grigore Rosu](https://github.com/grosu), [video playlist](https://www.youtube.com/watch?v=eSaIKHQOo4c&list=PLx_U8qR-tMtLQEDPvVk1y9gTIdUIWGaQd)
* (https://runtimeverification.com/) : Company formed from K Framework people.
Runtime Verification Inc. is currently developing three core products:
* RV-Predict is a predictive runtime analysis tool focused on automatically
detecting concurrency errors in your programs.
* RV-Monitor is a development methodology and library generation tool allowing
for the monitoring and enforcement of user-selected properties at runtime.
* RV-Match is a tool allowing for exhaustive runtime verification to be performed
symbolically on all possible program paths, proving certain properties correct
for all possible executions of a given program.
* [Viper](https://www.pm.inf.ethz.ch/research/viper.html) : Viper (Verification Infrastructure for Permission-based Reasoning) is a language and suite of tools developed at ETH Zurich, providing an architecture on which new verification tools and prototypes can be developed simply and quickly. It comprises a novel intermediate verification language, also named Viper, and automatic verifiers for the language, as well as example front-end tools. The Viper toolset can be used to implement verification techniques for front-end programming languages via translations into the Viper language. ETH Zurich has built several verifiers on top of Viper, including the [Gobra](https://www.pm.inf.ethz.ch/research/gobra.html) verifier for Go, [Nagini](https://www.pm.inf.ethz.ch/research/nagini.html) for Python and [Prusti](https://www.pm.inf.ethz.ch/research/prusti.html) for Rust.
## Projects
* [seL4](https://sel4.systems/) : The world's first operating-system kernel with
an end-to-end proof of implementation correctness and security enforcement
is available as open source.
[[brochure](https://sel4.systems/Info/Docs/seL4-brochure.pdf)]
[[white paper](https://sel4.systems/Info/Docs/GD-NICTA-whitepaper.pdf)]
* [Certikos](http://flint.cs.yale.edu/certikos/) : Certified Kit Operating
System.
* **Certified OS Kernels**: Clean-slate design with end-to-end guarantees on
extensibility, security, and resilience.
Without Zero-Day Kernel Vulnerabilities.
* **Layered Approach**: Divides a complex
system into multiple certified abstraction layers, which are deep
specifications of their underlying implementations.
* **Languages and Tools**: New formal methods, languages, compilers and other
tools for developing, checking, and automating specs and proofs.
* [Compcert](http://compcert.inria.fr/) : The CompCert project investigates the
formal verification of realistic compilers usable for critical embedded
software. Such verified compilers come with a mathematical, machine-checked
proof that the generated executable code behaves exactly as prescribed by the
semantics of the source program.
[[C compiler](http://compcert.inria.fr/download.html)]
* [Bedrock](http://plv.csail.mit.edu/bedrock/)
[[tutorial pdf](http://plv.csail.mit.edu/bedrock/tutorial.pdf)] :
Bedrock is a library that turns Coq into a tool much like classical
verification systems (e.g., ESC, Boogie), but niftier. In particular,
Bedrock is:
* **Low-level**: You can verify programs that, for performance reasons or
otherwise, can't tolerate any abstraction beyond that associated with
assembly language.
* **Foundational**: The output of a Bedrock verification is a theorem whose
statement depends only on the predicates you choose to use in the key
specifications and on the operational semantics of a simple cross-platform
machine language. That is, you don't need to trust that the verification
framework is bug-free; rather, you need to trust the usual Coq proof-checker
and the formalization of the machine language semantics.
* **Higher-order**: Bedrock facilitates quite pleasant reasoning about code
pointers as data.
* **Computational**: Many useful functions are specified most effectively by
comparing with "reference implementations" in a pure functional language.
Bedrock supports that model, backed by the full expressive power of Coq's
usual programming language.
* **Structured**: Bedrock is an extensible programming language: any client
program may add new control flow constructs by providing their proof rules.
For example, adding high-level syntax for your own calling convention or
exception handling construct is relatively straightforward and does not
require tweaking the core library code.
* **Mostly automated**: Tactics automate verification condition generation
(in a form inspired by separation logic) and most of the process of
discharging those conditions. Many interesting programs can be verified with
zero manual proof effort, in stark contrast to most Coq developments.
* [HACMS](https://www.darpa.mil/program/high-assurance-cyber-military-systems)
: High-Assurance Cyber Military Systems (HACMS)
. Dr. Raymond Richards. Technology for cyber-physical systems,
functionally correct and satisfying appropriate safety and security properties.
Clean-slate, formal methods, semi-automated code synthesis from executable,
formal specifications. HACMS seeks a synthesizer capable of producing a
machine-checkable proof that generated code satisfies functional
specs as well as security and safety policies, and development to ensure
proofs are composable, allowing components.
[[more Darpa "formal" tagged links](https://www.darpa.mil/tag-list?tt=78)]
[[verigames-crowdsourced formal verification](https://www.darpa.mil/news-events/2013-12-04a)]
* [Genode](http://genode.org/) : Genode is a novel OS architecture that is
able to master complexity by applying a strict organizational structure to all
software components including device drivers, system services, and applications.
## Books
* [The Little Prover](https://mitpress.mit.edu/books/little-prover)
The Little Prover introduces inductive proofs as a way to determine facts about
computer programs.
* [Certified Programming with Dependent Types](http://adam.chlipala.net/cpdt/) by
Adam Chlipala. Textbook about practical engineering with the Coq proof assistant.
The focus is on building programs with proofs of correctness, using dependent
types and scripted proof automation.
[[Latest draft](http://adam.chlipala.net/cpdt/cpdt.pdf)]
* [Software Foundations](https://softwarefoundations.cis.upenn.edu/) by
Benjamin Pierce and others. The Software Foundations series is a broad
introduction to the mathematical underpinnings of reliable software.
* [Vol. 1:](https://softwarefoundations.cis.upenn.edu/lf-current/index.html)
[[read](https://softwarefoundations.cis.upenn.edu/lf-current/toc.html)]
[[download](https://softwarefoundations.cis.upenn.edu/lf-current/lf.tgz)]
Logical Foundations, serves as the entry-point to the series. It covers
functional programming, basic concepts of logic, computer-assisted theorem
proving,and Coq.
* [Vol. 2:](https://softwarefoundations.cis.upenn.edu/plf-current/index.html)
[[read](https://softwarefoundations.cis.upenn.edu/plf-current/toc.html)]
[[download](https://softwarefoundations.cis.upenn.edu/plf-current/plf.tgz)]
Programming Language Foundations, surveys the theory of programming languages,
including operational semantics, Hoare logic, and static type systems.
* [Vol. 3: Verified Functional Algorithms](https://softwarefoundations.cis.upenn.edu/vfa-current/index.html)
[[read](https://softwarefoundations.cis.upenn.edu/vfa-current/index.html)]
[[download](https://softwarefoundations.cis.upenn.edu/vfa-current/vfa.tgz)]
Learn how to specify and verify (prove the correctness of) sorting algorithms,
binary search trees, balanced binary search trees, and priority queues.
* HoTT : [Homotopy Type Theory: Univalent Foundations of Mathematics](https://homotopytypetheory.org/book/)
[[pdf](http://saunders.phil.cmu.edu/book/hott-online.pdfUnivalent)]
Foundations of Mathematics is Vladimir Voevodsky’s new program for a
comprehensive, computational foundation for mathematics based on the homotopical
interpretation of type theory. The type theoretic univalence axiom relates
propositional equality on the universe with homotopy equivalence of small types.
The program is currently being implemented with the help of the automated proof
assistant Coq.
* https://math-comp.github.io/mcb/
## Courses
* [DeepSpec Summer School](https://www.youtube.com/channel/UC5yB0ZRgc4A99ttkwer-dDw)
41 Videos about deep specification. Coq videos, examples, tutorials.
* Adam Chlipala Videos:
* 2017-12-29 CCC Presentation: [Coming Soon Machine-Checked Mathematical Proofs in Everyday Software and Hardware Development](https://media.ccc.de/v/34c3-9105-coming_soon_machine-checked_mathematical_proofs_in_everyday_software_and_hardware_development)
* [Adam Chlipala Lecture 1, OPLSS 2015](https://www.youtube.com/watch?v=ORKAy_CHDYM)
* [Bedrock: A Software Development Ecosystem Inside a Proof Assistant](https://www.youtube.com/watch?v=BSyrp-iYBMo)
* [CACM August 2016 - Ur/Web: A Simple Model for Programming the Web](https://www.youtube.com/watch?v=J3XI6-aZZXk)
* [Proof engineering Adam Chlipala](https://www.youtube.com/watch?v=yXLeyANzAC4)
* [2015 Coq Proof Assistant and Its Applications to Programming-Language Semantics](https://www.youtube.com/playlist?list=PLt7hcIEdZLAnO7AawDQkHwE7RtwPDOFEc)
* [Type-Drive Development in Idris - Edwin Brady](https://www.youtube.com/watch?v=X36ye-1x_HQ)
* [Benjamin Pierce - Software Foundations Course](https://www.youtube.com/playlist?list=PLGCr8P_YncjUT7gXUVJWSoefQ40gTOz89)
* [Learning Automated Theorem Proving](https://cs.stackexchange.com/questions/820/learning-automated-theorem-proving) : Stackexchange post about learning
## More
* [Curry-Howard](https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence) :
Curry–Howard correspondence refers to the direct relationship between computer
programs and mathematical proofs. This is also the idea of "proofs as programs",
and "propositions (formulas)-as-types".
* [Hoare logic](https://en.wikipedia.org/wiki/Hoare_logic)
Hoare logic (also known as Floyd–Hoare logic or Hoare rules) is a formal system
with a set of logical rules for reasoning rigorously about the correctness of
computer programs.
* [Designing A Theorem Prover (Paulson, Cambridge, 1990)](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-192.pdf)
* [Rolf Rolles Program Synthesis in Reverse Engineering](https://www.youtube.com/watch?v=mFjSbxV_1vw)
...Assume you generate all possible programs...
* [The Open-Source seL4 Kernel. Military-Grade Security Through Mathematics - SFO17-417](https://www.youtube.com/watch?v=heSmrHzHcuM)
* [DARPA Hack Proof Drones](https://www.defensetech.org/2014/05/21/darpa-unveils-hack-proof-drone/)
Uses SEL4, other RTOS
* [Pentagon Wants Unhackable Helicopters](https://www.engadget.com/2015/03/16/pentagon-wants-unhackable-helicopters/)
* [Hacker-Proof Code Confirmed](https://www.quantamagazine.org/formal-verification-creates-hacker-proof-code-20160920/) :
Computer scientists can prove certain programs to be error-free with the same certainty that mathematicians prove theorems. The advances are being used to secure everything from unmanned drones to the internet.
* [CertiKOS enables creation of secure system kernels](http://www.zdnet.com/article/certikos-a-hacker-proof-os/) [[certikos project](http://flint.cs.yale.edu/certikos/)]
Computer system security stinks, because our
software is buggy and untestable in full. Great for cyber criminals, but not
for us. So why doesn't someone build a mathematically verified, secure,
concurrent kernel that can run on x86 and ARM? A team at Yale has.
* [seL4 Is Free – What Does This Mean For You?](https://www.youtube.com/watch?v=lRndE7rSXiI)
* [From L3 to seL4 what have we learnt in 20 years of L4 microkernels?](https://www.youtube.com/watch?v=RdoaFc5-1Rk)
* [seL4 introduction: Capability--based Access Model](https://www.youtube.com/watch?v=x3P6Y6VO0UI) <- Chinese, translation?
* [seL4 playlist](https://www.youtube.com/playlist?list=PL8UO9ZG39Nx43YCAKGCtj9Rb6p2_3utdc)
* [Creating drones that can't be hacked](https://www.youtube.com/watch?v=4oONdV5RYp8)
* [HACMS: Protecting Military Systems from Hackers](https://www.youtube.com/watch?v=OyqNpn6JpBk)
* [Formal Methods for Avionics Software Verification pt1](https://www.youtube.com/watch?v=tRtK4xOK-8o
)
* TODO: find Heartbleed Example with Agda Showing Proofs