YES 3.62 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
  ((rangeSize :: (Char,Char ->  Int) :: (Char,Char ->  Int)

module Main where
  import qualified Prelude



Replaced joker patterns by fresh variables and removed binding patterns.
Binding Reductions:
The bind variable of the following binding Pattern
r@(vv,vw)

is replaced by the following term
(vv,vw)

The bind variable of the following binding Pattern
b@(wu,wv)

is replaced by the following term
(wu,wv)



↳ HASKELL
  ↳ BR
HASKELL
      ↳ COR

mainModule Main
  ((rangeSize :: (Char,Char ->  Int) :: (Char,Char ->  Int)

module Main where
  import qualified Prelude



Cond Reductions:
The following Function with conditions
rangeSize (vv,vw)
 | null (range (vv,vw))
 = 0
 | otherwise
 = index (vv,vwvw + 1

is transformed to
rangeSize (vv,vw) = rangeSize2 (vv,vw)

rangeSize0 vv vw True = index (vv,vwvw + 1

rangeSize1 vv vw True = 0
rangeSize1 vv vw False = rangeSize0 vv vw otherwise

rangeSize2 (vv,vw) = rangeSize1 vv vw (null (range (vv,vw)))

The following Function with conditions
undefined 
 | False
 = undefined

is transformed to
undefined  = undefined1

undefined0 True = undefined

undefined1  = undefined0 False

The following Function with conditions
takeWhile p [] = []
takeWhile p (x : xs)
 | p x
 = x : takeWhile p xs
 | otherwise
 = []

is transformed to
takeWhile p [] = takeWhile3 p []
takeWhile p (x : xs) = takeWhile2 p (x : xs)

takeWhile1 p x xs True = x : takeWhile p xs
takeWhile1 p x xs False = takeWhile0 p x xs otherwise

takeWhile0 p x xs True = []

takeWhile2 p (x : xs) = takeWhile1 p x xs (p x)

takeWhile3 p [] = []
takeWhile3 xu xv = takeWhile2 xu xv

The following Function with conditions
index (wu,wvci
 | inRange (wu,wvci
 = fromEnum ci - fromEnum wu
 | otherwise
 = error []

is transformed to
index (wu,wvci = index2 (wu,wvci

index1 wu wv ci True = fromEnum ci - fromEnum wu
index1 wu wv ci False = index0 wu wv ci otherwise

index0 wu wv ci True = error []

index2 (wu,wvci = index1 wu wv ci (inRange (wu,wvci)



↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
HASKELL
          ↳ LetRed

mainModule Main
  ((rangeSize :: (Char,Char ->  Int) :: (Char,Char ->  Int)

module Main where
  import qualified Prelude



Let/Where Reductions:
The bindings of the following Let/Where expression
fromEnum c <= i && i <= fromEnum c'
where 
i  = fromEnum ci

are unpacked to the following functions on top level
inRangeI xx = fromEnum xx



↳ HASKELL
  ↳ BR
    ↳ HASKELL
      ↳ COR
        ↳ HASKELL
          ↳ LetRed
HASKELL
              ↳ NumRed

mainModule Main
  ((rangeSize :: (Char,Char ->  Int) :: (Char,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
          ↳ LetRed
            ↳ HASKELL
              ↳ NumRed
HASKELL
                  ↳ Narrow

mainModule Main
  (rangeSize :: (Char,Char ->  Int)

module Main where
  import qualified Prelude



Haskell To QDPs


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

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

new_primPlusInt(xy24, Succ(xy250)) → new_primPlusInt(xy24, xy250)

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
          ↳ LetRed
            ↳ HASKELL
              ↳ NumRed
                ↳ HASKELL
                  ↳ Narrow
                    ↳ AND
                      ↳ QDP
QDP
                        ↳ QDPSizeChangeProof
                      ↳ QDP
                      ↳ QDP
                      ↳ QDP

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

new_primPlusInt0(Succ(xy440), Succ(xy430)) → new_primPlusInt0(xy440, xy430)

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
          ↳ LetRed
            ↳ HASKELL
              ↳ NumRed
                ↳ HASKELL
                  ↳ Narrow
                    ↳ AND
                      ↳ QDP
                      ↳ QDP
QDP
                        ↳ QDPSizeChangeProof
                      ↳ QDP
                      ↳ QDP

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

new_primPlusInt1(xy43, xy44, Succ(xy450)) → new_primPlusInt1(xy43, xy44, xy450)

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
          ↳ LetRed
            ↳ HASKELL
              ↳ NumRed
                ↳ HASKELL
                  ↳ Narrow
                    ↳ AND
                      ↳ QDP
                      ↳ QDP
                      ↳ QDP
QDP
                        ↳ QDPSizeChangeProof
                      ↳ QDP

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

new_primPlusInt2(xy35, xy36, Succ(xy370), Succ(xy380)) → new_primPlusInt2(xy35, xy36, xy370, xy380)

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
          ↳ LetRed
            ↳ HASKELL
              ↳ NumRed
                ↳ HASKELL
                  ↳ Narrow
                    ↳ AND
                      ↳ QDP
                      ↳ QDP
                      ↳ QDP
                      ↳ QDP
QDP
                        ↳ QDPSizeChangeProof

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

new_rangeSize1(xy17, xy18, Succ(xy190), Succ(xy200), ba) → new_rangeSize1(xy17, xy18, xy190, xy200, ba)

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: