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

YES H-Termination proof of ../tpdb/FP/full_haskell/List_isPrefixOf_2.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

((List.isPrefixOf :: [()] -> [()] -> Bool) :: [()] -> [()] -> Bool)

module List where

import qualified Main
import qualified Maybe
import qualified Prelude

isPrefixOf :: Eq a => [a] -> [a] -> Bool

isPrefixOf	[] _	=	True
isPrefixOf	_ []	=	False
isPrefixOf	(x : xs) (y : ys)	=	x == y && isPrefixOf xs ys

module Main where

import qualified List
import qualified Maybe
import qualified Prelude

module Maybe where

import qualified List
import qualified Main
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

mainModule Main

((List.isPrefixOf :: [()] -> [()] -> Bool) :: [()] -> [()] -> Bool)

module List where

import qualified Main
import qualified Maybe
import qualified Prelude

isPrefixOf :: Eq a => [a] -> [a] -> Bool

isPrefixOf	[] vw	=	True
isPrefixOf	vx []	=	False
isPrefixOf	(x : xs) (y : ys)	=	x == y && isPrefixOf xs ys

module Main where

import qualified List
import qualified Maybe
import qualified Prelude

module Maybe where

import qualified List
import qualified Main
import qualified Prelude

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

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

        ↳ HASKELL

          ↳ Narrow

mainModule Main

(List.isPrefixOf :: [()] -> [()] -> Bool)

module List where

import qualified Main
import qualified Maybe
import qualified Prelude

isPrefixOf :: Eq a => [a] -> [a] -> Bool

isPrefixOf	[] vw	=	True
isPrefixOf	vx []	=	False
isPrefixOf	(x : xs) (y : ys)	=	x == y && isPrefixOf xs ys

module Main where

import qualified List
import qualified Maybe
import qualified Prelude

module Maybe where

import qualified List
import qualified Main
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.isPrefixOf\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.isPrefixOf wu3\n",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.isPrefixOf wu3 wu4\n",fontsize=16,color="burlywood",shape="triangle"];20[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 5[label="",style="solid", color="burlywood", weight=3];
21[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="List.isPrefixOf (wu30 : wu31) wu4\n",fontsize=16,color="burlywood",shape="box"];22[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 22[label="",style="solid", color="burlywood", weight=9];
22 -> 7[label="",style="solid", color="burlywood", weight=3];
23[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 8[label="",style="solid", color="burlywood", weight=3];
6[label="List.isPrefixOf [] wu4\n",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="List.isPrefixOf (wu30 : wu31) (wu40 : wu41)\n",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="List.isPrefixOf (wu30 : wu31) []\n",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="True\n",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="red", weight=0];
10[label="wu30 == wu40 && List.isPrefixOf wu31 wu41\n",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
11[label="False\n",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0];
13[label="List.isPrefixOf wu31 wu41\n",fontsize=16,color="magenta"];13 -> 14[label="",style="dashed", color="magenta", weight=3];
13 -> 15[label="",style="dashed", color="magenta", weight=3];
12[label="wu30 == wu40 && wu5\n",fontsize=16,color="burlywood",shape="triangle"];26[label="wu30/()",fontsize=10,color="white",style="solid",shape="box"];12 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="wu41\n",fontsize=16,color="green",shape="box"];15[label="wu31\n",fontsize=16,color="green",shape="box"];16[label="() == wu40 && wu5\n",fontsize=16,color="burlywood",shape="box"];27[label="wu40/()",fontsize=10,color="white",style="solid",shape="box"];16 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 17[label="",style="solid", color="burlywood", weight=3];
17[label="() == () && wu5\n",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="True && wu5\n",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="wu5\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ QDPSizeChangeProof

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

new_isPrefixOf(:(wu30, wu31), :(wu40, wu41)) → new_isPrefixOf(wu31, wu41)

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_isPrefixOf(:(wu30, wu31), :(wu40, wu41)) → new_isPrefixOf(wu31, wu41)
The graph contains the following edges 1 > 1, 2 > 2