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

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_GTGT___1.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Main

(((>>) :: Monad c => c a -> c b -> c b) :: Monad c => c a -> c b -> c b)

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\_→q

is transformed to

gtGt0 q _ = q

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Main

(((>>) :: Monad c => c b -> c a -> c a) :: Monad c => c b -> c a -> c 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

(((>>) :: Monad b => b c -> b a -> b a) :: Monad b => b c -> b a -> b 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

((>>) :: Monad c => c b -> c a -> c a)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="(>>)\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="(>>) vy3\n",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="(>>) vy3 vy4\n",fontsize=16,color="blue",shape="triangle"];37[label=">> :: ([] a) -> ([] b) -> [] b",fontsize=10,color="white",style="solid",shape="box"];4 -> 37[label="",style="solid", color="blue", weight=9];
37 -> 5[label="",style="solid", color="blue", weight=3];
38[label=">> :: (Maybe a) -> (Maybe b) -> Maybe b",fontsize=10,color="white",style="solid",shape="box"];4 -> 38[label="",style="solid", color="blue", weight=9];
38 -> 6[label="",style="solid", color="blue", weight=3];
39[label=">> :: (IO a) -> (IO b) -> IO b",fontsize=10,color="white",style="solid",shape="box"];4 -> 39[label="",style="solid", color="blue", weight=9];
39 -> 7[label="",style="solid", color="blue", weight=3];
5[label="(>>) vy3 vy4\n",fontsize=16,color="black",shape="box"];5 -> 8[label="",style="solid", color="black", weight=3];
6[label="(>>) vy3 vy4\n",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="(>>) vy3 vy4\n",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="vy3 >>= gtGt0 vy4\n",fontsize=16,color="burlywood",shape="triangle"];40[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];8 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 11[label="",style="solid", color="burlywood", weight=3];
41[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 12[label="",style="solid", color="burlywood", weight=3];
9[label="vy3 >>= gtGt0 vy4\n",fontsize=16,color="burlywood",shape="box"];42[label="vy3/Nothing",fontsize=10,color="white",style="solid",shape="box"];9 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 13[label="",style="solid", color="burlywood", weight=3];
43[label="vy3/Just vy30",fontsize=10,color="white",style="solid",shape="box"];9 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 14[label="",style="solid", color="burlywood", weight=3];
10[label="vy3 >>= gtGt0 vy4\n",fontsize=16,color="black",shape="box"];10 -> 15[label="",style="solid", color="black", weight=3];
11[label="vy30 : vy31 >>= gtGt0 vy4\n",fontsize=16,color="black",shape="box"];11 -> 16[label="",style="solid", color="black", weight=3];
12[label="[] >>= gtGt0 vy4\n",fontsize=16,color="black",shape="box"];12 -> 17[label="",style="solid", color="black", weight=3];
13[label="Nothing >>= gtGt0 vy4\n",fontsize=16,color="black",shape="box"];13 -> 18[label="",style="solid", color="black", weight=3];
14[label="Just vy30 >>= gtGt0 vy4\n",fontsize=16,color="black",shape="box"];14 -> 19[label="",style="solid", color="black", weight=3];
15[label="primbindIO vy3 (gtGt0 vy4)\n",fontsize=16,color="black",shape="box"];15 -> 20[label="",style="solid", color="black", weight=3];
16 -> 21[label="",style="dashed", color="red", weight=0];
16[label="gtGt0 vy4 vy30 ++ (vy31 >>= gtGt0 vy4)\n",fontsize=16,color="magenta"];16 -> 22[label="",style="dashed", color="magenta", weight=3];
17[label="[]\n",fontsize=16,color="green",shape="box"];18[label="Nothing\n",fontsize=16,color="green",shape="box"];19[label="gtGt0 vy4 vy30\n",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="terminator vy3 (gtGt0 vy4)\n",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
22 -> 8[label="",style="dashed", color="red", weight=0];
22[label="vy31 >>= gtGt0 vy4\n",fontsize=16,color="magenta"];22 -> 25[label="",style="dashed", color="magenta", weight=3];
21[label="gtGt0 vy4 vy30 ++ vy5\n",fontsize=16,color="black",shape="triangle"];21 -> 26[label="",style="solid", color="black", weight=3];
23[label="vy4\n",fontsize=16,color="green",shape="box"];24[label="ter0m vy3 (gtGt0 vy4)\n",fontsize=16,color="green",shape="box"];24 -> 27[label="",style="dashed", color="green", weight=3];
24 -> 28[label="",style="dashed", color="green", weight=3];
25[label="vy31\n",fontsize=16,color="green",shape="box"];26[label="vy4 ++ vy5\n",fontsize=16,color="burlywood",shape="triangle"];46[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];26 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 29[label="",style="solid", color="burlywood", weight=3];
47[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];26 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 30[label="",style="solid", color="burlywood", weight=3];
27[label="vy3\n",fontsize=16,color="green",shape="box"];28[label="gtGt0 vy4\n",fontsize=16,color="grey",shape="box"];28 -> 31[label="",style="dashed", color="grey", weight=3];
29[label="(vy40 : vy41) ++ vy5\n",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3];
30[label="[] ++ vy5\n",fontsize=16,color="black",shape="box"];30 -> 33[label="",style="solid", color="black", weight=3];
31[label="gtGt0 vy4 vy6\n",fontsize=16,color="black",shape="triangle"];31 -> 34[label="",style="solid", color="black", weight=3];
32[label="vy40 : vy41 ++ vy5\n",fontsize=16,color="green",shape="box"];32 -> 35[label="",style="dashed", color="green", weight=3];
33[label="vy5\n",fontsize=16,color="green",shape="box"];34[label="vy4\n",fontsize=16,color="green",shape="box"];35 -> 26[label="",style="dashed", color="red", weight=0];
35[label="vy41 ++ vy5\n",fontsize=16,color="magenta"];35 -> 36[label="",style="dashed", color="magenta", weight=3];
36[label="vy41\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_gtGtEs(:(vy30, vy31), vy4, h, ba) → new_gtGtEs(vy31, vy4, h, 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_gtGtEs(:(vy30, vy31), vy4, h, ba) → new_gtGtEs(vy31, vy4, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_psPs(:(vy40, vy41), vy5, h) → new_psPs(vy41, vy5, h)

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

new_psPs(:(vy40, vy41), vy5, h) → new_psPs(vy41, vy5, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3