# Short theorem prover

Theorem prover in 625 characters of Haskell

```
import Monad;import Maybe;import List
infixr 9 ?
(?)=(:>);z=Just;y=True
data P=A Integer|P:>P deriving(Read,Show,Eq)
[a,b,c,d,e,f]=map A[1,3..11]
g=h(?).(A.)
h f z(A x)=z x
h f z(x:>y)=h f z x`f`h f z y
i p(A i)j=p&&i==j
i p(a:>b)j=i y a j||i y b j
j(A l)s(A k)|i False s l=Nothing|l==k=z s|y=z$A k
j f@(A i)s(a:>b)=liftM2(?)(j f s a)(j f s b)
j f s@(A i)p=j s f p
j(a:>b)(c:>d)p=let u=j a c in join$liftM3 j(u b)(u d)(u p)
l x=g(toInteger.fromJust.flip elemIndex (h union (:[]) x))x
m=(a?a:)$map l$catMaybes$liftM2(uncurry.j.g(*2))m[((((a?b?c)
?(a?b)?a?c)?(d?e?d)?f),f),((a?b),(c?a)?c?b)]
main=readLn>>=print.(`elem`m)
```

You are not expected to understand that, so here is the explanation:

First, we need a type of prepositions. Each A constructor represents a prepositional variable (a, b, c, etc), and `:>`

represents implication. Thus, for example, `(A 0 :> A 0)`

is the axiom of tautology in classical logic.

```
type U = Integer
data P = A U | P:>P deriving(Read,Show,Eq)
infixr 9 :>
```

To prove theorems, we need axioms and deduction rules. This theorem prover is based on the Curry-Howard-Lambek correspondence applied to the programming language Jot (http://ling.ucsd.edu/~barker/Iota/#Goedel). Thus, we have a single basic combinator and two deduction rules, corresponding to partial application of the base case and two combinators of Jot.

```
axiom = A 0:>A 0
rules = let[a,b,c,d,e,f]=map A[1,3..11]in[ (((a:>b:>c):>(a:>b):>a:>c):>(d:>e:>d):>f):>f, (a:>b):>(c:>a):>c:>b]
```

We will also define foogomorphisms on the data structure:

```
pmap :: (U -> U) -> P -> P
pmap f = pfold (:>) (A .f)
pfold :: (a -> a -> a) -> (U -> a) -> P -> a
pfold f z (A x) = z x
pfold f z (x:>y) = pfold f z x`f`pfold f z y
```

In order to avoid infinite types (which are not intrinsically dangerous in a programming language but wreak havoc in logic because terms such as `fix a. (a -> b)`

correspond to statements such as "this statement is false"), we check whether the replaced variable is mentioned in the replacing term:

```
cnt p(A i) j = p && i == j
cnt p(a:>b) j = cnt True a j || cnt True b j
```

The deduction steps are performed by a standard unification routine:

```
unify :: P -> P -> P -> Maybe P
unify (A i) s (A j) | cnt False s i = Nothing
| i == j = Just $ s
| otherwise = Just $ A j
unify f@(A i) s (a:>b) = liftM2 (:>) (unify f s a) (unify f s b)
unify f s@(A i) pl = unify s f pl
unify (a:>b) (c:>d) pl = let u = unify a c in join $ liftM3 unify (u b) (u d) (u pl)
```

We need to renumber terms to prevent name conflicts.

```
fan (A x) = A (x*2)
fan (x:>y)= fan x:> fan y
```

Make a deduction given a rule, by setting the LHS of the rule equal to the state and taking the RHS of the rule.

```
deduce t r = let (a:>b)=r in unify (fan t) a b
```

Canonicalize the numbers in the rule, thus allowing matching.

```
canon :: P -> P
canon x = pmap (toInteger . fromJust . flip elemIndex ( allvars x )) x
allvars :: P -> [U]
allvars = pfold union (:[])
```

Given this, we can lazily construct the list of all true statements:

```
facts = nub $ (axiom:) $ map canon $ catMaybes $ liftM2 deduce facts rules
main=readLn>>=print.(`elem`facts)
```