YES 1.19
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
| ((eltsFM_GE :: FiniteMap Ordering a -> Ordering -> [a]) :: FiniteMap Ordering a -> Ordering -> [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
|
| eltsFM_GE :: Ord b => FiniteMap b a -> b -> [a]
eltsFM_GE | fm fr | = | foldFM_GE (\key elt rest ->elt : rest) [] fr fm |
|
| foldFM_GE :: Ord b => (b -> c -> a -> a) -> a -> b -> FiniteMap b c -> 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
|
Lambda Reductions:
The following Lambda expression
\keyeltrest→elt : rest
is transformed to
eltsFM_GE0 | key elt rest | = elt : rest |
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
mainModule FiniteMap
| ((eltsFM_GE :: FiniteMap Ordering a -> Ordering -> [a]) :: FiniteMap Ordering a -> Ordering -> [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 b a) where
|
| eltsFM_GE :: Ord a => FiniteMap a b -> a -> [b]
eltsFM_GE | fm fr | = | foldFM_GE eltsFM_GE0 [] fr fm |
|
|
eltsFM_GE0 | key elt rest | = | elt : rest |
|
| foldFM_GE :: Ord b => (b -> c -> a -> a) -> a -> b -> FiniteMap b c -> 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
| ((eltsFM_GE :: FiniteMap Ordering a -> Ordering -> [a]) :: FiniteMap Ordering a -> Ordering -> [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 b a) where
|
| eltsFM_GE :: Ord b => FiniteMap b a -> b -> [a]
eltsFM_GE | fm fr | = | foldFM_GE eltsFM_GE0 [] fr fm |
|
|
eltsFM_GE0 | key elt rest | = | elt : rest |
|
| 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 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_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
compare | x y |
| | x == y | |
| | x <= y | |
| | otherwise | |
|
is transformed to
compare | x y | = compare3 x y |
compare1 | x y True | = LT |
compare1 | x y False | = compare0 x y otherwise |
compare2 | x y True | = EQ |
compare2 | x y False | = compare1 x y (x <= y) |
compare3 | x y | = compare2 x y (x == y) |
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
mainModule FiniteMap
| (eltsFM_GE :: FiniteMap Ordering a -> Ordering -> [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
|
| eltsFM_GE :: Ord b => FiniteMap b a -> b -> [a]
eltsFM_GE | fm fr | = | foldFM_GE eltsFM_GE0 [] fr fm |
|
|
eltsFM_GE0 | key elt rest | = | elt : rest |
|
| 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_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, LT, wz331, wz332, wz333, wz334, h) → new_foldFM_GE(wz13, wz334, h)
new_foldFM_GE1(wz13, EQ, wz331, wz332, wz333, wz334, h) → new_foldFM_GE(wz13, wz334, h)
new_foldFM_GE1(wz13, GT, wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(wz331, new_foldFM_GE2(wz13, wz334, h), wz333, h)
new_foldFM_GE(wz13, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE1(wz13, wz3340, wz3341, wz3342, wz3343, wz3344, h)
new_foldFM_GE1(wz13, GT, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE1(wz13, wz3340, wz3341, wz3342, wz3343, wz3344, h)
new_foldFM_GE0(wz31, wz7, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE1(new_eltsFM_GE0(wz31, wz7, h), wz330, wz331, wz332, wz333, wz334, h)
The TRS R consists of the following rules:
new_eltsFM_GE0(wz31, wz7, h) → :(wz31, wz7)
new_foldFM_GE3(wz31, wz7, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE10(new_eltsFM_GE0(wz31, wz7, h), wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE2(wz13, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE10(wz13, wz3340, wz3341, wz3342, wz3343, wz3344, h)
new_foldFM_GE10(wz13, EQ, wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, wz334, h)
new_foldFM_GE10(wz13, LT, wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz13, wz334, h)
new_foldFM_GE10(wz13, GT, wz331, wz332, wz333, wz334, h) → new_foldFM_GE3(wz331, new_foldFM_GE2(wz13, wz334, h), wz333, h)
new_foldFM_GE3(wz31, wz7, EmptyFM, h) → new_eltsFM_GE0(wz31, wz7, h)
new_foldFM_GE2(wz13, EmptyFM, h) → wz13
The set Q consists of the following terms:
new_eltsFM_GE0(x0, x1, x2)
new_foldFM_GE3(x0, x1, EmptyFM, x2)
new_foldFM_GE2(x0, EmptyFM, x1)
new_foldFM_GE10(x0, GT, x1, x2, x3, x4, x5)
new_foldFM_GE3(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_foldFM_GE10(x0, EQ, x1, x2, x3, x4, x5)
new_foldFM_GE2(x0, Branch(x1, x2, x3, x4, x5), x6)
new_foldFM_GE10(x0, LT, 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_GE(wz13, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE1(wz13, 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_GE0(wz31, wz7, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE1(new_eltsFM_GE0(wz31, wz7, 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_GE1(wz13, GT, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE1(wz13, 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_GE1(wz13, GT, wz331, wz332, wz333, wz334, h) → new_foldFM_GE0(wz331, new_foldFM_GE2(wz13, wz334, h), wz333, h)
The graph contains the following edges 3 >= 1, 5 >= 3, 7 >= 4
- new_foldFM_GE1(wz13, EQ, wz331, wz332, wz333, wz334, h) → new_foldFM_GE(wz13, wz334, h)
The graph contains the following edges 1 >= 1, 6 >= 2, 7 >= 3
- new_foldFM_GE1(wz13, LT, wz331, wz332, wz333, wz334, h) → new_foldFM_GE(wz13, 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
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_foldFM_GE4(wz16, Branch(EQ, wz331, wz332, wz333, wz334), h) → new_foldFM_GE4(wz16, wz334, h)
new_foldFM_GE4(wz16, Branch(GT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE4(new_eltsFM_GE0(wz331, new_foldFM_GE5(wz16, wz334, h), h), wz333, h)
new_foldFM_GE4(wz16, Branch(LT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE4(wz16, wz334, h)
new_foldFM_GE4(wz16, Branch(GT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE4(wz16, wz334, h)
new_foldFM_GE4(wz16, Branch(EQ, wz331, wz332, wz333, wz334), h) → new_foldFM_GE4(new_eltsFM_GE00(wz331, new_foldFM_GE5(wz16, wz334, h), h), wz333, h)
The TRS R consists of the following rules:
new_eltsFM_GE0(wz31, wz7, h) → :(wz31, wz7)
new_foldFM_GE5(wz16, Branch(LT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE5(wz16, wz334, h)
new_eltsFM_GE00(wz31, wz6, h) → :(wz31, wz6)
new_foldFM_GE5(wz16, Branch(EQ, wz331, wz332, wz333, wz334), h) → new_foldFM_GE5(new_eltsFM_GE00(wz331, new_foldFM_GE5(wz16, wz334, h), h), wz333, h)
new_foldFM_GE5(wz16, EmptyFM, h) → wz16
new_foldFM_GE5(wz16, Branch(GT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE5(new_eltsFM_GE0(wz331, new_foldFM_GE5(wz16, wz334, h), h), wz333, h)
The set Q consists of the following terms:
new_eltsFM_GE0(x0, x1, x2)
new_foldFM_GE5(x0, EmptyFM, x1)
new_eltsFM_GE00(x0, x1, x2)
new_foldFM_GE5(x0, Branch(GT, x1, x2, x3, x4), x5)
new_foldFM_GE5(x0, Branch(LT, x1, x2, x3, x4), x5)
new_foldFM_GE5(x0, Branch(EQ, 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_GE4(wz16, Branch(GT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE4(new_eltsFM_GE0(wz331, new_foldFM_GE5(wz16, wz334, h), h), wz333, h)
The graph contains the following edges 2 > 2, 3 >= 3
- new_foldFM_GE4(wz16, Branch(EQ, wz331, wz332, wz333, wz334), h) → new_foldFM_GE4(wz16, wz334, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE4(wz16, Branch(GT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE4(wz16, wz334, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE4(wz16, Branch(LT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE4(wz16, wz334, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE4(wz16, Branch(EQ, wz331, wz332, wz333, wz334), h) → new_foldFM_GE4(new_eltsFM_GE00(wz331, new_foldFM_GE5(wz16, wz334, h), h), wz333, h)
The graph contains the following edges 2 > 2, 3 >= 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_GE6(wz14, Branch(LT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(new_eltsFM_GE01(wz331, new_foldFM_GE7(wz14, wz334, h), h), wz333, h)
new_foldFM_GE6(wz14, Branch(GT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(wz14, wz334, h)
new_foldFM_GE6(wz14, Branch(LT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(wz14, wz334, h)
new_foldFM_GE6(wz14, Branch(EQ, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(new_eltsFM_GE00(wz331, new_foldFM_GE7(wz14, wz334, h), h), wz333, h)
new_foldFM_GE6(wz14, Branch(EQ, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(wz14, wz334, h)
new_foldFM_GE6(wz14, Branch(GT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(new_eltsFM_GE0(wz331, new_foldFM_GE7(wz14, wz334, h), h), wz333, h)
The TRS R consists of the following rules:
new_eltsFM_GE0(wz31, wz7, h) → :(wz31, wz7)
new_foldFM_GE7(wz14, Branch(EQ, wz331, wz332, wz333, wz334), h) → new_foldFM_GE7(new_eltsFM_GE00(wz331, new_foldFM_GE7(wz14, wz334, h), h), wz333, h)
new_foldFM_GE7(wz14, Branch(LT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE7(new_eltsFM_GE01(wz331, new_foldFM_GE7(wz14, wz334, h), h), wz333, h)
new_eltsFM_GE00(wz31, wz6, h) → :(wz31, wz6)
new_foldFM_GE7(wz14, EmptyFM, h) → wz14
new_eltsFM_GE01(wz31, wz15, h) → :(wz31, wz15)
new_foldFM_GE7(wz14, Branch(GT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE7(new_eltsFM_GE0(wz331, new_foldFM_GE7(wz14, wz334, h), h), wz333, h)
The set Q consists of the following terms:
new_eltsFM_GE01(x0, x1, x2)
new_eltsFM_GE0(x0, x1, x2)
new_foldFM_GE7(x0, Branch(GT, x1, x2, x3, x4), x5)
new_foldFM_GE7(x0, EmptyFM, x1)
new_eltsFM_GE00(x0, x1, x2)
new_foldFM_GE7(x0, Branch(EQ, x1, x2, x3, x4), x5)
new_foldFM_GE7(x0, Branch(LT, 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_GE6(wz14, Branch(GT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(wz14, wz334, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE6(wz14, Branch(LT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(new_eltsFM_GE01(wz331, new_foldFM_GE7(wz14, wz334, h), h), wz333, h)
The graph contains the following edges 2 > 2, 3 >= 3
- new_foldFM_GE6(wz14, Branch(EQ, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(new_eltsFM_GE00(wz331, new_foldFM_GE7(wz14, wz334, h), h), wz333, h)
The graph contains the following edges 2 > 2, 3 >= 3
- new_foldFM_GE6(wz14, Branch(LT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(wz14, wz334, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE6(wz14, Branch(GT, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(new_eltsFM_GE0(wz331, new_foldFM_GE7(wz14, wz334, h), h), wz333, h)
The graph contains the following edges 2 > 2, 3 >= 3
- new_foldFM_GE6(wz14, Branch(EQ, wz331, wz332, wz333, wz334), h) → new_foldFM_GE6(wz14, wz334, 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
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_foldFM_GE8(GT, Branch(EQ, wz31, wz32, wz33, wz34), h) → new_foldFM_GE8(GT, wz34, h)
new_foldFM_GE8(GT, Branch(GT, wz31, wz32, wz33, wz34), h) → new_foldFM_GE8(GT, wz34, h)
new_foldFM_GE8(GT, Branch(LT, wz31, wz32, wz33, wz34), h) → new_foldFM_GE8(GT, 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_GE8(GT, Branch(EQ, wz31, wz32, wz33, wz34), h) → new_foldFM_GE8(GT, wz34, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_foldFM_GE8(GT, Branch(GT, wz31, wz32, wz33, wz34), h) → new_foldFM_GE8(GT, wz34, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3
- new_foldFM_GE8(GT, Branch(LT, wz31, wz32, wz33, wz34), h) → new_foldFM_GE8(GT, wz34, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3