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
  import qualified Prelude



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
  import qualified Prelude



Cond Reductions:
The following Function with conditions
undefined 
 | False
 = undefined

is transformed to
undefined  = undefined1

undefined0 True = undefined

undefined1  = undefined0 False

The following Function with conditions
absReal x
 | x >= 0
 = x
 | otherwise
 = `negate` x

is transformed to
absReal x = absReal2 x

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
  import qualified Prelude



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
  (abs :: Num a => a  ->  a)

module Main where
  import qualified Prelude



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: