YES 4.0600000000000005
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
|  | ((abs :: Num a => a  ->  a) :: Num a => a  ->  a) | 
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
mainModule Main
|  | ((abs :: Num a => a  ->  a) :: Num a => 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
is transformed to
| absReal0 | x True | = `negate` x | 
| absReal1 | x True | = x | 
| absReal1 | x False | = absReal0 x otherwise | 
| absReal2 | x | = absReal1 x (x >= 0) | 
↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
        ↳ HASKELL
          ↳ NumRed
mainModule Main
|  | ((abs :: Num a => a  ->  a) :: Num a => a  ->  a) | 
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
module Main where
Haskell To QDPs
↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
        ↳ HASKELL
          ↳ NumRed
            ↳ HASKELL
              ↳ Narrow
                ↳ QDP
                  ↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_primMulNat(Succ(vx31000)) → new_primMulNat(vx31000)
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(vx31000)) → new_primMulNat(vx31000)
 The graph contains the following edges 1 > 1