YES 3.967 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/FiniteMap.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:



HASKELL
  ↳ LR

mainModule FiniteMap
  ((keysFM_LE :: FiniteMap Int a  ->  Int  ->  [Int]) :: FiniteMap Int a  ->  Int  ->  [Int])

module FiniteMap where
  import qualified Maybe
import qualified Prelude

  data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b


  instance (Eq a, Eq b) => Eq (FiniteMap b a) where 

  foldFM_LE :: Ord a => (a  ->  b  ->  c  ->  c ->  c  ->  a  ->  FiniteMap a b  ->  c
foldFM_LE k z fr EmptyFM z
foldFM_LE k z fr (Branch key elt _ fm_l fm_r
 | key <= fr = 
foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
 | otherwise = 
foldFM_LE k z fr fm_l

  keysFM_LE :: Ord b => FiniteMap b a  ->  b  ->  [b]
keysFM_LE fm fr foldFM_LE (\key elt rest ->key : rest) [] fr fm


module Maybe where
  import qualified FiniteMap
import qualified Prelude



Lambda Reductions:
The following Lambda expression
\keyeltrestkey : rest

is transformed to
keysFM_LE0 key elt rest = key : rest



↳ HASKELL
  ↳ LR
HASKELL
      ↳ BR

mainModule FiniteMap
  ((keysFM_LE :: FiniteMap Int a  ->  Int  ->  [Int]) :: FiniteMap Int a  ->  Int  ->  [Int])

module FiniteMap where
  import qualified Maybe
import qualified Prelude

  data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a


  instance (Eq a, Eq b) => Eq (FiniteMap b a) where 

  foldFM_LE :: Ord b => (b  ->  c  ->  a  ->  a ->  a  ->  b  ->  FiniteMap b c  ->  a
foldFM_LE k z fr EmptyFM z
foldFM_LE k z fr (Branch key elt _ fm_l fm_r
 | key <= fr = 
foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
 | otherwise = 
foldFM_LE k z fr fm_l

  keysFM_LE :: Ord a => FiniteMap a b  ->  a  ->  [a]
keysFM_LE fm fr foldFM_LE keysFM_LE0 [] fr fm

  
keysFM_LE0 key elt rest key : rest


module Maybe where
  import qualified FiniteMap
import qualified Prelude



Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ BR
HASKELL
          ↳ COR

mainModule FiniteMap
  ((keysFM_LE :: FiniteMap Int a  ->  Int  ->  [Int]) :: FiniteMap Int a  ->  Int  ->  [Int])

module FiniteMap where
  import qualified Maybe
import qualified Prelude

  data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b


  instance (Eq a, Eq b) => Eq (FiniteMap a b) where 

  foldFM_LE :: Ord a => (a  ->  c  ->  b  ->  b ->  b  ->  a  ->  FiniteMap a c  ->  b
foldFM_LE k z fr EmptyFM z
foldFM_LE k z fr (Branch key elt vw fm_l fm_r
 | key <= fr = 
foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
 | otherwise = 
foldFM_LE k z fr fm_l

  keysFM_LE :: Ord b => FiniteMap b a  ->  b  ->  [b]
keysFM_LE fm fr foldFM_LE keysFM_LE0 [] fr fm

  
keysFM_LE0 key elt rest key : rest


module Maybe where
  import qualified FiniteMap
import qualified Prelude



Cond Reductions:
The following Function with conditions
foldFM_LE k z fr EmptyFM = z
foldFM_LE k z fr (Branch key elt vw fm_l fm_r)
 | key <= fr
 = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
 | otherwise
 = foldFM_LE k z fr fm_l

is transformed to
foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM
foldFM_LE k z fr (Branch key elt vw fm_l fm_r) = foldFM_LE2 k z fr (Branch key elt vw fm_l fm_r)

foldFM_LE0 k z fr key elt vw fm_l fm_r True = foldFM_LE k z fr fm_l

foldFM_LE1 k z fr key elt vw fm_l fm_r True = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
foldFM_LE1 k z fr key elt vw fm_l fm_r False = foldFM_LE0 k z fr key elt vw fm_l fm_r otherwise

foldFM_LE2 k z fr (Branch key elt vw fm_l fm_r) = foldFM_LE1 k z fr key elt vw fm_l fm_r (key <= fr)

foldFM_LE3 k z fr EmptyFM = z
foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy

The following Function with conditions
undefined 
 | False
 = undefined

is transformed to
undefined  = undefined1

undefined0 True = undefined

undefined1  = undefined0 False



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
HASKELL
              ↳ Narrow

mainModule FiniteMap
  (keysFM_LE :: FiniteMap Int a  ->  Int  ->  [Int])

module FiniteMap where
  import qualified Maybe
import qualified Prelude

  data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b


  instance (Eq a, Eq b) => Eq (FiniteMap b a) where 

  foldFM_LE :: Ord a => (a  ->  c  ->  b  ->  b ->  b  ->  a  ->  FiniteMap a c  ->  b
foldFM_LE k z fr EmptyFM foldFM_LE3 k z fr EmptyFM
foldFM_LE k z fr (Branch key elt vw fm_l fm_rfoldFM_LE2 k z fr (Branch key elt vw fm_l fm_r)

  
foldFM_LE0 k z fr key elt vw fm_l fm_r True foldFM_LE k z fr fm_l

  
foldFM_LE1 k z fr key elt vw fm_l fm_r True foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
foldFM_LE1 k z fr key elt vw fm_l fm_r False foldFM_LE0 k z fr key elt vw fm_l fm_r otherwise

  
foldFM_LE2 k z fr (Branch key elt vw fm_l fm_rfoldFM_LE1 k z fr key elt vw fm_l fm_r (key <= fr)

  
foldFM_LE3 k z fr EmptyFM z
foldFM_LE3 wv ww wx wy foldFM_LE2 wv ww wx wy

  keysFM_LE :: Ord b => FiniteMap b a  ->  b  ->  [b]
keysFM_LE fm fr foldFM_LE keysFM_LE0 [] fr fm

  
keysFM_LE0 key elt rest key : rest


module Maybe where
  import qualified FiniteMap
import qualified Prelude



Haskell To QDPs


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

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

new_foldFM_LE(wz147, wz125, Branch(Pos(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) → new_foldFM_LE(wz147, wz125, wz1303, h)
new_foldFM_LE(wz147, wz125, Branch(Neg(Zero), wz1301, wz1302, wz1303, wz1304), h) → new_foldFM_LE(wz147, wz125, wz1303, h)
new_foldFM_LE(wz147, wz125, Branch(Pos(Zero), wz1301, wz1302, wz1303, wz1304), h) → new_foldFM_LE(wz147, wz125, wz1303, h)
new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Succ(wz2640), ba) → new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, wz2630, wz2640, ba)
new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) → new_foldFM_LE(wz256, wz257, wz261, ba)
new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Zero, ba) → new_foldFM_LE(wz256, wz257, wz261, ba)
new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) → new_foldFM_LE(wz256, wz257, wz261, ba)
new_foldFM_LE(wz147, wz125, Branch(Neg(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) → new_foldFM_LE1(wz147, wz125, wz130000, wz1301, wz1302, wz1303, wz1304, Succ(wz125), Succ(wz130000), h)
new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Zero, ba) → new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba)
new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) → new_foldFM_LE(new_keysFM_LE0(wz258, wz259, new_foldFM_LE0(wz256, wz257, wz261, ba), ba), wz257, wz262, ba)
new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) → new_foldFM_LE(new_keysFM_LE0(wz258, wz259, new_foldFM_LE0(wz256, wz257, wz261, ba), ba), wz257, wz262, ba)

The TRS R consists of the following rules:

new_foldFM_LE0(wz147, wz125, EmptyFM, h) → wz147
new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Zero, ba) → new_foldFM_LE0(wz256, wz257, wz261, ba)
new_foldFM_LE0(wz147, wz125, Branch(Neg(Zero), wz1301, wz1302, wz1303, wz1304), h) → new_foldFM_LE0(wz147, wz125, wz1303, h)
new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Zero, ba) → new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba)
new_keysFM_LE0(wz3000, wz31, wz5, bb) → :(Neg(Succ(wz3000)), wz5)
new_foldFM_LE0(wz147, wz125, Branch(Neg(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) → new_foldFM_LE11(wz147, wz125, wz130000, wz1301, wz1302, wz1303, wz1304, Succ(wz125), Succ(wz130000), h)
new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Succ(wz2640), ba) → new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, wz2630, wz2640, ba)
new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) → new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba)
new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) → new_foldFM_LE0(new_keysFM_LE0(wz258, wz259, new_foldFM_LE0(wz256, wz257, wz261, ba), ba), wz257, wz262, ba)
new_foldFM_LE0(wz147, wz125, Branch(Pos(Zero), wz1301, wz1302, wz1303, wz1304), h) → new_foldFM_LE0(wz147, wz125, wz1303, h)
new_foldFM_LE0(wz147, wz125, Branch(Pos(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) → new_foldFM_LE0(wz147, wz125, wz1303, h)

The set Q consists of the following terms:

new_foldFM_LE0(x0, x1, Branch(Neg(Zero), x2, x3, x4, x5), x6)
new_foldFM_LE0(x0, x1, Branch(Pos(Zero), x2, x3, x4, x5), x6)
new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9)
new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8)
new_foldFM_LE0(x0, x1, Branch(Neg(Succ(x2)), x3, x4, x5, x6), x7)
new_foldFM_LE0(x0, x1, EmptyFM, x2)
new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8)
new_keysFM_LE0(x0, x1, x2, x3)
new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7)
new_foldFM_LE0(x0, x1, Branch(Pos(Succ(x2)), x3, x4, x5, x6), x7)
new_foldFM_LE12(x0, x1, x2, x3, x4, x5, x6, x7)

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

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

new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) → new_foldFM_LE3(new_keysFM_LE0(wz34000, wz341, new_foldFM_LE5(wz20, wz343, h), h), wz344, h)
new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) → new_foldFM_LE3(wz20, wz343, h)
new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE2(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
new_foldFM_LE3(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) → new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h)
new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) → new_foldFM_LE6(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h)
new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE2(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h)
new_foldFM_LE2(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) → new_foldFM_LE3(wz20, wz343, h)
new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) → new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h)
new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) → new_foldFM_LE4(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h)
new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) → new_foldFM_LE3(wz20, wz343, h)

The TRS R consists of the following rules:

new_foldFM_LE20(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) → new_foldFM_LE5(new_keysFM_LE0(wz34000, wz341, new_foldFM_LE5(wz20, wz343, h), h), wz344, h)
new_foldFM_LE20(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) → new_foldFM_LE7(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h)
new_keysFM_LE0(wz3000, wz31, wz5, h) → :(Neg(Succ(wz3000)), wz5)
new_foldFM_LE5(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) → new_foldFM_LE20(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h)
new_foldFM_LE7(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE20(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h)
new_foldFM_LE20(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) → new_foldFM_LE8(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h)
new_foldFM_LE8(wz31, wz7, EmptyFM, h) → new_foldFM_LE30(new_keysFM_LE00(wz31, wz7, h), h)
new_foldFM_LE8(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE20(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
new_keysFM_LE00(wz31, wz6, h) → :(Pos(Zero), wz6)
new_keysFM_LE01(wz31, wz8, h) → :(Neg(Zero), wz8)
new_foldFM_LE30(wz19, h) → wz19
new_foldFM_LE7(wz31, wz10, EmptyFM, h) → new_foldFM_LE30(new_keysFM_LE01(wz31, wz10, h), h)
new_foldFM_LE20(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) → new_foldFM_LE5(wz20, wz343, h)
new_foldFM_LE5(wz20, EmptyFM, h) → new_foldFM_LE30(wz20, h)

The set Q consists of the following terms:

new_foldFM_LE8(x0, x1, EmptyFM, x2)
new_foldFM_LE20(x0, Neg(Succ(x1)), x2, x3, x4, x5, x6)
new_foldFM_LE8(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_keysFM_LE00(x0, x1, x2)
new_foldFM_LE7(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_keysFM_LE0(x0, x1, x2, x3)
new_foldFM_LE5(x0, Branch(x1, x2, x3, x4, x5), x6)
new_foldFM_LE7(x0, x1, EmptyFM, x2)
new_foldFM_LE30(x0, x1)
new_foldFM_LE20(x0, Pos(Zero), x1, x2, x3, x4, x5)
new_foldFM_LE20(x0, Pos(Succ(x1)), x2, x3, x4, x5, x6)
new_keysFM_LE01(x0, x1, x2)
new_foldFM_LE5(x0, EmptyFM, x1)
new_foldFM_LE20(x0, Neg(Zero), x1, x2, x3, x4, x5)

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

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

new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba)
new_foldFM_LE16(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE13(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba)
new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(wz13, Zero, wz343, ba)
new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE18(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba)
new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) → new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h)
new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) → new_foldFM_LE9(wz189, Succ(wz190), wz194, h)
new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) → new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h)
new_foldFM_LE9(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) → new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba)
new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba)
new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) → new_foldFM_LE16(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba)
new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE17(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba)
new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(wz13, Zero, wz343, ba)
new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(wz13, Succ(wz400), wz343, ba)
new_foldFM_LE16(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(wz13, Zero, wz343, ba)
new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) → new_foldFM_LE16(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba)
new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) → new_foldFM_LE9(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h)
new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, ba) → new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba)
new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(wz13, Succ(wz400), wz343, ba)
new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) → new_foldFM_LE9(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h)
new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) → new_foldFM_LE9(wz189, Succ(wz190), wz194, h)
new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) → new_foldFM_LE9(wz189, Succ(wz190), wz194, h)
new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba)

The TRS R consists of the following rules:

new_keysFM_LE02(wz191, wz192, wz198, h) → :(Pos(Succ(wz191)), wz198)
new_foldFM_LE110(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba)
new_foldFM_LE14(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) → new_foldFM_LE110(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba)
new_foldFM_LE22(wz31, wz6, EmptyFM, ba) → new_keysFM_LE00(wz31, wz6, ba)
new_foldFM_LE22(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) → new_foldFM_LE110(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba)
new_keysFM_LE0(wz3000, wz31, wz5, ba) → :(Neg(Succ(wz3000)), wz5)
new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) → new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h)
new_foldFM_LE110(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE111(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba)
new_foldFM_LE110(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE22(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba)
new_foldFM_LE110(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE21(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba)
new_foldFM_LE110(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba)
new_foldFM_LE14(wz13, wz40, EmptyFM, ba) → wz13
new_foldFM_LE21(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) → new_foldFM_LE110(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba)
new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) → new_foldFM_LE19(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h)
new_keysFM_LE00(wz31, wz6, ba) → :(Pos(Zero), wz6)
new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) → new_foldFM_LE19(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h)
new_keysFM_LE01(wz31, wz8, ba) → :(Neg(Zero), wz8)
new_foldFM_LE19(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) → new_foldFM_LE14(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h)
new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) → new_foldFM_LE14(wz189, Succ(wz190), wz194, h)
new_foldFM_LE110(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE14(wz13, Zero, wz343, ba)
new_foldFM_LE110(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE14(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba)
new_foldFM_LE21(wz3000, wz31, wz5, wz40, EmptyFM, ba) → new_keysFM_LE0(wz3000, wz31, wz5, ba)

The set Q consists of the following terms:

new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8)
new_foldFM_LE21(x0, x1, x2, x3, Branch(x4, x5, x6, x7, x8), x9)
new_foldFM_LE14(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_foldFM_LE110(x0, Zero, Neg(Zero), x1, x2, x3, x4, x5)
new_foldFM_LE22(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_foldFM_LE110(x0, Succ(x1), Pos(Succ(x2)), x3, x4, x5, x6, x7)
new_keysFM_LE00(x0, x1, x2)
new_keysFM_LE02(x0, x1, x2, x3)
new_foldFM_LE110(x0, Zero, Pos(Succ(x1)), x2, x3, x4, x5, x6)
new_foldFM_LE14(x0, x1, EmptyFM, x2)
new_foldFM_LE110(x0, Succ(x1), Pos(Zero), x2, x3, x4, x5, x6)
new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7)
new_keysFM_LE0(x0, x1, x2, x3)
new_keysFM_LE01(x0, x1, x2)
new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9)
new_foldFM_LE110(x0, x1, Neg(Succ(x2)), x3, x4, x5, x6, x7)
new_foldFM_LE22(x0, x1, EmptyFM, x2)
new_foldFM_LE19(x0, x1, x2, x3, x4, x5, x6, x7)
new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8)
new_foldFM_LE110(x0, Zero, Pos(Zero), x1, x2, x3, x4, x5)
new_foldFM_LE110(x0, Succ(x1), Neg(Zero), x2, x3, x4, x5, x6)
new_foldFM_LE21(x0, x1, x2, x3, EmptyFM, x4)

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
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ Narrow
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
QDP
                    ↳ DependencyGraphProof

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

new_foldFM_LE23(Pos(Zero), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Zero), wz33, h)
new_foldFM_LE23(Neg(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(Zero), wz33, h)
new_foldFM_LE23(Pos(Succ(wz400)), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Succ(wz400)), wz33, h)
new_foldFM_LE23(Neg(wz40), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(wz40), wz33, h)
new_foldFM_LE23(Neg(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(Zero), wz33, h)
new_foldFM_LE23(Neg(Zero), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(Zero), wz33, h)
new_foldFM_LE23(Pos(Succ(wz400)), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Succ(wz400)), wz33, h)
new_foldFM_LE23(Pos(wz40), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(wz40), wz33, h)
new_foldFM_LE23(Pos(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Zero), wz33, h)
new_foldFM_LE23(Pos(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Zero), wz33, h)

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.

↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ Narrow
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ AND
QDP
                          ↳ QDPSizeChangeProof
                        ↳ QDP

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

new_foldFM_LE23(Neg(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(Zero), wz33, h)
new_foldFM_LE23(Neg(wz40), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(wz40), wz33, h)
new_foldFM_LE23(Neg(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(Zero), wz33, h)
new_foldFM_LE23(Neg(Zero), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(Zero), wz33, h)

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

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

new_foldFM_LE23(Pos(Zero), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Zero), wz33, h)
new_foldFM_LE23(Pos(Succ(wz400)), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Succ(wz400)), wz33, h)
new_foldFM_LE23(Pos(Succ(wz400)), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Succ(wz400)), wz33, h)
new_foldFM_LE23(Pos(wz40), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(wz40), wz33, h)
new_foldFM_LE23(Pos(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Zero), wz33, h)
new_foldFM_LE23(Pos(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Zero), wz33, h)

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: