YES 23.71
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
| ((toLower :: Char -> Char) :: Char -> Char) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((toLower :: Char -> Char) :: Char -> Char) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
The following Function with conditions
toLower | c |
| | isUpper c |
= | toEnum (fromEnum c - fromEnum 'A' + fromEnum 'a') |
|
| | otherwise | |
|
is transformed to
toLower1 | c True | = toEnum (fromEnum c - fromEnum 'A' + fromEnum 'a') |
toLower1 | c False | = toLower0 c otherwise |
toLower2 | c | = toLower1 c (isUpper c) |
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
mainModule Main
| ((toLower :: Char -> Char) :: Char -> Char) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
mainModule Main
| (toLower :: Char -> Char) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primIntToChar(Succ(vx990), Succ(vx9800)) → new_primIntToChar(vx990, vx9800)
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_primIntToChar(Succ(vx990), Succ(vx9800)) → new_primIntToChar(vx990, vx9800)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat(Succ(vx9700), Succ(vx990)) → new_primPlusNat(vx9700, vx990)
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_primPlusNat(Succ(vx9700), Succ(vx990)) → new_primPlusNat(vx9700, vx990)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primIntToChar0(Succ(vx970), Succ(vx980), vx99) → new_primIntToChar0(vx970, vx980, vx99)
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_primIntToChar0(Succ(vx970), Succ(vx980), vx99) → new_primIntToChar0(vx970, vx980, vx99)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_toLower1(vx112, Succ(vx1130), Succ(vx1140)) → new_toLower1(vx112, vx1130, vx1140)
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_toLower1(vx112, Succ(vx1130), Succ(vx1140)) → new_toLower1(vx112, vx1130, vx1140)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_toLower10(vx107, Succ(vx1080), Succ(vx1090), vx110) → new_toLower10(vx107, vx1080, vx1090, vx110)
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_toLower10(vx107, Succ(vx1080), Succ(vx1090), vx110) → new_toLower10(vx107, vx1080, vx1090, vx110)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_toLower11(vx101, Succ(vx1020), Succ(vx1030), vx104, vx105) → new_toLower11(vx101, vx1020, vx1030, vx104, vx105)
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_toLower11(vx101, Succ(vx1020), Succ(vx1030), vx104, vx105) → new_toLower11(vx101, vx1020, vx1030, vx104, vx105)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_toLower12(vx80, Succ(vx810), Succ(vx820), vx83, vx84, vx85) → new_toLower12(vx80, vx810, vx820, vx83, vx84, vx85)
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_toLower12(vx80, Succ(vx810), Succ(vx820), vx83, vx84, vx85) → new_toLower12(vx80, vx810, vx820, vx83, vx84, vx85)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_toLower13(vx72, Succ(vx730), Succ(vx740), vx75, vx76, vx77, vx78) → new_toLower13(vx72, vx730, vx740, vx75, vx76, vx77, vx78)
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_toLower13(vx72, Succ(vx730), Succ(vx740), vx75, vx76, vx77, vx78) → new_toLower13(vx72, vx730, vx740, vx75, vx76, vx77, vx78)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_toLower14(vx38, Succ(vx390), Succ(vx400), vx41, vx42, vx43, vx44, vx45) → new_toLower14(vx38, vx390, vx400, vx41, vx42, vx43, vx44, vx45)
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_toLower14(vx38, Succ(vx390), Succ(vx400), vx41, vx42, vx43, vx44, vx45) → new_toLower14(vx38, vx390, vx400, vx41, vx42, vx43, vx44, vx45)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8