Ecosyste.ms: Awesome

An open API service indexing awesome lists of open source software.

Awesome Lists | Featured Topics | Projects

https://github.com/joom/hezarfen

a theorem prover for intuitionistic propositional logic in Idris, with metaprogramming features
https://github.com/joom/hezarfen

automated-theorem-provers idris mathematical-logic metaprogramming theorem-prover

Last synced: about 1 month ago
JSON representation

a theorem prover for intuitionistic propositional logic in Idris, with metaprogramming features

Awesome Lists containing this project

README 

        
# Hezarfen



An Idris implementation of a theorem prover for [Roy Dyckhoff's

LJT](https://scholar.google.com/scholar?cluster=14155439887209124225), a sequent

calculus for propositional intuitionistic logic that is decidable and does

_not_ need loop checking. Initially ported from [Ayberk Tosun's Standard ML

implementation](https://github.com/ayberkt/sequents).



The main goal of the project is to make use of the [elaborator

reflection](http://docs.idris-lang.org/en/latest/reference/elaborator-reflection.html).

Similar to [Lennart Augustsson's Djinn](https://github.com/augustss/djinn), a

theorem prover that generates Haskell expressions, Hezarfen generates Idris expressions.

Unlike Djinn, Hezarfen is not a standalone program, it is a library that

generates Idris expressions of the type `Raw`, one of the types used for the

inner representation of the core language of Idris. This means these

expressions can easily be spliced into your programs. Hezarfen provides a

tactic that lets you do this:



```idris

f2 : (a -> b) -> (b -> c) -> (c -> d) -> a -> d

f2 = %runElab hezarfenExpr

```



However, instead of creating proof terms, you can also create definitions that

are more readable when you print their definitions.



```idris

f2 : (a -> b) -> (b -> c) -> (c -> d) -> a -> d

%runElab (hezarfen `{f2})

```



Or with the more readable syntax extension:



```idris

demorgan3 : Either (Not p) (Not q) -> Not (p, q)

derive demorgan3



contrapositive : (p -> q) -> (Not q -> Not p)

derive contrapositive

```



It can also make use of your existing lemmas:



```idris

evenOrOdd : (n : Nat) -> Either (Even n) (Odd n)

... -- some definition of an existing lemma



oddOrEven : (n : Nat) -> Either (Odd n) (Even n)

%runElab (hezarfen' `{oddOrEven} !(add [`{evenOrOdd}]))



-- something more complex, but passing the constructors for Even and Odd

-- using the more readable syntax

evenOrOddSS : (n : Nat) -> Either (Even (S (S n))) (Odd (S (S n)))

obtain evenOrOddSS from [`{evenOrOdd}, `{EvenSS}, `{OddSS}]

```



We also have a Coq-style hint database system that lets us keep a list of hint names that will be used to prove theorems. To use the hints in proofs, change `derive` to `derive'` and `obtain` to `obtain'`. Then the names in the hint database will be automatically added to the context in which the theorem prover runs.



```idris

hint evenOrOdd

hint EvenSS

hint OddSS



evenOrOddSS : (n : Nat) -> Either (Even (S (S n))) (Odd (S (S n)))

derive' evenOrOddSS

```



The even/odd example is beyond the logic Hezarfen is trying to decide. `Even n` and `Even (S (S n))` happen to be one function away, namely `EvenSS`.



Hezarfen attempts to prove a tiny bit more than propositional intuitionistic logic,

especially when it comes to equalities and `Dec`.

Even though this part is a bit ad hoc, it can currently prove things of this nature:



```idris

eqDec : x = y -> Dec (y = x)

derive eqDec



decCongB : x = y -> Dec (not x = not y)

derive decCongB

```



For details, look at [Examples.idr](https://github.com/joom/hezarfen/blob/master/Examples.idr).



## Edit-Time Tactics



The original purpose of Hezarfen was to be a part of a master's thesis on

"edit-time tactics", meaning that we would be able to run it from the editor.

Then it would be an alternative to the built-in proof search in Idris.

[Here](http://cattheory.com/editTimeTacticsDraft.pdf) is the draft of the

thesis. And below you can see how it works in the editor:



[![Screencast of how Hezarfen works in Emacs](https://asciinema.org/a/rrwboxAr2VdiUQsZov0RDinJw.png)](https://asciinema.org/a/rrwboxAr2VdiUQsZov0RDinJw)



The feature that allows this has not landed on the Idris compiler or the Idris

mode yet, but it should be merged soon!



## Future Work



Some support for deriving terms with type classes can be implemented, à la Djinn.



One of the end goals of Hezarfen is to generate proofs that are easy to read:



* Fresh variable names should be readable. Currently there is a hacky `fresh` function in [Prover.idr](https://github.com/joom/hezarfen/blob/master/Hezarfen/Prover.idr) that does that.

* There is [already some work on simplifying the proof terms](https://github.com/joom/hezarfen/blob/master/Hezarfen/Simplify.idr) generated by Hezarfen. There are some tricks Hezarfen can learn from Haskell's [pointfree](https://hackage.haskell.org/package/pointfree) package. ([web version](http://pointfree.io/))

* Currently Hezarfen primarily generates expressions as proofs. However when we are writing functions, we often use function definitions instead of lambda terms, and we pattern match on pairs in the function definition, instead of writing `let a = fst p in let b = snd p in ...` or `case p of (x, y) => ...` to do projections. There is [some work](https://github.com/joom/hezarfen/blob/master/Hezarfen/FunDefn.idr) on translating these proof terms to readable function definitions.



The definition it generates looks like this:

```idris

> :printdef contrapositive

contrapositive : (p -> q) -> Not q -> Not p

contrapositive c d = void . d . c

```



As opposed to proof term:

```idris

contrapositive : (p -> q) -> Not q -> Not p

contrapositive = \i20, j20 => void . j20 . i20

```



***



*hezarfen* (/hezaɾfæn/, sounds like "has are fan") is a Turkish word that means

polymath, literally "a thousand sciences".