/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) IFR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 24 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) AND
    (9) QDP
        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
        (11) AND
            (12) QDP
                (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (14) YES
            (15) QDP
                (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (17) YES
    (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy _ _ [] = [];
deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;

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

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

(1) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if eq x y then ys else y : deleteBy eq x ys"
is transformed to
"deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;
"

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy _ _ [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

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

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

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

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy vy vz [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

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

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

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

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

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy vy vz [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

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

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="(List.\\)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="wu3 (List.\\)",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="wu3 (List.\\) wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="foldl (flip List.delete) wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];38[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 6[label="",style="solid", color="burlywood", weight=3];
39[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="foldl (flip List.delete) wu3 (wu40 : wu41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl (flip List.delete) wu3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8 -> 5[label="",style="dashed", color="red", weight=0];
8[label="foldl (flip List.delete) (flip List.delete wu3 wu40) wu41",fontsize=16,color="magenta"];8 -> 10[label="",style="dashed", color="magenta", weight=3];
8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="wu3",fontsize=16,color="green",shape="box"];10[label="flip List.delete wu3 wu40",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="wu41",fontsize=16,color="green",shape="box"];12[label="List.delete wu40 wu3",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="List.deleteBy (==) wu40 wu3",fontsize=16,color="burlywood",shape="triangle"];40[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];13 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 14[label="",style="solid", color="burlywood", weight=3];
41[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 15[label="",style="solid", color="burlywood", weight=3];
14[label="List.deleteBy (==) wu40 (wu30 : wu31)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="List.deleteBy (==) wu40 []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="List.deleteBy0 wu31 wu30 (==) wu40 ((==) wu40 wu30)",fontsize=16,color="burlywood",shape="box"];42[label="wu40/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 18[label="",style="solid", color="burlywood", weight=3];
43[label="wu40/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 19[label="",style="solid", color="burlywood", weight=3];
17[label="[]",fontsize=16,color="green",shape="box"];18[label="List.deleteBy0 wu31 wu30 (==) False ((==) False wu30)",fontsize=16,color="burlywood",shape="box"];44[label="wu30/False",fontsize=10,color="white",style="solid",shape="box"];18 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 20[label="",style="solid", color="burlywood", weight=3];
45[label="wu30/True",fontsize=10,color="white",style="solid",shape="box"];18 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 21[label="",style="solid", color="burlywood", weight=3];
19[label="List.deleteBy0 wu31 wu30 (==) True ((==) True wu30)",fontsize=16,color="burlywood",shape="box"];46[label="wu30/False",fontsize=10,color="white",style="solid",shape="box"];19 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 22[label="",style="solid", color="burlywood", weight=3];
47[label="wu30/True",fontsize=10,color="white",style="solid",shape="box"];19 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 23[label="",style="solid", color="burlywood", weight=3];
20[label="List.deleteBy0 wu31 False (==) False ((==) False False)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="List.deleteBy0 wu31 True (==) False ((==) False True)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="List.deleteBy0 wu31 False (==) True ((==) True False)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="List.deleteBy0 wu31 True (==) True ((==) True True)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="List.deleteBy0 wu31 False (==) False True",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="List.deleteBy0 wu31 True (==) False False",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="List.deleteBy0 wu31 False (==) True False",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="List.deleteBy0 wu31 True (==) True True",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="wu31",fontsize=16,color="green",shape="box"];29[label="True : List.deleteBy (==) False wu31",fontsize=16,color="green",shape="box"];29 -> 32[label="",style="dashed", color="green", weight=3];
30[label="False : List.deleteBy (==) True wu31",fontsize=16,color="green",shape="box"];30 -> 33[label="",style="dashed", color="green", weight=3];
31[label="wu31",fontsize=16,color="green",shape="box"];32 -> 13[label="",style="dashed", color="red", weight=0];
32[label="List.deleteBy (==) False wu31",fontsize=16,color="magenta"];32 -> 34[label="",style="dashed", color="magenta", weight=3];
32 -> 35[label="",style="dashed", color="magenta", weight=3];
33 -> 13[label="",style="dashed", color="red", weight=0];
33[label="List.deleteBy (==) True wu31",fontsize=16,color="magenta"];33 -> 36[label="",style="dashed", color="magenta", weight=3];
33 -> 37[label="",style="dashed", color="magenta", weight=3];
34[label="wu31",fontsize=16,color="green",shape="box"];35[label="False",fontsize=16,color="green",shape="box"];36[label="wu31",fontsize=16,color="green",shape="box"];37[label="True",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_deleteBy(True, :(False, wu31)) -> new_deleteBy(True, wu31)
   new_deleteBy(False, :(True, wu31)) -> new_deleteBy(False, wu31)

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

(10) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------

(11)
Complex Obligation (AND)

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

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

   new_deleteBy(False, :(True, wu31)) -> new_deleteBy(False, wu31)

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_deleteBy(False, :(True, wu31)) -> new_deleteBy(False, wu31)
The graph contains the following edges 1 >= 1, 2 > 2


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

(14)
YES

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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_deleteBy(True, :(False, wu31)) -> new_deleteBy(True, wu31)

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

(16) 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_deleteBy(True, :(False, wu31)) -> new_deleteBy(True, wu31)
The graph contains the following edges 1 >= 1, 2 > 2


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

(17)
YES

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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy0(wu40, wu3), wu41)

The TRS R consists of the following rules:

   new_deleteBy0(wu40, []) -> []
   new_deleteBy0(True, :(False, wu31)) -> :(False, new_deleteBy0(True, wu31))
   new_deleteBy0(True, :(True, wu31)) -> wu31
   new_deleteBy0(False, :(True, wu31)) -> :(True, new_deleteBy0(False, wu31))
   new_deleteBy0(False, :(False, wu31)) -> wu31

The set Q consists of the following terms:

   new_deleteBy0(False, :(True, x0))
   new_deleteBy0(x0, [])
   new_deleteBy0(True, :(False, x0))
   new_deleteBy0(True, :(True, x0))
   new_deleteBy0(False, :(False, x0))

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

(19) 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_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy0(wu40, wu3), wu41)
The graph contains the following edges 2 > 2


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

(20)
YES