H-Termination proof of ../tpdb/FP/full_haskell/Prelude_sequence

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_sequence_2.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Main

((sequence :: [[a]] -> [[a]]) :: [[a]] -> [[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 :: [[a]] -> [[a]]) :: [[a]] -> [[a]])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

mainModule Main

((sequence :: [[a]] -> [[a]]) :: [[a]] -> [[a]])

module Main where

import qualified Prelude

Num Reduction:All numbers are transformed to thier corresponding representation with Succ, Pred and Zero.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule Main

(sequence :: [[a]] -> [[a]])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="sequence\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="sequence vx3\n",fontsize=16,color="burlywood",shape="triangle"];48[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 4[label="",style="solid", color="burlywood", weight=3];
49[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="sequence (vx30 : vx31)\n",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
5[label="sequence []\n",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="vx30 >>= sequence1 vx31\n",fontsize=16,color="burlywood",shape="triangle"];50[label="vx30/vx300 : vx301",fontsize=10,color="white",style="solid",shape="box"];6 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 8[label="",style="solid", color="burlywood", weight=3];
51[label="vx30/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 9[label="",style="solid", color="burlywood", weight=3];
7[label="return []\n",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="vx300 : vx301 >>= sequence1 vx31\n",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="[] >>= sequence1 vx31\n",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="[] : []\n",fontsize=16,color="green",shape="box"];11 -> 13[label="",style="dashed", color="red", weight=0];
11[label="sequence1 vx31 vx300 ++ (vx301 >>= sequence1 vx31)\n",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
12[label="[]\n",fontsize=16,color="green",shape="box"];14 -> 6[label="",style="dashed", color="red", weight=0];
14[label="vx301 >>= sequence1 vx31\n",fontsize=16,color="magenta"];14 -> 15[label="",style="dashed", color="magenta", weight=3];
13[label="sequence1 vx31 vx300 ++ vx4\n",fontsize=16,color="black",shape="triangle"];13 -> 16[label="",style="solid", color="black", weight=3];
15[label="vx301\n",fontsize=16,color="green",shape="box"];16 -> 17[label="",style="dashed", color="red", weight=0];
16[label="(sequence vx31 >>= sequence0 vx300) ++ vx4\n",fontsize=16,color="magenta"];16 -> 18[label="",style="dashed", color="magenta", weight=3];
18 -> 3[label="",style="dashed", color="red", weight=0];
18[label="sequence vx31\n",fontsize=16,color="magenta"];18 -> 19[label="",style="dashed", color="magenta", weight=3];
17[label="(vx5 >>= sequence0 vx300) ++ vx4\n",fontsize=16,color="burlywood",shape="triangle"];56[label="vx5/vx50 : vx51",fontsize=10,color="white",style="solid",shape="box"];17 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 20[label="",style="solid", color="burlywood", weight=3];
57[label="vx5/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 21[label="",style="solid", color="burlywood", weight=3];
19[label="vx31\n",fontsize=16,color="green",shape="box"];20[label="(vx50 : vx51 >>= sequence0 vx300) ++ vx4\n",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
21[label="([] >>= sequence0 vx300) ++ vx4\n",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="(sequence0 vx300 vx50 ++ (vx51 >>= sequence0 vx300)) ++ vx4\n",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
23[label="[] ++ vx4\n",fontsize=16,color="black",shape="triangle"];23 -> 25[label="",style="solid", color="black", weight=3];
24[label="(return (vx300 : vx50) ++ (vx51 >>= sequence0 vx300)) ++ vx4\n",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="vx4\n",fontsize=16,color="green",shape="box"];26[label="(((vx300 : vx50) : []) ++ (vx51 >>= sequence0 vx300)) ++ vx4\n",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3];
27 -> 28[label="",style="dashed", color="red", weight=0];
27[label="((vx300 : vx50) : [] ++ (vx51 >>= sequence0 vx300)) ++ vx4\n",fontsize=16,color="magenta"];27 -> 29[label="",style="dashed", color="magenta", weight=3];
29 -> 23[label="",style="dashed", color="red", weight=0];
29[label="[] ++ (vx51 >>= sequence0 vx300)\n",fontsize=16,color="magenta"];29 -> 30[label="",style="dashed", color="magenta", weight=3];
28[label="((vx300 : vx50) : vx6) ++ vx4\n",fontsize=16,color="black",shape="triangle"];28 -> 31[label="",style="solid", color="black", weight=3];
30[label="vx51 >>= sequence0 vx300\n",fontsize=16,color="burlywood",shape="triangle"];60[label="vx51/vx510 : vx511",fontsize=10,color="white",style="solid",shape="box"];30 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 32[label="",style="solid", color="burlywood", weight=3];
61[label="vx51/[]",fontsize=10,color="white",style="solid",shape="box"];30 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 33[label="",style="solid", color="burlywood", weight=3];
31[label="(vx300 : vx50) : vx6 ++ vx4\n",fontsize=16,color="green",shape="box"];31 -> 34[label="",style="dashed", color="green", weight=3];
32[label="vx510 : vx511 >>= sequence0 vx300\n",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
33[label="[] >>= sequence0 vx300\n",fontsize=16,color="black",shape="box"];33 -> 36[label="",style="solid", color="black", weight=3];
34[label="vx6 ++ vx4\n",fontsize=16,color="burlywood",shape="triangle"];62[label="vx6/vx60 : vx61",fontsize=10,color="white",style="solid",shape="box"];34 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 37[label="",style="solid", color="burlywood", weight=3];
63[label="vx6/[]",fontsize=10,color="white",style="solid",shape="box"];34 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 38[label="",style="solid", color="burlywood", weight=3];
35 -> 34[label="",style="dashed", color="red", weight=0];
35[label="sequence0 vx300 vx510 ++ (vx511 >>= sequence0 vx300)\n",fontsize=16,color="magenta"];35 -> 39[label="",style="dashed", color="magenta", weight=3];
35 -> 40[label="",style="dashed", color="magenta", weight=3];
36[label="[]\n",fontsize=16,color="green",shape="box"];37[label="(vx60 : vx61) ++ vx4\n",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="[] ++ vx4\n",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="sequence0 vx300 vx510\n",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
40 -> 30[label="",style="dashed", color="red", weight=0];
40[label="vx511 >>= sequence0 vx300\n",fontsize=16,color="magenta"];40 -> 44[label="",style="dashed", color="magenta", weight=3];
41[label="vx60 : vx61 ++ vx4\n",fontsize=16,color="green",shape="box"];41 -> 45[label="",style="dashed", color="green", weight=3];
42[label="vx4\n",fontsize=16,color="green",shape="box"];43[label="return (vx300 : vx510)\n",fontsize=16,color="black",shape="box"];43 -> 46[label="",style="solid", color="black", weight=3];
44[label="vx511\n",fontsize=16,color="green",shape="box"];45 -> 34[label="",style="dashed", color="red", weight=0];
45[label="vx61 ++ vx4\n",fontsize=16,color="magenta"];45 -> 47[label="",style="dashed", color="magenta", weight=3];
46[label="(vx300 : vx510) : []\n",fontsize=16,color="green",shape="box"];47[label="vx61\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

new_gtGtEs(:(vx510, vx511), vx300, h) → new_gtGtEs(vx511, vx300, 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_gtGtEs(:(vx510, vx511), vx300, h) → new_gtGtEs(vx511, vx300, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_gtGtEs0(:(vx300, vx301), vx31, h) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, h), h)
new_psPs0(vx31, vx300, vx4, h) → new_sequence(vx31, h)
new_sequence(:(:(vx300, vx301), vx31), h) → new_gtGtEs0(vx301, vx31, h)
new_gtGtEs0(:(vx300, vx301), vx31, h) → new_gtGtEs0(vx301, vx31, h)
new_sequence(:(:(vx300, vx301), vx31), h) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, h), h)

The TRS R consists of the following rules:

new_gtGtEs1([], vx31, h) → []
new_psPs3(vx31, vx300, vx4, h) → new_psPs4(new_sequence0(vx31, h), vx300, vx4, h)
new_psPs1(vx300, vx50, vx6, vx4, h) → :(:(vx300, vx50), new_psPs2(vx6, vx4, h))
new_psPs5(vx4, h) → vx4
new_psPs2([], vx4, h) → vx4
new_gtGtEs1(:(vx300, vx301), vx31, h) → new_psPs3(vx31, vx300, new_gtGtEs1(vx301, vx31, h), h)
new_gtGtEs2(:(vx510, vx511), vx300, h) → new_psPs2(:(:(vx300, vx510), []), new_gtGtEs2(vx511, vx300, h), h)
new_sequence0([], h) → :([], [])
new_psPs2(:(vx60, vx61), vx4, h) → :(vx60, new_psPs2(vx61, vx4, h))
new_psPs4(:(vx50, vx51), vx300, vx4, h) → new_psPs1(vx300, vx50, new_psPs5(new_gtGtEs2(vx51, vx300, h), h), vx4, h)
new_psPs4([], vx300, vx4, h) → new_psPs5(vx4, h)
new_sequence0(:(vx30, vx31), h) → new_gtGtEs1(vx30, vx31, h)
new_gtGtEs2([], vx300, h) → []

The set Q consists of the following terms:

new_psPs4(:(x0, x1), x2, x3, x4)
new_psPs1(x0, x1, x2, x3, x4)
new_sequence0(:(x0, x1), x2)
new_psPs5(x0, x1)
new_gtGtEs1([], x0, x1)
new_psPs2([], x0, x1)
new_psPs3(x0, x1, x2, x3)
new_sequence0([], x0)
new_gtGtEs1(:(x0, x1), x2, x3)
new_psPs4([], x0, x1, x2)
new_gtGtEs2(:(x0, x1), x2, x3)
new_gtGtEs2([], x0, x1)
new_psPs2(:(x0, x1), x2, x3)

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, h) → new_sequence(vx31, h)
The graph contains the following edges 1 >= 1, 4 >= 2
new_sequence(:(:(vx300, vx301), vx31), h) → new_gtGtEs0(vx301, vx31, h)
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3
new_gtGtEs0(:(vx300, vx301), vx31, h) → new_gtGtEs0(vx301, vx31, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
new_sequence(:(:(vx300, vx301), vx31), h) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, h), h)
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 4
new_gtGtEs0(:(vx300, vx301), vx31, h) → new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, h), h)
The graph contains the following edges 2 >= 1, 1 > 2, 3 >= 4

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_psPs(:(vx60, vx61), vx4, h) → new_psPs(vx61, vx4, h)

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

new_psPs(:(vx60, vx61), vx4, h) → new_psPs(vx61, vx4, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3