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

MAYBE 1.58 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

((zipWithM :: (a -> b -> [c]) -> [a] -> [b] -> [[c]]) :: (a -> b -> [c]) -> [a] -> [b] -> [[c]])

module Monad where

import qualified Maybe
import qualified Prelude

zipWithM :: Monad a => (b -> c -> a d) -> [b] -> [c] -> a [d]

zipWithM

f xs ys

sequence (zipWith f xs ys)

module Maybe where

import qualified Monad
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\xs→return (x : xs)

is transformed to

sequence0 x xs = return (x : xs)

The following Lambda expression

\x→sequence cs >>= sequence0 x

is transformed to

sequence1 cs x = sequence cs >>= sequence0 x

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Monad

((zipWithM :: (a -> c -> [b]) -> [a] -> [c] -> [[b]]) :: (a -> c -> [b]) -> [a] -> [c] -> [[b]])

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

zipWithM :: Monad a => (b -> c -> a d) -> [b] -> [c] -> a [d]

zipWithM

f xs ys

sequence (zipWith f xs ys)

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Monad

((zipWithM :: (c -> b -> [a]) -> [c] -> [b] -> [[a]]) :: (c -> b -> [a]) -> [c] -> [b] -> [[a]])

module Monad where

import qualified Maybe
import qualified Prelude

zipWithM :: Monad c => (d -> a -> c b) -> [d] -> [a] -> c [b]

zipWithM

f xs ys

sequence (zipWith f xs ys)

module Maybe where

import qualified Monad
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

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

              ↳ Narrow

mainModule Monad

(zipWithM :: (a -> b -> [c]) -> [a] -> [b] -> [[c]])

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

zipWithM :: Monad d => (a -> c -> d b) -> [a] -> [c] -> d [b]

zipWithM

f xs ys

sequence (zipWith f xs ys)

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="zipWithM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="zipWithM wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="zipWithM wv3 wv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="zipWithM wv3 wv4 wv5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence (zipWith wv3 wv4 wv5)",fontsize=16,color="burlywood",shape="triangle"];63[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];6 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 7[label="",style="solid", color="burlywood", weight=3];
64[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="sequence (zipWith wv3 (wv40 : wv41) wv5)",fontsize=16,color="burlywood",shape="box"];65[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];7 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 9[label="",style="solid", color="burlywood", weight=3];
66[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="sequence (zipWith wv3 [] wv5)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="sequence (zipWith wv3 (wv40 : wv41) (wv50 : wv51))",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="sequence (zipWith wv3 (wv40 : wv41) [])",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
11[label="sequence []",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3];
12[label="sequence (wv3 wv40 wv50 : zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
13 -> 11[label="",style="dashed", color="red", weight=0];
13[label="sequence []",fontsize=16,color="magenta"];14[label="return []",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15 -> 17[label="",style="dashed", color="red", weight=0];
15[label="wv3 wv40 wv50 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];15 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="[] : []",fontsize=16,color="green",shape="box"];18[label="wv3 wv40 wv50",fontsize=16,color="green",shape="box"];18 -> 23[label="",style="dashed", color="green", weight=3];
18 -> 24[label="",style="dashed", color="green", weight=3];
17[label="wv6 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="burlywood",shape="triangle"];69[label="wv6/wv60 : wv61",fontsize=10,color="white",style="solid",shape="box"];17 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 21[label="",style="solid", color="burlywood", weight=3];
70[label="wv6/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 22[label="",style="solid", color="burlywood", weight=3];
23[label="wv40",fontsize=16,color="green",shape="box"];24[label="wv50",fontsize=16,color="green",shape="box"];21[label="wv60 : wv61 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="[] >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
25 -> 27[label="",style="dashed", color="red", weight=0];
25[label="sequence1 (zipWith wv3 wv41 wv51) wv60 ++ (wv61 >>= sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="magenta"];25 -> 28[label="",style="dashed", color="magenta", weight=3];
26[label="[]",fontsize=16,color="green",shape="box"];28 -> 17[label="",style="dashed", color="red", weight=0];
28[label="wv61 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];28 -> 29[label="",style="dashed", color="magenta", weight=3];
27[label="sequence1 (zipWith wv3 wv41 wv51) wv60 ++ wv7",fontsize=16,color="black",shape="triangle"];27 -> 30[label="",style="solid", color="black", weight=3];
29[label="wv61",fontsize=16,color="green",shape="box"];30 -> 31[label="",style="dashed", color="red", weight=0];
30[label="(sequence (zipWith wv3 wv41 wv51) >>= sequence0 wv60) ++ wv7",fontsize=16,color="magenta"];30 -> 32[label="",style="dashed", color="magenta", weight=3];
32 -> 6[label="",style="dashed", color="red", weight=0];
32[label="sequence (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];32 -> 33[label="",style="dashed", color="magenta", weight=3];
32 -> 34[label="",style="dashed", color="magenta", weight=3];
31[label="(wv8 >>= sequence0 wv60) ++ wv7",fontsize=16,color="burlywood",shape="triangle"];75[label="wv8/wv80 : wv81",fontsize=10,color="white",style="solid",shape="box"];31 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 35[label="",style="solid", color="burlywood", weight=3];
76[label="wv8/[]",fontsize=10,color="white",style="solid",shape="box"];31 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 36[label="",style="solid", color="burlywood", weight=3];
33[label="wv41",fontsize=16,color="green",shape="box"];34[label="wv51",fontsize=16,color="green",shape="box"];35[label="(wv80 : wv81 >>= sequence0 wv60) ++ wv7",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="([] >>= sequence0 wv60) ++ wv7",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="(sequence0 wv60 wv80 ++ (wv81 >>= sequence0 wv60)) ++ wv7",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="[] ++ wv7",fontsize=16,color="black",shape="triangle"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="(return (wv60 : wv80) ++ (wv81 >>= sequence0 wv60)) ++ wv7",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40[label="wv7",fontsize=16,color="green",shape="box"];41[label="(((wv60 : wv80) : []) ++ (wv81 >>= sequence0 wv60)) ++ wv7",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
42 -> 43[label="",style="dashed", color="red", weight=0];
42[label="((wv60 : wv80) : [] ++ (wv81 >>= sequence0 wv60)) ++ wv7",fontsize=16,color="magenta"];42 -> 44[label="",style="dashed", color="magenta", weight=3];
44 -> 38[label="",style="dashed", color="red", weight=0];
44[label="[] ++ (wv81 >>= sequence0 wv60)",fontsize=16,color="magenta"];44 -> 45[label="",style="dashed", color="magenta", weight=3];
43[label="((wv60 : wv80) : wv9) ++ wv7",fontsize=16,color="black",shape="triangle"];43 -> 46[label="",style="solid", color="black", weight=3];
45[label="wv81 >>= sequence0 wv60",fontsize=16,color="burlywood",shape="triangle"];79[label="wv81/wv810 : wv811",fontsize=10,color="white",style="solid",shape="box"];45 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 47[label="",style="solid", color="burlywood", weight=3];
80[label="wv81/[]",fontsize=10,color="white",style="solid",shape="box"];45 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 48[label="",style="solid", color="burlywood", weight=3];
46[label="(wv60 : wv80) : wv9 ++ wv7",fontsize=16,color="green",shape="box"];46 -> 49[label="",style="dashed", color="green", weight=3];
47[label="wv810 : wv811 >>= sequence0 wv60",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3];
48[label="[] >>= sequence0 wv60",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3];
49[label="wv9 ++ wv7",fontsize=16,color="burlywood",shape="triangle"];81[label="wv9/wv90 : wv91",fontsize=10,color="white",style="solid",shape="box"];49 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 52[label="",style="solid", color="burlywood", weight=3];
82[label="wv9/[]",fontsize=10,color="white",style="solid",shape="box"];49 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 53[label="",style="solid", color="burlywood", weight=3];
50 -> 49[label="",style="dashed", color="red", weight=0];
50[label="sequence0 wv60 wv810 ++ (wv811 >>= sequence0 wv60)",fontsize=16,color="magenta"];50 -> 54[label="",style="dashed", color="magenta", weight=3];
50 -> 55[label="",style="dashed", color="magenta", weight=3];
51[label="[]",fontsize=16,color="green",shape="box"];52[label="(wv90 : wv91) ++ wv7",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="[] ++ wv7",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54 -> 45[label="",style="dashed", color="red", weight=0];
54[label="wv811 >>= sequence0 wv60",fontsize=16,color="magenta"];54 -> 58[label="",style="dashed", color="magenta", weight=3];
55[label="sequence0 wv60 wv810",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3];
56[label="wv90 : wv91 ++ wv7",fontsize=16,color="green",shape="box"];56 -> 60[label="",style="dashed", color="green", weight=3];
57[label="wv7",fontsize=16,color="green",shape="box"];58[label="wv811",fontsize=16,color="green",shape="box"];59[label="return (wv60 : wv810)",fontsize=16,color="black",shape="box"];59 -> 61[label="",style="solid", color="black", weight=3];
60 -> 49[label="",style="dashed", color="red", weight=0];
60[label="wv91 ++ wv7",fontsize=16,color="magenta"];60 -> 62[label="",style="dashed", color="magenta", weight=3];
61[label="(wv60 : wv810) : []",fontsize=16,color="green",shape="box"];62[label="wv91",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ 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(:(wv90, wv91), wv7, h) → new_psPs(wv91, wv7, 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(:(wv90, wv91), wv7, h) → new_psPs(wv91, wv7, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ 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(:(wv810, wv811), wv60, h) → new_gtGtEs(wv811, wv60, h)

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

new_gtGtEs(:(wv810, wv811), wv60, h) → new_gtGtEs(wv811, wv60, 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

                    ↳ UsableRulesProof

              ↳ Narrow

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

new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) → new_psPs0(wv3, wv41, wv51, h, ba, bb)
new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
new_psPs0(wv3, wv41, wv51, h, ba, bb) → new_sequence(wv3, wv41, wv51, h, ba, bb)

The TRS R consists of the following rules:

new_psPs3(wv60, wv80, wv9, wv7, h) → :(:(wv60, wv80), new_psPs1(wv9, wv7, h))
new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) → new_psPs5(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
new_psPs1([], wv7, h) → wv7
new_psPs2(:(wv80, wv81), wv60, wv7, h) → new_psPs3(wv60, wv80, new_psPs4(new_gtGtEs2(wv81, wv60, h), h), wv7, h)
new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) → []
new_gtGtEs2(:(wv810, wv811), wv60, h) → new_psPs1(:(:(wv60, wv810), []), new_gtGtEs2(wv811, wv60, h), h)
new_gtGtEs2([], wv60, h) → []
new_psPs1(:(wv90, wv91), wv7, h) → :(wv90, new_psPs1(wv91, wv7, h))
new_psPs2([], wv60, wv7, h) → new_psPs4(wv7, h)
new_psPs5(wv3, wv41, wv51, wv60, wv7, h, ba, bb) → new_psPs2(new_sequence0(wv3, wv41, wv51, h, ba, bb), wv60, wv7, h)
new_psPs4(wv7, h) → wv7

The set Q consists of the following terms:

new_gtGtEs1([], x0, x1, x2, x3, x4, x5)
new_psPs1(:(x0, x1), x2, x3)
new_psPs2([], x0, x1, x2)
new_psPs5(x0, x1, x2, x3, x4, x5, x6, x7)
new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs1([], x0, x1)
new_psPs3(x0, x1, x2, x3, x4)
new_psPs4(x0, x1)
new_gtGtEs2([], x0, x1)
new_gtGtEs2(:(x0, x1), x2, x3)
new_psPs2(:(x0, x1), x2, x3, x4)

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

      ↳ 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_gtGtEs0(wv3, wv41, wv51, h, ba, bb) → new_psPs0(wv3, wv41, wv51, h, ba, bb)
new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
new_psPs0(wv3, wv41, wv51, h, ba, bb) → new_sequence(wv3, wv41, wv51, h, ba, bb)

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

new_gtGtEs1([], x0, x1, x2, x3, x4, x5)
new_psPs1(:(x0, x1), x2, x3)
new_psPs2([], x0, x1, x2)
new_psPs5(x0, x1, x2, x3, x4, x5, x6, x7)
new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs1([], x0, x1)
new_psPs3(x0, x1, x2, x3, x4)
new_psPs4(x0, x1)
new_gtGtEs2([], x0, x1)
new_gtGtEs2(:(x0, x1), x2, x3)
new_psPs2(:(x0, x1), x2, x3, x4)

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_gtGtEs1([], x0, x1, x2, x3, x4, x5)
new_psPs1(:(x0, x1), x2, x3)
new_psPs2([], x0, x1, x2)
new_psPs5(x0, x1, x2, x3, x4, x5, x6, x7)
new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs1([], x0, x1)
new_psPs3(x0, x1, x2, x3, x4)
new_psPs4(x0, x1)
new_gtGtEs2([], x0, x1)
new_gtGtEs2(:(x0, x1), x2, x3)
new_psPs2(:(x0, x1), x2, x3, x4)

↳ HASKELL

  ↳ LR

    ↳ 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_sequence(wv3, :(wv40, wv41), :(wv50, wv51), h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) → new_psPs0(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
new_psPs0(wv3, wv41, wv51, h, ba, bb) → new_sequence(wv3, wv41, wv51, h, ba, bb)

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_sequence(wv3, :(wv40, wv41), :(wv50, wv51), h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) → new_psPs0(wv3, wv41, wv51, h, ba, bb)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(:(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(new_gtGtEs0(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₁ + 2·x₂ + 2·x₃ + x₄ + x₅ + x₆
POL(new_psPs0(x₁, x₂, x₃, x₄, x₅, x₆)) = x₁ + x₂ + 2·x₃ + x₄ + x₅ + x₆
POL(new_sequence(x₁, x₂, x₃, x₄, x₅, x₆)) = x₁ + x₂ + x₃ + x₄ + x₅ + x₆

↳ HASKELL

  ↳ LR

    ↳ 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_gtGtEs0(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
new_psPs0(wv3, wv41, wv51, h, ba, bb) → new_sequence(wv3, wv41, wv51, h, ba, bb)

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 1 less node.

↳ HASKELL

  ↳ LR

    ↳ 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(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)

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(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)

The TRS R consists of the following rules:none

s = new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) evaluates to t =new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)

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(wv3, wv41, wv51, h, ba, bb) to new_gtGtEs0(wv3, wv41, wv51, h, ba, bb).

Haskell To QDPs