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

MAYBE 1.686 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 :: Monad c => (d -> a -> c b) -> [d] -> [a] -> c [b]) :: Monad c => (d -> a -> c b) -> [d] -> [a] -> c [b])

module Monad where

import qualified Maybe
import qualified Prelude

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

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

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

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

module Monad where

import qualified Maybe
import qualified Prelude

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

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 :: Monad a => (d -> b -> a c) -> [d] -> [b] -> a [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 {
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20 -> 27[label="",style="dashed", color="red", weight=0];
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26[label="wv3 wv40 wv50",fontsize=16,color="green",shape="box"];26 -> 31[label="",style="dashed", color="green", weight=3];
26 -> 32[label="",style="dashed", color="green", weight=3];
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130 -> 33[label="",style="solid", color="burlywood", weight=3];
131[label="wv6/Just wv60",fontsize=10,color="white",style="solid",shape="box"];25 -> 131[label="",style="solid", color="burlywood", weight=9];
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27[label="wv7 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="burlywood",shape="triangle"];132[label="wv7/wv70 : wv71",fontsize=10,color="white",style="solid",shape="box"];27 -> 132[label="",style="solid", color="burlywood", weight=9];
132 -> 37[label="",style="solid", color="burlywood", weight=3];
133[label="wv7/[]",fontsize=10,color="white",style="solid",shape="box"];27 -> 133[label="",style="solid", color="burlywood", weight=9];
133 -> 38[label="",style="solid", color="burlywood", weight=3];
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30[label="terminator []",fontsize=16,color="black",shape="box"];30 -> 42[label="",style="solid", color="black", weight=3];
31[label="wv40",fontsize=16,color="green",shape="box"];32[label="wv50",fontsize=16,color="green",shape="box"];33[label="Nothing >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];33 -> 43[label="",style="solid", color="black", weight=3];
34[label="Just wv60 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];34 -> 44[label="",style="solid", color="black", weight=3];
39[label="wv40",fontsize=16,color="green",shape="box"];40[label="wv50",fontsize=16,color="green",shape="box"];37[label="wv70 : wv71 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];37 -> 45[label="",style="solid", color="black", weight=3];
38[label="[] >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];38 -> 46[label="",style="solid", color="black", weight=3];
41[label="terminator (wv3 wv40 wv50) (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3];
42[label="ter0m []",fontsize=16,color="green",shape="box"];42 -> 48[label="",style="dashed", color="green", weight=3];
43[label="Nothing",fontsize=16,color="green",shape="box"];44[label="sequence1 (zipWith wv3 wv41 wv51) wv60",fontsize=16,color="black",shape="box"];44 -> 49[label="",style="solid", color="black", weight=3];
45 -> 50[label="",style="dashed", color="red", weight=0];
45[label="sequence1 (zipWith wv3 wv41 wv51) wv70 ++ (wv71 >>= sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="magenta"];45 -> 51[label="",style="dashed", color="magenta", weight=3];
46[label="[]",fontsize=16,color="green",shape="box"];47[label="ter1m (wv3 wv40 wv50) (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="green",shape="box"];47 -> 52[label="",style="dashed", color="green", weight=3];
47 -> 53[label="",style="dashed", color="green", weight=3];
48[label="[]",fontsize=16,color="green",shape="box"];49 -> 54[label="",style="dashed", color="red", weight=0];
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55[label="sequence (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];55 -> 61[label="",style="dashed", color="magenta", weight=3];
55 -> 62[label="",style="dashed", color="magenta", weight=3];
54[label="wv9 >>= sequence0 wv60",fontsize=16,color="burlywood",shape="triangle"];138[label="wv9/Nothing",fontsize=10,color="white",style="solid",shape="box"];54 -> 138[label="",style="solid", color="burlywood", weight=9];
138 -> 63[label="",style="solid", color="burlywood", weight=3];
139[label="wv9/Just wv90",fontsize=10,color="white",style="solid",shape="box"];54 -> 139[label="",style="solid", color="burlywood", weight=9];
139 -> 64[label="",style="solid", color="burlywood", weight=3];
56[label="wv71",fontsize=16,color="green",shape="box"];57 -> 65[label="",style="dashed", color="red", weight=0];
57[label="(sequence (zipWith wv3 wv41 wv51) >>= sequence0 wv70) ++ wv8",fontsize=16,color="magenta"];57 -> 66[label="",style="dashed", color="magenta", weight=3];
58[label="wv40",fontsize=16,color="green",shape="box"];59[label="wv50",fontsize=16,color="green",shape="box"];60[label="sequence1 (zipWith wv3 wv41 wv51) wv10",fontsize=16,color="black",shape="triangle"];60 -> 67[label="",style="solid", color="black", weight=3];
61[label="wv41",fontsize=16,color="green",shape="box"];62[label="wv51",fontsize=16,color="green",shape="box"];63[label="Nothing >>= sequence0 wv60",fontsize=16,color="black",shape="box"];63 -> 68[label="",style="solid", color="black", weight=3];
64[label="Just wv90 >>= sequence0 wv60",fontsize=16,color="black",shape="box"];64 -> 69[label="",style="solid", color="black", weight=3];
66 -> 6[label="",style="dashed", color="red", weight=0];
66[label="sequence (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];66 -> 70[label="",style="dashed", color="magenta", weight=3];
66 -> 71[label="",style="dashed", color="magenta", weight=3];
65[label="(wv11 >>= sequence0 wv70) ++ wv8",fontsize=16,color="burlywood",shape="triangle"];142[label="wv11/wv110 : wv111",fontsize=10,color="white",style="solid",shape="box"];65 -> 142[label="",style="solid", color="burlywood", weight=9];
142 -> 72[label="",style="solid", color="burlywood", weight=3];
143[label="wv11/[]",fontsize=10,color="white",style="solid",shape="box"];65 -> 143[label="",style="solid", color="burlywood", weight=9];
143 -> 73[label="",style="solid", color="burlywood", weight=3];
67 -> 74[label="",style="dashed", color="red", weight=0];
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68[label="Nothing",fontsize=16,color="green",shape="box"];69[label="sequence0 wv60 wv90",fontsize=16,color="black",shape="box"];69 -> 76[label="",style="solid", color="black", weight=3];
70[label="wv41",fontsize=16,color="green",shape="box"];71[label="wv51",fontsize=16,color="green",shape="box"];72[label="(wv110 : wv111 >>= sequence0 wv70) ++ wv8",fontsize=16,color="black",shape="box"];72 -> 77[label="",style="solid", color="black", weight=3];
73[label="([] >>= sequence0 wv70) ++ wv8",fontsize=16,color="black",shape="box"];73 -> 78[label="",style="solid", color="black", weight=3];
75 -> 6[label="",style="dashed", color="red", weight=0];
75[label="sequence (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];75 -> 79[label="",style="dashed", color="magenta", weight=3];
75 -> 80[label="",style="dashed", color="magenta", weight=3];
74[label="wv12 >>= sequence0 wv10",fontsize=16,color="black",shape="triangle"];74 -> 81[label="",style="solid", color="black", weight=3];
76[label="return (wv60 : wv90)",fontsize=16,color="black",shape="box"];76 -> 82[label="",style="solid", color="black", weight=3];
77[label="(sequence0 wv70 wv110 ++ (wv111 >>= sequence0 wv70)) ++ wv8",fontsize=16,color="black",shape="box"];77 -> 83[label="",style="solid", color="black", weight=3];
78[label="[] ++ wv8",fontsize=16,color="black",shape="triangle"];78 -> 84[label="",style="solid", color="black", weight=3];
79[label="wv41",fontsize=16,color="green",shape="box"];80[label="wv51",fontsize=16,color="green",shape="box"];81[label="primbindIO wv12 (sequence0 wv10)",fontsize=16,color="black",shape="box"];81 -> 85[label="",style="solid", color="black", weight=3];
82[label="Just (wv60 : wv90)",fontsize=16,color="green",shape="box"];83[label="(return (wv70 : wv110) ++ (wv111 >>= sequence0 wv70)) ++ wv8",fontsize=16,color="black",shape="box"];83 -> 86[label="",style="solid", color="black", weight=3];
84[label="wv8",fontsize=16,color="green",shape="box"];85[label="terminator wv12 (sequence0 wv10)",fontsize=16,color="black",shape="box"];85 -> 87[label="",style="solid", color="black", weight=3];
86[label="(((wv70 : wv110) : []) ++ (wv111 >>= sequence0 wv70)) ++ wv8",fontsize=16,color="black",shape="box"];86 -> 88[label="",style="solid", color="black", weight=3];
87[label="ter2m wv12 (sequence0 wv10)",fontsize=16,color="green",shape="box"];87 -> 89[label="",style="dashed", color="green", weight=3];
87 -> 90[label="",style="dashed", color="green", weight=3];
88 -> 91[label="",style="dashed", color="red", weight=0];
88[label="((wv70 : wv110) : [] ++ (wv111 >>= sequence0 wv70)) ++ wv8",fontsize=16,color="magenta"];88 -> 92[label="",style="dashed", color="magenta", weight=3];
89[label="wv12",fontsize=16,color="green",shape="box"];90[label="sequence0 wv10",fontsize=16,color="grey",shape="box"];90 -> 93[label="",style="dashed", color="grey", weight=3];
92 -> 78[label="",style="dashed", color="red", weight=0];
92[label="[] ++ (wv111 >>= sequence0 wv70)",fontsize=16,color="magenta"];92 -> 94[label="",style="dashed", color="magenta", weight=3];
91[label="((wv70 : wv110) : wv13) ++ wv8",fontsize=16,color="black",shape="triangle"];91 -> 95[label="",style="solid", color="black", weight=3];
93[label="sequence0 wv10 wv14",fontsize=16,color="black",shape="triangle"];93 -> 96[label="",style="solid", color="black", weight=3];
94[label="wv111 >>= sequence0 wv70",fontsize=16,color="burlywood",shape="triangle"];148[label="wv111/wv1110 : wv1111",fontsize=10,color="white",style="solid",shape="box"];94 -> 148[label="",style="solid", color="burlywood", weight=9];
148 -> 97[label="",style="solid", color="burlywood", weight=3];
149[label="wv111/[]",fontsize=10,color="white",style="solid",shape="box"];94 -> 149[label="",style="solid", color="burlywood", weight=9];
149 -> 98[label="",style="solid", color="burlywood", weight=3];
95[label="(wv70 : wv110) : wv13 ++ wv8",fontsize=16,color="green",shape="box"];95 -> 99[label="",style="dashed", color="green", weight=3];
96[label="return (wv10 : wv14)",fontsize=16,color="black",shape="box"];96 -> 100[label="",style="solid", color="black", weight=3];
97[label="wv1110 : wv1111 >>= sequence0 wv70",fontsize=16,color="black",shape="box"];97 -> 101[label="",style="solid", color="black", weight=3];
98[label="[] >>= sequence0 wv70",fontsize=16,color="black",shape="box"];98 -> 102[label="",style="solid", color="black", weight=3];
99[label="wv13 ++ wv8",fontsize=16,color="burlywood",shape="triangle"];150[label="wv13/wv130 : wv131",fontsize=10,color="white",style="solid",shape="box"];99 -> 150[label="",style="solid", color="burlywood", weight=9];
150 -> 103[label="",style="solid", color="burlywood", weight=3];
151[label="wv13/[]",fontsize=10,color="white",style="solid",shape="box"];99 -> 151[label="",style="solid", color="burlywood", weight=9];
151 -> 104[label="",style="solid", color="burlywood", weight=3];
100[label="primretIO (wv10 : wv14)",fontsize=16,color="black",shape="box"];100 -> 105[label="",style="solid", color="black", weight=3];
101 -> 99[label="",style="dashed", color="red", weight=0];
101[label="sequence0 wv70 wv1110 ++ (wv1111 >>= sequence0 wv70)",fontsize=16,color="magenta"];101 -> 106[label="",style="dashed", color="magenta", weight=3];
101 -> 107[label="",style="dashed", color="magenta", weight=3];
102[label="[]",fontsize=16,color="green",shape="box"];103[label="(wv130 : wv131) ++ wv8",fontsize=16,color="black",shape="box"];103 -> 108[label="",style="solid", color="black", weight=3];
104[label="[] ++ wv8",fontsize=16,color="black",shape="box"];104 -> 109[label="",style="solid", color="black", weight=3];
105[label="terminator (wv10 : wv14)",fontsize=16,color="black",shape="box"];105 -> 110[label="",style="solid", color="black", weight=3];
106[label="sequence0 wv70 wv1110",fontsize=16,color="black",shape="box"];106 -> 111[label="",style="solid", color="black", weight=3];
107 -> 94[label="",style="dashed", color="red", weight=0];
107[label="wv1111 >>= sequence0 wv70",fontsize=16,color="magenta"];107 -> 112[label="",style="dashed", color="magenta", weight=3];
108[label="wv130 : wv131 ++ wv8",fontsize=16,color="green",shape="box"];108 -> 113[label="",style="dashed", color="green", weight=3];
109[label="wv8",fontsize=16,color="green",shape="box"];110[label="ter3m (wv10 : wv14)",fontsize=16,color="green",shape="box"];110 -> 114[label="",style="dashed", color="green", weight=3];
111[label="return (wv70 : wv1110)",fontsize=16,color="black",shape="box"];111 -> 115[label="",style="solid", color="black", weight=3];
112[label="wv1111",fontsize=16,color="green",shape="box"];113 -> 99[label="",style="dashed", color="red", weight=0];
113[label="wv131 ++ wv8",fontsize=16,color="magenta"];113 -> 116[label="",style="dashed", color="magenta", weight=3];
114[label="wv10 : wv14",fontsize=16,color="green",shape="box"];115[label="(wv70 : wv1110) : []",fontsize=16,color="green",shape="box"];116[label="wv131",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(:(wv130, wv131), wv8, h) → new_psPs(wv131, wv8, 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(:(wv130, wv131), wv8, h) → new_psPs(wv131, wv8, 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(:(wv1110, wv1111), wv70, h) → new_gtGtEs(wv1111, wv70, h)

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

new_gtGtEs(:(wv1110, wv1111), wv70, h) → new_gtGtEs(wv1111, wv70, 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

                    ↳ DependencyGraphProof

              ↳ Narrow

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

new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) → new_sequence(wv3, wv41, wv51, ty_Maybe, h, ba, bb)
new_sequence1(wv3, wv41, wv51, wv10, h, ba, bb) → new_sequence(wv3, wv41, wv51, ty_IO, h, ba, bb)
new_psPs0(wv3, wv41, wv51, h, ba, bb) → new_sequence(wv3, wv41, wv51, ty_[], h, ba, bb)
new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_[], h, ba, bb) → new_gtGtEs1(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_psPs0(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs1(wv3, wv41, wv51, h, ba, bb)
new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_Maybe, h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_IO, h, ba, bb) → new_sequence(wv3, wv41, wv51, ty_IO, h, ba, bb)

The TRS R consists of the following rules:

new_gtGtEs3([], wv70, h) → []
new_psPs4(:(wv110, wv111), wv70, wv8, h) → new_psPs5(wv70, wv110, new_psPs1(new_gtGtEs3(wv111, wv70, h), h), wv8, h)
new_psPs5(wv70, wv110, wv13, wv8, h) → :(:(wv70, wv110), new_psPs2(wv13, wv8, h))
new_gtGtEs2(:(wv70, wv71), wv3, wv41, wv51, h, ba, bb) → new_psPs3(wv3, wv41, wv51, wv70, new_gtGtEs2(wv71, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
new_psPs2(:(wv130, wv131), wv8, h) → :(wv130, new_psPs2(wv131, wv8, h))
new_gtGtEs3(:(wv1110, wv1111), wv70, h) → new_psPs2(:(:(wv70, wv1110), []), new_gtGtEs3(wv1111, wv70, h), h)
new_psPs1(wv8, h) → wv8
new_psPs3(wv3, wv41, wv51, wv70, wv8, h, ba, bb) → new_psPs4(new_sequence0(wv3, wv41, wv51, ty_[], h, ba, bb), wv70, wv8, h)
new_psPs2([], wv8, h) → wv8
new_gtGtEs2([], wv3, wv41, wv51, h, ba, bb) → []
new_psPs4([], wv70, wv8, h) → new_psPs1(wv8, h)

The set Q consists of the following terms:

new_gtGtEs3([], x0, x1)
new_gtGtEs2([], x0, x1, x2, x3, x4, x5)
new_gtGtEs2(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs5(x0, x1, x2, x3, x4)
new_psPs2(:(x0, x1), x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5, x6, x7)
new_psPs2([], x0, x1)
new_psPs1(x0, x1)
new_psPs4([], x0, x1, x2)
new_psPs4(:(x0, x1), x2, x3, x4)
new_gtGtEs3(:(x0, x1), x2, x3)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 1 less node.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ UsableRulesProof

                        ↳ QDP

                        ↳ QDP

              ↳ Narrow

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

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

The TRS R consists of the following rules:

new_gtGtEs3([], wv70, h) → []
new_psPs4(:(wv110, wv111), wv70, wv8, h) → new_psPs5(wv70, wv110, new_psPs1(new_gtGtEs3(wv111, wv70, h), h), wv8, h)
new_psPs5(wv70, wv110, wv13, wv8, h) → :(:(wv70, wv110), new_psPs2(wv13, wv8, h))
new_gtGtEs2(:(wv70, wv71), wv3, wv41, wv51, h, ba, bb) → new_psPs3(wv3, wv41, wv51, wv70, new_gtGtEs2(wv71, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
new_psPs2(:(wv130, wv131), wv8, h) → :(wv130, new_psPs2(wv131, wv8, h))
new_gtGtEs3(:(wv1110, wv1111), wv70, h) → new_psPs2(:(:(wv70, wv1110), []), new_gtGtEs3(wv1111, wv70, h), h)
new_psPs1(wv8, h) → wv8
new_psPs3(wv3, wv41, wv51, wv70, wv8, h, ba, bb) → new_psPs4(new_sequence0(wv3, wv41, wv51, ty_[], h, ba, bb), wv70, wv8, h)
new_psPs2([], wv8, h) → wv8
new_gtGtEs2([], wv3, wv41, wv51, h, ba, bb) → []
new_psPs4([], wv70, wv8, h) → new_psPs1(wv8, h)

The set Q consists of the following terms:

new_gtGtEs3([], x0, x1)
new_gtGtEs2([], x0, x1, x2, x3, x4, x5)
new_gtGtEs2(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs5(x0, x1, x2, x3, x4)
new_psPs2(:(x0, x1), x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5, x6, x7)
new_psPs2([], x0, x1)
new_psPs1(x0, x1)
new_psPs4([], x0, x1, x2)
new_psPs4(:(x0, x1), x2, x3, x4)
new_gtGtEs3(:(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

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

                        ↳ QDP

                        ↳ QDP

              ↳ Narrow

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

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

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

new_gtGtEs3([], x0, x1)
new_gtGtEs2([], x0, x1, x2, x3, x4, x5)
new_gtGtEs2(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs5(x0, x1, x2, x3, x4)
new_psPs2(:(x0, x1), x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5, x6, x7)
new_psPs2([], x0, x1)
new_psPs1(x0, x1)
new_psPs4([], x0, x1, x2)
new_psPs4(:(x0, x1), x2, x3, x4)
new_gtGtEs3(:(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_gtGtEs3([], x0, x1)
new_gtGtEs2([], x0, x1, x2, x3, x4, x5)
new_gtGtEs2(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs5(x0, x1, x2, x3, x4)
new_psPs2(:(x0, x1), x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5, x6, x7)
new_psPs2([], x0, x1)
new_psPs1(x0, x1)
new_psPs4([], x0, x1, x2)
new_psPs4(:(x0, x1), x2, x3, x4)
new_gtGtEs3(:(x0, x1), x2, x3)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

                                ↳ QDP

                                  ↳ QDPSizeChangeProof

                        ↳ QDP

                        ↳ QDP

              ↳ Narrow

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

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

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

new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_IO, h, ba, bb) → new_sequence(wv3, wv41, wv51, ty_IO, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

                        ↳ QDP

              ↳ Narrow

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

new_psPs0(wv3, wv41, wv51, h, ba, bb) → new_sequence(wv3, wv41, wv51, ty_[], h, ba, bb)
new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_[], h, ba, bb) → new_gtGtEs1(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_psPs0(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs1(wv3, wv41, wv51, h, ba, bb)

The TRS R consists of the following rules:

new_gtGtEs3([], wv70, h) → []
new_psPs4(:(wv110, wv111), wv70, wv8, h) → new_psPs5(wv70, wv110, new_psPs1(new_gtGtEs3(wv111, wv70, h), h), wv8, h)
new_psPs5(wv70, wv110, wv13, wv8, h) → :(:(wv70, wv110), new_psPs2(wv13, wv8, h))
new_gtGtEs2(:(wv70, wv71), wv3, wv41, wv51, h, ba, bb) → new_psPs3(wv3, wv41, wv51, wv70, new_gtGtEs2(wv71, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
new_psPs2(:(wv130, wv131), wv8, h) → :(wv130, new_psPs2(wv131, wv8, h))
new_gtGtEs3(:(wv1110, wv1111), wv70, h) → new_psPs2(:(:(wv70, wv1110), []), new_gtGtEs3(wv1111, wv70, h), h)
new_psPs1(wv8, h) → wv8
new_psPs3(wv3, wv41, wv51, wv70, wv8, h, ba, bb) → new_psPs4(new_sequence0(wv3, wv41, wv51, ty_[], h, ba, bb), wv70, wv8, h)
new_psPs2([], wv8, h) → wv8
new_gtGtEs2([], wv3, wv41, wv51, h, ba, bb) → []
new_psPs4([], wv70, wv8, h) → new_psPs1(wv8, h)

The set Q consists of the following terms:

new_gtGtEs3([], x0, x1)
new_gtGtEs2([], x0, x1, x2, x3, x4, x5)
new_gtGtEs2(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs5(x0, x1, x2, x3, x4)
new_psPs2(:(x0, x1), x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5, x6, x7)
new_psPs2([], x0, x1)
new_psPs1(x0, x1)
new_psPs4([], x0, x1, x2)
new_psPs4(:(x0, x1), x2, x3, x4)
new_gtGtEs3(:(x0, x1), x2, x3)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

                        ↳ QDP

              ↳ Narrow

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

new_psPs0(wv3, wv41, wv51, h, ba, bb) → new_sequence(wv3, wv41, wv51, ty_[], h, ba, bb)
new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_[], h, ba, bb) → new_gtGtEs1(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_psPs0(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs1(wv3, wv41, wv51, h, ba, bb)

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

new_gtGtEs3([], x0, x1)
new_gtGtEs2([], x0, x1, x2, x3, x4, x5)
new_gtGtEs2(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs5(x0, x1, x2, x3, x4)
new_psPs2(:(x0, x1), x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5, x6, x7)
new_psPs2([], x0, x1)
new_psPs1(x0, x1)
new_psPs4([], x0, x1, x2)
new_psPs4(:(x0, x1), x2, x3, x4)
new_gtGtEs3(:(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_gtGtEs3([], x0, x1)
new_gtGtEs2([], x0, x1, x2, x3, x4, x5)
new_gtGtEs2(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs5(x0, x1, x2, x3, x4)
new_psPs2(:(x0, x1), x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5, x6, x7)
new_psPs2([], x0, x1)
new_psPs1(x0, x1)
new_psPs4([], x0, x1, x2)
new_psPs4(:(x0, x1), x2, x3, x4)
new_gtGtEs3(:(x0, x1), x2, x3)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

                                ↳ QDP

                                  ↳ UsableRulesReductionPairsProof

                        ↳ QDP

              ↳ Narrow

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

new_psPs0(wv3, wv41, wv51, h, ba, bb) → new_sequence(wv3, wv41, wv51, ty_[], h, ba, bb)
new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_[], h, ba, bb) → new_gtGtEs1(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_psPs0(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs1(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), ty_[], h, ba, bb) → new_gtGtEs1(wv3, wv41, wv51, h, ba, bb)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(:(x₁, x₂)) = x₁ + x₂
POL(new_gtGtEs1(x₁, x₂, x₃, x₄, x₅, x₆)) = x₁ + 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, x₇)) = x₁ + x₂ + 2·x₃ + 2·x₄ + x₅ + x₆ + x₇
POL(ty_[]) = 0

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

                                ↳ QDP

                                  ↳ UsableRulesReductionPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                        ↳ QDP

              ↳ Narrow

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

new_psPs0(wv3, wv41, wv51, h, ba, bb) → new_sequence(wv3, wv41, wv51, ty_[], h, ba, bb)
new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_psPs0(wv3, wv41, wv51, h, ba, bb)
new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs1(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 2 less nodes.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

                                ↳ QDP

                                  ↳ UsableRulesReductionPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ NonTerminationProof

                        ↳ QDP

              ↳ Narrow

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

new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs1(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_gtGtEs1(wv3, wv41, wv51, h, ba, bb) → new_gtGtEs1(wv3, wv41, wv51, h, ba, bb)

The TRS R consists of the following rules:none

s = new_gtGtEs1(wv3, wv41, wv51, h, ba, bb) evaluates to t =new_gtGtEs1(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_gtGtEs1(wv3, wv41, wv51, h, ba, bb) to new_gtGtEs1(wv3, wv41, wv51, h, ba, bb).

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

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

The TRS R consists of the following rules:

new_gtGtEs3([], wv70, h) → []
new_psPs4(:(wv110, wv111), wv70, wv8, h) → new_psPs5(wv70, wv110, new_psPs1(new_gtGtEs3(wv111, wv70, h), h), wv8, h)
new_psPs5(wv70, wv110, wv13, wv8, h) → :(:(wv70, wv110), new_psPs2(wv13, wv8, h))
new_gtGtEs2(:(wv70, wv71), wv3, wv41, wv51, h, ba, bb) → new_psPs3(wv3, wv41, wv51, wv70, new_gtGtEs2(wv71, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
new_psPs2(:(wv130, wv131), wv8, h) → :(wv130, new_psPs2(wv131, wv8, h))
new_gtGtEs3(:(wv1110, wv1111), wv70, h) → new_psPs2(:(:(wv70, wv1110), []), new_gtGtEs3(wv1111, wv70, h), h)
new_psPs1(wv8, h) → wv8
new_psPs3(wv3, wv41, wv51, wv70, wv8, h, ba, bb) → new_psPs4(new_sequence0(wv3, wv41, wv51, ty_[], h, ba, bb), wv70, wv8, h)
new_psPs2([], wv8, h) → wv8
new_gtGtEs2([], wv3, wv41, wv51, h, ba, bb) → []
new_psPs4([], wv70, wv8, h) → new_psPs1(wv8, h)

The set Q consists of the following terms:

new_gtGtEs3([], x0, x1)
new_gtGtEs2([], x0, x1, x2, x3, x4, x5)
new_gtGtEs2(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs5(x0, x1, x2, x3, x4)
new_psPs2(:(x0, x1), x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5, x6, x7)
new_psPs2([], x0, x1)
new_psPs1(x0, x1)
new_psPs4([], x0, x1, x2)
new_psPs4(:(x0, x1), x2, x3, x4)
new_gtGtEs3(:(x0, x1), x2, x3)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

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

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

new_gtGtEs3([], x0, x1)
new_gtGtEs2([], x0, x1, x2, x3, x4, x5)
new_gtGtEs2(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs5(x0, x1, x2, x3, x4)
new_psPs2(:(x0, x1), x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5, x6, x7)
new_psPs2([], x0, x1)
new_psPs1(x0, x1)
new_psPs4([], x0, x1, x2)
new_psPs4(:(x0, x1), x2, x3, x4)
new_gtGtEs3(:(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_gtGtEs3([], x0, x1)
new_gtGtEs2([], x0, x1, x2, x3, x4, x5)
new_gtGtEs2(:(x0, x1), x2, x3, x4, x5, x6, x7)
new_psPs5(x0, x1, x2, x3, x4)
new_psPs2(:(x0, x1), x2, x3)
new_psPs3(x0, x1, x2, x3, x4, x5, x6, x7)
new_psPs2([], x0, x1)
new_psPs1(x0, x1)
new_psPs4([], x0, x1, x2)
new_psPs4(:(x0, x1), x2, x3, x4)
new_gtGtEs3(:(x0, x1), x2, x3)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

                                ↳ QDP

                                  ↳ QDPSizeChangeProof

              ↳ Narrow

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

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

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

new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_Maybe, h, ba, bb) → new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 5 >= 4, 6 >= 5, 7 >= 6
new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) → new_sequence(wv3, wv41, wv51, ty_Maybe, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 5, 5 >= 6, 6 >= 7

Haskell To QDPs