/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) AND
    (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (18) 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;
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;
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;
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;
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;
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;
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.intersectBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.intersectBy vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.intersectBy vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="List.intersectBy vz3 vz4 vz5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="concatMap (List.intersectBy0 vz3 vz5) vz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="concat . map (List.intersectBy0 vz3 vz5)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="concat (map (List.intersectBy0 vz3 vz5) vz4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr (++) [] (map (List.intersectBy0 vz3 vz5) vz4)",fontsize=16,color="burlywood",shape="triangle"];46[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];9 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 10[label="",style="solid", color="burlywood", weight=3];
47[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="foldr (++) [] (map (List.intersectBy0 vz3 vz5) (vz40 : vz41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr (++) [] (map (List.intersectBy0 vz3 vz5) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr (++) [] (List.intersectBy0 vz3 vz5 vz40 : map (List.intersectBy0 vz3 vz5) vz41)",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 vz3 vz5 vz40 foldr (++) [] (map (List.intersectBy0 vz3 vz5) vz41)",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 vz3 vz5) vz41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="(++) List.intersectBy0 vz3 vz5 vz40 vz6",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
18[label="vz41",fontsize=16,color="green",shape="box"];19[label="(++) List.intersectBy00 vz3 vz5 vz40 vz6",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20[label="(++) List.intersectBy000 vz40 (any (vz3 vz40) vz5) vz6",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="(++) List.intersectBy000 vz40 (or . map (vz3 vz40)) vz6",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="(++) List.intersectBy000 vz40 (or (map (vz3 vz40) vz5)) vz6",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="(++) List.intersectBy000 vz40 (foldr (||) False (map (vz3 vz40) vz5)) vz6",fontsize=16,color="burlywood",shape="triangle"];48[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];23 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 24[label="",style="solid", color="burlywood", weight=3];
49[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];23 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 25[label="",style="solid", color="burlywood", weight=3];
24[label="(++) List.intersectBy000 vz40 (foldr (||) False (map (vz3 vz40) (vz50 : vz51))) vz6",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="(++) List.intersectBy000 vz40 (foldr (||) False (map (vz3 vz40) [])) vz6",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="(++) List.intersectBy000 vz40 (foldr (||) False (vz3 vz40 vz50 : map (vz3 vz40) vz51)) vz6",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="(++) List.intersectBy000 vz40 (foldr (||) False []) vz6",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28 -> 30[label="",style="dashed", color="red", weight=0];
28[label="(++) List.intersectBy000 vz40 ((||) vz3 vz40 vz50 foldr (||) False (map (vz3 vz40) vz51)) vz6",fontsize=16,color="magenta"];28 -> 31[label="",style="dashed", color="magenta", weight=3];
29[label="(++) List.intersectBy000 vz40 False vz6",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3];
31[label="vz3 vz40 vz50",fontsize=16,color="green",shape="box"];31 -> 37[label="",style="dashed", color="green", weight=3];
31 -> 38[label="",style="dashed", color="green", weight=3];
30[label="(++) List.intersectBy000 vz40 ((||) vz7 foldr (||) False (map (vz3 vz40) vz51)) vz6",fontsize=16,color="burlywood",shape="triangle"];50[label="vz7/False",fontsize=10,color="white",style="solid",shape="box"];30 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 35[label="",style="solid", color="burlywood", weight=3];
51[label="vz7/True",fontsize=10,color="white",style="solid",shape="box"];30 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 36[label="",style="solid", color="burlywood", weight=3];
32[label="(++) [] vz6",fontsize=16,color="black",shape="triangle"];32 -> 39[label="",style="solid", color="black", weight=3];
37[label="vz40",fontsize=16,color="green",shape="box"];38[label="vz50",fontsize=16,color="green",shape="box"];35[label="(++) List.intersectBy000 vz40 ((||) False foldr (||) False (map (vz3 vz40) vz51)) vz6",fontsize=16,color="black",shape="box"];35 -> 40[label="",style="solid", color="black", weight=3];
36[label="(++) List.intersectBy000 vz40 ((||) True foldr (||) False (map (vz3 vz40) vz51)) vz6",fontsize=16,color="black",shape="box"];36 -> 41[label="",style="solid", color="black", weight=3];
39[label="vz6",fontsize=16,color="green",shape="box"];40 -> 23[label="",style="dashed", color="red", weight=0];
40[label="(++) List.intersectBy000 vz40 (foldr (||) False (map (vz3 vz40) vz51)) vz6",fontsize=16,color="magenta"];40 -> 42[label="",style="dashed", color="magenta", weight=3];
41[label="(++) List.intersectBy000 vz40 True vz6",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
42[label="vz51",fontsize=16,color="green",shape="box"];43[label="(++) (vz40 : []) vz6",fontsize=16,color="black",shape="box"];43 -> 44[label="",style="solid", color="black", weight=3];
44[label="vz40 : [] ++ vz6",fontsize=16,color="green",shape="box"];44 -> 45[label="",style="dashed", color="green", weight=3];
45 -> 32[label="",style="dashed", color="red", weight=0];
45[label="[] ++ vz6",fontsize=16,color="magenta"];}

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

(12)
Complex Obligation (AND)

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

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

   new_psPs0(vz40, vz3, :(vz50, vz51), vz6, ba) -> new_psPs(vz40, vz3, vz51, vz6, ba)
   new_psPs(vz40, vz3, vz51, vz6, ba) -> new_psPs0(vz40, vz3, vz51, vz6, ba)

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_psPs(vz40, vz3, vz51, vz6, ba) -> new_psPs0(vz40, vz3, vz51, vz6, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5


*new_psPs0(vz40, vz3, :(vz50, vz51), vz6, ba) -> new_psPs(vz40, vz3, vz51, vz6, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


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

(15)
YES

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

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

   new_foldr(vz3, vz5, :(vz40, vz41), ba) -> new_foldr(vz3, vz5, vz41, ba)

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

(17) 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(vz3, vz5, :(vz40, vz41), ba) -> new_foldr(vz3, vz5, vz41, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


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

(18)
YES