Hezarfen

An Idris implementation of a theorem prover for Roy Dyckhoff's LJT, 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.

The main goal of the project is to make use of the elaborator reflection. Similar to Lennart Augustsson's 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:

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.

f2 : (a -> b) -> (b -> c) -> (c -> d) -> a -> d
%runElab (hezarfen `{f2})

Or with the more readable syntax extension:

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:

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.

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:

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

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

For details, look at 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 is the draft of the thesis. And below you can see how it works in the editor:

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 that does that.
• There is already some work on simplifying the proof terms generated by Hezarfen. There are some tricks Hezarfen can learn from Haskell's pointfree package. (web version)
• 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 on translating these proof terms to readable function definitions.

The definition it generates looks like this:

> :printdef contrapositive
contrapositive : (p -> q) -> Not q -> Not p
contrapositive c d = void . d . c

As opposed to proof term:

contrapositive : (p -> q) -> Not q -> Not p
contrapositive = \i20, j20 => void . j20 . i20

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hezarfen (/hezaɾfæn/, sounds like "has are fan") is a Turkish word that means polymath, literally "a thousand sciences".