YES 0.994
H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/List.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:
↳ HASKELL
↳ IFR
mainModule List
| ((delete :: Ordering -> [Ordering] -> [Ordering]) :: Ordering -> [Ordering] -> [Ordering]) |
module List where
| import qualified Maybe import qualified Prelude
|
| delete :: Eq a => a -> [a] -> [a]
|
| deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]
deleteBy | _ _ [] | = | [] |
deleteBy | eq x (y : ys) | = | if x `eq` y then ys else y : deleteBy eq x ys |
|
module Maybe where
| import qualified List import qualified Prelude
|
If Reductions:
The following If expression
if eq x y then ys else y : deleteBy eq x ys
is transformed to
deleteBy0 | ys y eq x True | = ys |
deleteBy0 | ys y eq x False | = y : deleteBy eq x ys |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule List
| ((delete :: Ordering -> [Ordering] -> [Ordering]) :: Ordering -> [Ordering] -> [Ordering]) |
module List where
| import qualified Maybe import qualified Prelude
|
| delete :: Eq a => a -> [a] -> [a]
|
| deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]
deleteBy | _ _ [] | = | [] |
deleteBy | eq x (y : ys) | = | deleteBy0 ys y eq x (x `eq` y) |
|
|
deleteBy0 | ys y eq x True | = | ys |
deleteBy0 | ys y eq x False | = | y : deleteBy eq x ys |
|
module Maybe where
| import qualified List import qualified Prelude
|
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule List
| ((delete :: Ordering -> [Ordering] -> [Ordering]) :: Ordering -> [Ordering] -> [Ordering]) |
module List where
| import qualified Maybe import qualified Prelude
|
| delete :: Eq a => a -> [a] -> [a]
|
| deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]
deleteBy | vw vx [] | = | [] |
deleteBy | eq x (y : ys) | = | deleteBy0 ys y eq x (x `eq` y) |
|
|
deleteBy0 | ys y eq x True | = | ys |
deleteBy0 | ys y eq x False | = | y : deleteBy eq x ys |
|
module Maybe where
| import qualified List import qualified Prelude
|
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
mainModule List
| (delete :: Ordering -> [Ordering] -> [Ordering]) |
module List where
| import qualified Maybe import qualified Prelude
|
| delete :: Eq a => a -> [a] -> [a]
|
| deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]
deleteBy | vw vx [] | = | [] |
deleteBy | eq x (y : ys) | = | deleteBy0 ys y eq x (x `eq` y) |
|
|
deleteBy0 | ys y eq x True | = | ys |
deleteBy0 | ys y eq x False | = | y : deleteBy eq x ys |
|
module Maybe where
| import qualified List import qualified Prelude
|
Haskell To QDPs
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
new_deleteBy(LT, :(EQ, wu41)) → new_deleteBy(LT, wu41)
new_deleteBy(EQ, :(GT, wu41)) → new_deleteBy(EQ, wu41)
new_deleteBy(GT, :(EQ, wu41)) → new_deleteBy(GT, wu41)
new_deleteBy(EQ, :(LT, wu41)) → new_deleteBy(EQ, wu41)
new_deleteBy(GT, :(LT, wu41)) → new_deleteBy(GT, wu41)
new_deleteBy(LT, :(GT, wu41)) → new_deleteBy(LT, wu41)
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 3 SCCs.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ ATransformationProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_deleteBy(GT, :(EQ, wu41)) → new_deleteBy(GT, wu41)
new_deleteBy(GT, :(LT, wu41)) → new_deleteBy(GT, wu41)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We have applied the A-Transformation [17] to get from an applicative problem to a standard problem.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ ATransformationProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GT(LT(wu41)) → GT(wu41)
GT(EQ(wu41)) → GT(wu41)
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:
- GT(LT(wu41)) → GT(wu41)
The graph contains the following edges 1 > 1
- GT(EQ(wu41)) → GT(wu41)
The graph contains the following edges 1 > 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ ATransformationProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_deleteBy(LT, :(EQ, wu41)) → new_deleteBy(LT, wu41)
new_deleteBy(LT, :(GT, wu41)) → new_deleteBy(LT, wu41)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We have applied the A-Transformation [17] to get from an applicative problem to a standard problem.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ ATransformationProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
LT1(GT1(wu41)) → LT1(wu41)
LT1(EQ(wu41)) → LT1(wu41)
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:
- LT1(GT1(wu41)) → LT1(wu41)
The graph contains the following edges 1 > 1
- LT1(EQ(wu41)) → LT1(wu41)
The graph contains the following edges 1 > 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ ATransformationProof
Q DP problem:
The TRS P consists of the following rules:
new_deleteBy(EQ, :(GT, wu41)) → new_deleteBy(EQ, wu41)
new_deleteBy(EQ, :(LT, wu41)) → new_deleteBy(EQ, wu41)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We have applied the A-Transformation [17] to get from an applicative problem to a standard problem.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ ATransformationProof
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
EQ1(GT1(wu41)) → EQ1(wu41)
EQ1(LT(wu41)) → EQ1(wu41)
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:
- EQ1(LT(wu41)) → EQ1(wu41)
The graph contains the following edges 1 > 1
- EQ1(GT1(wu41)) → EQ1(wu41)
The graph contains the following edges 1 > 1