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

YES 0.849 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Main

((sequence :: Monad b => [b a] -> b [a]) :: Monad b => [b a] -> b [a])

module Main where

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 Main

((sequence :: Monad a => [a b] -> a [b]) :: Monad a => [a b] -> a [b])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

((sequence :: Monad b => [b a] -> b [a]) :: Monad b => [b a] -> b [a])

module Main where

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

mainModule Main

(sequence :: Monad a => [a b] -> a [b])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="sequence\n  Monad a",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="sequence vx3\n  Monad a",fontsize=16,color="burlywood",shape="triangle"];94[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 4[label="",style="solid", color="burlywood", weight=3];
95[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="sequence (vx30 : vx31)\n  Monad a",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
5[label="sequence []\n  Monad a",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="vx30 >>= sequence1 vx31\n  Monad a",fontsize=16,color="blue",shape="box"];96[label=">>= :: ([] a) -> (a -> [] ([] a)) -> [] ([] a)",fontsize=10,color="white",style="solid",shape="box"];6 -> 96[label="",style="solid", color="blue", weight=9];
96 -> 8[label="",style="solid", color="blue", weight=3];
97[label=">>= :: (Maybe a) -> (a -> Maybe ([] a)) -> Maybe ([] a)",fontsize=10,color="white",style="solid",shape="box"];6 -> 97[label="",style="solid", color="blue", weight=9];
97 -> 9[label="",style="solid", color="blue", weight=3];
98[label=">>= :: (IO a) -> (a -> IO ([] a)) -> IO ([] a)",fontsize=10,color="white",style="solid",shape="box"];6 -> 98[label="",style="solid", color="blue", weight=9];
98 -> 10[label="",style="solid", color="blue", weight=3];
7[label="return []\n  Monad a",fontsize=16,color="blue",shape="box"];99[label="return :: ([] a) -> [] ([] a)",fontsize=10,color="white",style="solid",shape="box"];7 -> 99[label="",style="solid", color="blue", weight=9];
99 -> 11[label="",style="solid", color="blue", weight=3];
100[label="return :: ([] a) -> Maybe ([] a)",fontsize=10,color="white",style="solid",shape="box"];7 -> 100[label="",style="solid", color="blue", weight=9];
100 -> 12[label="",style="solid", color="blue", weight=3];
101[label="return :: ([] a) -> IO ([] a)",fontsize=10,color="white",style="solid",shape="box"];7 -> 101[label="",style="solid", color="blue", weight=9];
101 -> 13[label="",style="solid", color="blue", weight=3];
8[label="vx30 >>= sequence1 vx31",fontsize=16,color="burlywood",shape="triangle"];102[label="vx30/vx300 : vx301",fontsize=10,color="white",style="solid",shape="box"];8 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 14[label="",style="solid", color="burlywood", weight=3];
103[label="vx30/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 15[label="",style="solid", color="burlywood", weight=3];
9[label="vx30 >>= sequence1 vx31",fontsize=16,color="burlywood",shape="box"];104[label="vx30/Nothing",fontsize=10,color="white",style="solid",shape="box"];9 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 16[label="",style="solid", color="burlywood", weight=3];
105[label="vx30/Just vx300",fontsize=10,color="white",style="solid",shape="box"];9 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 17[label="",style="solid", color="burlywood", weight=3];
10[label="vx30 >>= sequence1 vx31",fontsize=16,color="black",shape="box"];10 -> 18[label="",style="solid", color="black", weight=3];
11[label="return []",fontsize=16,color="black",shape="box"];11 -> 19[label="",style="solid", color="black", weight=3];
12[label="return []",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
13[label="return []",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="vx300 : vx301 >>= sequence1 vx31",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="[] >>= sequence1 vx31",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
16[label="Nothing >>= sequence1 vx31",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
17[label="Just vx300 >>= sequence1 vx31",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
18[label="primbindIO vx30 (sequence1 vx31)",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
19[label="[] : []",fontsize=16,color="green",shape="box"];20[label="Just []",fontsize=16,color="green",shape="box"];21[label="primretIO []",fontsize=16,color="black",shape="box"];21 -> 27[label="",style="solid", color="black", weight=3];
22 -> 28[label="",style="dashed", color="red", weight=0];
22[label="sequence1 vx31 vx300 ++ (vx301 >>= sequence1 vx31)",fontsize=16,color="magenta"];22 -> 29[label="",style="dashed", color="magenta", weight=3];
23[label="[]",fontsize=16,color="green",shape="box"];24[label="Nothing",fontsize=16,color="green",shape="box"];25[label="sequence1 vx31 vx300",fontsize=16,color="black",shape="box"];25 -> 30[label="",style="solid", color="black", weight=3];
26[label="terminator vx30 (sequence1 vx31)",fontsize=16,color="black",shape="box"];26 -> 31[label="",style="solid", color="black", weight=3];
27[label="terminator []",fontsize=16,color="black",shape="box"];27 -> 32[label="",style="solid", color="black", weight=3];
29 -> 8[label="",style="dashed", color="red", weight=0];
29[label="vx301 >>= sequence1 vx31",fontsize=16,color="magenta"];29 -> 33[label="",style="dashed", color="magenta", weight=3];
28[label="sequence1 vx31 vx300 ++ vx4",fontsize=16,color="black",shape="triangle"];28 -> 34[label="",style="solid", color="black", weight=3];
30 -> 35[label="",style="dashed", color="red", weight=0];
30[label="sequence vx31 >>= sequence0 vx300",fontsize=16,color="magenta"];30 -> 36[label="",style="dashed", color="magenta", weight=3];
31[label="ter0m vx30 (sequence1 vx31)",fontsize=16,color="green",shape="box"];31 -> 37[label="",style="dashed", color="green", weight=3];
31 -> 38[label="",style="dashed", color="green", weight=3];
32[label="ter1m []",fontsize=16,color="green",shape="box"];32 -> 39[label="",style="dashed", color="green", weight=3];
33[label="vx301",fontsize=16,color="green",shape="box"];34 -> 40[label="",style="dashed", color="red", weight=0];
34[label="(sequence vx31 >>= sequence0 vx300) ++ vx4",fontsize=16,color="magenta"];34 -> 41[label="",style="dashed", color="magenta", weight=3];
36 -> 3[label="",style="dashed", color="red", weight=0];
36[label="sequence vx31",fontsize=16,color="magenta"];36 -> 42[label="",style="dashed", color="magenta", weight=3];
35[label="vx5 >>= sequence0 vx300",fontsize=16,color="burlywood",shape="triangle"];111[label="vx5/Nothing",fontsize=10,color="white",style="solid",shape="box"];35 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 43[label="",style="solid", color="burlywood", weight=3];
112[label="vx5/Just vx50",fontsize=10,color="white",style="solid",shape="box"];35 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 44[label="",style="solid", color="burlywood", weight=3];
37[label="vx30",fontsize=16,color="green",shape="box"];38[label="sequence1 vx31",fontsize=16,color="grey",shape="box"];38 -> 45[label="",style="dashed", color="grey", weight=3];
39[label="[]",fontsize=16,color="green",shape="box"];41 -> 3[label="",style="dashed", color="red", weight=0];
41[label="sequence vx31",fontsize=16,color="magenta"];41 -> 46[label="",style="dashed", color="magenta", weight=3];
40[label="(vx6 >>= sequence0 vx300) ++ vx4",fontsize=16,color="burlywood",shape="triangle"];114[label="vx6/vx60 : vx61",fontsize=10,color="white",style="solid",shape="box"];40 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 47[label="",style="solid", color="burlywood", weight=3];
115[label="vx6/[]",fontsize=10,color="white",style="solid",shape="box"];40 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 48[label="",style="solid", color="burlywood", weight=3];
42[label="vx31",fontsize=16,color="green",shape="box"];43[label="Nothing >>= sequence0 vx300",fontsize=16,color="black",shape="box"];43 -> 49[label="",style="solid", color="black", weight=3];
44[label="Just vx50 >>= sequence0 vx300",fontsize=16,color="black",shape="box"];44 -> 50[label="",style="solid", color="black", weight=3];
45[label="sequence1 vx31 vx7",fontsize=16,color="black",shape="triangle"];45 -> 51[label="",style="solid", color="black", weight=3];
46[label="vx31",fontsize=16,color="green",shape="box"];47[label="(vx60 : vx61 >>= sequence0 vx300) ++ vx4",fontsize=16,color="black",shape="box"];47 -> 52[label="",style="solid", color="black", weight=3];
48[label="([] >>= sequence0 vx300) ++ vx4",fontsize=16,color="black",shape="box"];48 -> 53[label="",style="solid", color="black", weight=3];
49[label="Nothing",fontsize=16,color="green",shape="box"];50[label="sequence0 vx300 vx50",fontsize=16,color="black",shape="box"];50 -> 54[label="",style="solid", color="black", weight=3];
51 -> 55[label="",style="dashed", color="red", weight=0];
51[label="sequence vx31 >>= sequence0 vx7",fontsize=16,color="magenta"];51 -> 56[label="",style="dashed", color="magenta", weight=3];
52[label="(sequence0 vx300 vx60 ++ (vx61 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="black",shape="box"];52 -> 57[label="",style="solid", color="black", weight=3];
53[label="[] ++ vx4",fontsize=16,color="black",shape="triangle"];53 -> 58[label="",style="solid", color="black", weight=3];
54[label="return (vx300 : vx50)",fontsize=16,color="black",shape="box"];54 -> 59[label="",style="solid", color="black", weight=3];
56 -> 3[label="",style="dashed", color="red", weight=0];
56[label="sequence vx31",fontsize=16,color="magenta"];56 -> 60[label="",style="dashed", color="magenta", weight=3];
55[label="vx8 >>= sequence0 vx7",fontsize=16,color="black",shape="triangle"];55 -> 61[label="",style="solid", color="black", weight=3];
57[label="(return (vx300 : vx60) ++ (vx61 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="black",shape="box"];57 -> 62[label="",style="solid", color="black", weight=3];
58[label="vx4",fontsize=16,color="green",shape="box"];59[label="Just (vx300 : vx50)",fontsize=16,color="green",shape="box"];60[label="vx31",fontsize=16,color="green",shape="box"];61[label="primbindIO vx8 (sequence0 vx7)",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
62[label="(((vx300 : vx60) : []) ++ (vx61 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="black",shape="box"];62 -> 64[label="",style="solid", color="black", weight=3];
63[label="terminator vx8 (sequence0 vx7)",fontsize=16,color="black",shape="box"];63 -> 65[label="",style="solid", color="black", weight=3];
64 -> 66[label="",style="dashed", color="red", weight=0];
64[label="((vx300 : vx60) : [] ++ (vx61 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="magenta"];64 -> 67[label="",style="dashed", color="magenta", weight=3];
65[label="ter2m vx8 (sequence0 vx7)",fontsize=16,color="green",shape="box"];65 -> 68[label="",style="dashed", color="green", weight=3];
65 -> 69[label="",style="dashed", color="green", weight=3];
67 -> 53[label="",style="dashed", color="red", weight=0];
67[label="[] ++ (vx61 >>= sequence0 vx300)",fontsize=16,color="magenta"];67 -> 70[label="",style="dashed", color="magenta", weight=3];
66[label="((vx300 : vx60) : vx9) ++ vx4",fontsize=16,color="black",shape="triangle"];66 -> 71[label="",style="solid", color="black", weight=3];
68[label="vx8",fontsize=16,color="green",shape="box"];69[label="sequence0 vx7",fontsize=16,color="grey",shape="box"];69 -> 72[label="",style="dashed", color="grey", weight=3];
70[label="vx61 >>= sequence0 vx300",fontsize=16,color="burlywood",shape="triangle"];120[label="vx61/vx610 : vx611",fontsize=10,color="white",style="solid",shape="box"];70 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 73[label="",style="solid", color="burlywood", weight=3];
121[label="vx61/[]",fontsize=10,color="white",style="solid",shape="box"];70 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 74[label="",style="solid", color="burlywood", weight=3];
71[label="(vx300 : vx60) : vx9 ++ vx4",fontsize=16,color="green",shape="box"];71 -> 75[label="",style="dashed", color="green", weight=3];
72[label="sequence0 vx7 vx10",fontsize=16,color="black",shape="triangle"];72 -> 76[label="",style="solid", color="black", weight=3];
73[label="vx610 : vx611 >>= sequence0 vx300",fontsize=16,color="black",shape="box"];73 -> 77[label="",style="solid", color="black", weight=3];
74[label="[] >>= sequence0 vx300",fontsize=16,color="black",shape="box"];74 -> 78[label="",style="solid", color="black", weight=3];
75[label="vx9 ++ vx4",fontsize=16,color="burlywood",shape="triangle"];122[label="vx9/vx90 : vx91",fontsize=10,color="white",style="solid",shape="box"];75 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 79[label="",style="solid", color="burlywood", weight=3];
123[label="vx9/[]",fontsize=10,color="white",style="solid",shape="box"];75 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 80[label="",style="solid", color="burlywood", weight=3];
76[label="return (vx7 : vx10)",fontsize=16,color="black",shape="box"];76 -> 81[label="",style="solid", color="black", weight=3];
77 -> 75[label="",style="dashed", color="red", weight=0];
77[label="sequence0 vx300 vx610 ++ (vx611 >>= sequence0 vx300)",fontsize=16,color="magenta"];77 -> 82[label="",style="dashed", color="magenta", weight=3];
77 -> 83[label="",style="dashed", color="magenta", weight=3];
78[label="[]",fontsize=16,color="green",shape="box"];79[label="(vx90 : vx91) ++ vx4",fontsize=16,color="black",shape="box"];79 -> 84[label="",style="solid", color="black", weight=3];
80[label="[] ++ vx4",fontsize=16,color="black",shape="box"];80 -> 85[label="",style="solid", color="black", weight=3];
81[label="primretIO (vx7 : vx10)",fontsize=16,color="black",shape="box"];81 -> 86[label="",style="solid", color="black", weight=3];
82[label="sequence0 vx300 vx610",fontsize=16,color="black",shape="box"];82 -> 87[label="",style="solid", color="black", weight=3];
83 -> 70[label="",style="dashed", color="red", weight=0];
83[label="vx611 >>= sequence0 vx300",fontsize=16,color="magenta"];83 -> 88[label="",style="dashed", color="magenta", weight=3];
84[label="vx90 : vx91 ++ vx4",fontsize=16,color="green",shape="box"];84 -> 89[label="",style="dashed", color="green", weight=3];
85[label="vx4",fontsize=16,color="green",shape="box"];86[label="terminator (vx7 : vx10)",fontsize=16,color="black",shape="box"];86 -> 90[label="",style="solid", color="black", weight=3];
87[label="return (vx300 : vx610)",fontsize=16,color="black",shape="box"];87 -> 91[label="",style="solid", color="black", weight=3];
88[label="vx611",fontsize=16,color="green",shape="box"];89 -> 75[label="",style="dashed", color="red", weight=0];
89[label="vx91 ++ vx4",fontsize=16,color="magenta"];89 -> 92[label="",style="dashed", color="magenta", weight=3];
90[label="ter3m (vx7 : vx10)",fontsize=16,color="green",shape="box"];90 -> 93[label="",style="dashed", color="green", weight=3];
91[label="(vx300 : vx610) : []",fontsize=16,color="green",shape="box"];92[label="vx91",fontsize=16,color="green",shape="box"];93[label="vx7 : vx10",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

new_psPs(:(vx90, vx91), vx4, ba) → new_psPs(vx91, vx4, ba)

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(:(vx90, vx91), vx4, ba) → new_psPs(vx91, vx4, ba)
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

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

new_gtGtEs(:(vx610, vx611), vx300, ba) → new_gtGtEs(vx611, vx300, ba)

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

new_gtGtEs(:(vx610, vx611), vx300, ba) → new_gtGtEs(vx611, vx300, ba)
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

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

new_gtGtEs0(:(vx300, vx301), vx31, ba) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)
new_sequence1(vx31, vx7, ba) → new_sequence(vx31, ty_IO, ba)
new_psPs0(vx31, vx300, vx4, ba) → new_sequence(vx31, ty_[], ba)
new_sequence(:(:(vx300, vx301), vx31), ty_[], ba) → new_gtGtEs0(vx301, vx31, ba)
new_gtGtEs0(:(vx300, vx301), vx31, ba) → new_gtGtEs0(vx301, vx31, ba)
new_sequence(:(:(vx300, vx301), vx31), ty_[], ba) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)
new_sequence(:(vx30, vx31), ty_IO, ba) → new_sequence(vx31, ty_IO, ba)
new_sequence(:(Just(vx300), vx31), ty_Maybe, ba) → new_sequence(vx31, ty_Maybe, ba)

The TRS R consists of the following rules:

new_psPs4([], vx4, ba) → vx4
new_gtGtEs1([], vx31, ba) → []
new_psPs2(vx31, vx300, vx4, ba) → new_psPs3(new_sequence0(vx31, ty_[], ba), vx300, vx4, ba)
new_psPs3([], vx300, vx4, ba) → new_psPs1(vx4, ba)
new_psPs1(vx4, ba) → vx4
new_psPs5(vx300, vx60, vx9, vx4, ba) → :(:(vx300, vx60), new_psPs4(vx9, vx4, ba))
new_gtGtEs2([], vx300, ba) → []
new_gtGtEs2(:(vx610, vx611), vx300, ba) → new_psPs4(:(:(vx300, vx610), []), new_gtGtEs2(vx611, vx300, ba), ba)
new_psPs4(:(vx90, vx91), vx4, ba) → :(vx90, new_psPs4(vx91, vx4, ba))
new_psPs3(:(vx60, vx61), vx300, vx4, ba) → new_psPs5(vx300, vx60, new_psPs1(new_gtGtEs2(vx61, vx300, ba), ba), vx4, ba)
new_gtGtEs1(:(vx300, vx301), vx31, ba) → new_psPs2(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)

The set Q consists of the following terms:

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

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

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

new_sequence(:(Just(vx300), vx31), ty_Maybe, ba) → new_sequence(vx31, ty_Maybe, ba)

The TRS R consists of the following rules:

new_psPs4([], vx4, ba) → vx4
new_gtGtEs1([], vx31, ba) → []
new_psPs2(vx31, vx300, vx4, ba) → new_psPs3(new_sequence0(vx31, ty_[], ba), vx300, vx4, ba)
new_psPs3([], vx300, vx4, ba) → new_psPs1(vx4, ba)
new_psPs1(vx4, ba) → vx4
new_psPs5(vx300, vx60, vx9, vx4, ba) → :(:(vx300, vx60), new_psPs4(vx9, vx4, ba))
new_gtGtEs2([], vx300, ba) → []
new_gtGtEs2(:(vx610, vx611), vx300, ba) → new_psPs4(:(:(vx300, vx610), []), new_gtGtEs2(vx611, vx300, ba), ba)
new_psPs4(:(vx90, vx91), vx4, ba) → :(vx90, new_psPs4(vx91, vx4, ba))
new_psPs3(:(vx60, vx61), vx300, vx4, ba) → new_psPs5(vx300, vx60, new_psPs1(new_gtGtEs2(vx61, vx300, ba), ba), vx4, ba)
new_gtGtEs1(:(vx300, vx301), vx31, ba) → new_psPs2(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)

The set Q consists of the following terms:

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

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

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

new_sequence(:(Just(vx300), vx31), ty_Maybe, ba) → new_sequence(vx31, ty_Maybe, ba)

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

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

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

                                ↳ QDP

                                  ↳ QDPSizeChangeProof

                        ↳ QDP

                        ↳ QDP

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

new_sequence(:(Just(vx300), vx31), ty_Maybe, ba) → new_sequence(vx31, ty_Maybe, ba)

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

new_sequence(:(Just(vx300), vx31), ty_Maybe, ba) → new_sequence(vx31, ty_Maybe, ba)
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

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

                        ↳ QDP

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

new_sequence(:(vx30, vx31), ty_IO, ba) → new_sequence(vx31, ty_IO, ba)

The TRS R consists of the following rules:

new_psPs4([], vx4, ba) → vx4
new_gtGtEs1([], vx31, ba) → []
new_psPs2(vx31, vx300, vx4, ba) → new_psPs3(new_sequence0(vx31, ty_[], ba), vx300, vx4, ba)
new_psPs3([], vx300, vx4, ba) → new_psPs1(vx4, ba)
new_psPs1(vx4, ba) → vx4
new_psPs5(vx300, vx60, vx9, vx4, ba) → :(:(vx300, vx60), new_psPs4(vx9, vx4, ba))
new_gtGtEs2([], vx300, ba) → []
new_gtGtEs2(:(vx610, vx611), vx300, ba) → new_psPs4(:(:(vx300, vx610), []), new_gtGtEs2(vx611, vx300, ba), ba)
new_psPs4(:(vx90, vx91), vx4, ba) → :(vx90, new_psPs4(vx91, vx4, ba))
new_psPs3(:(vx60, vx61), vx300, vx4, ba) → new_psPs5(vx300, vx60, new_psPs1(new_gtGtEs2(vx61, vx300, ba), ba), vx4, ba)
new_gtGtEs1(:(vx300, vx301), vx31, ba) → new_psPs2(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

                        ↳ QDP

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

new_sequence(:(vx30, vx31), ty_IO, ba) → new_sequence(vx31, ty_IO, ba)

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

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

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

                                ↳ QDP

                                  ↳ QDPSizeChangeProof

                        ↳ QDP

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

new_sequence(:(vx30, vx31), ty_IO, ba) → new_sequence(vx31, ty_IO, ba)

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

new_sequence(:(vx30, vx31), ty_IO, ba) → new_sequence(vx31, ty_IO, ba)
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

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

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

new_gtGtEs0(:(vx300, vx301), vx31, ba) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)
new_psPs0(vx31, vx300, vx4, ba) → new_sequence(vx31, ty_[], ba)
new_sequence(:(:(vx300, vx301), vx31), ty_[], ba) → new_gtGtEs0(vx301, vx31, ba)
new_gtGtEs0(:(vx300, vx301), vx31, ba) → new_gtGtEs0(vx301, vx31, ba)
new_sequence(:(:(vx300, vx301), vx31), ty_[], ba) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)

The TRS R consists of the following rules:

new_psPs4([], vx4, ba) → vx4
new_gtGtEs1([], vx31, ba) → []
new_psPs2(vx31, vx300, vx4, ba) → new_psPs3(new_sequence0(vx31, ty_[], ba), vx300, vx4, ba)
new_psPs3([], vx300, vx4, ba) → new_psPs1(vx4, ba)
new_psPs1(vx4, ba) → vx4
new_psPs5(vx300, vx60, vx9, vx4, ba) → :(:(vx300, vx60), new_psPs4(vx9, vx4, ba))
new_gtGtEs2([], vx300, ba) → []
new_gtGtEs2(:(vx610, vx611), vx300, ba) → new_psPs4(:(:(vx300, vx610), []), new_gtGtEs2(vx611, vx300, ba), ba)
new_psPs4(:(vx90, vx91), vx4, ba) → :(vx90, new_psPs4(vx91, vx4, ba))
new_psPs3(:(vx60, vx61), vx300, vx4, ba) → new_psPs5(vx300, vx60, new_psPs1(new_gtGtEs2(vx61, vx300, ba), ba), vx4, ba)
new_gtGtEs1(:(vx300, vx301), vx31, ba) → new_psPs2(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

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

new_gtGtEs0(:(vx300, vx301), vx31, ba) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)
new_psPs0(vx31, vx300, vx4, ba) → new_sequence(vx31, ty_[], ba)
new_sequence(:(:(vx300, vx301), vx31), ty_[], ba) → new_gtGtEs0(vx301, vx31, ba)
new_gtGtEs0(:(vx300, vx301), vx31, ba) → new_gtGtEs0(vx301, vx31, ba)
new_sequence(:(:(vx300, vx301), vx31), ty_[], ba) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)

The TRS R consists of the following rules:

new_gtGtEs1([], vx31, ba) → []
new_gtGtEs1(:(vx300, vx301), vx31, ba) → new_psPs2(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)
new_psPs2(vx31, vx300, vx4, ba) → new_psPs3(new_sequence0(vx31, ty_[], ba), vx300, vx4, ba)

The set Q consists of the following terms:

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

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_gtGtEs2(:(x0, x1), x2, x3)
new_psPs5(x0, x1, x2, x3, x4)
new_gtGtEs2([], x0, x1)
new_psPs4(:(x0, x1), x2, x3)
new_psPs1(x0, x1)
new_psPs4([], x0, x1)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ QReductionProof

                                ↳ QDP

                                  ↳ QDPSizeChangeProof

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

new_gtGtEs0(:(vx300, vx301), vx31, ba) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)
new_psPs0(vx31, vx300, vx4, ba) → new_sequence(vx31, ty_[], ba)
new_sequence(:(:(vx300, vx301), vx31), ty_[], ba) → new_gtGtEs0(vx301, vx31, ba)
new_gtGtEs0(:(vx300, vx301), vx31, ba) → new_gtGtEs0(vx301, vx31, ba)
new_sequence(:(:(vx300, vx301), vx31), ty_[], ba) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)

The TRS R consists of the following rules:

new_gtGtEs1([], vx31, ba) → []
new_gtGtEs1(:(vx300, vx301), vx31, ba) → new_psPs2(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)
new_psPs2(vx31, vx300, vx4, ba) → new_psPs3(new_sequence0(vx31, ty_[], ba), vx300, vx4, ba)

The set Q consists of the following terms:

new_gtGtEs1(:(x0, x1), x2, x3)
new_psPs2(x0, x1, x2, x3)
new_psPs3(:(x0, x1), x2, x3, x4)
new_psPs3([], x0, x1, x2)
new_gtGtEs1([], x0, x1)

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_psPs0(vx31, vx300, vx4, ba) → new_sequence(vx31, ty_[], ba)
The graph contains the following edges 1 >= 1, 4 >= 3
new_sequence(:(:(vx300, vx301), vx31), ty_[], ba) → new_gtGtEs0(vx301, vx31, ba)
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3
new_gtGtEs0(:(vx300, vx301), vx31, ba) → new_gtGtEs0(vx301, vx31, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
new_sequence(:(:(vx300, vx301), vx31), ty_[], ba) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 4
new_gtGtEs0(:(vx300, vx301), vx31, ba) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, ba), ba)
The graph contains the following edges 2 >= 1, 1 > 2, 3 >= 4