/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files


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YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) QDP
(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(8) YES


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(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(1) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"takeWhile p [] = [];
takeWhile p (x : xs)|p xx : takeWhile p xs|otherwise[];
"
is transformed to
"takeWhile p [] = takeWhile3 p [];
takeWhile p (x : xs) = takeWhile2 p (x : xs);
"
"takeWhile0 p x xs True = [];
"
"takeWhile1 p x xs True = x : takeWhile p xs;
takeWhile1 p x xs False = takeWhile0 p x xs otherwise;
"
"takeWhile2 p (x : xs) = takeWhile1 p x xs (p x);
"
"takeWhile3 p [] = [];
takeWhile3 vz wu = takeWhile2 vz wu;
"
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

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(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="takeWhile",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="takeWhile wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="takeWhile wv3 wv4",fontsize=16,color="burlywood",shape="triangle"];23[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];4 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 5[label="",style="solid", color="burlywood", weight=3];
24[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="takeWhile wv3 (wv40 : wv41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="takeWhile wv3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="takeWhile2 wv3 (wv40 : wv41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="takeWhile3 wv3 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9 -> 11[label="",style="dashed", color="red", weight=0];
9[label="takeWhile1 wv3 wv40 wv41 (wv3 wv40)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
10[label="[]",fontsize=16,color="green",shape="box"];12[label="wv3 wv40",fontsize=16,color="green",shape="box"];12 -> 16[label="",style="dashed", color="green", weight=3];
11[label="takeWhile1 wv3 wv40 wv41 wv5",fontsize=16,color="burlywood",shape="triangle"];25[label="wv5/False",fontsize=10,color="white",style="solid",shape="box"];11 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 14[label="",style="solid", color="burlywood", weight=3];
26[label="wv5/True",fontsize=10,color="white",style="solid",shape="box"];11 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 15[label="",style="solid", color="burlywood", weight=3];
16[label="wv40",fontsize=16,color="green",shape="box"];14[label="takeWhile1 wv3 wv40 wv41 False",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
15[label="takeWhile1 wv3 wv40 wv41 True",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
17[label="takeWhile0 wv3 wv40 wv41 otherwise",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="wv40 : takeWhile wv3 wv41",fontsize=16,color="green",shape="box"];18 -> 20[label="",style="dashed", color="green", weight=3];
19[label="takeWhile0 wv3 wv40 wv41 True",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20 -> 4[label="",style="dashed", color="red", weight=0];
20[label="takeWhile wv3 wv41",fontsize=16,color="magenta"];20 -> 22[label="",style="dashed", color="magenta", weight=3];
21[label="[]",fontsize=16,color="green",shape="box"];22[label="wv41",fontsize=16,color="green",shape="box"];}

----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_takeWhile1(wv3, wv40, wv41, h) -> new_takeWhile(wv3, wv41, h)
   new_takeWhile(wv3, :(wv40, wv41), h) -> new_takeWhile1(wv3, wv40, wv41, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(7) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_takeWhile(wv3, :(wv40, wv41), h) -> new_takeWhile1(wv3, wv40, wv41, h)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4


*new_takeWhile1(wv3, wv40, wv41, h) -> new_takeWhile(wv3, wv41, h)
The graph contains the following edges 1 >= 1, 3 >= 2, 4 >= 3


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(8)
YES