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

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_sequence__4.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Main

((sequence_ :: [IO a] -> IO ()) :: [IO a] -> IO ())

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

((sequence_ :: [IO a] -> IO ()) :: [IO a] -> IO ())

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_ :: [IO a] -> IO ()) :: [IO a] -> IO ())

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_ :: [IO a] -> IO ())

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_ vy3\n",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldr (>>) (return ()) vy3\n",fontsize=16,color="burlywood",shape="triangle"];24[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 5[label="",style="solid", color="burlywood", weight=3];
25[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldr (>>) (return ()) (vy30 : vy31)\n",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldr (>>) (return ()) []\n",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7 -> 9[label="",style="dashed", color="red", weight=0];
7[label="(>>) vy30 foldr (>>) (return ()) vy31\n",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="return ()\n",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
10 -> 4[label="",style="dashed", color="red", weight=0];
10[label="foldr (>>) (return ()) vy31\n",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
9[label="(>>) vy30 vy4\n",fontsize=16,color="black",shape="triangle"];9 -> 13[label="",style="solid", color="black", weight=3];
11[label="primretIO ()\n",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
12[label="vy31\n",fontsize=16,color="green",shape="box"];13[label="vy30 >>= gtGt0 vy4\n",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="terminator ()\n",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="primbindIO vy30 (gtGt0 vy4)\n",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="ter0m ()\n",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3];
17[label="terminator vy30 (gtGt0 vy4)\n",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="()\n",fontsize=16,color="green",shape="box"];19[label="ter1m vy30 (gtGt0 vy4)\n",fontsize=16,color="green",shape="box"];19 -> 20[label="",style="dashed", color="green", weight=3];
19 -> 21[label="",style="dashed", color="green", weight=3];
20[label="vy30\n",fontsize=16,color="green",shape="box"];21[label="gtGt0 vy4\n",fontsize=16,color="grey",shape="box"];21 -> 22[label="",style="dashed", color="grey", weight=3];
22[label="gtGt0 vy4 vy5\n",fontsize=16,color="black",shape="triangle"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="vy4\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

new_foldr(:(vy30, vy31), h) → new_foldr(vy31, 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_foldr(:(vy30, vy31), h) → new_foldr(vy31, h)
The graph contains the following edges 1 > 1, 2 >= 2