YES 206.952
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
| ((isAlpha :: Char -> Bool) :: Char -> Bool) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((isAlpha :: Char -> Bool) :: Char -> Bool) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
mainModule Main
| ((isAlpha :: Char -> Bool) :: Char -> Bool) |
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
| (isAlpha :: Char -> Bool) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_not(Succ(vx2550), Succ(vx2560)) → new_not(vx2550, vx2560)
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_not(Succ(vx2550), Succ(vx2560)) → new_not(vx2550, vx2560)
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
Q DP problem:
The TRS P consists of the following rules:
new_not0(Succ(vx840), Succ(vx870)) → new_not0(vx840, vx870)
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_not0(Succ(vx840), Succ(vx870)) → new_not0(vx840, vx870)
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
Q DP problem:
The TRS P consists of the following rules:
new_pePe(Succ(vx18300000000), Succ(vx18600000000), vx211, vx204) → new_pePe(vx18300000000, vx18600000000, vx211, vx204)
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_pePe(Succ(vx18300000000), Succ(vx18600000000), vx211, vx204) → new_pePe(vx18300000000, vx18600000000, vx211, vx204)
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
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_pePe0(Succ(vx1440), Succ(vx1450), vx146, vx147, vx148) → new_pePe0(vx1440, vx1450, vx146, vx147, vx148)
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_pePe0(Succ(vx1440), Succ(vx1450), vx146, vx147, vx148) → new_pePe0(vx1440, vx1450, vx146, vx147, vx148)
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
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_pePe1(Succ(vx820), Succ(vx830), vx84, vx85, vx86, vx87) → new_pePe1(vx820, vx830, vx84, vx85, vx86, vx87)
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_pePe1(Succ(vx820), Succ(vx830), vx84, vx85, vx86, vx87) → new_pePe1(vx820, vx830, vx84, vx85, vx86, vx87)
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
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_pePe2(Succ(vx740), Succ(vx750), vx76, vx77, vx78, vx79, vx80) → new_pePe2(vx740, vx750, vx76, vx77, vx78, vx79, vx80)
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_pePe2(Succ(vx740), Succ(vx750), vx76, vx77, vx78, vx79, vx80) → new_pePe2(vx740, vx750, vx76, vx77, vx78, vx79, vx80)
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
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_pePe3(Succ(vx380), Succ(vx390), vx40, vx41, vx42, vx43, vx44, vx45) → new_pePe3(vx380, vx390, vx40, 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_pePe3(Succ(vx380), Succ(vx390), vx40, vx41, vx42, vx43, vx44, vx45) → new_pePe3(vx380, vx390, vx40, 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