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

YES 0.865 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

  ↳ LR

mainModule List

((intersect :: [()] -> [()] -> [()]) :: [()] -> [()] -> [()])

module List where

import qualified Maybe
import qualified Prelude

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

intersect

intersectBy (==)

intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

intersectBy

eq xs ys

concatMap (\vv2 ->

case	vv2 of
	x	->	if any (eq x) ys then x : [] else []
	_	->	[]

) xs

module Maybe where

import qualified List
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\vv2→

case vv2 of
x → if any (eq x) ys then x : [] else []

_ → []

is transformed to

intersectBy0 eq ys vv2 =

case vv2 of
x → if any (eq x) ys then x : [] else []

_ → []

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

mainModule List

((intersect :: [()] -> [()] -> [()]) :: [()] -> [()] -> [()])

module List where

import qualified Maybe
import qualified Prelude

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

intersect

intersectBy (==)

intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

intersectBy

eq xs ys

concatMap (intersectBy0 eq ys) xs

intersectBy0

eq ys vv2

case	vv2 of
	x	->	if any (eq x) ys then x : [] else []
	_	->	[]

module Maybe where

import qualified List
import qualified Prelude

Case Reductions:
The following Case expression

case vv2 of
x → if any (eq x) ys then x : [] else []

_ → []

is transformed to

intersectBy00 eq ys x = if any (eq x) ys then x : [] else []

intersectBy00 eq ys _ = []

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

mainModule List

((intersect :: [()] -> [()] -> [()]) :: [()] -> [()] -> [()])

module List where

import qualified Maybe
import qualified Prelude

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

intersect

intersectBy (==)

intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

intersectBy

eq xs ys

concatMap (intersectBy0 eq ys) xs

intersectBy0

eq ys vv2

intersectBy00 eq ys vv2

intersectBy00	eq ys x	=	if any (eq x) ys then x : [] else []
intersectBy00	eq ys _	=	[]

module Maybe where

import qualified List
import qualified Prelude

If Reductions:
The following If expression

if any (eq x) ys then x : [] else []

is transformed to

intersectBy000 x True = x : []

intersectBy000 x False = []

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

mainModule List

((intersect :: [()] -> [()] -> [()]) :: [()] -> [()] -> [()])

module List where

import qualified Maybe
import qualified Prelude

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

intersect

intersectBy (==)

intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

intersectBy

eq xs ys

concatMap (intersectBy0 eq ys) xs

intersectBy0

eq ys vv2

intersectBy00 eq ys vv2

intersectBy00	eq ys x	=	intersectBy000 x (any (eq x) ys)
intersectBy00	eq ys _	=	[]

intersectBy000	x True	=	x : []
intersectBy000	x False	=	[]

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

mainModule List

((intersect :: [()] -> [()] -> [()]) :: [()] -> [()] -> [()])

module List where

import qualified Maybe
import qualified Prelude

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

intersect

intersectBy (==)

intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

intersectBy

eq xs ys

concatMap (intersectBy0 eq ys) xs

intersectBy0

eq ys vv2

intersectBy00 eq ys vv2

intersectBy00	eq ys x	=	intersectBy000 x (any (eq x) ys)
intersectBy00	eq ys vw	=	[]

intersectBy000	x True	=	x : []
intersectBy000	x False	=	[]

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

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

                    ↳ HASKELL

                      ↳ Narrow

mainModule List

(intersect :: [()] -> [()] -> [()])

module List where

import qualified Maybe
import qualified Prelude

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

intersect

intersectBy (==)

intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

intersectBy

eq xs ys

concatMap (intersectBy0 eq ys) xs

intersectBy0

eq ys vv2

intersectBy00 eq ys vv2

intersectBy00	eq ys x	=	intersectBy000 x (any (eq x) ys)
intersectBy00	eq ys vw	=	[]

intersectBy000	x True	=	x : []
intersectBy000	x False	=	[]

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="intersect",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="intersect vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="intersect vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="intersectBy (==) vz3 vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="concatMap (intersectBy0 (==) vz4) vz3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="concat . map (intersectBy0 (==) vz4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="concat (map (intersectBy0 (==) vz4) vz3)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr (++) [] (map (intersectBy0 (==) vz4) vz3)",fontsize=16,color="burlywood",shape="triangle"];39[label="vz3/vz30 : vz31",fontsize=10,color="white",style="solid",shape="box"];9 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 10[label="",style="solid", color="burlywood", weight=3];
40[label="vz3/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="foldr (++) [] (map (intersectBy0 (==) vz4) (vz30 : vz31))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr (++) [] (map (intersectBy0 (==) vz4) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr (++) [] (intersectBy0 (==) vz4 vz30 : map (intersectBy0 (==) vz4) vz31)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14 -> 16[label="",style="dashed", color="red", weight=0];
14[label="(++) intersectBy0 (==) vz4 vz30 foldr (++) [] (map (intersectBy0 (==) vz4) vz31)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
15[label="[]",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
17[label="foldr (++) [] (map (intersectBy0 (==) vz4) vz31)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="(++) intersectBy0 (==) vz4 vz30 vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
18[label="vz31",fontsize=16,color="green",shape="box"];19[label="(++) intersectBy00 (==) vz4 vz30 vz5",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20[label="(++) intersectBy000 vz30 (any ((==) vz30) vz4) vz5",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="(++) intersectBy000 vz30 (or . map ((==) vz30)) vz5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="(++) intersectBy000 vz30 (or (map ((==) vz30) vz4)) vz5",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="(++) intersectBy000 vz30 (foldr (||) False (map ((==) vz30) vz4)) vz5",fontsize=16,color="burlywood",shape="box"];43[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];23 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 24[label="",style="solid", color="burlywood", weight=3];
44[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];23 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 25[label="",style="solid", color="burlywood", weight=3];
24[label="(++) intersectBy000 vz30 (foldr (||) False (map ((==) vz30) (vz40 : vz41))) vz5",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="(++) intersectBy000 vz30 (foldr (||) False (map ((==) vz30) [])) vz5",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="(++) intersectBy000 vz30 (foldr (||) False (((==) vz30 vz40) : map ((==) vz30) vz41)) vz5",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="(++) intersectBy000 vz30 (foldr (||) False []) vz5",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="(++) intersectBy000 vz30 ((||) (==) vz30 vz40 foldr (||) False (map ((==) vz30) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];45[label="vz30/()",fontsize=10,color="white",style="solid",shape="box"];28 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 30[label="",style="solid", color="burlywood", weight=3];
29[label="(++) intersectBy000 vz30 False vz5",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="(++) intersectBy000 () ((||) (==) () vz40 foldr (||) False (map ((==) ()) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];46[label="vz40/()",fontsize=10,color="white",style="solid",shape="box"];30 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 32[label="",style="solid", color="burlywood", weight=3];
31[label="(++) [] vz5",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3];
32[label="(++) intersectBy000 () ((||) (==) () () foldr (||) False (map ((==) ()) vz41)) vz5",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
33[label="vz5",fontsize=16,color="green",shape="box"];34[label="(++) intersectBy000 () ((||) True foldr (||) False (map ((==) ()) vz41)) vz5",fontsize=16,color="black",shape="box"];34 -> 35[label="",style="solid", color="black", weight=3];
35[label="(++) intersectBy000 () True vz5",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
36[label="(++) (() : []) vz5",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
37[label="() : [] ++ vz5",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
38 -> 31[label="",style="dashed", color="red", weight=0];
38[label="[] ++ vz5",fontsize=16,color="magenta"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ QDP

                          ↳ QDPSizeChangeProof

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

new_foldr(vz4, :(vz30, vz31)) → new_foldr(vz4, vz31)

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_foldr(vz4, :(vz30, vz31)) → new_foldr(vz4, vz31)
The graph contains the following edges 1 >= 1, 2 > 2

intersectBy00	eq ys x	= if any (eq x) ys then x : [] else []
intersectBy00	eq ys _	= []