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

YES 0.8310000000000001 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/List.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule List

((isPrefixOf :: [Char] -> [Char] -> Bool) :: [Char] -> [Char] -> Bool)

module List where

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 Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule List

((isPrefixOf :: [Char] -> [Char] -> Bool) :: [Char] -> [Char] -> Bool)

module List where

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 Maybe where

import qualified List
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

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule List

(isPrefixOf :: [Char] -> [Char] -> Bool)

module List where

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 Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="isPrefixOf",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="isPrefixOf wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="isPrefixOf wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];34[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];4 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 5[label="",style="solid", color="burlywood", weight=3];
35[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="isPrefixOf (wu30 : wu31) wu4",fontsize=16,color="burlywood",shape="box"];36[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 7[label="",style="solid", color="burlywood", weight=3];
37[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 8[label="",style="solid", color="burlywood", weight=3];
6[label="isPrefixOf [] wu4",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="isPrefixOf (wu30 : wu31) (wu40 : wu41)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="isPrefixOf (wu30 : wu31) []",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="True",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="red", weight=0];
10[label="wu30 == wu40 && isPrefixOf wu31 wu41",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
11[label="False",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0];
13[label="isPrefixOf wu31 wu41",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",fontsize=16,color="black",shape="triangle"];12 -> 16[label="",style="solid", color="black", weight=3];
14[label="wu31",fontsize=16,color="green",shape="box"];15[label="wu41",fontsize=16,color="green",shape="box"];16[label="primEqChar wu30 wu40 && wu5",fontsize=16,color="burlywood",shape="box"];40[label="wu30/Char wu300",fontsize=10,color="white",style="solid",shape="box"];16 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 17[label="",style="solid", color="burlywood", weight=3];
17[label="primEqChar (Char wu300) wu40 && wu5",fontsize=16,color="burlywood",shape="box"];41[label="wu40/Char wu400",fontsize=10,color="white",style="solid",shape="box"];17 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 18[label="",style="solid", color="burlywood", weight=3];
18[label="primEqChar (Char wu300) (Char wu400) && wu5",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="primEqNat wu300 wu400 && wu5",fontsize=16,color="burlywood",shape="triangle"];42[label="wu300/Succ wu3000",fontsize=10,color="white",style="solid",shape="box"];19 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 20[label="",style="solid", color="burlywood", weight=3];
43[label="wu300/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 21[label="",style="solid", color="burlywood", weight=3];
20[label="primEqNat (Succ wu3000) wu400 && wu5",fontsize=16,color="burlywood",shape="box"];44[label="wu400/Succ wu4000",fontsize=10,color="white",style="solid",shape="box"];20 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 22[label="",style="solid", color="burlywood", weight=3];
45[label="wu400/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 23[label="",style="solid", color="burlywood", weight=3];
21[label="primEqNat Zero wu400 && wu5",fontsize=16,color="burlywood",shape="box"];46[label="wu400/Succ wu4000",fontsize=10,color="white",style="solid",shape="box"];21 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 24[label="",style="solid", color="burlywood", weight=3];
47[label="wu400/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 25[label="",style="solid", color="burlywood", weight=3];
22[label="primEqNat (Succ wu3000) (Succ wu4000) && wu5",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="primEqNat (Succ wu3000) Zero && wu5",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="primEqNat Zero (Succ wu4000) && wu5",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="primEqNat Zero Zero && wu5",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26 -> 19[label="",style="dashed", color="red", weight=0];
26[label="primEqNat wu3000 wu4000 && wu5",fontsize=16,color="magenta"];26 -> 30[label="",style="dashed", color="magenta", weight=3];
26 -> 31[label="",style="dashed", color="magenta", weight=3];
27[label="False && wu5",fontsize=16,color="black",shape="triangle"];27 -> 32[label="",style="solid", color="black", weight=3];
28 -> 27[label="",style="dashed", color="red", weight=0];
28[label="False && wu5",fontsize=16,color="magenta"];29[label="True && wu5",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="wu4000",fontsize=16,color="green",shape="box"];31[label="wu3000",fontsize=16,color="green",shape="box"];32[label="False",fontsize=16,color="green",shape="box"];33[label="wu5",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

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

new_asAs(Succ(wu3000), Succ(wu4000), wu5) → new_asAs(wu3000, wu4000, wu5)

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_asAs(Succ(wu3000), Succ(wu4000), wu5) → new_asAs(wu3000, wu4000, wu5)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

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

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

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