YES 4.1
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
| ((fmToList_GE :: FiniteMap Int a -> Int -> [(Int,a)]) :: FiniteMap Int a -> Int -> [(Int,a)]) |
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
|
| fmToList_GE :: Ord a => FiniteMap a b -> a -> [(a,b)]
fmToList_GE | fm fr | = | foldFM_GE (\key elt rest ->(key,elt) : rest) [] fr fm |
|
| foldFM_GE :: Ord a => (a -> b -> c -> c) -> c -> a -> FiniteMap a b -> 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 | = |
|
|
|
module Maybe where
| import qualified FiniteMap import qualified Prelude
|
Lambda Reductions:
The following Lambda expression
\keyeltrest→(key,elt) : rest
is transformed to
fmToList_GE0 | key elt rest | = (key,elt) : rest |
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
mainModule FiniteMap
| ((fmToList_GE :: FiniteMap Int a -> Int -> [(Int,a)]) :: FiniteMap Int a -> Int -> [(Int,a)]) |
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
|
| fmToList_GE :: Ord a => FiniteMap a b -> a -> [(a,b)]
fmToList_GE | fm fr | = | foldFM_GE fmToList_GE0 [] fr fm |
|
|
fmToList_GE0 | key elt rest | = | (key,elt) : rest |
|
| foldFM_GE :: Ord c => (c -> b -> a -> a) -> a -> c -> FiniteMap c b -> a
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 | = |
|
|
|
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
| ((fmToList_GE :: FiniteMap Int a -> Int -> [(Int,a)]) :: FiniteMap Int a -> Int -> [(Int,a)]) |
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
|
| fmToList_GE :: Ord b => FiniteMap b a -> b -> [(b,a)]
fmToList_GE | fm fr | = | foldFM_GE fmToList_GE0 [] fr fm |
|
|
fmToList_GE0 | key elt rest | = | (key,elt) : rest |
|
| 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 | = |
|
|
|
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 | |
|
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_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_GE0 | k z fr key elt vw fm_l fm_r True | = foldFM_GE k z fr fm_r |
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
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
mainModule FiniteMap
| (fmToList_GE :: FiniteMap Int a -> Int -> [(Int,a)]) |
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 a b) where
|
| fmToList_GE :: Ord b => FiniteMap b a -> b -> [(b,a)]
fmToList_GE | fm fr | = | foldFM_GE fmToList_GE0 [] fr fm |
|
|
fmToList_GE0 | key elt rest | = | (key,elt) : rest |
|
| foldFM_GE :: Ord a => (a -> b -> c -> c) -> c -> a -> FiniteMap a b -> c
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 |
|
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_GE1(wz13, Succ(wz400), Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Succ(wz400), wz334, h)
new_foldFM_GE1(wz13, Zero, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_fmToList_GE00(wz331, new_foldFM_GE0(wz13, Zero, wz334, h), h), Zero, wz333, 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, Zero, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Zero, wz334, h)
new_foldFM_GE10(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Succ(wz2190), Zero, ba) → new_foldFM_GE2(new_fmToList_GE02(wz214, wz215, new_foldFM_GE0(wz212, Succ(wz213), wz218, ba), ba), Succ(wz213), wz217, ba)
new_foldFM_GE1(wz13, Zero, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_fmToList_GE0(wz331, new_foldFM_GE0(wz13, Zero, wz334, h), h), Zero, wz333, h)
new_foldFM_GE1(wz13, Zero, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Zero, wz334, h)
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_GE1(wz13, Succ(wz400), Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_fmToList_GE00(wz331, new_foldFM_GE0(wz13, Succ(wz400), wz334, h), h), Succ(wz400), wz333, 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_GE11(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba) → new_foldFM_GE2(new_fmToList_GE02(wz214, wz215, new_foldFM_GE0(wz212, Succ(wz213), wz218, ba), ba), Succ(wz213), wz217, ba)
new_foldFM_GE(wz3000, wz31, wz5, wz40, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE1(new_fmToList_GE01(wz3000, wz31, wz5, h), wz40, wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE11(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba) → new_foldFM_GE2(wz212, Succ(wz213), wz218, ba)
new_foldFM_GE1(wz13, Succ(wz400), Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_fmToList_GE0(wz331, new_foldFM_GE0(wz13, Succ(wz400), wz334, h), h), Succ(wz400), wz333, h)
The TRS R consists of the following rules:
new_foldFM_GE13(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Zero, Zero, ba) → new_foldFM_GE12(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba)
new_foldFM_GE13(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Zero, Succ(wz2200), ba) → new_foldFM_GE0(wz212, Succ(wz213), wz218, ba)
new_foldFM_GE13(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Succ(wz2190), Zero, ba) → new_foldFM_GE12(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba)
new_fmToList_GE01(wz3000, wz31, wz5, h) → :(@2(Pos(Succ(wz3000)), wz31), wz5)
new_foldFM_GE14(wz13, Succ(wz400), Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(new_fmToList_GE00(wz331, new_foldFM_GE0(wz13, Succ(wz400), wz334, h), h), Succ(wz400), wz333, h)
new_foldFM_GE0(wz13, wz40, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE14(wz13, wz40, wz3340, wz3341, wz3342, wz3343, wz3344, h)
new_foldFM_GE14(wz13, Succ(wz400), Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(new_fmToList_GE0(wz331, new_foldFM_GE0(wz13, Succ(wz400), wz334, h), h), Succ(wz400), wz333, h)
new_fmToList_GE00(wz31, wz9, h) → :(@2(Neg(Zero), wz31), wz9)
new_foldFM_GE14(wz13, Zero, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(new_fmToList_GE0(wz331, new_foldFM_GE0(wz13, Zero, wz334, h), h), Zero, wz333, h)
new_foldFM_GE14(wz13, Succ(wz400), Neg(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE13(wz13, wz400, wz33000, wz331, wz332, wz333, wz334, wz400, wz33000, h)
new_fmToList_GE0(wz31, wz6, h) → :(@2(Pos(Zero), wz31), wz6)
new_foldFM_GE14(wz13, Zero, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(new_fmToList_GE00(wz331, new_foldFM_GE0(wz13, Zero, wz334, h), h), Zero, wz333, h)
new_foldFM_GE3(wz3000, wz31, wz5, wz40, EmptyFM, h) → new_fmToList_GE01(wz3000, wz31, wz5, h)
new_foldFM_GE3(wz3000, wz31, wz5, wz40, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE14(new_fmToList_GE01(wz3000, wz31, wz5, h), wz40, wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE14(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_GE13(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Succ(wz2190), Succ(wz2200), ba) → new_foldFM_GE13(wz212, wz213, wz214, wz215, wz216, wz217, wz218, wz2190, wz2200, ba)
new_fmToList_GE02(wz214, wz215, wz267, ba) → :(@2(Neg(Succ(wz214)), wz215), wz267)
new_foldFM_GE0(wz13, wz40, EmptyFM, h) → wz13
new_foldFM_GE14(wz13, Zero, Neg(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(wz13, Zero, wz334, h)
new_foldFM_GE12(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba) → new_foldFM_GE0(new_fmToList_GE02(wz214, wz215, new_foldFM_GE0(wz212, Succ(wz213), wz218, ba), ba), Succ(wz213), wz217, ba)
The set Q consists of the following terms:
new_foldFM_GE14(x0, Zero, Pos(Zero), x1, x2, x3, x4, x5)
new_foldFM_GE0(x0, x1, EmptyFM, x2)
new_foldFM_GE14(x0, Zero, Neg(Zero), x1, x2, x3, x4, x5)
new_foldFM_GE14(x0, Zero, Neg(Succ(x1)), x2, x3, x4, x5, x6)
new_fmToList_GE02(x0, x1, x2, x3)
new_foldFM_GE13(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7)
new_fmToList_GE00(x0, x1, x2)
new_foldFM_GE14(x0, Succ(x1), Neg(Succ(x2)), x3, x4, x5, x6, x7)
new_foldFM_GE3(x0, x1, x2, x3, EmptyFM, x4)
new_foldFM_GE3(x0, x1, x2, x3, Branch(x4, x5, x6, x7, x8), x9)
new_foldFM_GE14(x0, Succ(x1), Neg(Zero), x2, x3, x4, x5, x6)
new_foldFM_GE12(x0, x1, x2, x3, x4, x5, x6, x7)
new_foldFM_GE13(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8)
new_fmToList_GE01(x0, x1, x2, x3)
new_foldFM_GE14(x0, x1, Pos(Succ(x2)), x3, x4, x5, x6, x7)
new_foldFM_GE13(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8)
new_foldFM_GE13(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9)
new_foldFM_GE0(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_foldFM_GE14(x0, Succ(x1), Pos(Zero), x2, x3, x4, x5, x6)
new_fmToList_GE0(x0, x1, x2)
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:
- new_foldFM_GE2(wz13, wz40, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE1(wz13, wz40, wz3340, wz3341, wz3342, wz3343, wz3344, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 3 > 7, 4 >= 8
- 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)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 > 8, 9 > 9, 10 >= 10
- 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)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 2 > 8, 3 > 9, 8 >= 10
- 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)
The graph contains the following edges 1 >= 1, 2 >= 2, 7 > 3, 7 > 4, 7 > 5, 7 > 6, 7 > 7, 8 >= 8
- new_foldFM_GE(wz3000, wz31, wz5, wz40, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE1(new_fmToList_GE01(wz3000, wz31, wz5, h), wz40, wz330, wz331, wz332, wz333, wz334, h)
The graph contains the following edges 4 >= 2, 5 > 3, 5 > 4, 5 > 5, 5 > 6, 5 > 7, 6 >= 8
- new_foldFM_GE10(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Zero, Zero, ba) → new_foldFM_GE11(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 10 >= 8
- 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)
The graph contains the following edges 3 > 1, 4 >= 2, 2 >= 4, 6 >= 5, 8 >= 6
- new_foldFM_GE10(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Succ(wz2190), Zero, ba) → new_foldFM_GE2(new_fmToList_GE02(wz214, wz215, new_foldFM_GE0(wz212, Succ(wz213), wz218, ba), ba), Succ(wz213), wz217, ba)
The graph contains the following edges 6 >= 3, 10 >= 4
- new_foldFM_GE10(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Zero, Succ(wz2200), ba) → new_foldFM_GE2(wz212, Succ(wz213), wz218, ba)
The graph contains the following edges 1 >= 1, 7 >= 3, 10 >= 4
- new_foldFM_GE10(wz212, wz213, wz214, wz215, wz216, wz217, wz218, Succ(wz2190), Zero, ba) → new_foldFM_GE2(wz212, Succ(wz213), wz218, ba)
The graph contains the following edges 1 >= 1, 7 >= 3, 10 >= 4
- new_foldFM_GE1(wz13, Succ(wz400), Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Succ(wz400), wz334, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 7 >= 3, 8 >= 4
- new_foldFM_GE1(wz13, Zero, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_fmToList_GE00(wz331, new_foldFM_GE0(wz13, Zero, wz334, h), h), Zero, wz333, h)
The graph contains the following edges 2 >= 2, 3 > 2, 6 >= 3, 8 >= 4
- new_foldFM_GE1(wz13, Zero, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Zero, wz334, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 7 >= 3, 8 >= 4
- new_foldFM_GE1(wz13, Zero, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_fmToList_GE0(wz331, new_foldFM_GE0(wz13, Zero, wz334, h), h), Zero, wz333, h)
The graph contains the following edges 2 >= 2, 3 > 2, 6 >= 3, 8 >= 4
- new_foldFM_GE1(wz13, Zero, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Zero, wz334, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 7 >= 3, 8 >= 4
- new_foldFM_GE1(wz13, Zero, Neg(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Zero, wz334, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 7 >= 3, 8 >= 4
- new_foldFM_GE1(wz13, Succ(wz400), Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_fmToList_GE00(wz331, new_foldFM_GE0(wz13, Succ(wz400), wz334, h), h), Succ(wz400), wz333, h)
The graph contains the following edges 2 >= 2, 6 >= 3, 8 >= 4
- new_foldFM_GE1(wz13, Succ(wz400), Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, Succ(wz400), wz334, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 7 >= 3, 8 >= 4
- new_foldFM_GE1(wz13, Succ(wz400), Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(new_fmToList_GE0(wz331, new_foldFM_GE0(wz13, Succ(wz400), wz334, h), h), Succ(wz400), wz333, h)
The graph contains the following edges 2 >= 2, 6 >= 3, 8 >= 4
- new_foldFM_GE11(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba) → new_foldFM_GE2(new_fmToList_GE02(wz214, wz215, new_foldFM_GE0(wz212, Succ(wz213), wz218, ba), ba), Succ(wz213), wz217, ba)
The graph contains the following edges 6 >= 3, 8 >= 4
- new_foldFM_GE11(wz212, wz213, wz214, wz215, wz216, wz217, wz218, ba) → new_foldFM_GE2(wz212, Succ(wz213), wz218, ba)
The graph contains the following edges 1 >= 1, 7 >= 3, 8 >= 4
↳ 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_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_GE4(wz3000, wz31, wz11, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE20(new_fmToList_GE01(wz3000, wz31, wz11, h), wz330, wz331, wz332, wz333, wz334, 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_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_GE7(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE20(new_fmToList_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE8(wz31, wz9, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE20(new_fmToList_GE00(wz31, wz9, h), wz330, wz331, wz332, wz333, wz334, h)
The TRS R consists of the following rules:
new_fmToList_GE01(wz3000, wz31, wz5, h) → :(@2(Pos(Succ(wz3000)), wz31), wz5)
new_foldFM_GE21(wz17, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE9(wz331, new_foldFM_GE5(wz17, wz334, h), wz333, h)
new_fmToList_GE00(wz31, wz9, h) → :(@2(Neg(Zero), wz31), wz9)
new_foldFM_GE30(wz16, h) → wz16
new_foldFM_GE5(wz17, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE21(wz17, wz3340, wz3341, wz3342, wz3343, wz3344, h)
new_fmToList_GE0(wz31, wz6, h) → :(@2(Pos(Zero), wz31), wz6)
new_foldFM_GE21(wz17, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE16(wz331, new_foldFM_GE5(wz17, wz334, h), wz333, h)
new_foldFM_GE21(wz17, Neg(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE5(wz17, wz334, h)
new_foldFM_GE15(wz3000, wz31, wz11, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE21(new_fmToList_GE01(wz3000, wz31, wz11, h), wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE9(wz31, wz9, EmptyFM, h) → new_foldFM_GE30(new_fmToList_GE00(wz31, wz9, h), h)
new_foldFM_GE21(wz17, Pos(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE15(wz33000, wz331, new_foldFM_GE5(wz17, wz334, h), wz333, h)
new_foldFM_GE16(wz31, wz6, EmptyFM, h) → new_foldFM_GE30(new_fmToList_GE0(wz31, wz6, h), h)
new_foldFM_GE5(wz17, EmptyFM, h) → new_foldFM_GE30(wz17, h)
new_foldFM_GE9(wz31, wz9, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE21(new_fmToList_GE00(wz31, wz9, h), wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE16(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE21(new_fmToList_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE15(wz3000, wz31, wz11, EmptyFM, h) → new_foldFM_GE30(new_fmToList_GE01(wz3000, wz31, wz11, h), h)
The set Q consists of the following terms:
new_foldFM_GE15(x0, x1, x2, EmptyFM, x3)
new_foldFM_GE9(x0, x1, EmptyFM, x2)
new_foldFM_GE30(x0, x1)
new_fmToList_GE00(x0, x1, x2)
new_foldFM_GE5(x0, EmptyFM, x1)
new_foldFM_GE21(x0, Pos(Zero), x1, x2, x3, x4, x5)
new_foldFM_GE9(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_fmToList_GE01(x0, x1, x2, x3)
new_foldFM_GE5(x0, Branch(x1, x2, x3, x4, x5), x6)
new_foldFM_GE15(x0, x1, x2, Branch(x3, x4, x5, x6, x7), x8)
new_foldFM_GE16(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_foldFM_GE16(x0, x1, EmptyFM, x2)
new_foldFM_GE21(x0, Pos(Succ(x1)), x2, x3, x4, x5, x6)
new_foldFM_GE21(x0, Neg(Zero), x1, x2, x3, x4, x5)
new_foldFM_GE21(x0, Neg(Succ(x1)), x2, x3, x4, x5, x6)
new_fmToList_GE0(x0, x1, x2)
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:
- 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)
The graph contains the following edges 2 > 1, 3 >= 2, 5 >= 4, 7 >= 5
- 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)
The graph contains the following edges 1 >= 1, 6 > 2, 6 > 3, 6 > 4, 6 > 5, 6 > 6, 7 >= 7
- new_foldFM_GE4(wz3000, wz31, wz11, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE20(new_fmToList_GE01(wz3000, wz31, wz11, h), wz330, wz331, wz332, wz333, wz334, h)
The graph contains the following edges 4 > 2, 4 > 3, 4 > 4, 4 > 5, 4 > 6, 5 >= 7
- new_foldFM_GE6(wz17, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE20(wz17, wz3340, wz3341, wz3342, wz3343, wz3344, h)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4, 2 > 5, 2 > 6, 3 >= 7
- new_foldFM_GE7(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE20(new_fmToList_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h)
The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7
- new_foldFM_GE8(wz31, wz9, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE20(new_fmToList_GE00(wz31, wz9, h), wz330, wz331, wz332, wz333, wz334, h)
The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7
- new_foldFM_GE20(wz17, Pos(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE7(wz331, new_foldFM_GE5(wz17, wz334, h), wz333, h)
The graph contains the following edges 3 >= 1, 5 >= 3, 7 >= 4
- new_foldFM_GE20(wz17, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE8(wz331, new_foldFM_GE5(wz17, wz334, h), wz333, h)
The graph contains the following edges 3 >= 1, 5 >= 3, 7 >= 4
- new_foldFM_GE20(wz17, Neg(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE6(wz17, wz334, h)
The graph contains the following edges 1 >= 1, 6 >= 2, 7 >= 3
- new_foldFM_GE20(wz17, Pos(Succ(wz33000)), wz331, wz332, wz333, wz334, h) → new_foldFM_GE6(wz17, wz334, h)
The graph contains the following edges 1 >= 1, 6 >= 2, 7 >= 3
- new_foldFM_GE20(wz17, Neg(Zero), wz331, wz332, wz333, wz334, h) → new_foldFM_GE6(wz17, wz334, h)
The graph contains the following edges 1 >= 1, 6 >= 2, 7 >= 3
↳ 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_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), Zero, h) → new_foldFM_GE18(new_fmToList_GE01(wz287, wz288, new_foldFM_GE19(wz285, wz286, wz291, h), h), wz286, wz290, 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_GE110(wz285, wz286, wz287, wz288, wz289, wz290, wz291, h) → new_foldFM_GE18(new_fmToList_GE01(wz287, wz288, new_foldFM_GE19(wz285, wz286, wz291, h), h), wz286, wz290, h)
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)
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_GE19(wz133, wz111, Branch(Neg(Succ(wz115000)), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE19(wz133, wz111, wz1154, ba)
new_fmToList_GE01(wz3000, wz31, wz5, bb) → :(@2(Pos(Succ(wz3000)), wz31), wz5)
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_GE112(wz285, wz286, wz287, wz288, wz289, wz290, wz291, h) → new_foldFM_GE19(new_fmToList_GE01(wz287, wz288, new_foldFM_GE19(wz285, wz286, wz291, h), h), wz286, wz290, h)
new_foldFM_GE111(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Zero, Succ(wz2930), h) → new_foldFM_GE19(wz285, wz286, wz291, h)
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_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_fmToList_GE01(x0, x1, x2, x3)
new_foldFM_GE111(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8)
new_foldFM_GE112(x0, x1, x2, x3, x4, x5, x6, x7)
new_foldFM_GE111(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8)
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:
- new_foldFM_GE17(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Zero, Zero, h) → new_foldFM_GE110(wz285, wz286, wz287, wz288, wz289, wz290, wz291, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 10 >= 8
- 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)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 > 8, 9 > 9, 10 >= 10
- 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)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 3 > 7, 3 > 8, 4 >= 10
- new_foldFM_GE18(wz133, wz111, Branch(Neg(Zero), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE18(wz133, wz111, wz1154, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- new_foldFM_GE18(wz133, wz111, Branch(Pos(Zero), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE18(wz133, wz111, wz1154, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- new_foldFM_GE18(wz133, wz111, Branch(Neg(Succ(wz115000)), wz1151, wz1152, wz1153, wz1154), ba) → new_foldFM_GE18(wz133, wz111, wz1154, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- new_foldFM_GE17(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Succ(wz2920), Zero, h) → new_foldFM_GE18(new_fmToList_GE01(wz287, wz288, new_foldFM_GE19(wz285, wz286, wz291, h), h), wz286, wz290, h)
The graph contains the following edges 2 >= 2, 6 >= 3, 10 >= 4
- new_foldFM_GE17(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Zero, Succ(wz2930), h) → new_foldFM_GE18(wz285, wz286, wz291, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 7 >= 3, 10 >= 4
- new_foldFM_GE17(wz285, wz286, wz287, wz288, wz289, wz290, wz291, Succ(wz2920), Zero, h) → new_foldFM_GE18(wz285, wz286, wz291, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 7 >= 3, 10 >= 4
- new_foldFM_GE110(wz285, wz286, wz287, wz288, wz289, wz290, wz291, h) → new_foldFM_GE18(wz285, wz286, wz291, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 7 >= 3, 8 >= 4
- new_foldFM_GE110(wz285, wz286, wz287, wz288, wz289, wz290, wz291, h) → new_foldFM_GE18(new_fmToList_GE01(wz287, wz288, new_foldFM_GE19(wz285, wz286, wz291, h), h), wz286, wz290, h)
The graph contains the following edges 2 >= 2, 6 >= 3, 8 >= 4
↳ 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:
- new_foldFM_GE22(Neg(wz40), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(wz40), wz34, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE22(Neg(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Zero), wz34, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3
- new_foldFM_GE22(Neg(Zero), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Zero), wz34, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE22(Neg(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Zero), wz34, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE22(Neg(Succ(wz400)), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Succ(wz400)), wz34, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE22(Neg(Succ(wz400)), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Neg(Succ(wz400)), wz34, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
↳ 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:
- new_foldFM_GE22(Pos(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(Zero), wz34, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE22(Pos(wz40), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(wz40), wz34, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE22(Pos(Zero), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(Zero), wz34, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE22(Pos(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_GE22(Pos(Zero), wz34, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3