YES 0.866
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_LE :: FiniteMap Bool a -> Bool -> [(Bool,a)]) :: FiniteMap Bool a -> Bool -> [(Bool,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)
|
| fmToList_LE :: Ord b => FiniteMap b a -> b -> [(b,a)]
fmToList_LE | fm fr | = | foldFM_LE (\key elt rest ->(key,elt) : rest) [] fr fm |
|
| foldFM_LE :: Ord c => (c -> a -> b -> b) -> b -> c -> FiniteMap c a -> b
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 | = |
|
|
|
module Maybe where
| import qualified FiniteMap import qualified Prelude
|
Lambda Reductions:
The following Lambda expression
\keyeltrest→(key,elt) : rest
is transformed to
fmToList_LE0 | key elt rest | = (key,elt) : rest |
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
mainModule FiniteMap
| ((fmToList_LE :: FiniteMap Bool a -> Bool -> [(Bool,a)]) :: FiniteMap Bool a -> Bool -> [(Bool,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)
|
| fmToList_LE :: Ord b => FiniteMap b a -> b -> [(b,a)]
fmToList_LE | fm fr | = | foldFM_LE fmToList_LE0 [] fr fm |
|
|
fmToList_LE0 | key elt rest | = | (key,elt) : rest |
|
| 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 _ fm_l fm_r) | |
| | key <= fr | = |
foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r |
|
| | 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_LE :: FiniteMap Bool a -> Bool -> [(Bool,a)]) :: FiniteMap Bool a -> Bool -> [(Bool,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)
|
| fmToList_LE :: Ord b => FiniteMap b a -> b -> [(b,a)]
fmToList_LE | fm fr | = | foldFM_LE fmToList_LE0 [] fr fm |
|
|
fmToList_LE0 | key elt rest | = | (key,elt) : rest |
|
| 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 | = |
|
|
|
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
| (fmToList_LE :: FiniteMap Bool a -> Bool -> [(Bool,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)
|
| fmToList_LE :: Ord b => FiniteMap b a -> b -> [(b,a)]
fmToList_LE | fm fr | = | foldFM_LE fmToList_LE0 [] fr fm |
|
|
fmToList_LE0 | key elt rest | = | (key,elt) : rest |
|
| 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 |
|
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
Q DP problem:
The TRS P consists of the following rules:
new_foldFM_LE2(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE1(new_fmToList_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
new_foldFM_LE1(wz9, False, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) → new_foldFM_LE1(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
new_foldFM_LE1(wz9, True, wz341, wz342, wz343, wz344, h) → new_foldFM_LE3(wz9, wz343, h)
new_foldFM_LE(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE1(new_fmToList_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h)
new_foldFM_LE3(wz9, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) → new_foldFM_LE1(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
new_foldFM_LE1(wz9, True, wz341, wz342, wz343, wz344, h) → new_foldFM_LE2(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
new_foldFM_LE1(wz9, False, wz341, wz342, wz343, wz344, h) → new_foldFM_LE(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
The TRS R consists of the following rules:
new_foldFM_LE5(wz31, wz7, EmptyFM, h) → new_fmToList_LE00(wz31, wz7, h)
new_foldFM_LE5(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE10(new_fmToList_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
new_fmToList_LE0(wz31, wz5, h) → :(@2(False, wz31), wz5)
new_foldFM_LE0(wz9, EmptyFM, h) → wz9
new_foldFM_LE10(wz9, False, wz341, wz342, wz343, wz344, h) → new_foldFM_LE4(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
new_foldFM_LE0(wz9, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) → new_foldFM_LE10(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
new_foldFM_LE4(wz31, wz6, EmptyFM, h) → new_fmToList_LE0(wz31, wz6, h)
new_foldFM_LE4(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE10(new_fmToList_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h)
new_fmToList_LE00(wz31, wz7, h) → :(@2(True, wz31), wz7)
new_foldFM_LE10(wz9, True, wz341, wz342, wz343, wz344, h) → new_foldFM_LE5(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
The set Q consists of the following terms:
new_foldFM_LE10(x0, True, x1, x2, x3, x4, x5)
new_fmToList_LE00(x0, x1, x2)
new_foldFM_LE0(x0, EmptyFM, x1)
new_foldFM_LE4(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
new_foldFM_LE5(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_fmToList_LE0(x0, x1, x2)
new_foldFM_LE10(x0, False, x1, x2, x3, x4, x5)
new_foldFM_LE5(x0, x1, EmptyFM, x2)
new_foldFM_LE4(x0, x1, EmptyFM, 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_LE1(wz9, True, wz341, wz342, wz343, wz344, h) → new_foldFM_LE2(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4
- new_foldFM_LE1(wz9, False, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) → new_foldFM_LE1(wz9, 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_LE3(wz9, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) → new_foldFM_LE1(wz9, 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_LE1(wz9, False, wz341, wz342, wz343, wz344, h) → new_foldFM_LE(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4
- new_foldFM_LE1(wz9, True, wz341, wz342, wz343, wz344, h) → new_foldFM_LE3(wz9, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3
- new_foldFM_LE2(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE1(new_fmToList_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_LE(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE1(new_fmToList_LE0(wz31, wz6, 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
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_foldFM_LE11(wz8, False, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) → new_foldFM_LE11(wz8, wz3430, wz3431, wz3432, wz3433, wz3434, h)
new_foldFM_LE11(wz8, False, wz341, wz342, wz343, wz344, h) → new_foldFM_LE6(wz341, new_foldFM_LE7(wz8, wz343, h), wz344, h)
new_foldFM_LE6(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE11(new_fmToList_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
new_foldFM_LE8(wz8, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) → new_foldFM_LE11(wz8, wz3430, wz3431, wz3432, wz3433, wz3434, h)
new_foldFM_LE11(wz8, True, wz341, wz342, wz343, wz344, h) → new_foldFM_LE8(wz8, wz343, h)
The TRS R consists of the following rules:
new_fmToList_LE0(wz31, wz5, h) → :(@2(False, wz31), wz5)
new_foldFM_LE12(wz8, False, wz341, wz342, wz343, wz344, h) → new_foldFM_LE9(wz341, new_foldFM_LE7(wz8, wz343, h), wz344, h)
new_foldFM_LE7(wz8, EmptyFM, h) → wz8
new_foldFM_LE7(wz8, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) → new_foldFM_LE12(wz8, wz3430, wz3431, wz3432, wz3433, wz3434, h)
new_foldFM_LE9(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE12(new_fmToList_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
new_foldFM_LE12(wz8, True, wz341, wz342, wz343, wz344, h) → new_foldFM_LE7(wz8, wz343, h)
new_foldFM_LE9(wz31, wz5, EmptyFM, h) → new_fmToList_LE0(wz31, wz5, h)
The set Q consists of the following terms:
new_foldFM_LE7(x0, EmptyFM, x1)
new_foldFM_LE7(x0, Branch(x1, x2, x3, x4, x5), x6)
new_foldFM_LE12(x0, True, x1, x2, x3, x4, x5)
new_fmToList_LE0(x0, x1, x2)
new_foldFM_LE9(x0, x1, EmptyFM, x2)
new_foldFM_LE9(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_foldFM_LE12(x0, False, 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_LE11(wz8, False, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) → new_foldFM_LE11(wz8, 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_LE6(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE11(new_fmToList_LE0(wz31, wz5, 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_LE8(wz8, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) → new_foldFM_LE11(wz8, 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_LE11(wz8, False, wz341, wz342, wz343, wz344, h) → new_foldFM_LE6(wz341, new_foldFM_LE7(wz8, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4
- new_foldFM_LE11(wz8, True, wz341, wz342, wz343, wz344, h) → new_foldFM_LE8(wz8, 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
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
new_foldFM_LE13(False, Branch(False, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(False, wz33, h)
new_foldFM_LE13(False, Branch(True, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(False, wz33, h)
new_foldFM_LE13(True, Branch(False, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(True, wz33, h)
new_foldFM_LE13(True, Branch(True, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(True, 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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_foldFM_LE13(True, Branch(False, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(True, wz33, h)
new_foldFM_LE13(True, Branch(True, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(True, 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_LE13(True, Branch(True, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(True, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3
- new_foldFM_LE13(True, Branch(False, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(True, 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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_foldFM_LE13(False, Branch(False, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(False, wz33, h)
new_foldFM_LE13(False, Branch(True, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(False, 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_LE13(False, Branch(True, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(False, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_LE13(False, Branch(False, wz31, wz32, wz33, wz34), h) → new_foldFM_LE13(False, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3