YES 0.66
H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:
↳ HASKELL
↳ BR
mainModule Main
| ((takeWhile :: (a -> Bool) -> [a] -> [a]) :: (a -> Bool) -> [a] -> [a]) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((takeWhile :: (a -> Bool) -> [a] -> [a]) :: (a -> Bool) -> [a] -> [a]) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
The following Function with conditions
takeWhile | p [] | = [] |
takeWhile | p (x : xs) | |
is transformed to
takeWhile | p [] | = takeWhile3 p [] |
takeWhile | p (x : xs) | = takeWhile2 p (x : xs) |
takeWhile1 | p x xs True | = x : takeWhile p xs |
takeWhile1 | p x xs False | = takeWhile0 p x xs otherwise |
takeWhile0 | p x xs True | = [] |
takeWhile2 | p (x : xs) | = takeWhile1 p x xs (p x) |
takeWhile3 | p [] | = [] |
takeWhile3 | vz wu | = takeWhile2 vz wu |
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
mainModule Main
| (takeWhile :: (a -> Bool) -> [a] -> [a]) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_takeWhile(wv3, :(wv40, wv41), ba) → new_takeWhile1(wv3, wv40, wv41, ba)
new_takeWhile1(wv3, wv40, wv41, ba) → new_takeWhile(wv3, wv41, ba)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_takeWhile1(wv3, wv40, wv41, ba) → new_takeWhile(wv3, wv41, ba)
The graph contains the following edges 1 >= 1, 3 >= 2, 4 >= 3
- new_takeWhile(wv3, :(wv40, wv41), ba) → new_takeWhile1(wv3, wv40, wv41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4