YES 0.929
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
| ((maximum :: [Char] -> Char) :: [Char] -> Char) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((maximum :: [Char] -> Char) :: [Char] -> Char) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
max1 | x y True | = y |
max1 | x y False | = max0 x y otherwise |
max2 | x y | = max1 x y (x <= y) |
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
mainModule Main
| (maximum :: [Char] -> Char) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_max1(vx26, vx27, Succ(vx280), Succ(vx290)) → new_max1(vx26, vx27, vx280, vx290)
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_max1(vx26, vx27, Succ(vx280), Succ(vx290)) → new_max1(vx26, vx27, vx280, vx290)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_foldl(vx30, :(vx310, vx311)) → new_foldl(new_max10(vx30, vx310), vx311)
The TRS R consists of the following rules:
new_max11(vx26, vx27, Zero, Succ(vx290)) → new_max12(vx26, vx27)
new_max12(vx26, vx27) → Char(Succ(vx27))
new_max11(vx26, vx27, Zero, Zero) → new_max12(vx26, vx27)
new_max11(vx26, vx27, Succ(vx280), Zero) → Char(Succ(vx26))
new_max10(Char(Succ(vx3000)), Char(Zero)) → Char(Succ(vx3000))
new_max10(Char(Zero), Char(Succ(vx31000))) → Char(Succ(vx31000))
new_max10(Char(Zero), Char(Zero)) → Char(Zero)
new_max10(Char(Succ(vx3000)), Char(Succ(vx31000))) → new_max11(vx3000, vx31000, vx3000, vx31000)
new_max11(vx26, vx27, Succ(vx280), Succ(vx290)) → new_max11(vx26, vx27, vx280, vx290)
The set Q consists of the following terms:
new_max11(x0, x1, Zero, Zero)
new_max11(x0, x1, Succ(x2), Zero)
new_max10(Char(Zero), Char(Zero))
new_max11(x0, x1, Succ(x2), Succ(x3))
new_max11(x0, x1, Zero, Succ(x2))
new_max10(Char(Succ(x0)), Char(Succ(x1)))
new_max10(Char(Succ(x0)), Char(Zero))
new_max10(Char(Zero), Char(Succ(x0)))
new_max12(x0, x1)
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_foldl(vx30, :(vx310, vx311)) → new_foldl(new_max10(vx30, vx310), vx311)
The graph contains the following edges 2 > 2