YES 2.168 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 :: Int  ->  [Int ->  [Int]) :: Int  ->  [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 :: Int  ->  [Int ->  [Int]) :: Int  ->  [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 :: Int  ->  [Int ->  [Int]) :: Int  ->  [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 :: Int  ->  [Int ->  [Int]) :: Int  ->  [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 :: Int  ->  [Int ->  [Int]) :: Int  ->  [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 :: Int  ->  [Int ->  [Int]) :: Int  ->  [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 :: Int  ->  [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

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

new_foldr0(wy41110, wy41111, wy23, wy22) → new_foldr(wy41111, wy22)
new_foldr(:(wy41110, wy41111), wy19) → new_foldr0(wy41110, wy41111, new_primPlusNat(wy19), new_primPlusNat(wy19))

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_primPlusNat0(Succ(wy1100)) → Succ(wy1100)
new_primPlusNat(Succ(wy110)) → Succ(Succ(new_primPlusNat0(wy110)))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

new_primPlusNat0(Succ(x0))
new_primPlusNat(Zero)
new_primPlusNat0(Zero)
new_primPlusNat(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
                                ↳ QDPSizeChangeProof
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP

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

new_foldr1(:(wy41110, wy41111), wy18) → new_foldr2(wy41110, wy41111, new_primPlusNat(wy18), new_primPlusNat(wy18))
new_foldr2(wy41110, wy41111, wy21, wy20) → new_foldr1(wy41111, wy20)

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_primPlusNat0(Succ(wy1100)) → Succ(wy1100)
new_primPlusNat(Succ(wy110)) → Succ(Succ(new_primPlusNat0(wy110)))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

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

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

new_psPs(wy10, Succ(wy30000), Succ(wy41100000), wy15) → new_psPs(wy10, wy30000, wy41100000, wy15)

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
                                ↳ QDPSizeChangeProof
                              ↳ QDP
                              ↳ QDP

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

new_foldr4(wy300, Neg(Zero), wy4111, wy8, wy7) → new_foldr3(wy300, wy4111, wy8)
new_foldr4(wy300, Neg(Succ(wy411000)), wy4111, wy8, wy7) → new_foldr3(wy300, wy4111, wy8)
new_foldr3(wy300, :(wy41110, wy41111), wy8) → new_foldr4(wy300, wy41110, wy41111, new_primPlusNat(wy8), new_primPlusNat(wy8))
new_foldr4(wy300, Pos(wy41100), wy4111, wy8, wy7) → new_foldr3(wy300, wy4111, wy8)

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_primPlusNat0(Succ(wy1100)) → Succ(wy1100)
new_primPlusNat(Succ(wy110)) → Succ(Succ(new_primPlusNat0(wy110)))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

new_primPlusNat0(Succ(x0))
new_primPlusNat(Zero)
new_primPlusNat0(Zero)
new_primPlusNat(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
                                ↳ QDPSizeChangeProof
                              ↳ QDP

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

new_foldr6(wy300, Neg(wy41100), wy4111, wy10, wy9) → new_foldr5(wy300, wy4111, wy10)
new_foldr6(wy300, Pos(Succ(wy411000)), wy4111, wy10, wy9) → new_foldr5(wy300, wy4111, wy10)
new_foldr6(wy300, Pos(Zero), wy4111, wy10, wy9) → new_foldr5(wy300, wy4111, wy10)
new_foldr5(wy300, :(wy41110, wy41111), wy10) → new_foldr6(wy300, wy41110, wy41111, new_primPlusNat(wy10), new_primPlusNat(wy10))

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_primPlusNat0(Succ(wy1100)) → Succ(wy1100)
new_primPlusNat(Succ(wy110)) → Succ(Succ(new_primPlusNat0(wy110)))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

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

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

new_psPs0(Succ(wy30000), Succ(wy400000), wy5) → new_psPs0(wy30000, wy400000, 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: