YES 3.814 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_GE :: 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_GE :: Ord b => (b  ->  a  ->  c  ->  c ->  c  ->  b  ->  FiniteMap b a  ->  c
foldFM_GE k z fr EmptyFM z
foldFM_GE k z fr (Branch key elt _ fm_l fm_r
 | key >= fr = 
foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
 | otherwise = 
foldFM_GE k z fr fm_r

  keysFM_GE :: Ord a => FiniteMap a b  ->  a  ->  [a]
keysFM_GE fm fr foldFM_GE (\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_GE0 key elt rest = key : rest



↳ HASKELL
  ↳ LR
HASKELL
      ↳ BR

mainModule FiniteMap
  ((keysFM_GE :: 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_GE :: Ord c => (c  ->  a  ->  b  ->  b ->  b  ->  c  ->  FiniteMap c a  ->  b
foldFM_GE k z fr EmptyFM z
foldFM_GE k z fr (Branch key elt _ fm_l fm_r
 | key >= fr = 
foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
 | otherwise = 
foldFM_GE k z fr fm_r

  keysFM_GE :: Ord b => FiniteMap b a  ->  b  ->  [b]
keysFM_GE fm fr foldFM_GE keysFM_GE0 [] fr fm

  
keysFM_GE0 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_GE :: 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_GE :: Ord c => (c  ->  a  ->  b  ->  b ->  b  ->  c  ->  FiniteMap c a  ->  b
foldFM_GE k z fr EmptyFM z
foldFM_GE k z fr (Branch key elt vw fm_l fm_r
 | key >= fr = 
foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
 | otherwise = 
foldFM_GE k z fr fm_r

  keysFM_GE :: Ord a => FiniteMap a b  ->  a  ->  [a]
keysFM_GE fm fr foldFM_GE keysFM_GE0 [] fr fm

  
keysFM_GE0 key elt rest key : rest


module Maybe where
  import qualified FiniteMap
import qualified Prelude



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

is transformed to
foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM
foldFM_GE k z fr (Branch key elt vw fm_l fm_r) = foldFM_GE2 k z fr (Branch key elt vw fm_l fm_r)

foldFM_GE0 k z fr key elt vw fm_l fm_r True = foldFM_GE k z fr fm_r

foldFM_GE1 k z fr key elt vw fm_l fm_r True = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
foldFM_GE1 k z fr key elt vw fm_l fm_r False = foldFM_GE0 k z fr key elt vw fm_l fm_r otherwise

foldFM_GE2 k z fr (Branch key elt vw fm_l fm_r) = foldFM_GE1 k z fr key elt vw fm_l fm_r (key >= fr)

foldFM_GE3 k z fr EmptyFM = z
foldFM_GE3 wv ww wx wy = foldFM_GE2 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_GE :: 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_GE :: Ord c => (c  ->  b  ->  a  ->  a ->  a  ->  c  ->  FiniteMap c b  ->  a
foldFM_GE k z fr EmptyFM foldFM_GE3 k z fr EmptyFM
foldFM_GE k z fr (Branch key elt vw fm_l fm_rfoldFM_GE2 k z fr (Branch key elt vw fm_l fm_r)

  
foldFM_GE0 k z fr key elt vw fm_l fm_r True foldFM_GE k z fr fm_r

  
foldFM_GE1 k z fr key elt vw fm_l fm_r True foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
foldFM_GE1 k z fr key elt vw fm_l fm_r False foldFM_GE0 k z fr key elt vw fm_l fm_r otherwise

  
foldFM_GE2 k z fr (Branch key elt vw fm_l fm_rfoldFM_GE1 k z fr key elt vw fm_l fm_r (key >= fr)

  
foldFM_GE3 k z fr EmptyFM z
foldFM_GE3 wv ww wx wy foldFM_GE2 wv ww wx wy

  keysFM_GE :: Ord b => FiniteMap b a  ->  b  ->  [b]
keysFM_GE fm fr foldFM_GE keysFM_GE0 [] fr fm

  
keysFM_GE0 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_GE10(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Succ(wz2190), Zero, ba) → new_foldFM_GE2(new_keysFM_GE02(wz214, wz215, new_foldFM_GE0(wz212, Succ(wz213), wz218, ba), ba), Succ(wz213), wz217, ba)
new_foldFM_GE1(wz13, Succ(wz400), Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Succ(wz400), wz334, h)
new_foldFM_GE10(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Succ(wz2190), Succ(wz2200), ba) → new_foldFM_GE10(wz212, wz213, wz214, wz215, wz216, wz217, wz218, wz2190, wz2200, ba)
new_foldFM_GE1(wz13, Succ(wz400), Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_keysFM_GE0(wz331, new_foldFM_GE0(wz13, Succ(wz400), wz334, h), h), Succ(wz400), wz333, h)
new_foldFM_GE1(wz13, Zero, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Zero, wz334, h)
new_foldFM_GE1(wz13, Zero, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_keysFM_GE0(wz331, new_foldFM_GE0(wz13, Zero, wz334, h), h), Zero, wz333, h)
new_foldFM_GE(wz3000, wz31, wz5, wz40, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE1(new_keysFM_GE01(wz3000, wz31, wz5, h), wz40, wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE1(wz13, Zero, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Zero, wz334, h)
new_foldFM_GE11(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba) → new_foldFM_GE2(new_keysFM_GE02(wz214, wz215, new_foldFM_GE0(wz212, Succ(wz213), wz218, ba), ba), Succ(wz213), wz217, ba)
new_foldFM_GE10(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Zero, Succ(wz2200), ba) → new_foldFM_GE2(wz212, Succ(wz213), wz218, ba)
new_foldFM_GE10(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Succ(wz2190), Zero, ba) → new_foldFM_GE2(wz212, Succ(wz213), wz218, ba)
new_foldFM_GE1(wz13, Zero, Neg(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Zero, wz334, h)
new_foldFM_GE2(wz13, wz40, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE1(wz13, wz40, wz3340, wz3341, wz3342, wz3343, wz3344, h)
new_foldFM_GE1(wz13, Succ(wz400), Neg(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE10(wz13, wz400, wz33000, wz331, wz332, wz333, wz334, wz400, wz33000, h)
new_foldFM_GE1(wz13, Succ(wz400), Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Succ(wz400), wz334, h)
new_foldFM_GE1(wz13, wz40, Pos(Succ(wz33000)), wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE1(wz13, wz40, wz3340, wz3341, wz3342, wz3343, wz3344, h)
new_foldFM_GE1(wz13, wz40, Pos(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE(wz33000, wz331, new_foldFM_GE0(wz13, wz40, wz334, h), wz40, wz333, h)
new_foldFM_GE10(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Zero, Zero, ba) → new_foldFM_GE11(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba)
new_foldFM_GE1(wz13, Succ(wz400), Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_keysFM_GE00(wz331, new_foldFM_GE0(wz13, Succ(wz400), wz334, h), h), Succ(wz400), wz333, h)
new_foldFM_GE1(wz13, Zero, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_keysFM_GE00(wz331, new_foldFM_GE0(wz13, Zero, wz334, h), h), Zero, wz333, h)
new_foldFM_GE11(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba) → new_foldFM_GE2(wz212, Succ(wz213), wz218, ba)

The TRS R consists of the following rules:

new_foldFM_GE12(wz13, Zero, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(new_keysFM_GE00(wz331, new_foldFM_GE0(wz13, Zero, wz334, h), h), Zero, wz333, h)
new_foldFM_GE0(wz13, wz40, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE12(wz13, wz40, wz3340, wz3341, wz3342, wz3343, wz3344, h)
new_keysFM_GE02(wz214, wz215, wz267, ba) → :(Neg(Succ(wz214)), wz267)
new_foldFM_GE12(wz13, Succ(wz400), Neg(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE14(wz13, wz400, wz33000, wz331, wz332, wz333, wz334, wz400, wz33000, h)
new_foldFM_GE14(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Succ(wz2190), Succ(wz2200), ba) → new_foldFM_GE14(wz212, wz213, wz214, wz215, wz216, wz217, wz218, wz2190, wz2200, ba)
new_foldFM_GE13(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba) → new_foldFM_GE0(new_keysFM_GE02(wz214, wz215, new_foldFM_GE0(wz212, Succ(wz213), wz218, ba), ba), Succ(wz213), wz217, ba)
new_foldFM_GE12(wz13, Zero, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(new_keysFM_GE0(wz331, new_foldFM_GE0(wz13, Zero, wz334, h), h), Zero, wz333, h)
new_foldFM_GE12(wz13, Zero, Neg(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(wz13, Zero, wz334, h)
new_keysFM_GE01(wz3000, wz31, wz5, h) → :(Pos(Succ(wz3000)), wz5)
new_foldFM_GE12(wz13, Succ(wz400), Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(new_keysFM_GE00(wz331, new_foldFM_GE0(wz13, Succ(wz400), wz334, h), h), Succ(wz400), wz333, h)
new_foldFM_GE3(wz3000, wz31, wz5, wz40, EmptyFM, h) → new_keysFM_GE01(wz3000, wz31, wz5, h)
new_foldFM_GE14(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Succ(wz2190), Zero, ba) → new_foldFM_GE13(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba)
new_foldFM_GE14(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Zero, Succ(wz2200), ba) → new_foldFM_GE0(wz212, Succ(wz213), wz218, ba)
new_foldFM_GE3(wz3000, wz31, wz5, wz40, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE12(new_keysFM_GE01(wz3000, wz31, wz5, h), wz40, wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE14(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Zero, Zero, ba) → new_foldFM_GE13(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba)
new_keysFM_GE00(wz31, wz9, h) → :(Neg(Zero), wz9)
new_foldFM_GE12(wz13, wz40, Pos(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE3(wz33000, wz331, new_foldFM_GE0(wz13, wz40, wz334, h), wz40, wz333, h)
new_foldFM_GE0(wz13, wz40, EmptyFM, h) → wz13
new_foldFM_GE12(wz13, Succ(wz400), Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(new_keysFM_GE0(wz331, new_foldFM_GE0(wz13, Succ(wz400), wz334, h), h), Succ(wz400), wz333, h)
new_keysFM_GE0(wz31, wz6, h) → :(Pos(Zero), wz6)

The set Q consists of the following terms:

new_foldFM_GE0(x0, x1, EmptyFM, x2)
new_foldFM_GE12(x0, Succ(x1), Neg(Zero), x2, x3, x4, x5, x6)
new_foldFM_GE3(x0, x1, x2, x3, EmptyFM, x4)
new_foldFM_GE12(x0, Succ(x1), Neg(Succ(x2)), x3, x4, x5, x6, x7)
new_foldFM_GE12(x0, Zero, Neg(Succ(x1)), x2, x3, x4, x5, x6)
new_foldFM_GE14(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8)
new_foldFM_GE3(x0, x1, x2, x3, Branch(x4, x5, x6, x7, x8), x9)
new_keysFM_GE00(x0, x1, x2)
new_foldFM_GE14(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8)
new_foldFM_GE12(x0, x1, Pos(Succ(x2)), x3, x4, x5, x6, x7)
new_foldFM_GE12(x0, Zero, Neg(Zero), x1, x2, x3, x4, x5)
new_keysFM_GE01(x0, x1, x2, x3)
new_keysFM_GE0(x0, x1, x2)
new_foldFM_GE12(x0, Zero, Pos(Zero), x1, x2, x3, x4, x5)
new_foldFM_GE13(x0, x1, x2, x3, x4, x5, x6, x7)
new_foldFM_GE14(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7)
new_foldFM_GE12(x0, Succ(x1), Pos(Zero), x2, x3, x4, x5, x6)
new_foldFM_GE0(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_keysFM_GE02(x0, x1, x2, x3)
new_foldFM_GE14(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9)

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_GE4(wz3000, wz31, wz11, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE20(new_keysFM_GE01(wz3000, wz31, wz11, h), wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE6(wz17, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE20(wz17, wz3340, wz3341, wz3342, wz3343, wz3344, h)
new_foldFM_GE20(wz17, Pos(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE4(wz33000, wz331, new_foldFM_GE5(wz17, wz334, h), wz333, h)
new_foldFM_GE20(wz17, Pos(Zero), wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE20(wz17, wz3340, wz3341, wz3342, wz3343, wz3344, h)
new_foldFM_GE20(wz17, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE7(wz331, new_foldFM_GE5(wz17, wz334, h), wz333, h)
new_foldFM_GE20(wz17, Neg(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE6(wz17, wz334, h)
new_foldFM_GE20(wz17, Pos(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE6(wz17, wz334, h)
new_foldFM_GE7(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE20(new_keysFM_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE20(wz17, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE6(wz17, wz334, h)
new_foldFM_GE20(wz17, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE8(wz331, new_foldFM_GE5(wz17, wz334, h), wz333, h)
new_foldFM_GE8(wz31, wz9, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE20(new_keysFM_GE00(wz31, wz9, h), wz330, wz331, wz332, wz333, wz334, h)

The TRS R consists of the following rules:

new_foldFM_GE15(wz31, wz9, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE21(new_keysFM_GE00(wz31, wz9, h), wz330, wz331, wz332, wz333, wz334, h)
new_keysFM_GE01(wz3000, wz31, wz5, h) → :(Pos(Succ(wz3000)), wz5)
new_foldFM_GE30(wz16, h) → wz16
new_foldFM_GE21(wz17, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE9(wz331, new_foldFM_GE5(wz17, wz334, h), wz333, h)
new_foldFM_GE9(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE21(new_keysFM_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE5(wz17, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE21(wz17, wz3340, wz3341, wz3342, wz3343, wz3344, h)
new_foldFM_GE21(wz17, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE15(wz331, new_foldFM_GE5(wz17, wz334, h), wz333, h)
new_foldFM_GE9(wz31, wz6, EmptyFM, h) → new_foldFM_GE30(new_keysFM_GE0(wz31, wz6, h), h)
new_foldFM_GE21(wz17, Neg(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE5(wz17, wz334, h)
new_foldFM_GE21(wz17, Pos(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE16(wz33000, wz331, new_foldFM_GE5(wz17, wz334, h), wz333, h)
new_foldFM_GE5(wz17, EmptyFM, h) → new_foldFM_GE30(wz17, h)
new_keysFM_GE00(wz31, wz9, h) → :(Neg(Zero), wz9)
new_foldFM_GE16(wz3000, wz31, wz11, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE21(new_keysFM_GE01(wz3000, wz31, wz11, h), wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE15(wz31, wz9, EmptyFM, h) → new_foldFM_GE30(new_keysFM_GE00(wz31, wz9, h), h)
new_keysFM_GE0(wz31, wz6, h) → :(Pos(Zero), wz6)
new_foldFM_GE16(wz3000, wz31, wz11, EmptyFM, h) → new_foldFM_GE30(new_keysFM_GE01(wz3000, wz31, wz11, h), h)

The set Q consists of the following terms:

new_foldFM_GE30(x0, x1)
new_foldFM_GE15(x0, x1, EmptyFM, x2)
new_foldFM_GE9(x0, x1, EmptyFM, x2)
new_foldFM_GE16(x0, x1, x2, EmptyFM, x3)
new_foldFM_GE9(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_foldFM_GE5(x0, EmptyFM, x1)
new_keysFM_GE00(x0, x1, x2)
new_keysFM_GE01(x0, x1, x2, x3)
new_foldFM_GE21(x0, Pos(Zero), x1, x2, x3, x4, x5)
new_keysFM_GE0(x0, x1, x2)
new_foldFM_GE5(x0, Branch(x1, x2, x3, x4, x5), x6)
new_foldFM_GE21(x0, Pos(Succ(x1)), x2, x3, x4, x5, x6)
new_foldFM_GE15(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_foldFM_GE21(x0, Neg(Zero), x1, x2, x3, x4, x5)
new_foldFM_GE16(x0, x1, x2, Branch(x3, x4, x5, x6, x7), x8)
new_foldFM_GE21(x0, Neg(Succ(x1)), x2, x3, x4, x5, x6)

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_GE17(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Succ(wz2920), Zero, h) → new_foldFM_GE18(new_keysFM_GE01(wz287, wz288, new_foldFM_GE19(wz285, wz286, wz291, h), h), wz286, wz290, h)
new_foldFM_GE110(wz285, wz286, wz287, wz288, wz289, wz290, wz291, h) → new_foldFM_GE18(wz285, wz286, wz291, h)
new_foldFM_GE17(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Succ(wz2920), Succ(wz2930), h) → new_foldFM_GE17(wz285, wz286, wz287, wz288, wz289, wz290, wz291, wz2920, wz2930, h)
new_foldFM_GE18(wz133, wz111, Branch(Pos(Succ(wz115000)), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE17(wz133, wz111, wz115000, wz1151, wz1152, wz1153, wz1154, Succ(wz115000), Succ(wz111), ba)
new_foldFM_GE17(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Zero, Zero, h) → new_foldFM_GE110(wz285, wz286, wz287, wz288, wz289, wz290, wz291, h)
new_foldFM_GE18(wz133, wz111, Branch(Neg(Zero), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE18(wz133, wz111, wz1154, ba)
new_foldFM_GE18(wz133, wz111, Branch(Pos(Zero), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE18(wz133, wz111, wz1154, ba)
new_foldFM_GE17(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Succ(wz2920), Zero, h) → new_foldFM_GE18(wz285, wz286, wz291, h)
new_foldFM_GE17(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Zero, Succ(wz2930), h) → new_foldFM_GE18(wz285, wz286, wz291, h)
new_foldFM_GE18(wz133, wz111, Branch(Neg(Succ(wz115000)), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE18(wz133, wz111, wz1154, ba)
new_foldFM_GE110(wz285, wz286, wz287, wz288, wz289, wz290, wz291, h) → new_foldFM_GE18(new_keysFM_GE01(wz287, wz288, new_foldFM_GE19(wz285, wz286, wz291, h), h), wz286, wz290, h)

The TRS R consists of the following rules:

new_foldFM_GE111(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Succ(wz2920), Succ(wz2930), h) → new_foldFM_GE111(wz285, wz286, wz287, wz288, wz289, wz290, wz291, wz2920, wz2930, h)
new_foldFM_GE111(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Succ(wz2920), Zero, h) → new_foldFM_GE112(wz285, wz286, wz287, wz288, wz289, wz290, wz291, h)
new_foldFM_GE112(wz285, wz286, wz287, wz288, wz289, wz290, wz291, h) → new_foldFM_GE19(new_keysFM_GE01(wz287, wz288, new_foldFM_GE19(wz285, wz286, wz291, h), h), wz286, wz290, h)
new_foldFM_GE19(wz133, wz111, Branch(Neg(Succ(wz115000)), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE19(wz133, wz111, wz1154, ba)
new_foldFM_GE19(wz133, wz111, Branch(Pos(Succ(wz115000)), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE111(wz133, wz111, wz115000, wz1151, wz1152, wz1153, wz1154, Succ(wz115000), Succ(wz111), ba)
new_foldFM_GE19(wz133, wz111, EmptyFM, ba) → wz133
new_foldFM_GE111(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Zero, Succ(wz2930), h) → new_foldFM_GE19(wz285, wz286, wz291, h)
new_keysFM_GE01(wz3000, wz31, wz5, bb) → :(Pos(Succ(wz3000)), wz5)
new_foldFM_GE19(wz133, wz111, Branch(Pos(Zero), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE19(wz133, wz111, wz1154, ba)
new_foldFM_GE19(wz133, wz111, Branch(Neg(Zero), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE19(wz133, wz111, wz1154, ba)
new_foldFM_GE111(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Zero, Zero, h) → new_foldFM_GE112(wz285, wz286, wz287, wz288, wz289, wz290, wz291, h)

The set Q consists of the following terms:

new_foldFM_GE19(x0, x1, Branch(Neg(Zero), x2, x3, x4, x5), x6)
new_foldFM_GE19(x0, x1, EmptyFM, x2)
new_foldFM_GE19(x0, x1, Branch(Neg(Succ(x2)), x3, x4, x5, x6), x7)
new_keysFM_GE01(x0, x1, x2, x3)
new_foldFM_GE19(x0, x1, Branch(Pos(Zero), x2, x3, x4, x5), x6)
new_foldFM_GE111(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9)
new_foldFM_GE111(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7)
new_foldFM_GE111(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8)
new_foldFM_GE111(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8)
new_foldFM_GE112(x0, x1, x2, x3, x4, x5, x6, x7)
new_foldFM_GE19(x0, x1, Branch(Pos(Succ(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
                  ↳ QDP
QDP
                    ↳ DependencyGraphProof

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

new_foldFM_GE22(Pos(wz40), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(wz40), wz34, h)
new_foldFM_GE22(Neg(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Zero), wz34, h)
new_foldFM_GE22(Neg(Succ(wz400)), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Succ(wz400)), wz34, h)
new_foldFM_GE22(Neg(wz40), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(wz40), wz34, h)
new_foldFM_GE22(Neg(Succ(wz400)), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Succ(wz400)), wz34, h)
new_foldFM_GE22(Neg(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Zero), wz34, h)
new_foldFM_GE22(Neg(Zero), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Zero), wz34, h)
new_foldFM_GE22(Pos(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(Zero), wz34, h)
new_foldFM_GE22(Pos(Zero), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(Zero), wz34, h)
new_foldFM_GE22(Pos(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(Zero), wz34, 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_GE22(Neg(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Zero), wz34, h)
new_foldFM_GE22(Neg(Succ(wz400)), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Succ(wz400)), wz34, h)
new_foldFM_GE22(Neg(wz40), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(wz40), wz34, h)
new_foldFM_GE22(Neg(Succ(wz400)), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Succ(wz400)), wz34, h)
new_foldFM_GE22(Neg(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Zero), wz34, h)
new_foldFM_GE22(Neg(Zero), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Zero), wz34, 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_GE22(Pos(wz40), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(wz40), wz34, h)
new_foldFM_GE22(Pos(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(Zero), wz34, h)
new_foldFM_GE22(Pos(Zero), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(Zero), wz34, h)
new_foldFM_GE22(Pos(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(Zero), wz34, 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: