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 | = |
|
|
|
| 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
\keyeltrest→key : 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 | = |
|
|
|
| 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 | = |
|
|
|
| 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 | |
|
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
is transformed to
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_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 |
|
| 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:
- 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)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 3 > 7, 3 > 9, 4 >= 10
- 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)
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_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Zero, ba) → new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, 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_LE(wz147, wz125, Branch(Pos(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) → new_foldFM_LE(wz147, wz125, wz1303, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- new_foldFM_LE(wz147, wz125, Branch(Neg(Zero), wz1301, wz1302, wz1303, wz1304), h) → new_foldFM_LE(wz147, wz125, wz1303, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- new_foldFM_LE(wz147, wz125, Branch(Pos(Zero), wz1301, wz1302, wz1303, wz1304), h) → new_foldFM_LE(wz147, wz125, wz1303, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) → new_foldFM_LE(wz256, wz257, wz261, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 6 >= 3, 10 >= 4
- new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Zero, ba) → new_foldFM_LE(wz256, wz257, wz261, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 6 >= 3, 10 >= 4
- 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 graph contains the following edges 2 >= 2, 7 >= 3, 10 >= 4
- new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) → new_foldFM_LE(wz256, wz257, wz261, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 6 >= 3, 8 >= 4
- 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)
The graph contains the following edges 2 >= 2, 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_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:
- new_foldFM_LE3(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) → new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4, 2 > 5, 2 > 6, 3 >= 7
- 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)
The graph contains the following edges 1 >= 1, 5 > 2, 5 > 3, 5 > 4, 5 > 5, 5 > 6, 7 >= 7
- new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) → new_foldFM_LE4(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4
- new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) → new_foldFM_LE6(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4
- 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)
The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7
- 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)
The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7
- 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)
The graph contains the following edges 6 >= 2, 7 >= 3
- new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) → new_foldFM_LE3(wz20, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3
- new_foldFM_LE2(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) → new_foldFM_LE3(wz20, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3
- new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) → new_foldFM_LE3(wz20, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 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_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:
- new_foldFM_LE9(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) → new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 3 > 7, 4 >= 8
- 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)
The graph contains the following edges 3 > 3, 3 > 4, 3 > 5, 3 > 6, 3 > 7, 4 >= 8
- 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)
The graph contains the following edges 4 >= 2, 5 > 3, 5 > 4, 5 > 5, 5 > 6, 5 > 7, 6 >= 8
- 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)
The graph contains the following edges 1 >= 1, 2 >= 2, 6 > 3, 6 > 4, 6 > 5, 6 > 6, 6 > 7, 8 >= 8
- 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)
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_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)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 3 > 8, 2 > 9, 8 >= 10
- new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) → new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, 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_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE17(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba)
The graph contains the following edges 4 >= 1, 7 >= 3, 8 >= 4
- 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)
The graph contains the following edges 3 > 1, 4 >= 2, 2 >= 4, 7 >= 5, 8 >= 6
- 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)
The graph contains the following edges 7 >= 3, 10 >= 4
- new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) → new_foldFM_LE9(wz189, Succ(wz190), wz194, h)
The graph contains the following edges 1 >= 1, 6 >= 3, 10 >= 4
- new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) → new_foldFM_LE9(wz189, Succ(wz190), wz194, h)
The graph contains the following edges 1 >= 1, 6 >= 3, 10 >= 4
- new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) → new_foldFM_LE9(wz189, Succ(wz190), wz194, h)
The graph contains the following edges 1 >= 1, 6 >= 3, 8 >= 4
- 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)
The graph contains the following edges 7 >= 3, 8 >= 4
- new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(wz13, Zero, wz343, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 6 >= 3, 8 >= 4
- new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(wz13, Zero, wz343, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 6 >= 3, 8 >= 4
- new_foldFM_LE16(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(wz13, Zero, wz343, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 6 >= 3, 8 >= 4
- 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 graph contains the following edges 2 >= 2, 3 > 2, 7 >= 3, 8 >= 4
- 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)
The graph contains the following edges 2 >= 2, 7 >= 3, 8 >= 4
- 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)
The graph contains the following edges 2 >= 2, 7 >= 3, 8 >= 4
- new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(wz13, Succ(wz400), wz343, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 6 >= 3, 8 >= 4
- new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) → new_foldFM_LE9(wz13, Succ(wz400), wz343, ba)
The graph contains the following edges 1 >= 1, 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_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:
- new_foldFM_LE23(Neg(wz40), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(wz40), wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_LE23(Neg(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(Zero), wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_LE23(Neg(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(Zero), wz33, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3
- new_foldFM_LE23(Neg(Zero), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Neg(Zero), wz33, 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_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:
- new_foldFM_LE23(Pos(wz40), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(wz40), wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_LE23(Pos(Succ(wz400)), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Succ(wz400)), wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_LE23(Pos(Succ(wz400)), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Succ(wz400)), wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_LE23(Pos(Zero), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Zero), wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_LE23(Pos(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Zero), wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_LE23(Pos(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) → new_foldFM_LE23(Pos(Zero), wz33, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3