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


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


YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 20 ms]
(10) HASKELL
(11) Narrow [SOUND, 0 ms]
(12) AND
    (13) QDP
        (14) DependencyGraphProof [EQUIVALENT, 0 ms]
        (15) AND
            (16) QDP
                (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (18) YES
            (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"];271[label="vz3/vz30 : vz31",fontsize=10,color="white",style="solid",shape="box"];9 -> 271[label="",style="solid", color="burlywood", weight=9];
271 -> 10[label="",style="solid", color="burlywood", weight=3];
272[label="vz3/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 272[label="",style="solid", color="burlywood", weight=9];
272 -> 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="box"];273[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];23 -> 273[label="",style="solid", color="burlywood", weight=9];
273 -> 24[label="",style="solid", color="burlywood", weight=3];
274[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];23 -> 274[label="",style="solid", color="burlywood", weight=9];
274 -> 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="triangle"];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="black",shape="box"];28 -> 30[label="",style="solid", color="black", weight=3];
29[label="(++) List.intersectBy000 vz30 False vz5",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="(++) List.intersectBy000 vz30 ((||) primEqChar vz30 vz40 foldr (||) False (map (primEqChar vz30) vz41)) vz5",fontsize=16,color="burlywood",shape="triangle"];275[label="vz30/Char vz300",fontsize=10,color="white",style="solid",shape="box"];30 -> 275[label="",style="solid", color="burlywood", weight=9];
275 -> 32[label="",style="solid", color="burlywood", weight=3];
31[label="(++) [] vz5",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3];
32[label="(++) List.intersectBy000 (Char vz300) ((||) primEqChar (Char vz300) vz40 foldr (||) False (map (primEqChar (Char vz300)) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];276[label="vz40/Char vz400",fontsize=10,color="white",style="solid",shape="box"];32 -> 276[label="",style="solid", color="burlywood", weight=9];
276 -> 34[label="",style="solid", color="burlywood", weight=3];
33[label="vz5",fontsize=16,color="green",shape="box"];34[label="(++) List.intersectBy000 (Char vz300) ((||) primEqChar (Char vz300) (Char vz400) foldr (||) False (map (primEqChar (Char vz300)) vz41)) vz5",fontsize=16,color="black",shape="box"];34 -> 35[label="",style="solid", color="black", weight=3];
35[label="(++) List.intersectBy000 (Char vz300) ((||) primEqNat vz300 vz400 foldr (||) False (map (primEqChar (Char vz300)) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];277[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 277[label="",style="solid", color="burlywood", weight=9];
277 -> 36[label="",style="solid", color="burlywood", weight=3];
278[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 278[label="",style="solid", color="burlywood", weight=9];
278 -> 37[label="",style="solid", color="burlywood", weight=3];
36[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) primEqNat (Succ vz3000) vz400 foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];279[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];36 -> 279[label="",style="solid", color="burlywood", weight=9];
279 -> 38[label="",style="solid", color="burlywood", weight=3];
280[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 280[label="",style="solid", color="burlywood", weight=9];
280 -> 39[label="",style="solid", color="burlywood", weight=3];
37[label="(++) List.intersectBy000 (Char Zero) ((||) primEqNat Zero vz400 foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];281[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];37 -> 281[label="",style="solid", color="burlywood", weight=9];
281 -> 40[label="",style="solid", color="burlywood", weight=3];
282[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];37 -> 282[label="",style="solid", color="burlywood", weight=9];
282 -> 41[label="",style="solid", color="burlywood", weight=3];
38[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) primEqNat (Succ vz3000) (Succ vz4000) foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) primEqNat (Succ vz3000) Zero foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
40[label="(++) List.intersectBy000 (Char Zero) ((||) primEqNat Zero (Succ vz4000) foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
41[label="(++) List.intersectBy000 (Char Zero) ((||) primEqNat Zero Zero foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
42 -> 202[label="",style="dashed", color="red", weight=0];
42[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) primEqNat vz3000 vz4000 foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="magenta"];42 -> 203[label="",style="dashed", color="magenta", weight=3];
42 -> 204[label="",style="dashed", color="magenta", weight=3];
42 -> 205[label="",style="dashed", color="magenta", weight=3];
42 -> 206[label="",style="dashed", color="magenta", weight=3];
42 -> 207[label="",style="dashed", color="magenta", weight=3];
43[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) False foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="black",shape="triangle"];43 -> 48[label="",style="solid", color="black", weight=3];
44[label="(++) List.intersectBy000 (Char Zero) ((||) False foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="black",shape="box"];44 -> 49[label="",style="solid", color="black", weight=3];
45[label="(++) List.intersectBy000 (Char Zero) ((||) True foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="black",shape="box"];45 -> 50[label="",style="solid", color="black", weight=3];
203[label="vz3000",fontsize=16,color="green",shape="box"];204[label="vz4000",fontsize=16,color="green",shape="box"];205[label="vz3000",fontsize=16,color="green",shape="box"];206[label="vz41",fontsize=16,color="green",shape="box"];207[label="vz5",fontsize=16,color="green",shape="box"];202[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat vz11 vz12 foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="burlywood",shape="triangle"];283[label="vz11/Succ vz110",fontsize=10,color="white",style="solid",shape="box"];202 -> 283[label="",style="solid", color="burlywood", weight=9];
283 -> 248[label="",style="solid", color="burlywood", weight=3];
284[label="vz11/Zero",fontsize=10,color="white",style="solid",shape="box"];202 -> 284[label="",style="solid", color="burlywood", weight=9];
284 -> 249[label="",style="solid", color="burlywood", weight=3];
48[label="(++) List.intersectBy000 (Char (Succ vz3000)) (foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];285[label="vz41/vz410 : vz411",fontsize=10,color="white",style="solid",shape="box"];48 -> 285[label="",style="solid", color="burlywood", weight=9];
285 -> 55[label="",style="solid", color="burlywood", weight=3];
286[label="vz41/[]",fontsize=10,color="white",style="solid",shape="box"];48 -> 286[label="",style="solid", color="burlywood", weight=9];
286 -> 56[label="",style="solid", color="burlywood", weight=3];
49[label="(++) List.intersectBy000 (Char Zero) (foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];287[label="vz41/vz410 : vz411",fontsize=10,color="white",style="solid",shape="box"];49 -> 287[label="",style="solid", color="burlywood", weight=9];
287 -> 57[label="",style="solid", color="burlywood", weight=3];
288[label="vz41/[]",fontsize=10,color="white",style="solid",shape="box"];49 -> 288[label="",style="solid", color="burlywood", weight=9];
288 -> 58[label="",style="solid", color="burlywood", weight=3];
50[label="(++) List.intersectBy000 (Char Zero) True vz5",fontsize=16,color="black",shape="box"];50 -> 59[label="",style="solid", color="black", weight=3];
248[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat (Succ vz110) vz12 foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="burlywood",shape="box"];289[label="vz12/Succ vz120",fontsize=10,color="white",style="solid",shape="box"];248 -> 289[label="",style="solid", color="burlywood", weight=9];
289 -> 250[label="",style="solid", color="burlywood", weight=3];
290[label="vz12/Zero",fontsize=10,color="white",style="solid",shape="box"];248 -> 290[label="",style="solid", color="burlywood", weight=9];
290 -> 251[label="",style="solid", color="burlywood", weight=3];
249[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat Zero vz12 foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="burlywood",shape="box"];291[label="vz12/Succ vz120",fontsize=10,color="white",style="solid",shape="box"];249 -> 291[label="",style="solid", color="burlywood", weight=9];
291 -> 252[label="",style="solid", color="burlywood", weight=3];
292[label="vz12/Zero",fontsize=10,color="white",style="solid",shape="box"];249 -> 292[label="",style="solid", color="burlywood", weight=9];
292 -> 253[label="",style="solid", color="burlywood", weight=3];
55[label="(++) List.intersectBy000 (Char (Succ vz3000)) (foldr (||) False (map (primEqChar (Char (Succ vz3000))) (vz410 : vz411))) vz5",fontsize=16,color="black",shape="box"];55 -> 64[label="",style="solid", color="black", weight=3];
56[label="(++) List.intersectBy000 (Char (Succ vz3000)) (foldr (||) False (map (primEqChar (Char (Succ vz3000))) [])) vz5",fontsize=16,color="black",shape="box"];56 -> 65[label="",style="solid", color="black", weight=3];
57[label="(++) List.intersectBy000 (Char Zero) (foldr (||) False (map (primEqChar (Char Zero)) (vz410 : vz411))) vz5",fontsize=16,color="black",shape="box"];57 -> 66[label="",style="solid", color="black", weight=3];
58[label="(++) List.intersectBy000 (Char Zero) (foldr (||) False (map (primEqChar (Char Zero)) [])) vz5",fontsize=16,color="black",shape="box"];58 -> 67[label="",style="solid", color="black", weight=3];
59[label="(++) (Char Zero : []) vz5",fontsize=16,color="black",shape="box"];59 -> 68[label="",style="solid", color="black", weight=3];
250[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat (Succ vz110) (Succ vz120) foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="black",shape="box"];250 -> 254[label="",style="solid", color="black", weight=3];
251[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat (Succ vz110) Zero foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="black",shape="box"];251 -> 255[label="",style="solid", color="black", weight=3];
252[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat Zero (Succ vz120) foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="black",shape="box"];252 -> 256[label="",style="solid", color="black", weight=3];
253[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat Zero Zero foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="black",shape="box"];253 -> 257[label="",style="solid", color="black", weight=3];
64[label="(++) List.intersectBy000 (Char (Succ vz3000)) (foldr (||) False (primEqChar (Char (Succ vz3000)) vz410 : map (primEqChar (Char (Succ vz3000))) vz411)) vz5",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3];
65 -> 27[label="",style="dashed", color="red", weight=0];
65[label="(++) List.intersectBy000 (Char (Succ vz3000)) (foldr (||) False []) vz5",fontsize=16,color="magenta"];65 -> 75[label="",style="dashed", color="magenta", weight=3];
66[label="(++) List.intersectBy000 (Char Zero) (foldr (||) False (primEqChar (Char Zero) vz410 : map (primEqChar (Char Zero)) vz411)) vz5",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3];
67 -> 27[label="",style="dashed", color="red", weight=0];
67[label="(++) List.intersectBy000 (Char Zero) (foldr (||) False []) vz5",fontsize=16,color="magenta"];67 -> 77[label="",style="dashed", color="magenta", weight=3];
68[label="Char Zero : [] ++ vz5",fontsize=16,color="green",shape="box"];68 -> 78[label="",style="dashed", color="green", weight=3];
254 -> 202[label="",style="dashed", color="red", weight=0];
254[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat vz110 vz120 foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="magenta"];254 -> 258[label="",style="dashed", color="magenta", weight=3];
254 -> 259[label="",style="dashed", color="magenta", weight=3];
255 -> 43[label="",style="dashed", color="red", weight=0];
255[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) False foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="magenta"];255 -> 260[label="",style="dashed", color="magenta", weight=3];
255 -> 261[label="",style="dashed", color="magenta", weight=3];
255 -> 262[label="",style="dashed", color="magenta", weight=3];
256 -> 43[label="",style="dashed", color="red", weight=0];
256[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) False foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="magenta"];256 -> 263[label="",style="dashed", color="magenta", weight=3];
256 -> 264[label="",style="dashed", color="magenta", weight=3];
256 -> 265[label="",style="dashed", color="magenta", weight=3];
257[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) True foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="black",shape="box"];257 -> 266[label="",style="solid", color="black", weight=3];
74 -> 30[label="",style="dashed", color="red", weight=0];
74[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) primEqChar (Char (Succ vz3000)) vz410 foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz411)) vz5",fontsize=16,color="magenta"];74 -> 84[label="",style="dashed", color="magenta", weight=3];
74 -> 85[label="",style="dashed", color="magenta", weight=3];
74 -> 86[label="",style="dashed", color="magenta", weight=3];
75[label="Char (Succ vz3000)",fontsize=16,color="green",shape="box"];76 -> 30[label="",style="dashed", color="red", weight=0];
76[label="(++) List.intersectBy000 (Char Zero) ((||) primEqChar (Char Zero) vz410 foldr (||) False (map (primEqChar (Char Zero)) vz411)) vz5",fontsize=16,color="magenta"];76 -> 87[label="",style="dashed", color="magenta", weight=3];
76 -> 88[label="",style="dashed", color="magenta", weight=3];
76 -> 89[label="",style="dashed", color="magenta", weight=3];
77[label="Char Zero",fontsize=16,color="green",shape="box"];78 -> 31[label="",style="dashed", color="red", weight=0];
78[label="[] ++ vz5",fontsize=16,color="magenta"];258[label="vz110",fontsize=16,color="green",shape="box"];259[label="vz120",fontsize=16,color="green",shape="box"];260[label="vz13",fontsize=16,color="green",shape="box"];261[label="vz10",fontsize=16,color="green",shape="box"];262[label="vz14",fontsize=16,color="green",shape="box"];263[label="vz13",fontsize=16,color="green",shape="box"];264[label="vz10",fontsize=16,color="green",shape="box"];265[label="vz14",fontsize=16,color="green",shape="box"];266[label="(++) List.intersectBy000 (Char (Succ vz10)) True vz14",fontsize=16,color="black",shape="box"];266 -> 267[label="",style="solid", color="black", weight=3];
84[label="vz411",fontsize=16,color="green",shape="box"];85[label="vz410",fontsize=16,color="green",shape="box"];86[label="Char (Succ vz3000)",fontsize=16,color="green",shape="box"];87[label="vz411",fontsize=16,color="green",shape="box"];88[label="vz410",fontsize=16,color="green",shape="box"];89[label="Char Zero",fontsize=16,color="green",shape="box"];267[label="(++) (Char (Succ vz10) : []) vz14",fontsize=16,color="black",shape="box"];267 -> 268[label="",style="solid", color="black", weight=3];
268[label="Char (Succ vz10) : [] ++ vz14",fontsize=16,color="green",shape="box"];268 -> 269[label="",style="dashed", color="green", weight=3];
269 -> 31[label="",style="dashed", color="red", weight=0];
269[label="[] ++ vz14",fontsize=16,color="magenta"];269 -> 270[label="",style="dashed", color="magenta", weight=3];
270[label="vz14",fontsize=16,color="green",shape="box"];}

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

(12)
Complex Obligation (AND)

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

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

   new_psPs(vz10, Succ(vz110), Succ(vz120), vz13, vz14) -> new_psPs(vz10, vz110, vz120, vz13, vz14)
   new_psPs(vz10, Succ(vz110), Zero, vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
   new_psPs(vz10, Zero, Succ(vz120), vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
   new_psPs0(vz3000, :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, vz5)
   new_psPs1(Char(Zero), Char(Succ(vz4000)), :(vz410, vz411), vz5) -> new_psPs1(Char(Zero), vz410, vz411, vz5)
   new_psPs1(Char(Succ(vz3000)), Char(Succ(vz4000)), vz41, vz5) -> new_psPs(vz3000, vz3000, vz4000, vz41, vz5)
   new_psPs1(Char(Succ(vz3000)), Char(Zero), :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, vz5)

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

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

(15)
Complex Obligation (AND)

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

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

   new_psPs1(Char(Zero), Char(Succ(vz4000)), :(vz410, vz411), vz5) -> new_psPs1(Char(Zero), vz410, vz411, vz5)

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_psPs1(Char(Zero), Char(Succ(vz4000)), :(vz410, vz411), vz5) -> new_psPs1(Char(Zero), vz410, vz411, vz5)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3, 4 >= 4


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

(18)
YES

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

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

   new_psPs(vz10, Succ(vz110), Zero, vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
   new_psPs0(vz3000, :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, vz5)
   new_psPs1(Char(Succ(vz3000)), Char(Succ(vz4000)), vz41, vz5) -> new_psPs(vz3000, vz3000, vz4000, vz41, vz5)
   new_psPs(vz10, Succ(vz110), Succ(vz120), vz13, vz14) -> new_psPs(vz10, vz110, vz120, vz13, vz14)
   new_psPs(vz10, Zero, Succ(vz120), vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
   new_psPs1(Char(Succ(vz3000)), Char(Zero), :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, 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_psPs0(vz3000, :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, vz5)
The graph contains the following edges 2 > 2, 2 > 3, 3 >= 4


*new_psPs1(Char(Succ(vz3000)), Char(Succ(vz4000)), vz41, vz5) -> new_psPs(vz3000, vz3000, vz4000, vz41, vz5)
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 3 >= 4, 4 >= 5


*new_psPs(vz10, Succ(vz110), Succ(vz120), vz13, vz14) -> new_psPs(vz10, vz110, vz120, vz13, vz14)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5


*new_psPs1(Char(Succ(vz3000)), Char(Zero), :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, vz5)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3, 4 >= 4


*new_psPs(vz10, Succ(vz110), Zero, vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
The graph contains the following edges 1 >= 1, 4 >= 2, 5 >= 3


*new_psPs(vz10, Zero, Succ(vz120), vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
The graph contains the following edges 1 >= 1, 4 >= 2, 5 >= 3


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

(21)
YES

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

(22)
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.
----------------------------------------

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


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

(24)
YES