/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) LetRed [EQUIVALENT, 0 ms]
(8) HASKELL
(9) NumRed [SOUND, 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
    (19) QDP
        (20) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (21) YES


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM :: Monad a => Int  ->  a b  ->  a [b];
replicateM n x = sequence (replicate n x);

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\xs->return (x : xs)"
is transformed to
"sequence0 x xs = return (x : xs);
"
The following Lambda expression
"\x->sequence cs >>= sequence0 x"
is transformed to
"sequence1 cs x = sequence cs >>= sequence0 x;
"

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM :: Monad a => Int  ->  a b  ->  a [b];
replicateM n x = sequence (replicate n x);

}

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

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

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM :: Monad b => Int  ->  b a  ->  b [a];
replicateM n x = sequence (replicate n x);

}

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

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"
The following Function with conditions
"take n vx|n <= 0[];
take vy [] = [];
take n (x : xs) = x : take (n - 1) xs;
"
is transformed to
"take n vx = take3 n vx;
take vy [] = take1 vy [];
take n (x : xs) = take0 n (x : xs);
"
"take0 n (x : xs) = x : take (n - 1) xs;
"
"take1 vy [] = [];
take1 ww wx = take0 ww wx;
"
"take2 n vx True = [];
take2 n vx False = take1 n vx;
"
"take3 n vx = take2 n vx (n <= 0);
take3 wy wz = take1 wy wz;
"

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

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM :: Monad a => Int  ->  a b  ->  a [b];
replicateM n x = sequence (replicate n x);

}

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

(7) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"xs where {
xs  = x : xs;
} 
"
are unpacked to the following functions on top level
"repeatXs xu = xu : repeatXs xu;
"

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

(8)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM :: Monad b => Int  ->  b a  ->  b [a];
replicateM n x = sequence (replicate n x);

}

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

(9) NumRed (SOUND)
Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
----------------------------------------

(10)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM :: Monad a => Int  ->  a b  ->  a [b];
replicateM n x = sequence (replicate n x);

}

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

(11) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.replicateM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.replicateM xv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.replicateM xv3 xv4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="sequence (replicate xv3 xv4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence (take xv3 (repeat xv4))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="sequence (take3 xv3 (repeat xv4))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="sequence (take2 xv3 (repeat xv4) (xv3 <= Pos Zero))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="sequence (take2 xv3 (repeat xv4) (compare xv3 (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="sequence (take2 xv3 (repeat xv4) (not (compare xv3 (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="sequence (take2 xv3 (repeat xv4) (not (primCmpInt xv3 (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];315[label="xv3/Pos xv30",fontsize=10,color="white",style="solid",shape="box"];11 -> 315[label="",style="solid", color="burlywood", weight=9];
315 -> 12[label="",style="solid", color="burlywood", weight=3];
316[label="xv3/Neg xv30",fontsize=10,color="white",style="solid",shape="box"];11 -> 316[label="",style="solid", color="burlywood", weight=9];
316 -> 13[label="",style="solid", color="burlywood", weight=3];
12[label="sequence (take2 (Pos xv30) (repeat xv4) (not (primCmpInt (Pos xv30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];317[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];12 -> 317[label="",style="solid", color="burlywood", weight=9];
317 -> 14[label="",style="solid", color="burlywood", weight=3];
318[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 318[label="",style="solid", color="burlywood", weight=9];
318 -> 15[label="",style="solid", color="burlywood", weight=3];
13[label="sequence (take2 (Neg xv30) (repeat xv4) (not (primCmpInt (Neg xv30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];319[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];13 -> 319[label="",style="solid", color="burlywood", weight=9];
319 -> 16[label="",style="solid", color="burlywood", weight=3];
320[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 320[label="",style="solid", color="burlywood", weight=9];
320 -> 17[label="",style="solid", color="burlywood", weight=3];
14[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not (primCmpInt (Pos (Succ xv300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="sequence (take2 (Pos Zero) (repeat xv4) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) (not (primCmpInt (Neg (Succ xv300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="sequence (take2 (Neg Zero) (repeat xv4) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not (primCmpNat (Succ xv300) Zero == GT)))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="sequence (take2 (Pos Zero) (repeat xv4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) (not (LT == GT)))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="sequence (take2 (Neg Zero) (repeat xv4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not (GT == GT)))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="sequence (take2 (Pos Zero) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="sequence (take2 (Neg Zero) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not True))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="sequence (take2 (Pos Zero) (repeat xv4) True)",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) True)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="sequence (take2 (Neg Zero) (repeat xv4) True)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) False)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="sequence []",fontsize=16,color="black",shape="triangle"];31 -> 35[label="",style="solid", color="black", weight=3];
32 -> 31[label="",style="dashed", color="red", weight=0];
32[label="sequence []",fontsize=16,color="magenta"];33 -> 31[label="",style="dashed", color="red", weight=0];
33[label="sequence []",fontsize=16,color="magenta"];34[label="sequence (take1 (Pos (Succ xv300)) (repeat xv4))",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="return []",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="sequence (take1 (Pos (Succ xv300)) (repeatXs xv4))",fontsize=16,color="black",shape="triangle"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="[] : []",fontsize=16,color="green",shape="box"];38[label="sequence (take1 (Pos (Succ xv300)) (xv4 : repeatXs xv4))",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
39[label="sequence (take0 (Pos (Succ xv300)) (xv4 : repeatXs xv4))",fontsize=16,color="black",shape="box"];39 -> 40[label="",style="solid", color="black", weight=3];
40[label="sequence (xv4 : take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs xv4))",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
41[label="xv4 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs xv4))",fontsize=16,color="burlywood",shape="box"];321[label="xv4/xv40 : xv41",fontsize=10,color="white",style="solid",shape="box"];41 -> 321[label="",style="solid", color="burlywood", weight=9];
321 -> 42[label="",style="solid", color="burlywood", weight=3];
322[label="xv4/[]",fontsize=10,color="white",style="solid",shape="box"];41 -> 322[label="",style="solid", color="burlywood", weight=9];
322 -> 43[label="",style="solid", color="burlywood", weight=3];
42[label="xv40 : xv41 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)))",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
43[label="[] >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs []))",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
44[label="sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41))) xv40 ++ (xv41 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41))))",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
45[label="[]",fontsize=16,color="green",shape="box"];46 -> 111[label="",style="dashed", color="red", weight=0];
46[label="(sequence (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41))) >>= sequence0 xv40) ++ (xv41 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41))))",fontsize=16,color="magenta"];46 -> 112[label="",style="dashed", color="magenta", weight=3];
46 -> 113[label="",style="dashed", color="magenta", weight=3];
46 -> 114[label="",style="dashed", color="magenta", weight=3];
112 -> 277[label="",style="dashed", color="red", weight=0];
112[label="xv41 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)))",fontsize=16,color="magenta"];112 -> 278[label="",style="dashed", color="magenta", weight=3];
112 -> 279[label="",style="dashed", color="magenta", weight=3];
112 -> 280[label="",style="dashed", color="magenta", weight=3];
112 -> 281[label="",style="dashed", color="magenta", weight=3];
113[label="sequence (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)))",fontsize=16,color="black",shape="triangle"];113 -> 185[label="",style="solid", color="black", weight=3];
114[label="xv40",fontsize=16,color="green",shape="box"];111[label="(xv10 >>= sequence0 xv410) ++ xv8",fontsize=16,color="burlywood",shape="triangle"];323[label="xv10/xv100 : xv101",fontsize=10,color="white",style="solid",shape="box"];111 -> 323[label="",style="solid", color="burlywood", weight=9];
323 -> 186[label="",style="solid", color="burlywood", weight=3];
324[label="xv10/[]",fontsize=10,color="white",style="solid",shape="box"];111 -> 324[label="",style="solid", color="burlywood", weight=9];
324 -> 187[label="",style="solid", color="burlywood", weight=3];
278[label="xv300",fontsize=16,color="green",shape="box"];279[label="xv40",fontsize=16,color="green",shape="box"];280[label="xv41",fontsize=16,color="green",shape="box"];281[label="xv41",fontsize=16,color="green",shape="box"];277[label="xv13 >>= sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="burlywood",shape="triangle"];325[label="xv13/xv130 : xv131",fontsize=10,color="white",style="solid",shape="box"];277 -> 325[label="",style="solid", color="burlywood", weight=9];
325 -> 302[label="",style="solid", color="burlywood", weight=3];
326[label="xv13/[]",fontsize=10,color="white",style="solid",shape="box"];277 -> 326[label="",style="solid", color="burlywood", weight=9];
326 -> 303[label="",style="solid", color="burlywood", weight=3];
185[label="sequence (take3 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)))",fontsize=16,color="black",shape="box"];185 -> 190[label="",style="solid", color="black", weight=3];
186[label="(xv100 : xv101 >>= sequence0 xv410) ++ xv8",fontsize=16,color="black",shape="box"];186 -> 191[label="",style="solid", color="black", weight=3];
187[label="([] >>= sequence0 xv410) ++ xv8",fontsize=16,color="black",shape="box"];187 -> 192[label="",style="solid", color="black", weight=3];
302[label="xv130 : xv131 >>= sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="black",shape="box"];302 -> 304[label="",style="solid", color="black", weight=3];
303[label="[] >>= sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="black",shape="box"];303 -> 305[label="",style="solid", color="black", weight=3];
190[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)) (Pos (Succ xv300) - Pos (Succ Zero) <= Pos Zero))",fontsize=16,color="black",shape="box"];190 -> 194[label="",style="solid", color="black", weight=3];
191[label="(sequence0 xv410 xv100 ++ (xv101 >>= sequence0 xv410)) ++ xv8",fontsize=16,color="black",shape="box"];191 -> 195[label="",style="solid", color="black", weight=3];
192[label="[] ++ xv8",fontsize=16,color="black",shape="triangle"];192 -> 196[label="",style="solid", color="black", weight=3];
304 -> 221[label="",style="dashed", color="red", weight=0];
304[label="sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16))) xv130 ++ (xv131 >>= sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16))))",fontsize=16,color="magenta"];304 -> 306[label="",style="dashed", color="magenta", weight=3];
304 -> 307[label="",style="dashed", color="magenta", weight=3];
305[label="[]",fontsize=16,color="green",shape="box"];194[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)) (compare (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];194 -> 199[label="",style="solid", color="black", weight=3];
195[label="(return (xv410 : xv100) ++ (xv101 >>= sequence0 xv410)) ++ xv8",fontsize=16,color="black",shape="box"];195 -> 200[label="",style="solid", color="black", weight=3];
196[label="xv8",fontsize=16,color="green",shape="box"];306 -> 277[label="",style="dashed", color="red", weight=0];
306[label="xv131 >>= sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="magenta"];306 -> 308[label="",style="dashed", color="magenta", weight=3];
307[label="sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16))) xv130",fontsize=16,color="black",shape="box"];307 -> 309[label="",style="solid", color="black", weight=3];
221[label="xv11 ++ xv8",fontsize=16,color="burlywood",shape="triangle"];327[label="xv11/xv110 : xv111",fontsize=10,color="white",style="solid",shape="box"];221 -> 327[label="",style="solid", color="burlywood", weight=9];
327 -> 229[label="",style="solid", color="burlywood", weight=3];
328[label="xv11/[]",fontsize=10,color="white",style="solid",shape="box"];221 -> 328[label="",style="solid", color="burlywood", weight=9];
328 -> 230[label="",style="solid", color="burlywood", weight=3];
199[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)) (not (compare (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];199 -> 204[label="",style="solid", color="black", weight=3];
200[label="(((xv410 : xv100) : []) ++ (xv101 >>= sequence0 xv410)) ++ xv8",fontsize=16,color="black",shape="box"];200 -> 205[label="",style="solid", color="black", weight=3];
308[label="xv131",fontsize=16,color="green",shape="box"];309 -> 213[label="",style="dashed", color="red", weight=0];
309[label="sequence (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16))) >>= sequence0 xv130",fontsize=16,color="magenta"];309 -> 310[label="",style="dashed", color="magenta", weight=3];
309 -> 311[label="",style="dashed", color="magenta", weight=3];
229[label="(xv110 : xv111) ++ xv8",fontsize=16,color="black",shape="box"];229 -> 237[label="",style="solid", color="black", weight=3];
230[label="[] ++ xv8",fontsize=16,color="black",shape="box"];230 -> 238[label="",style="solid", color="black", weight=3];
204[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)) (not (primCmpInt (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];204 -> 208[label="",style="solid", color="black", weight=3];
205 -> 209[label="",style="dashed", color="red", weight=0];
205[label="((xv410 : xv100) : [] ++ (xv101 >>= sequence0 xv410)) ++ xv8",fontsize=16,color="magenta"];205 -> 210[label="",style="dashed", color="magenta", weight=3];
310 -> 113[label="",style="dashed", color="red", weight=0];
310[label="sequence (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="magenta"];310 -> 312[label="",style="dashed", color="magenta", weight=3];
310 -> 313[label="",style="dashed", color="magenta", weight=3];
310 -> 314[label="",style="dashed", color="magenta", weight=3];
311[label="xv130",fontsize=16,color="green",shape="box"];213[label="xv101 >>= sequence0 xv410",fontsize=16,color="burlywood",shape="triangle"];329[label="xv101/xv1010 : xv1011",fontsize=10,color="white",style="solid",shape="box"];213 -> 329[label="",style="solid", color="burlywood", weight=9];
329 -> 219[label="",style="solid", color="burlywood", weight=3];
330[label="xv101/[]",fontsize=10,color="white",style="solid",shape="box"];213 -> 330[label="",style="solid", color="burlywood", weight=9];
330 -> 220[label="",style="solid", color="burlywood", weight=3];
237[label="xv110 : xv111 ++ xv8",fontsize=16,color="green",shape="box"];237 -> 245[label="",style="dashed", color="green", weight=3];
238[label="xv8",fontsize=16,color="green",shape="box"];208[label="sequence (take2 (primMinusInt (Pos (Succ xv300)) (Pos (Succ Zero))) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusInt (Pos (Succ xv300)) (Pos (Succ Zero))) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];208 -> 212[label="",style="solid", color="black", weight=3];
210 -> 192[label="",style="dashed", color="red", weight=0];
210[label="[] ++ (xv101 >>= sequence0 xv410)",fontsize=16,color="magenta"];210 -> 213[label="",style="dashed", color="magenta", weight=3];
209[label="((xv410 : xv100) : xv11) ++ xv8",fontsize=16,color="black",shape="triangle"];209 -> 214[label="",style="solid", color="black", weight=3];
312[label="xv14",fontsize=16,color="green",shape="box"];313[label="xv16",fontsize=16,color="green",shape="box"];314[label="xv15",fontsize=16,color="green",shape="box"];219[label="xv1010 : xv1011 >>= sequence0 xv410",fontsize=16,color="black",shape="box"];219 -> 227[label="",style="solid", color="black", weight=3];
220[label="[] >>= sequence0 xv410",fontsize=16,color="black",shape="box"];220 -> 228[label="",style="solid", color="black", weight=3];
245 -> 221[label="",style="dashed", color="red", weight=0];
245[label="xv111 ++ xv8",fontsize=16,color="magenta"];245 -> 252[label="",style="dashed", color="magenta", weight=3];
212[label="sequence (take2 (primMinusNat (Succ xv300) (Succ Zero)) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat (Succ xv300) (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];212 -> 218[label="",style="solid", color="black", weight=3];
214[label="(xv410 : xv100) : xv11 ++ xv8",fontsize=16,color="green",shape="box"];214 -> 221[label="",style="dashed", color="green", weight=3];
227 -> 221[label="",style="dashed", color="red", weight=0];
227[label="sequence0 xv410 xv1010 ++ (xv1011 >>= sequence0 xv410)",fontsize=16,color="magenta"];227 -> 235[label="",style="dashed", color="magenta", weight=3];
227 -> 236[label="",style="dashed", color="magenta", weight=3];
228[label="[]",fontsize=16,color="green",shape="box"];252[label="xv111",fontsize=16,color="green",shape="box"];218[label="sequence (take2 (primMinusNat xv300 Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat xv300 Zero) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];331[label="xv300/Succ xv3000",fontsize=10,color="white",style="solid",shape="box"];218 -> 331[label="",style="solid", color="burlywood", weight=9];
331 -> 225[label="",style="solid", color="burlywood", weight=3];
332[label="xv300/Zero",fontsize=10,color="white",style="solid",shape="box"];218 -> 332[label="",style="solid", color="burlywood", weight=9];
332 -> 226[label="",style="solid", color="burlywood", weight=3];
235 -> 213[label="",style="dashed", color="red", weight=0];
235[label="xv1011 >>= sequence0 xv410",fontsize=16,color="magenta"];235 -> 243[label="",style="dashed", color="magenta", weight=3];
236[label="sequence0 xv410 xv1010",fontsize=16,color="black",shape="box"];236 -> 244[label="",style="solid", color="black", weight=3];
225[label="sequence (take2 (primMinusNat (Succ xv3000) Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat (Succ xv3000) Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];225 -> 233[label="",style="solid", color="black", weight=3];
226[label="sequence (take2 (primMinusNat Zero Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat Zero Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];226 -> 234[label="",style="solid", color="black", weight=3];
243[label="xv1011",fontsize=16,color="green",shape="box"];244[label="return (xv410 : xv1010)",fontsize=16,color="black",shape="box"];244 -> 251[label="",style="solid", color="black", weight=3];
233[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not (primCmpInt (Pos (Succ xv3000)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];233 -> 241[label="",style="solid", color="black", weight=3];
234[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];234 -> 242[label="",style="solid", color="black", weight=3];
251[label="(xv410 : xv1010) : []",fontsize=16,color="green",shape="box"];241[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not (primCmpNat (Succ xv3000) Zero == GT)))",fontsize=16,color="black",shape="box"];241 -> 249[label="",style="solid", color="black", weight=3];
242[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];242 -> 250[label="",style="solid", color="black", weight=3];
249[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not (GT == GT)))",fontsize=16,color="black",shape="box"];249 -> 257[label="",style="solid", color="black", weight=3];
250[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) (not False))",fontsize=16,color="black",shape="box"];250 -> 258[label="",style="solid", color="black", weight=3];
257[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not True))",fontsize=16,color="black",shape="box"];257 -> 262[label="",style="solid", color="black", weight=3];
258[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) True)",fontsize=16,color="black",shape="box"];258 -> 263[label="",style="solid", color="black", weight=3];
262[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) False)",fontsize=16,color="black",shape="box"];262 -> 267[label="",style="solid", color="black", weight=3];
263 -> 31[label="",style="dashed", color="red", weight=0];
263[label="sequence []",fontsize=16,color="magenta"];267 -> 36[label="",style="dashed", color="red", weight=0];
267[label="sequence (take1 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)))",fontsize=16,color="magenta"];267 -> 272[label="",style="dashed", color="magenta", weight=3];
267 -> 273[label="",style="dashed", color="magenta", weight=3];
272[label="xv3000",fontsize=16,color="green",shape="box"];273[label="xv40 : xv41",fontsize=16,color="green",shape="box"];}

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

(12)
Complex Obligation (AND)

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

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

   new_gtGtEs(:(xv1010, xv1011), xv410, h) -> new_gtGtEs(xv1011, xv410, h)

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_gtGtEs(:(xv1010, xv1011), xv410, h) -> new_gtGtEs(xv1011, xv410, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(15)
YES

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

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

   new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, h) -> new_sequence(xv14, xv15, xv16, h)
   new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, h) -> new_gtGtEs0(xv131, xv14, xv15, xv16, h)
   new_sequence(Succ(xv3000), xv40, xv41, ba) -> new_sequence0(xv3000, :(xv40, xv41), ba)
   new_sequence0(xv300, :(xv40, xv41), ba) -> new_gtGtEs0(xv41, xv300, xv40, xv41, ba)
   new_sequence0(Succ(xv3000), :(xv40, xv41), ba) -> new_sequence0(xv3000, :(xv40, xv41), 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_sequence(Succ(xv3000), xv40, xv41, ba) -> new_sequence0(xv3000, :(xv40, xv41), ba)
The graph contains the following edges 1 > 1, 4 >= 3


*new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, h) -> new_gtGtEs0(xv131, xv14, xv15, xv16, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5


*new_sequence0(xv300, :(xv40, xv41), ba) -> new_gtGtEs0(xv41, xv300, xv40, xv41, ba)
The graph contains the following edges 2 > 1, 1 >= 2, 2 > 3, 2 > 4, 3 >= 5


*new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, h) -> new_sequence(xv14, xv15, xv16, h)
The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4


*new_sequence0(Succ(xv3000), :(xv40, xv41), ba) -> new_sequence0(xv3000, :(xv40, xv41), ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(18)
YES

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

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

   new_psPs(:(xv110, xv111), xv8, h) -> new_psPs(xv111, xv8, h)

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(:(xv110, xv111), xv8, h) -> new_psPs(xv111, xv8, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(21)
YES