YES 4.128 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/List.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:



HASKELL
  ↳ LR

mainModule List
  ((elemIndices :: Ratio Int  ->  [Ratio Int ->  [Int]) :: Ratio Int  ->  [Ratio Int ->  [Int])

module List where
  import qualified Maybe
import qualified Prelude

  elemIndices :: Eq a => a  ->  [a ->  [Int]
elemIndices x findIndices (== x)

  findIndices :: (a  ->  Bool ->  [a ->  [Int]
findIndices p xs concatMap (\vv1 ->
case vv1 of
  (x,i)->  if p x then i : [] else []
  _-> []
) (zip xs (enumFrom 0))


module Maybe where
  import qualified List
import qualified Prelude



Lambda Reductions:
The following Lambda expression
\vv1
case vv1 of
 (x,i) → if p x then i : [] else []
 _ → []

is transformed to
findIndices0 p vv1 = 
case vv1 of
 (x,i) → if p x then i : [] else []
 _ → []

The following Lambda expression
\ab→(a,b)

is transformed to
zip0 a b = (a,b)



↳ HASKELL
  ↳ LR
HASKELL
      ↳ CR

mainModule List
  ((elemIndices :: Ratio Int  ->  [Ratio Int ->  [Int]) :: Ratio Int  ->  [Ratio Int ->  [Int])

module List where
  import qualified Maybe
import qualified Prelude

  elemIndices :: Eq a => a  ->  [a ->  [Int]
elemIndices x findIndices (== x)

  findIndices :: (a  ->  Bool ->  [a ->  [Int]
findIndices p xs concatMap (findIndices0 p) (zip xs (enumFrom 0))

  
findIndices0 p vv1 
case vv1 of
  (x,i)->  if p x then i : [] else []
  _-> []


module Maybe where
  import qualified List
import qualified Prelude



Case Reductions:
The following Case expression
case vv1 of
 (x,i) → if p x then i : [] else []
 _ → []

is transformed to
findIndices00 p (x,i) = if p x then i : [] else []
findIndices00 p _ = []



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
HASKELL
          ↳ IFR

mainModule List
  ((elemIndices :: Ratio Int  ->  [Ratio Int ->  [Int]) :: Ratio Int  ->  [Ratio Int ->  [Int])

module List where
  import qualified Maybe
import qualified Prelude

  elemIndices :: Eq a => a  ->  [a ->  [Int]
elemIndices x findIndices (== x)

  findIndices :: (a  ->  Bool ->  [a ->  [Int]
findIndices p xs concatMap (findIndices0 p) (zip xs (enumFrom 0))

  
findIndices0 p vv1 findIndices00 p vv1

  
findIndices00 p (x,i if p x then i : [] else []
findIndices00 p _ []


module Maybe where
  import qualified List
import qualified Prelude



If Reductions:
The following If expression
if p x then i : [] else []

is transformed to
findIndices000 i True = i : []
findIndices000 i False = []



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
HASKELL
              ↳ BR

mainModule List
  ((elemIndices :: Ratio Int  ->  [Ratio Int ->  [Int]) :: Ratio Int  ->  [Ratio Int ->  [Int])

module List where
  import qualified Maybe
import qualified Prelude

  elemIndices :: Eq a => a  ->  [a ->  [Int]
elemIndices x findIndices (== x)

  findIndices :: (a  ->  Bool ->  [a ->  [Int]
findIndices p xs concatMap (findIndices0 p) (zip xs (enumFrom 0))

  
findIndices0 p vv1 findIndices00 p vv1

  
findIndices00 p (x,ifindIndices000 i (p x)
findIndices00 p _ []

  
findIndices000 i True i : []
findIndices000 i False []


module Maybe where
  import qualified List
import qualified Prelude



Replaced joker patterns by fresh variables and removed binding patterns.

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

mainModule List
  ((elemIndices :: Ratio Int  ->  [Ratio Int ->  [Int]) :: Ratio Int  ->  [Ratio Int ->  [Int])

module List where
  import qualified Maybe
import qualified Prelude

  elemIndices :: Eq a => a  ->  [a ->  [Int]
elemIndices x findIndices (== x)

  findIndices :: (a  ->  Bool ->  [a ->  [Int]
findIndices p xs concatMap (findIndices0 p) (zip xs (enumFrom 0))

  
findIndices0 p vv1 findIndices00 p vv1

  
findIndices00 p (x,ifindIndices000 i (p x)
findIndices00 p vw []

  
findIndices000 i True i : []
findIndices000 i False []


module Maybe where
  import qualified List
import qualified Prelude



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

is transformed to
undefined  = undefined1

undefined0 True = undefined

undefined1  = undefined0 False



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
HASKELL
                      ↳ NumRed

mainModule List
  ((elemIndices :: Ratio Int  ->  [Ratio Int ->  [Int]) :: Ratio Int  ->  [Ratio Int ->  [Int])

module List where
  import qualified Maybe
import qualified Prelude

  elemIndices :: Eq a => a  ->  [a ->  [Int]
elemIndices x findIndices (== x)

  findIndices :: (a  ->  Bool ->  [a ->  [Int]
findIndices p xs concatMap (findIndices0 p) (zip xs (enumFrom 0))

  
findIndices0 p vv1 findIndices00 p vv1

  
findIndices00 p (x,ifindIndices000 i (p x)
findIndices00 p vw []

  
findIndices000 i True i : []
findIndices000 i False []


module Maybe where
  import qualified List
import qualified Prelude



Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.

↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
                    ↳ HASKELL
                      ↳ NumRed
HASKELL
                          ↳ Narrow

mainModule List
  (elemIndices :: Ratio Int  ->  [Ratio Int ->  [Int])

module List where
  import qualified Maybe
import qualified Prelude

  elemIndices :: Eq a => a  ->  [a ->  [Int]
elemIndices x findIndices (== x)

  findIndices :: (a  ->  Bool ->  [a ->  [Int]
findIndices p xs concatMap (findIndices0 p) (zip xs (enumFrom (Pos Zero)))

  
findIndices0 p vv1 findIndices00 p vv1

  
findIndices00 p (x,ifindIndices000 i (p x)
findIndices00 p vw []

  
findIndices000 i True i : []
findIndices000 i False []


module Maybe where
  import qualified List
import qualified Prelude



Haskell To QDPs


↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ 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_psPs(wy13, Succ(wy31000), Succ(wy41101000), wy20) → new_psPs(wy13, wy31000, wy41101000, wy20)

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
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ 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_foldr0(wy31, :%(Pos(Succ(wy4110000)), wy41101), :(wy41110, wy41111), wy13, wy14) → new_foldr0(wy31, wy41110, wy41111, new_primPlusNat(wy14), new_primPlusNat(wy14))
new_psPs1(wy13, Pos(Succ(wy3100)), Pos(Succ(wy4110100)), wy4111, wy14) → new_foldr(Pos(Succ(wy3100)), wy4111, wy14)
new_psPs1(wy13, Pos(Succ(wy3100)), Pos(Zero), wy4111, wy14) → new_foldr(Pos(Succ(wy3100)), wy4111, wy14)
new_psPs1(wy13, Pos(Zero), Neg(Zero), wy4111, wy14) → new_foldr(Pos(Zero), wy4111, wy14)
new_psPs1(wy13, Neg(Succ(wy3100)), Pos(wy411010), wy4111, wy14) → new_foldr(Neg(Succ(wy3100)), wy4111, wy14)
new_foldr0(Neg(Succ(wy3100)), :%(Pos(Zero), Pos(wy411010)), wy4111, wy13, wy14) → new_foldr(Neg(Succ(wy3100)), wy4111, wy14)
new_foldr(wy31, :(wy41110, wy41111), wy14) → new_foldr0(wy31, wy41110, wy41111, new_primPlusNat(wy14), new_primPlusNat(wy14))
new_foldr0(wy31, :%(Neg(Zero), wy41101), wy4111, wy13, wy14) → new_psPs1(wy13, wy31, wy41101, wy4111, wy14)
new_foldr0(Pos(Zero), :%(Pos(Zero), Pos(Succ(wy4110100))), wy4111, wy13, wy14) → new_foldr(Pos(Zero), wy4111, wy14)
new_foldr0(Pos(Zero), :%(Pos(Zero), Pos(Zero)), wy4111, wy13, wy14) → new_foldr(Pos(Zero), wy4111, wy14)
new_foldr0(Neg(Zero), :%(Pos(Zero), Pos(Zero)), wy4111, wy13, wy14) → new_foldr(Neg(Zero), wy4111, wy14)
new_psPs0(wy13, wy31, wy41101, :(wy41110, wy41111), wy14) → new_foldr0(wy31, wy41110, wy41111, new_primPlusNat(wy14), new_primPlusNat(wy14))
new_psPs1(wy13, Neg(Zero), Neg(Zero), wy4111, wy14) → new_foldr(Neg(Zero), wy4111, wy14)
new_psPs1(wy13, Neg(Zero), Neg(Succ(wy4110100)), wy4111, wy14) → new_foldr(Neg(Zero), wy4111, wy14)
new_psPs1(wy13, Neg(Zero), Pos(Zero), wy4111, wy14) → new_foldr(Neg(Zero), wy4111, wy14)
new_psPs1(wy13, Pos(Succ(wy3100)), Neg(wy411010), wy4111, wy14) → new_foldr(Pos(Succ(wy3100)), wy4111, wy14)
new_psPs1(wy13, Pos(Zero), Pos(Zero), wy4111, wy14) → new_foldr(Pos(Zero), wy4111, wy14)
new_psPs1(wy13, Pos(Zero), Pos(Succ(wy4110100)), wy4111, wy14) → new_foldr(Pos(Zero), wy4111, wy14)
new_foldr0(Pos(Succ(wy3100)), :%(Pos(Zero), Neg(wy411010)), wy4111, wy13, wy14) → new_foldr(Pos(Succ(wy3100)), wy4111, wy14)
new_psPs1(wy13, Neg(Zero), Pos(Succ(wy4110100)), wy4111, wy14) → new_foldr(Neg(Zero), wy4111, wy14)
new_foldr0(Pos(Zero), :%(Pos(Zero), Neg(Succ(wy4110100))), wy4111, wy13, wy14) → new_foldr(Pos(Zero), wy4111, wy14)
new_foldr0(Pos(Succ(wy3100)), :%(Pos(Zero), Pos(Zero)), wy4111, wy13, wy14) → new_foldr(Pos(Succ(wy3100)), wy4111, wy14)
new_foldr0(Pos(Zero), :%(Pos(Zero), Neg(Zero)), wy4111, wy13, wy14) → new_foldr(Pos(Zero), wy4111, wy14)
new_foldr0(Pos(Succ(wy3100)), :%(Pos(Zero), Pos(Succ(wy4110100))), wy4111, wy13, wy14) → new_foldr(Pos(Succ(wy3100)), wy4111, wy14)
new_foldr0(Neg(Succ(wy3100)), :%(Pos(Zero), Neg(Succ(wy4110100))), wy4111, wy13, wy14) → new_foldr(Neg(Succ(wy3100)), wy4111, wy14)
new_foldr0(wy31, :%(Neg(Succ(wy4110000)), wy41101), wy4111, wy13, wy14) → new_psPs0(wy13, wy31, wy41101, wy4111, wy14)
new_foldr0(Neg(Succ(wy3100)), :%(Pos(Zero), Neg(Zero)), wy4111, wy13, wy14) → new_foldr(Neg(Succ(wy3100)), wy4111, wy14)
new_psPs1(wy13, Neg(Succ(wy3100)), Neg(Zero), wy4111, wy14) → new_foldr(Neg(Succ(wy3100)), wy4111, wy14)
new_foldr0(Neg(Zero), :%(Pos(Zero), Neg(Succ(wy4110100))), wy4111, wy13, wy14) → new_foldr(Neg(Zero), wy4111, wy14)
new_foldr0(Neg(Zero), :%(Pos(Zero), Neg(Zero)), wy4111, wy13, wy14) → new_foldr(Neg(Zero), wy4111, wy14)
new_foldr0(Neg(Zero), :%(Pos(Zero), Pos(Succ(wy4110100))), wy4111, wy13, wy14) → new_foldr(Neg(Zero), wy4111, wy14)
new_psPs1(wy13, Pos(Zero), Neg(Succ(wy4110100)), wy4111, wy14) → new_foldr(Pos(Zero), wy4111, wy14)
new_psPs1(wy13, Neg(Succ(wy3100)), Neg(Succ(wy4110100)), wy4111, wy14) → new_foldr(Neg(Succ(wy3100)), wy4111, wy14)

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_primPlusNat0(Succ(wy1700)) → Succ(wy1700)
new_primPlusNat(Succ(wy170)) → Succ(Succ(new_primPlusNat0(wy170)))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

new_primPlusNat(Succ(x0))
new_primPlusNat(Zero)
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))

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
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ 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_psPs2(wy17, wy31, wy41101, :(wy41110, wy41111), wy18) → new_foldr2(wy31, wy41110, new_primPlusNat(wy18), wy41111, new_primPlusNat(wy18))
new_foldr2(Neg(Succ(wy3100)), :%(Pos(Zero), Pos(wy411010)), wy17, wy4111, wy18) → new_foldr1(Neg(Succ(wy3100)), wy4111, wy18)
new_psPs3(wy17, Pos(Zero), Neg(Succ(wy4110100)), wy4111, wy18) → new_foldr1(Pos(Zero), wy4111, wy18)
new_foldr2(Pos(Succ(wy3100)), :%(Pos(Zero), Pos(Succ(wy4110100))), wy17, wy4111, wy18) → new_foldr1(Pos(Succ(wy3100)), wy4111, wy18)
new_foldr1(wy31, :(wy41110, wy41111), wy18) → new_foldr2(wy31, wy41110, new_primPlusNat(wy18), wy41111, new_primPlusNat(wy18))
new_foldr2(Pos(Zero), :%(Pos(Zero), Neg(Succ(wy4110100))), wy17, wy4111, wy18) → new_foldr1(Pos(Zero), wy4111, wy18)
new_foldr2(Neg(Zero), :%(Pos(Zero), Neg(Succ(wy4110100))), wy17, wy4111, wy18) → new_foldr1(Neg(Zero), wy4111, wy18)
new_foldr2(Neg(Zero), :%(Pos(Zero), Neg(Zero)), wy17, wy4111, wy18) → new_foldr1(Neg(Zero), wy4111, wy18)
new_psPs3(wy17, Pos(Zero), Pos(Succ(wy4110100)), wy4111, wy18) → new_foldr1(Pos(Zero), wy4111, wy18)
new_foldr2(wy31, :%(Neg(Zero), wy41101), wy17, wy4111, wy18) → new_psPs3(wy17, wy31, wy41101, wy4111, wy18)
new_foldr2(wy31, :%(Neg(Succ(wy4110000)), wy41101), wy17, wy4111, wy18) → new_psPs2(wy17, wy31, wy41101, wy4111, wy18)
new_foldr2(Neg(Zero), :%(Pos(Zero), Pos(Succ(wy4110100))), wy17, wy4111, wy18) → new_foldr1(Neg(Zero), wy4111, wy18)
new_psPs3(wy17, Neg(Succ(wy3100)), Pos(wy411010), wy4111, wy18) → new_foldr1(Neg(Succ(wy3100)), wy4111, wy18)
new_foldr2(Pos(Zero), :%(Pos(Zero), Pos(Succ(wy4110100))), wy17, wy4111, wy18) → new_foldr1(Pos(Zero), wy4111, wy18)
new_foldr2(Neg(Succ(wy3100)), :%(Pos(Zero), Neg(Zero)), wy17, wy4111, wy18) → new_foldr1(Neg(Succ(wy3100)), wy4111, wy18)
new_foldr2(Neg(Succ(wy3100)), :%(Pos(Zero), Neg(Succ(wy4110100))), wy17, wy4111, wy18) → new_foldr1(Neg(Succ(wy3100)), wy4111, wy18)
new_psPs3(wy17, Pos(Zero), Pos(Zero), wy4111, wy18) → new_foldr1(Pos(Zero), wy4111, wy18)
new_foldr2(Pos(Zero), :%(Pos(Zero), Pos(Zero)), wy17, wy4111, wy18) → new_foldr1(Pos(Zero), wy4111, wy18)
new_foldr2(Pos(Succ(wy3100)), :%(Pos(Zero), Pos(Zero)), wy17, wy4111, wy18) → new_foldr1(Pos(Succ(wy3100)), wy4111, wy18)
new_psPs3(wy17, Pos(Succ(wy3100)), Pos(Succ(wy4110100)), wy4111, wy18) → new_foldr1(Pos(Succ(wy3100)), wy4111, wy18)
new_psPs3(wy17, Neg(Succ(wy3100)), Neg(Succ(wy4110100)), wy4111, wy18) → new_foldr1(Neg(Succ(wy3100)), wy4111, wy18)
new_foldr2(Neg(Zero), :%(Pos(Zero), Pos(Zero)), wy17, wy4111, wy18) → new_foldr1(Neg(Zero), wy4111, wy18)
new_psPs3(wy17, Neg(Zero), Pos(Zero), wy4111, wy18) → new_foldr1(Neg(Zero), wy4111, wy18)
new_foldr2(wy31, :%(Pos(Succ(wy4110000)), wy41101), wy17, :(wy41110, wy41111), wy18) → new_foldr2(wy31, wy41110, new_primPlusNat(wy18), wy41111, new_primPlusNat(wy18))
new_psPs3(wy17, Neg(Succ(wy3100)), Neg(Zero), wy4111, wy18) → new_foldr1(Neg(Succ(wy3100)), wy4111, wy18)
new_psPs3(wy17, Neg(Zero), Neg(Zero), wy4111, wy18) → new_foldr1(Neg(Zero), wy4111, wy18)
new_psPs3(wy17, Pos(Succ(wy3100)), Pos(Zero), wy4111, wy18) → new_foldr1(Pos(Succ(wy3100)), wy4111, wy18)
new_psPs3(wy17, Pos(Succ(wy3100)), Neg(wy411010), wy4111, wy18) → new_foldr1(Pos(Succ(wy3100)), wy4111, wy18)
new_psPs3(wy17, Pos(Zero), Neg(Zero), wy4111, wy18) → new_foldr1(Pos(Zero), wy4111, wy18)
new_psPs3(wy17, Neg(Zero), Pos(Succ(wy4110100)), wy4111, wy18) → new_foldr1(Neg(Zero), wy4111, wy18)
new_foldr2(Pos(Zero), :%(Pos(Zero), Neg(Zero)), wy17, wy4111, wy18) → new_foldr1(Pos(Zero), wy4111, wy18)
new_foldr2(Pos(Succ(wy3100)), :%(Pos(Zero), Neg(wy411010)), wy17, wy4111, wy18) → new_foldr1(Pos(Succ(wy3100)), wy4111, wy18)
new_psPs3(wy17, Neg(Zero), Neg(Succ(wy4110100)), wy4111, wy18) → new_foldr1(Neg(Zero), wy4111, wy18)

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_primPlusNat0(Succ(wy1700)) → Succ(wy1700)
new_primPlusNat(Succ(wy170)) → Succ(Succ(new_primPlusNat0(wy170)))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

new_primPlusNat(Succ(x0))
new_primPlusNat(Zero)
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))

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
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ 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_psPs4(wy11, Succ(wy300000), Succ(wy411000000), wy31, wy41101, wy19) → new_psPs4(wy11, wy300000, wy411000000, wy31, wy41101, wy19)

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
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ 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_foldr3(wy3000, wy31, :%(Neg(Zero), wy41101), wy11, wy4111, wy12) → new_psPs5(wy11, wy31, wy41101, wy3000, wy4111, wy12)
new_psPs5(wy11, wy31, wy41101, wy3000, :(wy41110, wy41111), wy12) → new_foldr3(wy3000, wy31, wy41110, new_primPlusNat(wy12), wy41111, new_primPlusNat(wy12))
new_foldr3(wy3000, wy31, :%(Pos(wy411000), wy41101), wy11, :(wy41110, wy41111), wy12) → new_foldr3(wy3000, wy31, wy41110, new_primPlusNat(wy12), wy41111, new_primPlusNat(wy12))
new_foldr4(wy3000, wy31, :(wy41110, wy41111), wy12) → new_foldr3(wy3000, wy31, wy41110, new_primPlusNat(wy12), wy41111, new_primPlusNat(wy12))
new_foldr3(wy3000, wy31, :%(Neg(Succ(wy4110000)), wy41101), wy11, wy4111, wy12) → new_foldr4(wy3000, wy31, wy4111, wy12)

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_primPlusNat0(Succ(wy1700)) → Succ(wy1700)
new_primPlusNat(Succ(wy170)) → Succ(Succ(new_primPlusNat0(wy170)))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

new_primPlusNat(Succ(x0))
new_primPlusNat(Zero)
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))

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
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ 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_foldr5(wy3000, wy31, :(wy41110, wy41111), wy16) → new_psPs6(wy3000, wy31, wy41110, new_primPlusNat(wy16), wy41111, new_primPlusNat(wy16))
new_psPs7(wy15, wy31, wy41101, wy3000, :(wy41110, wy41111), wy16) → new_psPs6(wy3000, wy31, wy41110, new_primPlusNat(wy16), wy41111, new_primPlusNat(wy16))
new_psPs6(wy3000, wy31, :%(Pos(Succ(wy4110000)), wy41101), wy15, wy4111, wy16) → new_foldr5(wy3000, wy31, wy4111, wy16)
new_psPs6(wy3000, wy31, :%(Neg(wy411000), wy41101), wy15, :(wy41110, wy41111), wy16) → new_psPs6(wy3000, wy31, wy41110, new_primPlusNat(wy16), wy41111, new_primPlusNat(wy16))
new_psPs6(wy3000, wy31, :%(Pos(Zero), wy41101), wy15, wy4111, wy16) → new_psPs7(wy15, wy31, wy41101, wy3000, wy4111, wy16)

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_primPlusNat0(Succ(wy1700)) → Succ(wy1700)
new_primPlusNat(Succ(wy170)) → Succ(Succ(new_primPlusNat0(wy170)))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

new_primPlusNat(Succ(x0))
new_primPlusNat(Zero)
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))

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
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ 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_psPs8(Succ(wy31000), Succ(wy401000), wy6) → new_psPs8(wy31000, wy401000, wy6)

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
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ 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_psPs9(Succ(wy300000), Succ(wy4000000), wy31, wy401, wy5) → new_psPs9(wy300000, wy4000000, wy31, wy401, wy5)

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: