# Epigram (programming language)

Epigram is a functional programming language with dependent types. Epigram also refers to the IDE usually packaged with the language. Epigram's type system is strong enough to express program specifications. 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.

Paradigm Functional Conor McBrideJames McKinna Unmaintained 2004 1 / October 11, 2006 strong, static, dependent Cross-platform: Linux, Windows, Mac OS X MIT[1] ALF Agda, Idris

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 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, Windows and Mac OS X.

It is currently unmaintained, and version 2, which was intended to implement Observational Type Theory, was never officially released but exists in GitHub. The design of Epigram and Epigram 2 have inspired the development Agda,[citation needed] Idris,[citation needed] and Coq.[citation needed]

## Syntax

Epigram uses a two-dimensional, natural deduction style syntax, with a LaTeX version and an ASCII version. Here are some examples from The Epigram Tutorial:

### Examples

#### The natural numbers

The following declaration defines the natural numbers:

     (         !       (          !   (  n : Nat  !
data !---------! where !----------! ; !-----------!
! Nat : * )       !zero : Nat)   !suc n : Nat)

The declaration says that Nat is a type with kind * (i.e., it is a simple type) and two constructors: zero and suc. The constructor suc takes a single Nat argument and returns a Nat. This is equivalent to the Haskell declaration "data Nat = Zero | Suc Nat".

In LaTeX, the code is displayed as:

${\displaystyle {\underline {\mathrm {data} }}\;\left({\frac {}{{\mathsf {Nat}}:\star }}\right)\;{\underline {\mathrm {where} }}\;\left({\frac {}{{\mathsf {zero}}:{\mathsf {Nat}}}}\right)\;;\;\left({\frac {n:{\mathsf {Nat}}}{{\mathsf {suc}}\ n:{\mathsf {Nat}}}}\right)}$

The horizontal-line notation can be read as "assuming (what is on the top) is true, we can infer that (what is on the bottom) is true." For example, "assuming n is of type Nat, then suc n is of type Nat." If nothing is on the top, then the bottom statement is always true: "zero is of type Nat (in all cases)."

#### Recursion on naturals

${\displaystyle {\mathsf {NatInd}}:{\begin{matrix}\forall P:{\mathsf {Nat}}\rightarrow \star \Rightarrow P\ {\mathsf {zero}}\rightarrow \\(\forall n:{\mathsf {Nat}}\Rightarrow P\ n\rightarrow P\ ({\mathsf {suc}}\ n))\rightarrow \\\forall n:{\mathsf {Nat}}\Rightarrow P\ n\end{matrix}}}$

${\displaystyle {\mathsf {NatInd}}\ P\ mz\ ms\ {\mathsf {zero}}\equiv mz}$

${\displaystyle {\mathsf {NatInd}}\ P\ mz\ ms\ ({\mathsf {suc}}\ n)\equiv ms\ n\ (NatInd\ P\ mz\ ms\ n)}$

...And in ASCII:

NatInd : all P : Nat -> * => P zero ->
(all n : Nat => P n -> P (suc n)) ->
all n : Nat => P n
NatInd P mz ms zero => mz
NatInd P mz ms (suc n) => ms n (NatInd P mz ms n)

 ${\displaystyle {\mathsf {plus}}\ x\ y\Leftarrow {\underline {\mathrm {rec} }}\ x\ \{}$ ${\displaystyle {\mathsf {plus}}\ x\ y\Leftarrow {\underline {\mathrm {case} }}\ x\ \{}$ ${\displaystyle {\mathsf {plus\ zero}}\ y\Rightarrow y}$ ${\displaystyle \quad \quad {\mathsf {plus}}\ ({\mathsf {suc}}\ x)\ y\Rightarrow suc\ ({\mathsf {plus}}\ x\ y)\ \}\ \}}$

...And in ASCII:

plus x y <= rec x {
plus x y <= case x {
plus zero y => y
plus (suc x) y => suc (plus x y)
}
}

## Dependent types

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 ${\displaystyle \star }$ ; types are arbitrary expressions of type ${\displaystyle \star }$ , and type equivalence is defined in terms of the types' normal forms. Second, it has a dependent function type; instead of ${\displaystyle P\rightarrow Q}$ , ${\displaystyle \forall x:P\Rightarrow Q}$ , where ${\displaystyle x}$  is bound in ${\displaystyle Q}$  to the value that the function's argument (of type ${\displaystyle P}$ ) eventually takes.

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.