YES 2.428
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
↳ IFR
mainModule Main
| ((showSigned :: (Ratio Int -> [Char] -> [Char]) -> Int -> Ratio Int -> [Char] -> [Char]) :: (Ratio Int -> [Char] -> [Char]) -> Int -> Ratio Int -> [Char] -> [Char]) |
module Main where
If Reductions:
The following If expression
if b then (showChar '(') . p . showChar ')' else p
is transformed to
showParen0 | p True | = (showChar '(') . p . showChar ')' |
showParen0 | p False | = p |
The following If expression
if x < 0 then showParen (p > 6) ((showChar '-') . showPos (`negate` x)) else showPos x
is transformed to
showSigned0 | p showPos x True | = showParen (p > 6) ((showChar '-') . showPos (`negate` x)) |
showSigned0 | p showPos x False | = showPos x |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
| ((showSigned :: (Ratio Int -> [Char] -> [Char]) -> Int -> Ratio Int -> [Char] -> [Char]) :: (Ratio Int -> [Char] -> [Char]) -> Int -> Ratio Int -> [Char] -> [Char]) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((showSigned :: (Ratio Int -> [Char] -> [Char]) -> Int -> Ratio Int -> [Char] -> [Char]) :: (Ratio Int -> [Char] -> [Char]) -> Int -> Ratio Int -> [Char] -> [Char]) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
mainModule Main
| ((showSigned :: (Ratio Int -> [Char] -> [Char]) -> Int -> Ratio Int -> [Char] -> [Char]) :: (Ratio Int -> [Char] -> [Char]) -> Int -> Ratio Int -> [Char] -> [Char]) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
mainModule Main
| (showSigned :: (Ratio Int -> [Char] -> [Char]) -> Int -> Ratio Int -> [Char] -> [Char]) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_showParen0(vx115, vx116, vx117, Succ(vx113000), Succ(vx1140), vx118) → new_showParen0(vx115, vx116, vx117, vx113000, vx1140, vx118)
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_showParen0(vx115, vx116, vx117, Succ(vx113000), Succ(vx1140), vx118) → new_showParen0(vx115, vx116, vx117, vx113000, vx1140, vx118)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5, 6 >= 6
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_primMulNat(Succ(vx50000)) → new_primMulNat(vx50000)
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_primMulNat(Succ(vx50000)) → new_primMulNat(vx50000)
The graph contains the following edges 1 > 1