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

MAYBE 1.719 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/Monad.hs
H-Termination of the given Haskell-Program with start terms could not be shown:

↳ HASKELL

  ↳ LR

mainModule Monad

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

module Monad where

import qualified Maybe
import qualified Prelude

filterM :: Monad a => (b -> a Bool) -> [b] -> a [b]

filterM	_ []	=	return []
filterM	p (x : xs)	=	p x >>= (\flg ->filterM p xs >>= (\ys ->return ( if flg then x : ys else ys)))

module Maybe where

import qualified Monad
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\ys→return (if flg then x : ys else ys)

is transformed to

filterM0 flg x ys = return (if flg then x : ys else ys)

The following Lambda expression

\flg→filterM p xs >>= filterM0 flg x

is transformed to

filterM1 p xs x flg = filterM p xs >>= filterM0 flg x

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

mainModule Monad

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

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

filterM :: Monad b => (a -> b Bool) -> [a] -> b [a]

filterM	_ []	=	return []
filterM	p (x : xs)	=	p x >>= filterM1 p xs x

filterM0

flg x ys

return ( if flg then x : ys else ys)

filterM1

p xs x flg

filterM p xs >>= filterM0 flg x

If Reductions:
The following If expression

if flg then x : ys else ys

is transformed to

filterM00 x ys True = x : ys

filterM00 x ys False = ys

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

mainModule Monad

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

module Monad where

import qualified Maybe
import qualified Prelude

filterM :: Monad a => (b -> a Bool) -> [b] -> a [b]

filterM	_ []	=	return []
filterM	p (x : xs)	=	p x >>= filterM1 p xs x

filterM0

flg x ys

return (filterM00 x ys flg)

filterM00	x ys True	=	x : ys
filterM00	x ys False	=	ys

filterM1

p xs x flg

filterM p xs >>= filterM0 flg x

module Maybe where

import qualified Monad
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

mainModule Monad

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

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

filterM :: Monad b => (a -> b Bool) -> [a] -> b [a]

filterM	vw []	=	return []
filterM	p (x : xs)	=	p x >>= filterM1 p xs x

filterM0

flg x ys

return (filterM00 x ys flg)

filterM00	x ys True	=	x : ys
filterM00	x ys False	=	ys

filterM1

p xs x flg

filterM p xs >>= filterM0 flg x

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                  ↳ Narrow

mainModule Monad

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

module Monad where

import qualified Maybe
import qualified Prelude

filterM :: Monad b => (a -> b Bool) -> [a] -> b [a]

filterM	vw []	=	return []
filterM	p (x : xs)	=	p x >>= filterM1 p xs x

filterM0

flg x ys

return (filterM00 x ys flg)

filterM00	x ys True	=	x : ys
filterM00	x ys False	=	ys

filterM1

p xs x flg

filterM p xs >>= filterM0 flg x

module Maybe where

import qualified Monad
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="filterM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="filterM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="filterM vz3 vz4",fontsize=16,color="burlywood",shape="triangle"];62[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];4 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 5[label="",style="solid", color="burlywood", weight=3];
63[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="filterM vz3 (vz40 : vz41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="filterM vz3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7 -> 9[label="",style="dashed", color="red", weight=0];
7[label="vz3 vz40 >>= filterM1 vz3 vz41 vz40",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8 -> 56[label="",style="dashed", color="red", weight=0];
8[label="return []",fontsize=16,color="magenta"];8 -> 57[label="",style="dashed", color="magenta", weight=3];
10[label="vz3 vz40",fontsize=16,color="green",shape="box"];10 -> 15[label="",style="dashed", color="green", weight=3];
9[label="vz5 >>= filterM1 vz3 vz41 vz40",fontsize=16,color="burlywood",shape="triangle"];66[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];9 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 13[label="",style="solid", color="burlywood", weight=3];
67[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 14[label="",style="solid", color="burlywood", weight=3];
57[label="[]",fontsize=16,color="green",shape="box"];56[label="return vz9",fontsize=16,color="black",shape="triangle"];56 -> 59[label="",style="solid", color="black", weight=3];
15[label="vz40",fontsize=16,color="green",shape="box"];13[label="vz50 : vz51 >>= filterM1 vz3 vz41 vz40",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
14[label="[] >>= filterM1 vz3 vz41 vz40",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
59[label="vz9 : []",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="red", weight=0];
16[label="filterM1 vz3 vz41 vz40 vz50 ++ (vz51 >>= filterM1 vz3 vz41 vz40)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
17[label="[]",fontsize=16,color="green",shape="box"];19 -> 9[label="",style="dashed", color="red", weight=0];
19[label="vz51 >>= filterM1 vz3 vz41 vz40",fontsize=16,color="magenta"];19 -> 20[label="",style="dashed", color="magenta", weight=3];
18[label="filterM1 vz3 vz41 vz40 vz50 ++ vz6",fontsize=16,color="black",shape="triangle"];18 -> 21[label="",style="solid", color="black", weight=3];
20[label="vz51",fontsize=16,color="green",shape="box"];21 -> 22[label="",style="dashed", color="red", weight=0];
21[label="(filterM vz3 vz41 >>= filterM0 vz50 vz40) ++ vz6",fontsize=16,color="magenta"];21 -> 23[label="",style="dashed", color="magenta", weight=3];
23 -> 4[label="",style="dashed", color="red", weight=0];
23[label="filterM vz3 vz41",fontsize=16,color="magenta"];23 -> 24[label="",style="dashed", color="magenta", weight=3];
22[label="(vz7 >>= filterM0 vz50 vz40) ++ vz6",fontsize=16,color="burlywood",shape="triangle"];72[label="vz7/vz70 : vz71",fontsize=10,color="white",style="solid",shape="box"];22 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 25[label="",style="solid", color="burlywood", weight=3];
73[label="vz7/[]",fontsize=10,color="white",style="solid",shape="box"];22 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 26[label="",style="solid", color="burlywood", weight=3];
24[label="vz41",fontsize=16,color="green",shape="box"];25[label="(vz70 : vz71 >>= filterM0 vz50 vz40) ++ vz6",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="([] >>= filterM0 vz50 vz40) ++ vz6",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="(filterM0 vz50 vz40 vz70 ++ (vz71 >>= filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="[] ++ vz6",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3];
29[label="(return (filterM00 vz40 vz70 vz50) ++ (vz71 >>= filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="vz6",fontsize=16,color="green",shape="box"];31[label="((filterM00 vz40 vz70 vz50 : []) ++ (vz71 >>= filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
32 -> 33[label="",style="dashed", color="red", weight=0];
32[label="(filterM00 vz40 vz70 vz50 : [] ++ (vz71 >>= filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="magenta"];32 -> 34[label="",style="dashed", color="magenta", weight=3];
34 -> 28[label="",style="dashed", color="red", weight=0];
34[label="[] ++ (vz71 >>= filterM0 vz50 vz40)",fontsize=16,color="magenta"];34 -> 35[label="",style="dashed", color="magenta", weight=3];
33[label="(filterM00 vz40 vz70 vz50 : vz8) ++ vz6",fontsize=16,color="black",shape="triangle"];33 -> 36[label="",style="solid", color="black", weight=3];
35[label="vz71 >>= filterM0 vz50 vz40",fontsize=16,color="burlywood",shape="triangle"];76[label="vz71/vz710 : vz711",fontsize=10,color="white",style="solid",shape="box"];35 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 37[label="",style="solid", color="burlywood", weight=3];
77[label="vz71/[]",fontsize=10,color="white",style="solid",shape="box"];35 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 38[label="",style="solid", color="burlywood", weight=3];
36[label="filterM00 vz40 vz70 vz50 : vz8 ++ vz6",fontsize=16,color="green",shape="box"];36 -> 39[label="",style="dashed", color="green", weight=3];
36 -> 40[label="",style="dashed", color="green", weight=3];
37[label="vz710 : vz711 >>= filterM0 vz50 vz40",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="[] >>= filterM0 vz50 vz40",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="filterM00 vz40 vz70 vz50",fontsize=16,color="burlywood",shape="triangle"];78[label="vz50/False",fontsize=10,color="white",style="solid",shape="box"];39 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 43[label="",style="solid", color="burlywood", weight=3];
79[label="vz50/True",fontsize=10,color="white",style="solid",shape="box"];39 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 44[label="",style="solid", color="burlywood", weight=3];
40[label="vz8 ++ vz6",fontsize=16,color="burlywood",shape="triangle"];80[label="vz8/vz80 : vz81",fontsize=10,color="white",style="solid",shape="box"];40 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 45[label="",style="solid", color="burlywood", weight=3];
81[label="vz8/[]",fontsize=10,color="white",style="solid",shape="box"];40 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 46[label="",style="solid", color="burlywood", weight=3];
41 -> 40[label="",style="dashed", color="red", weight=0];
41[label="filterM0 vz50 vz40 vz710 ++ (vz711 >>= filterM0 vz50 vz40)",fontsize=16,color="magenta"];41 -> 47[label="",style="dashed", color="magenta", weight=3];
41 -> 48[label="",style="dashed", color="magenta", weight=3];
42[label="[]",fontsize=16,color="green",shape="box"];43[label="filterM00 vz40 vz70 False",fontsize=16,color="black",shape="box"];43 -> 49[label="",style="solid", color="black", weight=3];
44[label="filterM00 vz40 vz70 True",fontsize=16,color="black",shape="box"];44 -> 50[label="",style="solid", color="black", weight=3];
45[label="(vz80 : vz81) ++ vz6",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3];
46[label="[] ++ vz6",fontsize=16,color="black",shape="box"];46 -> 52[label="",style="solid", color="black", weight=3];
47[label="filterM0 vz50 vz40 vz710",fontsize=16,color="black",shape="box"];47 -> 53[label="",style="solid", color="black", weight=3];
48 -> 35[label="",style="dashed", color="red", weight=0];
48[label="vz711 >>= filterM0 vz50 vz40",fontsize=16,color="magenta"];48 -> 54[label="",style="dashed", color="magenta", weight=3];
49[label="vz70",fontsize=16,color="green",shape="box"];50[label="vz40 : vz70",fontsize=16,color="green",shape="box"];51[label="vz80 : vz81 ++ vz6",fontsize=16,color="green",shape="box"];51 -> 55[label="",style="dashed", color="green", weight=3];
52[label="vz6",fontsize=16,color="green",shape="box"];53 -> 56[label="",style="dashed", color="red", weight=0];
53[label="return (filterM00 vz40 vz710 vz50)",fontsize=16,color="magenta"];53 -> 58[label="",style="dashed", color="magenta", weight=3];
54[label="vz711",fontsize=16,color="green",shape="box"];55 -> 40[label="",style="dashed", color="red", weight=0];
55[label="vz81 ++ vz6",fontsize=16,color="magenta"];55 -> 60[label="",style="dashed", color="magenta", weight=3];
58 -> 39[label="",style="dashed", color="red", weight=0];
58[label="filterM00 vz40 vz710 vz50",fontsize=16,color="magenta"];58 -> 61[label="",style="dashed", color="magenta", weight=3];
60[label="vz81",fontsize=16,color="green",shape="box"];61[label="vz710",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

                  ↳ Narrow

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

new_psPs(:(vz80, vz81), vz6, h) → new_psPs(vz81, vz6, 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_psPs(:(vz80, vz81), vz6, h) → new_psPs(vz81, vz6, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

                  ↳ Narrow

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

new_gtGtEs(:(vz710, vz711), vz50, vz40, h) → new_gtGtEs(vz711, vz50, vz40, h)

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

new_gtGtEs(:(vz710, vz711), vz50, vz40, h) → new_gtGtEs(vz711, vz50, vz40, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                  ↳ Narrow

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

new_psPs0(vz3, vz41, vz40, h) → new_filterM(vz3, vz41, h)
new_filterM(vz3, :(vz40, vz41), h) → new_gtGtEs0(vz3, vz41, vz40, h)
new_gtGtEs0(vz3, vz41, vz40, h) → new_psPs0(vz3, vz41, vz40, h)
new_gtGtEs0(vz3, vz41, vz40, h) → new_gtGtEs0(vz3, vz41, vz40, h)

The TRS R consists of the following rules:

new_psPs1(vz3, vz41, vz40, vz50, vz6, h) → new_psPs2(new_filterM0(vz3, vz41, h), vz50, vz40, vz6, h)
new_filterM00(vz40, vz70, False, h) → vz70
new_psPs2([], vz50, vz40, vz6, h) → new_psPs4(vz6, h)
new_return(vz9, h) → :(vz9, [])
new_gtGtEs2(:(vz710, vz711), vz50, vz40, h) → new_psPs5(new_return(new_filterM00(vz40, vz710, vz50, h), h), new_gtGtEs2(vz711, vz50, vz40, h), h)
new_psPs5(:(vz80, vz81), vz6, h) → :(vz80, new_psPs5(vz81, vz6, h))
new_filterM00(vz40, vz70, True, h) → :(vz40, vz70)
new_psPs4(vz6, h) → vz6
new_gtGtEs1(:(vz50, vz51), vz3, vz41, vz40, h) → new_psPs1(vz3, vz41, vz40, vz50, new_gtGtEs1(vz51, vz3, vz41, vz40, h), h)
new_psPs5([], vz6, h) → vz6
new_psPs3(vz40, vz70, vz50, vz8, vz6, h) → :(new_filterM00(vz40, vz70, vz50, h), new_psPs5(vz8, vz6, h))
new_gtGtEs1([], vz3, vz41, vz40, h) → []
new_psPs2(:(vz70, vz71), vz50, vz40, vz6, h) → new_psPs3(vz40, vz70, vz50, new_psPs4(new_gtGtEs2(vz71, vz50, vz40, h), h), vz6, h)
new_gtGtEs2([], vz50, vz40, h) → []

The set Q consists of the following terms:

new_psPs1(x0, x1, x2, x3, x4, x5)
new_gtGtEs2([], x0, x1, x2)
new_return(x0, x1)
new_psPs5([], x0, x1)
new_gtGtEs1(:(x0, x1), x2, x3, x4, x5)
new_psPs2([], x0, x1, x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5)
new_psPs4(x0, x1)
new_gtGtEs2(:(x0, x1), x2, x3, x4)
new_psPs5(:(x0, x1), x2, x3)
new_psPs2(:(x0, x1), x2, x3, x4, x5)
new_filterM00(x0, x1, True, x2)
new_filterM00(x0, x1, False, x2)
new_gtGtEs1([], x0, x1, x2, x3)

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                  ↳ Narrow

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

new_psPs0(vz3, vz41, vz40, h) → new_filterM(vz3, vz41, h)
new_filterM(vz3, :(vz40, vz41), h) → new_gtGtEs0(vz3, vz41, vz40, h)
new_gtGtEs0(vz3, vz41, vz40, h) → new_psPs0(vz3, vz41, vz40, h)
new_gtGtEs0(vz3, vz41, vz40, h) → new_gtGtEs0(vz3, vz41, vz40, h)

R is empty.
The set Q consists of the following terms:

new_psPs1(x0, x1, x2, x3, x4, x5)
new_gtGtEs2([], x0, x1, x2)
new_return(x0, x1)
new_psPs5([], x0, x1)
new_gtGtEs1(:(x0, x1), x2, x3, x4, x5)
new_psPs2([], x0, x1, x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5)
new_psPs4(x0, x1)
new_gtGtEs2(:(x0, x1), x2, x3, x4)
new_psPs5(:(x0, x1), x2, x3)
new_psPs2(:(x0, x1), x2, x3, x4, x5)
new_filterM00(x0, x1, True, x2)
new_filterM00(x0, x1, False, x2)
new_gtGtEs1([], x0, x1, x2, x3)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

new_psPs1(x0, x1, x2, x3, x4, x5)
new_gtGtEs2([], x0, x1, x2)
new_return(x0, x1)
new_psPs5([], x0, x1)
new_gtGtEs1(:(x0, x1), x2, x3, x4, x5)
new_psPs2([], x0, x1, x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5)
new_psPs4(x0, x1)
new_gtGtEs2(:(x0, x1), x2, x3, x4)
new_psPs5(:(x0, x1), x2, x3)
new_psPs2(:(x0, x1), x2, x3, x4, x5)
new_filterM00(x0, x1, True, x2)
new_filterM00(x0, x1, False, x2)
new_gtGtEs1([], x0, x1, x2, x3)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                  ↳ Narrow

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

new_filterM(vz3, :(vz40, vz41), h) → new_gtGtEs0(vz3, vz41, vz40, h)
new_psPs0(vz3, vz41, vz40, h) → new_filterM(vz3, vz41, h)
new_gtGtEs0(vz3, vz41, vz40, h) → new_psPs0(vz3, vz41, vz40, h)
new_gtGtEs0(vz3, vz41, vz40, h) → new_gtGtEs0(vz3, vz41, vz40, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

new_filterM(vz3, :(vz40, vz41), h) → new_gtGtEs0(vz3, vz41, vz40, h)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(:(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(new_filterM(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(new_gtGtEs0(x₁, x₂, x₃, x₄)) = x₁ + 2·x₂ + 2·x₃ + x₄
POL(new_psPs0(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                  ↳ Narrow

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

new_psPs0(vz3, vz41, vz40, h) → new_filterM(vz3, vz41, h)
new_gtGtEs0(vz3, vz41, vz40, h) → new_psPs0(vz3, vz41, vz40, h)
new_gtGtEs0(vz3, vz41, vz40, h) → new_gtGtEs0(vz3, vz41, vz40, 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 1 SCC with 2 less nodes.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ NonTerminationProof

                  ↳ Narrow

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

new_gtGtEs0(vz3, vz41, vz40, h) → new_gtGtEs0(vz3, vz41, vz40, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

new_gtGtEs0(vz3, vz41, vz40, h) → new_gtGtEs0(vz3, vz41, vz40, h)

The TRS R consists of the following rules:none

s = new_gtGtEs0(vz3, vz41, vz40, h) evaluates to t =new_gtGtEs0(vz3, vz41, vz40, h)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_gtGtEs0(vz3, vz41, vz40, h) to new_gtGtEs0(vz3, vz41, vz40, h).

Haskell To QDPs