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| {{Infobox programming language
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| | name = Epigram
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| | logo =
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| | paradigm = [[Functional programming|Functional]]
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| | year = 2004
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| | designer = [[Conor McBride]] and<br/>James McKinna
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| | developer = ''Unmaintained''
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| | latest_release_version = 1
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| | latest release date ={{release date|mf=yes|2006|10|11}}
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| | typing = [[strong typing|strong]], [[static typing|static]], [[dependent typing|dependent]]
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| | implementations =
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| | dialects =
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| | influenced_by = [[ALF (theorem prover)|ALF]]
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| | influenced =
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| | operating_system = [[Cross-platform]]: [[Linux]], [[Microsoft Windows|Windows]], [[Mac OS X]]
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| | license = MIT<ref>http://code.google.com/p/epigram/</ref>
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| | file_ext =
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| }}
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| '''Epigram''' is the name of a [[functional programming language]] with [[dependent type]]s. ''Epigram'' also refers to the [[Integrated development environment|IDE]] usually packaged with the language. Epigram's [[type system]] is strong enough to express [[program specification]]s. The goal is to support a smooth transition from ordinary programming to integrated programs and proofs whose correctness can be checked and certified by the [[compiler]]. Epigram exploits the [[propositions as types principle]], and is based on [[intuitionistic type theory]].
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| The Epigram prototype was implemented by [[Conor McBride]] based on joint work with James McKinna. Its development is continued by the Epigram group in [[Nottingham]], [[Durham]], [[St Andrews]] and [[Royal Holloway]] in the [[United Kingdom|UK]]. The current experimental implementation of the Epigram system is freely available together with a user manual, a tutorial and some background material. The system has been used under [[Linux]], [[Microsoft Windows|Windows]] and [[Mac OS X]].
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| It is currently unmaintained, and version 2, which was intended to implement [[Observational Type Theory]], was never released.
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| == Syntax ==
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| Epigram uses a two-dimensional syntax, with a LaTeX version and an ASCII version. Here are some examples from ''The Epigram Tutorial'':
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| === Examples ===
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| ==== The natural numbers ====
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| The following declaration defines the [[natural numbers]]:
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| <pre> ( ! ( ! ( n : Nat !
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| data !---------! where !----------! ; !-----------!
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| ! Nat : * ) !zero : Nat) !suc n : Nat)</pre>
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| The declaration says that <code>Nat</code> is a type with [[Type system#Types of types|kind]] <code>*</code> (i.e., it is a simple type) and two constructors: <code>zero</code> and <code>suc</code>. The constructor <code>suc</code> takes a single <code>Nat</code> argument and returns a <code>Nat</code>. This is equivalent to the [[Haskell (programming language)|Haskell]] declaration "<code>data Nat = Zero | Suc Nat</code>".
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| In LaTeX, the code is displayed as:
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| <math>\underline{\mathrm{data}} \; \left(\frac{}{\mathsf{Nat} : \star}\right) \; \underline{\mathrm{where}} \;
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| \left(\frac{}{\mathsf{zero} : \mathsf{Nat}}\right) \; ; \;
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| \left(\frac{n : \mathsf{Nat}}{\mathsf{suc}\ n : \mathsf{Nat}}\right)</math>
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| ==== Recursion on naturals ====
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| <math>\mathsf{NatInd} : \begin{matrix}
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| \forall P : \mathsf{Nat} \rightarrow \star \Rightarrow P\ \mathsf{zero} \rightarrow \\
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| (\forall n : \mathsf{Nat} \Rightarrow P\ n \rightarrow P\ (\mathsf{suc}\ n)) \rightarrow\\
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| \forall n : \mathsf{Nat} \Rightarrow P\ n
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| \end{matrix}</math>
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| <math>\mathsf{NatInd}\ P\ mz\ ms\ \mathsf{zero} \equiv mz</math>
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| <math>\mathsf{NatInd}\ P\ mz\ ms\ (\mathsf{suc}\ n) \equiv ms\ n\ (NatInd\ P\ mz\ ms\ n)</math>
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| ...And in ASCII:
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| <pre>NatInd : all P : Nat -> * => P zero ->
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| (all n : Nat => P n -> P (suc n)) ->
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| all n : Nat => P n
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| NatInd P mz ms zero => mz
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| NatInd P mz ms (suc n) => ms n (NatInd P mz ms n)</pre>
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| ==== Addition ====
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| {|
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| | <math>\mathsf{plus}\ x\ y \Leftarrow \underline{\mathrm{rec}}\ x\ \{</math>
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| |-
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| | <math>\mathsf{plus}\ x\ y \Leftarrow \underline{\mathrm{case}}\ x\ \{</math>
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| |-
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| | <math>\mathsf{plus\ zero}\ y \Rightarrow y</math>
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| |-
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| | <math>\quad\quad \mathsf{plus}\ (\mathsf{suc}\ x)\ y \Rightarrow suc\ (\mathsf{plus}\ x\ y)\ \}\ \}</math>
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| |}
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| ...And in ASCII: | |
| <pre>plus x y <= rec x {
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| plus x y <= case x {
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| plus zero y => y
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| plus (suc x) y => suc (plus x y)
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| }
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| }</pre>
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| == Dependent types ==
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| Epigram is essentially a [[typed lambda calculus]] with [[generalized algebraic data type]] extensions, except for two extensions. First, types are first-class entities, of type <math>\star</math>; types are arbitrary expressions of type <math>\star</math>, and type equivalence is defined in terms of the types' normal forms. Second, it has a dependent function type; instead of <math>P \rightarrow Q</math>, <math>\forall x : P \Rightarrow Q</math>, where <math>x</math> is bound in <math>Q</math> to the value that the function's argument (of type <math>P</math>) eventually takes.
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| Full dependent types, as implemented in Epigram, are a powerful abstraction. (Unlike in [[Dependent ML]], the value(s) depended upon may be of any valid type.) A sample of the new formal specification capabilities dependent types bring may be found in ''The Epigram Tutorial''.
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| == See also ==
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| *[[ALF (proof assistant)|Alf]], a proof assistant among the predecessors of Epigram.
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| ==Further reading==
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| *Conor McBride and James McKinna (2004), ''The view from the left'', Journal of Functional Programming
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| *Conor McBride (2004), ''The Epigram Prototype, a nod and two winks''
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| *Conor McBride (2004), ''The Epigram Tutorial''
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| *Thorsten Altenkirch, Conor McBride and James McKinna (2005), ''Why Dependent Types Matter''
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| == External links ==
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| *[http://www.macs.hw.ac.uk/~fairouz/projects/EffProClaLog.html EPSRC] on ALF, lego and related
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| == References ==
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| {{reflist}}
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| [[Category:Functional languages]]
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| [[Category:Dependently typed languages]]
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| [[Category:Proof assistants]]
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| [[Category:Discontinued programming languages]]
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