YES 3.478 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
  ↳ LR

mainModule Main
  ((pred :: Enum a => a  ->  a) :: Enum a => a  ->  a)

module Main where
  import qualified Prelude



Lambda Reductions:
The following Lambda expression
\(m,_)→m

is transformed to
m0 (m,_) = m

The following Lambda expression
\(_,r)→r

is transformed to
r0 (_,r) = r

The following Lambda expression
\(q,_)→q

is transformed to
q1 (q,_) = q



↳ HASKELL
  ↳ LR
HASKELL
      ↳ IFR

mainModule Main
  ((pred :: Enum a => a  ->  a) :: Enum a => a  ->  a)

module Main where
  import qualified Prelude



If Reductions:
The following If expression
if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero

is transformed to
primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y))
primDivNatS0 x y False = Zero

The following If expression
if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x

is transformed to
primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y)
primModNatS0 x y False = Succ x



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ IFR
HASKELL
          ↳ BR

mainModule Main
  ((pred :: Enum a => a  ->  a) :: Enum a => a  ->  a)

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
frac@(Double vy vz)

is replaced by the following term
Double vy vz

The bind variable of the following binding Pattern
frac@(Float ww wx)

is replaced by the following term
Float ww wx



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ IFR
        ↳ HASKELL
          ↳ BR
HASKELL
              ↳ COR

mainModule Main
  ((pred :: Enum a => a  ->  a) :: Enum 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
toEnum 0 = False
toEnum 1 = True

is transformed to
toEnum xz = toEnum3 xz
toEnum xy = toEnum1 xy

toEnum0 True xy = True

toEnum1 xy = toEnum0 (xy == 1) xy

toEnum2 True xz = False
toEnum2 yu yv = toEnum1 yv

toEnum3 xz = toEnum2 (xz == 0) xz
toEnum3 yw = toEnum1 yw

The following Function with conditions
toEnum 0 = ()

is transformed to
toEnum yx = toEnum5 yx

toEnum4 True yx = ()

toEnum5 yx = toEnum4 (yx == 0) yx

The following Function with conditions
toEnum 0 = LT
toEnum 1 = EQ
toEnum 2 = GT

is transformed to
toEnum zx = toEnum11 zx
toEnum yz = toEnum9 yz
toEnum yy = toEnum7 yy

toEnum6 True yy = GT

toEnum7 yy = toEnum6 (yy == 2) yy

toEnum8 True yz = EQ
toEnum8 zu zv = toEnum7 zv

toEnum9 yz = toEnum8 (yz == 1) yz
toEnum9 zw = toEnum7 zw

toEnum10 True zx = LT
toEnum10 zy zz = toEnum9 zz

toEnum11 zx = toEnum10 (zx == 0) zx
toEnum11 vuu = toEnum9 vuu



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ IFR
        ↳ HASKELL
          ↳ BR
            ↳ HASKELL
              ↳ COR
HASKELL
                  ↳ LetRed

mainModule Main
  ((pred :: Enum a => a  ->  a) :: Enum a => a  ->  a)

module Main where
  import qualified Prelude



Let/Where Reductions:
The bindings of the following Let/Where expression
m
where 
m  = m0 vu6
m0 (m,vv) = m
vu6  = properFraction x

are unpacked to the following functions on top level
truncateM vuv = truncateM0 vuv (truncateVu6 vuv)

truncateVu6 vuv = properFraction vuv

truncateM0 vuv (m,vv) = m

The bindings of the following Let/Where expression
(fromIntegral q,r :% y)
where 
q  = q1 vu30
q1 (q,xv) = q
r  = r0 vu30
r0 (xw,r) = r
vu30  = quotRem x y

are unpacked to the following functions on top level
properFractionR vuw vux = properFractionR0 vuw vux (properFractionVu30 vuw vux)

properFractionQ1 vuw vux (q,xv) = q

properFractionQ vuw vux = properFractionQ1 vuw vux (properFractionVu30 vuw vux)

properFractionR0 vuw vux (xw,r) = r

properFractionVu30 vuw vux = quotRem vuw vux



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ IFR
        ↳ HASKELL
          ↳ BR
            ↳ HASKELL
              ↳ COR
                ↳ HASKELL
                  ↳ LetRed
HASKELL
                      ↳ NumRed

mainModule Main
  ((pred :: Enum a => a  ->  a) :: Enum 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
  ↳ LR
    ↳ HASKELL
      ↳ IFR
        ↳ HASKELL
          ↳ BR
            ↳ HASKELL
              ↳ COR
                ↳ HASKELL
                  ↳ LetRed
                    ↳ HASKELL
                      ↳ NumRed
HASKELL
                          ↳ Narrow

mainModule Main
  (pred :: Enum a => a  ->  a)

module Main where
  import qualified Prelude



Haskell To QDPs


↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ IFR
        ↳ HASKELL
          ↳ BR
            ↳ HASKELL
              ↳ COR
                ↳ HASKELL
                  ↳ LetRed
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
QDP
                                ↳ DependencyGraphProof
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP

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

new_primDivNatS0(vuy135, vuy136, Zero, Zero) → new_primDivNatS00(vuy135, vuy136)
new_primDivNatS0(vuy135, vuy136, Succ(vuy1370), Succ(vuy1380)) → new_primDivNatS0(vuy135, vuy136, vuy1370, vuy1380)
new_primDivNatS(Succ(vuy1160), Succ(vuy1170), vuy118) → new_primDivNatS(vuy1160, vuy1170, vuy118)
new_primDivNatS(Succ(Succ(vuy11600)), Zero, Succ(vuy1180)) → new_primDivNatS0(vuy11600, vuy1180, vuy11600, vuy1180)
new_primDivNatS(Succ(Zero), Zero, Zero) → new_primDivNatS(Zero, Zero, Zero)
new_primDivNatS(Succ(Succ(vuy11600)), Zero, Zero) → new_primDivNatS(Succ(vuy11600), Zero, Zero)
new_primDivNatS00(vuy135, vuy136) → new_primDivNatS(Succ(vuy135), Succ(vuy136), Succ(vuy136))
new_primDivNatS0(vuy135, vuy136, Succ(vuy1370), Zero) → new_primDivNatS(Succ(vuy135), Succ(vuy136), Succ(vuy136))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.

↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ IFR
        ↳ HASKELL
          ↳ BR
            ↳ HASKELL
              ↳ COR
                ↳ HASKELL
                  ↳ LetRed
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ AND
QDP
                                      ↳ QDPSizeChangeProof
                                    ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP

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

new_primDivNatS(Succ(Succ(vuy11600)), Zero, Zero) → new_primDivNatS(Succ(vuy11600), Zero, Zero)

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
  ↳ LR
    ↳ HASKELL
      ↳ IFR
        ↳ HASKELL
          ↳ BR
            ↳ HASKELL
              ↳ COR
                ↳ HASKELL
                  ↳ LetRed
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ AND
                                    ↳ QDP
QDP
                                      ↳ QDPOrderProof
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP

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

new_primDivNatS0(vuy135, vuy136, Zero, Zero) → new_primDivNatS00(vuy135, vuy136)
new_primDivNatS0(vuy135, vuy136, Succ(vuy1370), Succ(vuy1380)) → new_primDivNatS0(vuy135, vuy136, vuy1370, vuy1380)
new_primDivNatS(Succ(vuy1160), Succ(vuy1170), vuy118) → new_primDivNatS(vuy1160, vuy1170, vuy118)
new_primDivNatS(Succ(Succ(vuy11600)), Zero, Succ(vuy1180)) → new_primDivNatS0(vuy11600, vuy1180, vuy11600, vuy1180)
new_primDivNatS00(vuy135, vuy136) → new_primDivNatS(Succ(vuy135), Succ(vuy136), Succ(vuy136))
new_primDivNatS0(vuy135, vuy136, Succ(vuy1370), Zero) → new_primDivNatS(Succ(vuy135), Succ(vuy136), Succ(vuy136))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


new_primDivNatS(Succ(vuy1160), Succ(vuy1170), vuy118) → new_primDivNatS(vuy1160, vuy1170, vuy118)
new_primDivNatS(Succ(Succ(vuy11600)), Zero, Succ(vuy1180)) → new_primDivNatS0(vuy11600, vuy1180, vuy11600, vuy1180)
The remaining pairs can at least be oriented weakly.

new_primDivNatS0(vuy135, vuy136, Zero, Zero) → new_primDivNatS00(vuy135, vuy136)
new_primDivNatS0(vuy135, vuy136, Succ(vuy1370), Succ(vuy1380)) → new_primDivNatS0(vuy135, vuy136, vuy1370, vuy1380)
new_primDivNatS00(vuy135, vuy136) → new_primDivNatS(Succ(vuy135), Succ(vuy136), Succ(vuy136))
new_primDivNatS0(vuy135, vuy136, Succ(vuy1370), Zero) → new_primDivNatS(Succ(vuy135), Succ(vuy136), Succ(vuy136))
Used ordering: Polynomial interpretation [25]:

POL(Succ(x1)) = 1 + x1   
POL(Zero) = 0   
POL(new_primDivNatS(x1, x2, x3)) = x1   
POL(new_primDivNatS0(x1, x2, x3, x4)) = 1 + x1   
POL(new_primDivNatS00(x1, x2)) = 1 + x1   

The following usable rules [17] were oriented: none



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ IFR
        ↳ HASKELL
          ↳ BR
            ↳ HASKELL
              ↳ COR
                ↳ HASKELL
                  ↳ LetRed
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ AND
                                    ↳ QDP
                                    ↳ QDP
                                      ↳ QDPOrderProof
QDP
                                          ↳ DependencyGraphProof
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP

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

new_primDivNatS0(vuy135, vuy136, Zero, Zero) → new_primDivNatS00(vuy135, vuy136)
new_primDivNatS0(vuy135, vuy136, Succ(vuy1370), Succ(vuy1380)) → new_primDivNatS0(vuy135, vuy136, vuy1370, vuy1380)
new_primDivNatS00(vuy135, vuy136) → new_primDivNatS(Succ(vuy135), Succ(vuy136), Succ(vuy136))
new_primDivNatS0(vuy135, vuy136, Succ(vuy1370), Zero) → new_primDivNatS(Succ(vuy135), Succ(vuy136), Succ(vuy136))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ IFR
        ↳ HASKELL
          ↳ BR
            ↳ HASKELL
              ↳ COR
                ↳ HASKELL
                  ↳ LetRed
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ AND
                                    ↳ QDP
                                    ↳ QDP
                                      ↳ QDPOrderProof
                                        ↳ QDP
                                          ↳ DependencyGraphProof
QDP
                                              ↳ QDPSizeChangeProof
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP

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

new_primDivNatS0(vuy135, vuy136, Succ(vuy1370), Succ(vuy1380)) → new_primDivNatS0(vuy135, vuy136, vuy1370, vuy1380)

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

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

new_primPlusNat(vuy111, vuy112, Succ(vuy1130), Succ(vuy1140)) → new_primPlusNat(vuy111, vuy112, vuy1130, vuy1140)

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

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

new_primPlusNat0(Succ(vuy640), Succ(vuy650), vuy66) → new_primPlusNat0(vuy640, vuy650, vuy66)

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

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

new_primPlusNat1(vuy51, vuy52, Succ(vuy530), Succ(vuy540)) → new_primPlusNat1(vuy51, vuy52, vuy530, vuy540)

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

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

new_primMinusNat(vuy95, vuy96, Succ(vuy970), Succ(vuy980)) → new_primMinusNat(vuy95, vuy96, vuy970, vuy980)

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

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

new_primMinusNat0(Succ(vuy580), Succ(vuy590), vuy60) → new_primMinusNat0(vuy580, vuy590, vuy60)

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

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

new_primMinusNat1(vuy38, vuy39, Succ(vuy400), Succ(vuy410)) → new_primMinusNat1(vuy38, vuy39, vuy400, vuy410)

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: