YES 4.7780000000000005 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
  ((digitToInt :: Char  ->  Int) :: Char  ->  Int)

module Main where
  import qualified Prelude



Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL
  ↳ BR
HASKELL
      ↳ COR

mainModule Main
  ((digitToInt :: Char  ->  Int) :: Char  ->  Int)

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
digitToInt c
 | isDigit c
 = fromEnum c - fromEnum '0'
 | c >= 'a' && c <= 'f'
 = fromEnum c - fromEnum 'a' + 10
 | c >= 'A' && c <= 'F'
 = fromEnum c - fromEnum 'A' + 10
 | otherwise
 = error []

is transformed to
digitToInt c = digitToInt4 c

digitToInt0 c True = error []

digitToInt1 c True = fromEnum c - fromEnum 'A' + 10
digitToInt1 c False = digitToInt0 c otherwise

digitToInt3 c True = fromEnum c - fromEnum '0'
digitToInt3 c False = digitToInt2 c (c >= 'a' && c <= 'f')

digitToInt2 c True = fromEnum c - fromEnum 'a' + 10
digitToInt2 c False = digitToInt1 c (c >= 'A' && c <= 'F')

digitToInt4 c = digitToInt3 c (isDigit c)



↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
HASKELL
          ↳ NumRed

mainModule Main
  ((digitToInt :: Char  ->  Int) :: Char  ->  Int)

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
  (digitToInt :: Char  ->  Int)

module Main where
  import qualified Prelude



Haskell To QDPs


↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
        ↳ HASKELL
          ↳ NumRed
            ↳ HASKELL
              ↳ Narrow
                ↳ AND
QDP
                    ↳ QDPSizeChangeProof
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_primMinusNat(Succ(vx1030), Succ(vx1040)) → new_primMinusNat(vx1030, vx1040)

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:



↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
        ↳ HASKELL
          ↳ NumRed
            ↳ HASKELL
              ↳ Narrow
                ↳ AND
                  ↳ QDP
QDP
                    ↳ QDPSizeChangeProof
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_primPlusNat(Succ(vx134000), Succ(vx1330)) → new_primPlusNat(vx134000, vx1330)

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:



↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
        ↳ HASKELL
          ↳ NumRed
            ↳ HASKELL
              ↳ Narrow
                ↳ AND
                  ↳ QDP
                  ↳ QDP
QDP
                    ↳ QDPSizeChangeProof
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_digitToInt1(vx136, Succ(vx1370), Succ(vx1380)) → new_digitToInt1(vx136, vx1370, vx1380)

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:



↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
        ↳ HASKELL
          ↳ NumRed
            ↳ HASKELL
              ↳ Narrow
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
QDP
                    ↳ QDPSizeChangeProof
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_digitToInt10(vx123, Succ(vx1240), Succ(vx1250), vx126) → new_digitToInt10(vx123, vx1240, vx1250, vx126)

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:



↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
        ↳ HASKELL
          ↳ NumRed
            ↳ HASKELL
              ↳ Narrow
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
QDP
                    ↳ QDPSizeChangeProof
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_digitToInt2(vx119, Succ(vx1200), Succ(vx1210)) → new_digitToInt2(vx119, vx1200, vx1210)

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:



↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
        ↳ HASKELL
          ↳ NumRed
            ↳ HASKELL
              ↳ Narrow
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
QDP
                    ↳ QDPSizeChangeProof
                  ↳ QDP
                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_digitToInt20(vx98, Succ(vx990), Succ(vx1000), vx101) → new_digitToInt20(vx98, vx990, vx1000, vx101)

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:



↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
        ↳ HASKELL
          ↳ NumRed
            ↳ HASKELL
              ↳ Narrow
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
QDP
                    ↳ QDPSizeChangeProof
                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_digitToInt3(vx89, Succ(vx900), Succ(vx910)) → new_digitToInt3(vx89, vx900, vx910)

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:



↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
        ↳ HASKELL
          ↳ NumRed
            ↳ HASKELL
              ↳ Narrow
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
QDP
                    ↳ QDPSizeChangeProof

Q DP problem:
The TRS P consists of the following rules:

new_digitToInt30(vx61, Succ(vx620), Succ(vx630), vx64) → new_digitToInt30(vx61, vx620, vx630, vx64)

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: