/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 19 ms]
(10) HASKELL
(11) Narrow [SOUND, 0 ms]
(12) AND
    (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) DependencyGraphProof [EQUIVALENT, 0 ms]
        (18) AND
            (19) QDP
                (20) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (21) YES
            (22) QDP
                (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (24) 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;
intersect :: Eq a => [a]  ->  [a]  ->  [a];
intersect = intersectBy (==);

intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
intersectBy eq xs ys = concatMap (\vv2 ->case vv2 of { 
x->  if any (eq x) ys then x : [] else []; 
_-> []; 
 } ) xs;

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

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\vv2->case vv2 of {
x  -> if any (eq x) ys then x : [] else [];
_  -> []}
"
is transformed to
"intersectBy0 eq ys vv2 = case vv2 of {
x  -> if any (eq x) ys then x : [] else [];
_  -> []}
;
"

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

(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;
intersect :: Eq a => [a]  ->  [a]  ->  [a];
intersect = intersectBy (==);

intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;

intersectBy0 eq ys vv2 = case vv2 of { 
x->  if any (eq x) ys then x : [] else []; 
_-> []; 
 } ;

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

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

(3) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case vv2 of {
x  -> if any (eq x) ys then x : [] else [];
_  -> []}
"
is transformed to
"intersectBy00 eq ys x = if any (eq x) ys then x : [] else [];
intersectBy00 eq ys _ = [];
"

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

(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;
intersect :: Eq a => [a]  ->  [a]  ->  [a];
intersect = intersectBy (==);

intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;

intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;

intersectBy00 eq ys x =  if any (eq x) ys then x : [] else [];
intersectBy00 eq ys _ = [];

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

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

(5) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if any (eq x) ys then x : [] else []"
is transformed to
"intersectBy000 x True = x : [];
intersectBy000 x 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;
intersect :: Eq a => [a]  ->  [a]  ->  [a];
intersect = intersectBy (==);

intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;

intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;

intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys);
intersectBy00 eq ys _ = [];

intersectBy000 x True = x : [];
intersectBy000 x False = [];

}
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;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
intersect :: Eq a => [a]  ->  [a]  ->  [a];
intersect = intersectBy (==);

intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;

intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;

intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys);
intersectBy00 eq ys vy = [];

intersectBy000 x True = x : [];
intersectBy000 x False = [];

}
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;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
intersect :: Eq a => [a]  ->  [a]  ->  [a];
intersect = intersectBy (==);

intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;

intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;

intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys);
intersectBy00 eq ys vy = [];

intersectBy000 x True = x : [];
intersectBy000 x False = [];

}
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.intersect",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.intersect vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.intersect vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="List.intersectBy (==) vz3 vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="concatMap (List.intersectBy0 (==) vz4) vz3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="concat . map (List.intersectBy0 (==) vz4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="concat (map (List.intersectBy0 (==) vz4) vz3)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) vz3)",fontsize=16,color="burlywood",shape="triangle"];56[label="vz3/vz30 : vz31",fontsize=10,color="white",style="solid",shape="box"];9 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 10[label="",style="solid", color="burlywood", weight=3];
57[label="vz3/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) (vz30 : vz31))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr (++) [] (List.intersectBy0 (==) vz4 vz30 : map (List.intersectBy0 (==) vz4) vz31)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14 -> 16[label="",style="dashed", color="red", weight=0];
14[label="(++) List.intersectBy0 (==) vz4 vz30 foldr (++) [] (map (List.intersectBy0 (==) vz4) vz31)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
15[label="[]",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
17[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) vz31)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="(++) List.intersectBy0 (==) vz4 vz30 vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
18[label="vz31",fontsize=16,color="green",shape="box"];19[label="(++) List.intersectBy00 (==) vz4 vz30 vz5",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20[label="(++) List.intersectBy000 vz30 (any ((==) vz30) vz4) vz5",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="(++) List.intersectBy000 vz30 (or . map ((==) vz30)) vz5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="(++) List.intersectBy000 vz30 (or (map ((==) vz30) vz4)) vz5",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="(++) List.intersectBy000 vz30 (foldr (||) False (map ((==) vz30) vz4)) vz5",fontsize=16,color="burlywood",shape="triangle"];58[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];23 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 24[label="",style="solid", color="burlywood", weight=3];
59[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];23 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 25[label="",style="solid", color="burlywood", weight=3];
24[label="(++) List.intersectBy000 vz30 (foldr (||) False (map ((==) vz30) (vz40 : vz41))) vz5",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="(++) List.intersectBy000 vz30 (foldr (||) False (map ((==) vz30) [])) vz5",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="(++) List.intersectBy000 vz30 (foldr (||) False (((==) vz30 vz40) : map ((==) vz30) vz41)) vz5",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="(++) List.intersectBy000 vz30 (foldr (||) False []) vz5",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="(++) List.intersectBy000 vz30 ((||) (==) vz30 vz40 foldr (||) False (map ((==) vz30) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];60[label="vz30/False",fontsize=10,color="white",style="solid",shape="box"];28 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 30[label="",style="solid", color="burlywood", weight=3];
61[label="vz30/True",fontsize=10,color="white",style="solid",shape="box"];28 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 31[label="",style="solid", color="burlywood", weight=3];
29[label="(++) List.intersectBy000 vz30 False vz5",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3];
30[label="(++) List.intersectBy000 False ((||) (==) False vz40 foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];62[label="vz40/False",fontsize=10,color="white",style="solid",shape="box"];30 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 33[label="",style="solid", color="burlywood", weight=3];
63[label="vz40/True",fontsize=10,color="white",style="solid",shape="box"];30 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 34[label="",style="solid", color="burlywood", weight=3];
31[label="(++) List.intersectBy000 True ((||) (==) True vz40 foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];64[label="vz40/False",fontsize=10,color="white",style="solid",shape="box"];31 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 35[label="",style="solid", color="burlywood", weight=3];
65[label="vz40/True",fontsize=10,color="white",style="solid",shape="box"];31 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 36[label="",style="solid", color="burlywood", weight=3];
32[label="(++) [] vz5",fontsize=16,color="black",shape="triangle"];32 -> 37[label="",style="solid", color="black", weight=3];
33[label="(++) List.intersectBy000 False ((||) (==) False False foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="black",shape="box"];33 -> 38[label="",style="solid", color="black", weight=3];
34[label="(++) List.intersectBy000 False ((||) (==) False True foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="black",shape="box"];34 -> 39[label="",style="solid", color="black", weight=3];
35[label="(++) List.intersectBy000 True ((||) (==) True False foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="black",shape="box"];35 -> 40[label="",style="solid", color="black", weight=3];
36[label="(++) List.intersectBy000 True ((||) (==) True True foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="black",shape="box"];36 -> 41[label="",style="solid", color="black", weight=3];
37[label="vz5",fontsize=16,color="green",shape="box"];38[label="(++) List.intersectBy000 False ((||) True foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="(++) List.intersectBy000 False ((||) False foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
40[label="(++) List.intersectBy000 True ((||) False foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
41[label="(++) List.intersectBy000 True ((||) True foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
42[label="(++) List.intersectBy000 False True vz5",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3];
43 -> 23[label="",style="dashed", color="red", weight=0];
43[label="(++) List.intersectBy000 False (foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="magenta"];43 -> 47[label="",style="dashed", color="magenta", weight=3];
43 -> 48[label="",style="dashed", color="magenta", weight=3];
44 -> 23[label="",style="dashed", color="red", weight=0];
44[label="(++) List.intersectBy000 True (foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="magenta"];44 -> 49[label="",style="dashed", color="magenta", weight=3];
44 -> 50[label="",style="dashed", color="magenta", weight=3];
45[label="(++) List.intersectBy000 True True vz5",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3];
46[label="(++) (False : []) vz5",fontsize=16,color="black",shape="box"];46 -> 52[label="",style="solid", color="black", weight=3];
47[label="False",fontsize=16,color="green",shape="box"];48[label="vz41",fontsize=16,color="green",shape="box"];49[label="True",fontsize=16,color="green",shape="box"];50[label="vz41",fontsize=16,color="green",shape="box"];51[label="(++) (True : []) vz5",fontsize=16,color="black",shape="box"];51 -> 53[label="",style="solid", color="black", weight=3];
52[label="False : [] ++ vz5",fontsize=16,color="green",shape="box"];52 -> 54[label="",style="dashed", color="green", weight=3];
53[label="True : [] ++ vz5",fontsize=16,color="green",shape="box"];53 -> 55[label="",style="dashed", color="green", weight=3];
54 -> 32[label="",style="dashed", color="red", weight=0];
54[label="[] ++ vz5",fontsize=16,color="magenta"];55 -> 32[label="",style="dashed", color="red", weight=0];
55[label="[] ++ vz5",fontsize=16,color="magenta"];}

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

(12)
Complex Obligation (AND)

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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldr(vz4, :(vz30, vz31)) -> new_foldr(vz4, vz31)

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

(14) 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_foldr(vz4, :(vz30, vz31)) -> new_foldr(vz4, vz31)
The graph contains the following edges 1 >= 1, 2 > 2


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

(15)
YES

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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(False, :(True, vz41), vz5) -> new_psPs(False, vz41, vz5)
   new_psPs(True, :(False, vz41), vz5) -> new_psPs(True, vz41, vz5)

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

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

(18)
Complex Obligation (AND)

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

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(True, :(False, vz41), vz5) -> new_psPs(True, vz41, vz5)

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

(20) 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_psPs(True, :(False, vz41), vz5) -> new_psPs(True, vz41, vz5)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(21)
YES

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

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(False, :(True, vz41), vz5) -> new_psPs(False, vz41, vz5)

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

(23) 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_psPs(False, :(True, vz41), vz5) -> new_psPs(False, vz41, vz5)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(24)
YES