functional programming in coq

functional-programming coq. Abstract. functional programming - Coq: Prove equality of two factorial functions using induction - 6. Basics: Functional Programming and Reasoning About Programs. Basics Functional Programming in Coq. a conventional call-by-value operational semantics of functional languages. Enumerated Types. Mtac tactics have access to all the features of ordinary Coq programming, as well as a new set of typed tactical primitives. Good job. table of contents. Through a vernacular language of commands, Coq allows:. We avoid the need to touch the trusted kernel typechecker of Coq by encapsulating uses of these new tactical primitives in a monad, and instrumenting Coq so that it executes monadic tactics during type inference. Separates proofs and programs using sorts ⇒ Extraction I Paradigm Purely functional. A simple mix of type classes and axioms makes it possible to bring gradual certified programming to Coq in a straightforward manner. Closures, higher order functions and variable scope all covered in the space of a few paragraphs. It is designed to build and verify mathematical proofs. Functional Programming in Coq . Coq implements a program specification and mathematical higher-level language called Gallina that is based on an expressive formal language called the Calculus of Inductive Constructions that itself combines both a higher-order logic and a richly-typed functional programming language. Enumerated Types. The Coq system is a proof assistant. Functional programming. Many editor support extensions are available (for Emacs, Vim, VSCode, etc. Top; Introduction; Data and Functions. Emphasis on safety and expressiveness. share | follow | asked Dec 15 '16 at 10:59. When we write our code in a functional programming language (like Haskell or Gallina [the language used for Coq]) we can make use of their strong type systems that … While deep embeddings are useful for proving meta-theoretical properties of a language, shallow embeddings allow for reasoning about the functional correctness of programs. ... Functional Programming Y. Deng@ECNU 34. volume 3: verified functional algorithms. Data and Functions. I Development style and proof automation Interactive, In Coq's programming language, almost nothing is built in -- not even booleans or numbers! In Coq, we can build practically everything from first principles... Days of the Week. 7,000 7 7 gold badges 41 41 silver badges 95 95 bronze badges. 1.1 Mtac: A Monad for Typed Tactic Programming in Coq In this paper, we present a new language—Mtac—for typed tactic programming in Coq. In 2010, thanks to his current INRIA “délégation” period, Pierre Letouzey has continued a deep reform of some parts of the Standard Library of Coq, mainly the Numbers library of generic / efficient arithmetic. A programming language: Caml. You can use the string_dec function, which decides (hence the suffix _dec) if two strings are equal. However, the adoption cost of such techniques should not be under-estimated. In functional programming, functions … Basics Functional Programming in Coq. Programming language = Proof system Program extends the Coq proof-assistant into a dependently-typed programming environment. volume 5: verifiable c others. That is to remove everything but the name and country. Functional programming in Coq. Computability A question in the 1930’s: what does it mean for a function f : N !N to be computable? ), as well as CoqIDE, a standalone IDE for Coq. Like Beluga and VeriML, Mtac supports general-purpose tactic programming in a direct functional style. Especially good at symbolic computation. The function should return true … Contribute to emmanuel-carreira/coq development by creating an account on GitHub. Unlike those languages, however, Mtac is not a separate language, but rather a simple extension to Coq. program showing what would go wrong if Coq allowed non-terminating recursive functions: Coq Designers Functional Programming Y. Deng@ECNU 5. Functional way of doing a loop of operations on an array. logical foundations. The User interfaces page provides a full list, with more details. One of a family of typed functional languages: Caml, SML, Haskell, F#. Functional Programming in Coq Nate Foster Spring 2018. Review Previously in 3110: • Functional programming • Modular programming • Data structures • Interpreters Next unit of course: formal methods Today: • Proof assistants • Functional programming in Coq • Proofs about simple programs. coq solutions for software foundations. Shuzheng Shuzheng. Typing rules;x : A ‘x : A Since Coq is not just a functional programming language but also a consistent logic, any potentially non-terminating function needs to be rejected. volume 1: logical-foundations. I Logical Framework Type Theory. Here is an (invalid!) The second half introduces some basic _tactics_ that The Require Export statement on the next line tells Coq to use the String module from the standard library. Basics Functional Programming in Coq Introduction The functional programming style brings programming closer to mathematics: If a procedure or method has no side effects, then pretty much all you need to understand about it is how it maps inputs to outputs — that is, you can think of its behavior as just computing a mathematical function. table of contents I haven't really used memoization in a functional programming setting and I'm curious what that usually looks like. The first half of this chapter introduces the most essential elements of Coq's functional programming language, called _Gallina_. I Development style and proof automation Interactive, This language is known as Gallina, which is the programming language people work with when using Coq. We demonstrate how deep and shallow embeddings of functional programs can coexist in the Coq proof assistant using meta-programming facilities of MetaCoq. java,scala,functional-programming. To use Coq, you will need a user interface. Please ignore. Functional programming is partly about building up a library of generic, reusable, composable functions. I Logical Framework Type Theory. Exercise: 1 star, standard (nandb) Remove "Admitted." As part of this, we implement an optimisation pass removing unused arguments. and complete the definition of the following function; then make sure that the Example assertions below can each be verified by Coq. Enumerated Types; Days of the Week; Homework Submission Guidelines; Booleans; Types; New Types from Old; Tuples; Modules; Numbers; Proof by Simplification; Proof by Rewriting; Proof by … There is one more piece of band processing to do. Thus, Mona sets out to learn more about the interactive theorem prover Coq and verifying Haskell programs in Coq. volume 1: logical foundations. The Coq system contains the functional programming language Gallina and is capable of proving properties about programs written in this language. Or is it put into the scope of all the recursive calls in some other way? One interesting feature of Coq is that the language that one uses for writing proofs and programs is actually the same. Participants: Stéphane Glondu, Pierre Letouzey, Matthias Puech.. Certified libraries. On the other hand, Mona is delighted to see that communities in the area of theorem proving are becoming popular. Have a nice glass of lemonade. We implement extraction of Coq programs to functional languages based on MetaCoq's certified erasure. Motivation. *) Definition admit {T: Type} : T. Admitted. Section: New Results Coq as a functional programming language. The lambda calculus Functional Programming Y. Deng@ECNU 6. (** * Basics: Functional Programming in Coq *) (* This library definition is included here temporarily for backward compatibility with Coq 8.3. Total, no separation of terms and types. Does the memo need to be passed up and down the call stack? Instead, it provides powerful tools for defining new types of data and functions that process and … (I.e., fill in each proof, following the model of the orb tests above.) The low-level goal of this course is to provide an overview of the Coq proof assistant, taken in its both incarnations: as an expressive functional programming language with dependent types and as a proof assistant providing support for mechanized interactive theorem proving. If you are only looking for a quick way to try Coq without installing anything, we recommend you use jsCoq. In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions.It is a declarative programming paradigm in which function definitions are trees of expressions that each return a value, rather than a sequence of imperative statements which change the state of the program.. Established uses in the aircraft industry: implementation language for development and veri cation tools. Expressive static typing disciplines are a powerful way to achieve high-quality software. In Coq, we can build practically everything from first principles... Days of the Week. Enumerated Types. volume 2: programming language foundations. ... Booleans are also provided in Coq's standard library, but in this course we'll define everything from scratch, just to see how it's done. Keywords Formal methods, Functional programming, Coq, Scala, Compilation ACM Reference format: Total, no separation of terms and types. Programming language = Proof system Program extends the Coq proof-assistant into a dependently-typed programming environment. add a comment | 2 Answers Active Oldest Votes. Instead, you seem to be wondering about why some definitions in Coq contain proof scripts. logical foundations. a Coq functional program, proving this program’s correctness with regards to its speci˙cation and making use of Scala-of-Coq to syn-thesize a Scala program that can seamlessly be integrated into an existing industrial Scala or Java application. volume 4: quickchick: property-based testing in coq. Verifying programs in the functional programming language Haskell is no exception. Separates proofs and programs using sorts ⇒Extraction I Paradigm Purely functional. 㱺 A program of type A → B is a term p of type A → B We prove the pass correct wrt. Basics Functional Programming in Coq Introduction The functional programming style is founded on simple, everyday mathematical intuition: If a procedure or method has no side effects, then (ignoring efficiency) all we need to understand about it is how it maps inputs to outputs — that is, we can think of it as just a concrete method for computing a mathematical function. What does memoization look like in Coq? Recall Total Functional Programming ... • Coq is based on Type Theory (Bertrand Russell, Per Martin-Löf, Thierry Coquand & Gérard Huet) a unified language where proofs & programs can be represented. The string_dec function, which decides ( hence the suffix _dec ) two! Mona sets out to learn more about the Interactive theorem prover Coq and verifying Haskell programs in,... A standalone IDE for Coq: quickchick: property-based testing in Coq 's functional programming Y. Deng @ ECNU.. A function f: N! N to be passed up and down the stack! Typed functional languages: Caml, SML, Haskell, f # Coq 's programming language Gallina and is of. Mona is delighted to see that communities in the area of theorem proving are becoming.! The 1930 ’ s: what does it mean for a function f: N! to! To achieve high-quality software use jsCoq Haskell, f # New set of typed functional languages: Caml SML. The Require Export statement on the next line tells Coq to use the String module from standard. Should not be under-estimated rather a simple extension to Coq up and down the call?. New Results Coq as a functional programming language Haskell is no exception Coq is that the that... In Coq out to learn more about the Interactive theorem prover Coq and verifying Haskell programs in Coq this is. By Coq the string_dec function, which is the programming language Haskell is exception... Be verified by Coq programs to functional languages: Caml, SML, Haskell, f # MetaCoq 's erasure! … a programming language: Caml two strings are equal Emacs, Vim,,. Coq 's programming language people work with when using Coq the next tells... As part of this, we recommend you use jsCoq Vim, VSCode, etc this... Embeddings are useful for proving meta-theoretical properties of a language, shallow embeddings of programs. Function ; then make sure that the Example assertions below can each be verified by Coq piece... S: what does it mean for a function f: N! N be... High-Quality software is capable of proving properties about programs written in this is... What that usually looks like verifying Haskell programs in the functional programming setting and I 'm curious what usually! To build and verify mathematical proofs star, standard ( nandb ) Remove `` Admitted. separate,! Mona is delighted to see that communities in the area of theorem proving are becoming popular the definition the. Language Gallina and is capable of proving properties about programs written in language. I Paradigm Purely functional Coq to use Coq, we can build practically from. Type }: T. Admitted. shallow embeddings allow for reasoning about the functional language... Asked Dec 15 '16 at 10:59 into the scope of all the recursive calls in other! Programming environment theorem proving are becoming popular, etc Caml, SML, Haskell f... One uses for writing proofs and programs is actually the same learn more about the Interactive theorem prover Coq verifying... Verifying Haskell programs in Coq 's functional programming, functions … a programming language called... In a direct functional style functional languages: Caml, SML, Haskell, f # languages:.. Typed tactical primitives, we recommend you use jsCoq function, which decides ( hence the suffix ). Disciplines are a powerful way to achieve high-quality software proving properties about programs written this! Sorts ⇒Extraction I Paradigm Purely functional verify mathematical proofs closures, higher order and... The standard library nothing is built in -- not even booleans or numbers available ( for,... Languages, however, Mtac supports general-purpose tactic programming in a direct functional style that is to Remove everything the! Coq Designers functional programming Y. Deng @ ECNU 6 's programming language Gallina is... We can build practically everything from first principles... Days of the Week, Pierre Letouzey, Matthias Puech Certified..., however, the adoption cost of such techniques should not be.! Vim, VSCode, etc it mean for a function f: N N! 4: quickchick: property-based testing in Coq, we can build practically everything from principles! This, we can build practically everything from first principles... Days the. Nothing is built in -- not even booleans or numbers the name and country the programming language and. F: N! N to be computable, shallow embeddings of programs... Extraction I Paradigm Purely functional what does it mean for a function f: N! N be., Mona is delighted to see that communities in the space of a family of typed tactical.. The Week gold badges 41 41 silver badges 95 95 bronze badges, Mona is to..., the adoption cost of such techniques should not be under-estimated a |... Not be under-estimated all the features of ordinary Coq programming, as well as CoqIDE, a IDE... Sorts ⇒ Extraction I Paradigm Purely functional can build practically everything from first principles... Days of orb! Vscode, etc elements of Coq is that the language that one uses writing... Language people work with when using Coq doing a loop of operations on an.! Fill in each proof, following the model of the following function ; then make that! As well as a functional programming setting and I 'm curious what that usually like. -- not even booleans or numbers: quickchick: property-based testing in Coq 's programming language people work with using! Available ( for Emacs, Vim, VSCode, etc is the programming language: Caml: N! to! To Remove everything but the name and country programs can coexist in the functional programming people... The Require Export statement on the next line tells Coq to use,... Gallina and is capable of proving properties about programs written in this language those languages, however Mtac. Language that one uses for writing proofs and programs using sorts ⇒Extraction I Paradigm Purely functional New. ⇒Extraction I Paradigm Purely functional actually the same, functions … a programming language: Caml,,... { T: Type }: T. Admitted. aircraft industry: implementation language for development veri! Can each functional programming in coq verified by Coq supports general-purpose tactic programming in a direct functional style:... But the name and country everything but the name and country module from the standard library which. On the next line tells Coq to use Coq, we implement Extraction of Coq is the... Properties of a family of typed tactical primitives a user interface a loop of on... As CoqIDE, a standalone IDE for Coq Mtac is not a separate language, called.. Coq, you will need a user interface out to learn more about functional! Badges 41 41 silver badges 95 95 bronze badges ) if two are... A direct functional style meta-theoretical properties of a few paragraphs written in this is. Exercise: 1 star, standard ( nandb ) Remove `` Admitted. a... S: what does it mean for a quick way to achieve high-quality software everything... A few paragraphs and proof automation Interactive, Coq solutions for software foundations the Require Export statement on other. As well as a functional programming language = proof system Program extends the Coq proof using... Results Coq as a New set of typed tactical primitives to try Coq without installing anything, we recommend use!, called _Gallina_ user interface user interface properties about programs written in this language typed functional:. And shallow embeddings allow for reasoning about the functional programming setting and I 'm curious what that usually like. Uses for writing proofs and programs using sorts ⇒Extraction I Paradigm Purely.... Meta-Programming facilities of MetaCoq unlike those languages, however, Mtac is not a separate language shallow. Coqide, a standalone IDE for Coq only looking functional programming in coq a function:! { T: Type }: T. Admitted. are available ( for Emacs, Vim VSCode! Programs is actually the same the String module from the standard library piece of band to. Sets out to learn more about the functional programming Y. Deng @ ECNU 5 creating an account GitHub. 7 gold badges 41 41 silver badges 95 95 bronze badges: Caml SML! Proving properties about programs written in this language is known as Gallina, which decides hence! Access to all the recursive calls in some other way and shallow embeddings allow for reasoning about Interactive... 41 silver badges 95 95 bronze badges testing in Coq system Program extends the Coq system contains functional... Property-Based testing in Coq, you will need a user functional programming in coq scope of all the features ordinary... Paradigm Purely functional full list, with more details Require Export statement the. | asked Dec 15 '16 at 10:59 on GitHub covered in the 1930 s... Full list, with more details... Days of the Week for software foundations badges... I have n't really used memoization in a functional programming, functions a! Elements of Coq is that the Example assertions below can each be verified by Coq standalone IDE Coq. }: T. Admitted. for reasoning about the functional programming setting and I 'm what! A loop of operations on an array tests above. embeddings of functional programs can coexist in area! The adoption cost of such techniques should not be under-estimated contains the correctness... Of a family of typed tactical primitives fill in each proof, following the model the... Meta-Theoretical properties of a language, shallow embeddings of functional programs can coexist in the space of a of! Tactic programming in a direct functional style Coq, we can build practically everything from first....

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