I'm playing with a little formalisation in Idris and I'm having some strange behaviour: high compilation time and CPU usage for a function.
The code is an regex pattern matching algorithm. First the regex definition:
data RegExp : Type where
Zero : RegExp
Eps : RegExp
Chr : Char -> RegExp
Cat : RegExp -> RegExp -> RegExp
Alt : RegExp -> RegExp -> RegExp
Star : RegExp -> RegExp
Comp : RegExp -> RegExp
Regex membership and non-membership are defined as the following mutually recursive data types:
mutual
data NotInRegExp : List Char -> RegExp -> Type where
NotInZero : NotInRegExp xs Zero
NotInEps : Not (xs = []) -> NotInRegExp xs Eps
NotInChr : Not (xs = [ c ]) -> NotInRegExp xs (Chr c)
NotInCat : zs = xs ++ ys -> (Either (NotInRegExp xs l)
((InRegExp xs l)
,(NotInRegExp ys r)))
-> NotInRegExp zs (Cat l r)
NotInAlt : NotInRegExp xs l -> NotInRegExp xs r -> NotInRegExp xs (Alt l r)
NotInStar : NotInRegExp xs Eps ->
NotInRegExp xs (Cat e (Star e)) ->
NotInRegExp xs (Star e)
NotInComp : InRegExp xs e -> NotInRegExp xs (Comp e)
data InRegExp : List Char -> RegExp -> Type where
InEps : InRegExp [] Eps
InChr : InRegExp [ a ] (Chr a)
InCat : InRegExp xs l ->
InRegExp ys r ->
zs = xs ++ ys ->
InRegExp zs (Cat l r)
InAltL : InRegExp xs l ->
InRegExp xs (Alt l r)
InAltR : InRegExp xs r ->
InRegExp xs (Alt l r)
InStar : InRegExp xs (Alt Eps (Cat e (Star e))) ->
InRegExp xs (Star e)
InComp : NotInRegExp xs e -> InRegExp xs (Comp e)
After these rather long definitions, I define a smart constructor for alternatives:
infixl 4 .|.
(.|.) : RegExp -> RegExp -> RegExp
Zero .|. e = e
e .|. Zero = e
e .|. e' = Alt e e'
Now, I want to prove that this smart constructor is sound and complete with respect to regex membership semantics. The proofs are almost straightforward induction / case analysis. But, one of these proofs is demanding a lot of time and CPU to compile (around 90% of CPU in Mac OS X El Capitan).
The offending function is:
altOptNotInComplete : NotInRegExp xs (Alt l r) -> NotInRegExp xs (l .|. r)
altOptNotInComplete {l = Zero} (NotInAlt x y) = y
altOptNotInComplete {l = Eps}{r = Zero} (NotInAlt x y) = x
altOptNotInComplete {l = Eps}{r = Eps} pr = pr
altOptNotInComplete {l = Eps}{r = (Chr x)} pr = pr
altOptNotInComplete {l = Eps}{r = (Cat x y)} pr = pr
altOptNotInComplete {l = Eps}{r = (Alt x y)} pr = pr
altOptNotInComplete {l = Eps}{r = (Star x)} pr = pr
altOptNotInComplete {l = Eps}{r = (Comp x)} pr = pr
altOptNotInComplete {l = (Chr x)}{r = Zero} (NotInAlt y z) = y
altOptNotInComplete {l = (Chr x)}{r = Eps} pr = pr
altOptNotInComplete {l = (Chr x)}{r = (Chr y)} pr = pr
altOptNotInComplete {l = (Chr x)}{r = (Cat y z)} pr = pr
altOptNotInComplete {l = (Chr x)}{r = (Alt y z)} pr = pr
altOptNotInComplete {l = (Chr x)}{r = (Star y)} pr = pr
altOptNotInComplete {l = (Chr x)}{r = (Comp y)} pr = pr
altOptNotInComplete {l = (Cat x y)}{r = Zero} (NotInAlt z w) = z
altOptNotInComplete {l = (Cat x y)}{r = Eps} pr = pr
altOptNotInComplete {l = (Cat x y)}{r = (Chr z)} pr = pr
altOptNotInComplete {l = (Cat x y)}{r = (Cat z w)} pr = pr
altOptNotInComplete {l = (Cat x y)}{r = (Alt z w)} pr = pr
altOptNotInComplete {l = (Cat x y)}{r = (Star z)} pr = pr
altOptNotInComplete {l = (Cat x y)}{r = (Comp z)} pr = pr
altOptNotInComplete {l = (Alt x y)}{r = Zero} (NotInAlt z w) = z
altOptNotInComplete {l = (Alt x y)}{r = Eps} pr = pr
altOptNotInComplete {l = (Alt x y)}{r = (Chr z)} pr = pr
altOptNotInComplete {l = (Alt x y)}{r = (Cat z w)} pr = pr
altOptNotInComplete {l = (Alt x y)}{r = (Alt z w)} pr = pr
altOptNotInComplete {l = (Alt x y)}{r = (Star z)} pr = pr
altOptNotInComplete {l = (Alt x y)}{r = (Comp z)} pr = pr
altOptNotInComplete {l = (Star x)}{r = Zero} (NotInAlt y z) = y
altOptNotInComplete {l = (Star x)}{r = Eps} pr = pr
altOptNotInComplete {l = (Star x)}{r = (Chr y)} pr = pr
altOptNotInComplete {l = (Star x)}{r = (Cat y z)} pr = pr
altOptNotInComplete {l = (Star x)}{r = (Alt y z)} pr = pr
altOptNotInComplete {l = (Star x)}{r = (Star y)} pr = pr
altOptNotInComplete {l = (Star x)}{r = (Comp y)} pr = pr
altOptNotInComplete {l = (Comp x)}{r = Zero} (NotInAlt y z) = y
altOptNotInComplete {l = (Comp x)}{r = Eps} pr = pr
altOptNotInComplete {l = (Comp x)}{r = (Chr y)} pr = pr
altOptNotInComplete {l = (Comp x)}{r = (Cat y z)} pr = pr
altOptNotInComplete {l = (Comp x)}{r = (Alt y z)} pr = pr
altOptNotInComplete {l = (Comp x)}{r = (Star y)} pr = pr
altOptNotInComplete {l = (Comp x)}{r = (Comp y)} pr = pr
I can't understand why this function is demanding so much CPU. Is there a way to "optimize" this code in order that compilation behaves normally?
The previous code is available at the following gist. I'm using Idris 0.10 on Mac Os X El Capitan.
Any clue is highly welcome.
