/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) HASKELL
(1) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) CR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) IFR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) BR [EQUIVALENT, 0 ms]
(8) HASKELL
(9) COR [EQUIVALENT, 0 ms]
(10) HASKELL
(11) Narrow [SOUND, 0 ms]
(12) QDP
(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(14) YES


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : _) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
find p = Maybe.listToMaybe . filter p;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\vu39->case vu39 of {
x  -> if p x then x : [] else [];
_  -> []}
"
is transformed to
"filter0 p vu39 = case vu39 of {
x  -> if p x then x : [] else [];
_  -> []}
;
"

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : _) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
find p = Maybe.listToMaybe . filter p;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

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

(3) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case vu39 of {
x  -> if p x then x : [] else [];
_  -> []}
"
is transformed to
"filter00 p x = if p x then x : [] else [];
filter00 p _ = [];
"

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

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : _) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
find p = Maybe.listToMaybe . filter p;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

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

(5) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if p x then x : [] else []"
is transformed to
"filter000 x True = x : [];
filter000 x False = [];
"

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

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : _) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
find p = Maybe.listToMaybe . filter p;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

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

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

(8)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : vz) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
find p = Maybe.listToMaybe . filter p;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

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

(9) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

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

(10)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : vz) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
find p = Maybe.listToMaybe . filter p;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

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

(11) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.find",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.find wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.find wu3 wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="Maybe.listToMaybe . filter wu3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="Maybe.listToMaybe (filter wu3 wu4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="Maybe.listToMaybe (concatMap (filter0 wu3) wu4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="Maybe.listToMaybe (concat . map (filter0 wu3))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="Maybe.listToMaybe (concat (map (filter0 wu3) wu4))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) wu4))",fontsize=16,color="burlywood",shape="triangle"];32[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];10 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 11[label="",style="solid", color="burlywood", weight=3];
33[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 12[label="",style="solid", color="burlywood", weight=3];
11[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) (wu40 : wu41)))",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) []))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="Maybe.listToMaybe (foldr (++) [] (filter0 wu3 wu40 : map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="Maybe.listToMaybe (foldr (++) [] [])",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="Maybe.listToMaybe ((++) filter0 wu3 wu40 foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="Maybe.listToMaybe []",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="Maybe.listToMaybe ((++) filter00 wu3 wu40 foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="Nothing",fontsize=16,color="green",shape="box"];19 -> 20[label="",style="dashed", color="red", weight=0];
19[label="Maybe.listToMaybe ((++) filter000 wu40 (wu3 wu40) foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
21[label="wu3 wu40",fontsize=16,color="green",shape="box"];21 -> 25[label="",style="dashed", color="green", weight=3];
20[label="Maybe.listToMaybe ((++) filter000 wu40 wu5 foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="burlywood",shape="triangle"];34[label="wu5/False",fontsize=10,color="white",style="solid",shape="box"];20 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 23[label="",style="solid", color="burlywood", weight=3];
35[label="wu5/True",fontsize=10,color="white",style="solid",shape="box"];20 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 24[label="",style="solid", color="burlywood", weight=3];
25[label="wu40",fontsize=16,color="green",shape="box"];23[label="Maybe.listToMaybe ((++) filter000 wu40 False foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="Maybe.listToMaybe ((++) filter000 wu40 True foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
26[label="Maybe.listToMaybe ((++) [] foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="Maybe.listToMaybe ((++) (wu40 : []) foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28 -> 10[label="",style="dashed", color="red", weight=0];
28[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="magenta"];28 -> 30[label="",style="dashed", color="magenta", weight=3];
29[label="Maybe.listToMaybe (wu40 : [] ++ foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="wu41",fontsize=16,color="green",shape="box"];31[label="Just wu40",fontsize=16,color="green",shape="box"];}

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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_listToMaybe0(wu3, :(wu40, wu41), ba) -> new_listToMaybe(wu40, wu3, wu41, ba)
   new_listToMaybe(wu40, wu3, wu41, ba) -> new_listToMaybe0(wu3, wu41, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) 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_listToMaybe(wu40, wu3, wu41, ba) -> new_listToMaybe0(wu3, wu41, ba)
The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3


*new_listToMaybe0(wu3, :(wu40, wu41), ba) -> new_listToMaybe(wu40, wu3, wu41, ba)
The graph contains the following edges 2 > 1, 1 >= 2, 2 > 3, 3 >= 4


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

(14)
YES