H-Termination proof of /home/matraf/haskell/eval_FullyBlown

YES 0.911 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 :: Bool -> [Bool] -> [Bool]) :: Bool -> [Bool] -> [Bool])

module List where

import qualified Maybe
import qualified Prelude

delete :: Eq a => a -> [a] -> [a]

delete

deleteBy (==)

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 :: Bool -> [Bool] -> [Bool]) :: Bool -> [Bool] -> [Bool])

module List where

import qualified Maybe
import qualified Prelude

delete :: Eq a => a -> [a] -> [a]

delete

deleteBy (==)

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 :: Bool -> [Bool] -> [Bool]) :: Bool -> [Bool] -> [Bool])

module List where

import qualified Maybe
import qualified Prelude

delete :: Eq a => a -> [a] -> [a]

delete

deleteBy (==)

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

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule List

(delete :: Bool -> [Bool] -> [Bool])

module List where

import qualified Maybe
import qualified Prelude

delete :: Eq a => a -> [a] -> [a]

delete

deleteBy (==)

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

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="delete",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="delete wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="delete wu3 wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="deleteBy (==) wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];30[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 6[label="",style="solid", color="burlywood", weight=3];
31[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="deleteBy (==) wu3 (wu40 : wu41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="deleteBy (==) wu3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="deleteBy0 wu41 wu40 (==) wu3 ((==) wu3 wu40)",fontsize=16,color="burlywood",shape="box"];32[label="wu3/False",fontsize=10,color="white",style="solid",shape="box"];8 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 10[label="",style="solid", color="burlywood", weight=3];
33[label="wu3/True",fontsize=10,color="white",style="solid",shape="box"];8 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 11[label="",style="solid", color="burlywood", weight=3];
9[label="[]",fontsize=16,color="green",shape="box"];10[label="deleteBy0 wu41 wu40 (==) False ((==) False wu40)",fontsize=16,color="burlywood",shape="box"];34[label="wu40/False",fontsize=10,color="white",style="solid",shape="box"];10 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 12[label="",style="solid", color="burlywood", weight=3];
35[label="wu40/True",fontsize=10,color="white",style="solid",shape="box"];10 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 13[label="",style="solid", color="burlywood", weight=3];
11[label="deleteBy0 wu41 wu40 (==) True ((==) True wu40)",fontsize=16,color="burlywood",shape="box"];36[label="wu40/False",fontsize=10,color="white",style="solid",shape="box"];11 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 14[label="",style="solid", color="burlywood", weight=3];
37[label="wu40/True",fontsize=10,color="white",style="solid",shape="box"];11 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 15[label="",style="solid", color="burlywood", weight=3];
12[label="deleteBy0 wu41 False (==) False ((==) False False)",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
13[label="deleteBy0 wu41 True (==) False ((==) False True)",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
14[label="deleteBy0 wu41 False (==) True ((==) True False)",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="deleteBy0 wu41 True (==) True ((==) True True)",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="deleteBy0 wu41 False (==) False True",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="deleteBy0 wu41 True (==) False False",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="deleteBy0 wu41 False (==) True False",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="deleteBy0 wu41 True (==) True True",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="wu41",fontsize=16,color="green",shape="box"];21[label="True : deleteBy (==) False wu41",fontsize=16,color="green",shape="box"];21 -> 24[label="",style="dashed", color="green", weight=3];
22[label="False : deleteBy (==) True wu41",fontsize=16,color="green",shape="box"];22 -> 25[label="",style="dashed", color="green", weight=3];
23[label="wu41",fontsize=16,color="green",shape="box"];24 -> 5[label="",style="dashed", color="red", weight=0];
24[label="deleteBy (==) False wu41",fontsize=16,color="magenta"];24 -> 26[label="",style="dashed", color="magenta", weight=3];
24 -> 27[label="",style="dashed", color="magenta", weight=3];
25 -> 5[label="",style="dashed", color="red", weight=0];
25[label="deleteBy (==) True wu41",fontsize=16,color="magenta"];25 -> 28[label="",style="dashed", color="magenta", weight=3];
25 -> 29[label="",style="dashed", color="magenta", weight=3];
26[label="wu41",fontsize=16,color="green",shape="box"];27[label="False",fontsize=16,color="green",shape="box"];28[label="wu41",fontsize=16,color="green",shape="box"];29[label="True",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

new_deleteBy(True, :(False, wu41)) → new_deleteBy(True, wu41)
new_deleteBy(False, :(True, wu41)) → new_deleteBy(False, 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 2 SCCs.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ ATransformationProof

                      ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_deleteBy(False, :(True, wu41)) → new_deleteBy(False, 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

Q DP problem:
The TRS P consists of the following rules:

False(True(wu41)) → False(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:

False(True(wu41)) → False(wu41)
The graph contains the following edges 1 > 1

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ ATransformationProof

Q DP problem:
The TRS P consists of the following rules:

new_deleteBy(True, :(False, wu41)) → new_deleteBy(True, 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

Q DP problem:
The TRS P consists of the following rules:

True1(False1(wu41)) → True1(wu41)

From the DPs we obtained the following set of size-change graphs:

True1(False1(wu41)) → True1(wu41)
The graph contains the following edges 1 > 1

deleteBy0	ys y eq x True	= ys
deleteBy0	ys y eq x False	= y : deleteBy eq x ys