/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 25 ms]
(4) HASKELL
(5) LetRed [EQUIVALENT, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) TransformationProof [EQUIVALENT, 0 ms]
(12) QDP
(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(14) 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;
elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
elem_by _ _ [] = False;
elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;

nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
nubBy eq l = nubBy' l [] where { 
nubBy' [] _ = [];
nubBy' (y : ys) xs  | elem_by eq y xs = nubBy' ys xs
 | otherwise = y : nubBy' ys (y : xs);
 };

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

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

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

(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;
elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
elem_by vy vz [] = False;
elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;

nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
nubBy eq l = nubBy' l [] where { 
nubBy' [] wu = [];
nubBy' (y : ys) xs  | elem_by eq y xs = nubBy' ys xs
 | otherwise = y : nubBy' ys (y : xs);
 };

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

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

(3) 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
"nubBy' [] wu = [];
nubBy' (y : ys) xs|elem_by eq y xsnubBy' ys xs|otherwisey : nubBy' ys (y : xs);
"
is transformed to
"nubBy' [] wu = nubBy'3 [] wu;
nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
"
"nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
"
"nubBy'1 y ys xs True = nubBy' ys xs;
nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
"
"nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
"
"nubBy'3 [] wu = [];
nubBy'3 wx wy = nubBy'2 wx wy;
"

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

(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;
elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
elem_by vy vz [] = False;
elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;

nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
nubBy eq l = nubBy' l [] where { 
nubBy' [] wu = nubBy'3 [] wu;
nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
nubBy'1 y ys xs True = nubBy' ys xs;
nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
nubBy'3 [] wu = [];
nubBy'3 wx wy = nubBy'2 wx wy;
 };

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

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

(5) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"nubBy' l [] where {
nubBy' [] wu = nubBy'3 [] wu;
nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
;
nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
;
nubBy'1 y ys xs True = nubBy' ys xs;
nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
;
nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
;
nubBy'3 [] wu = [];
nubBy'3 wx wy = nubBy'2 wx wy;
} 
"
are unpacked to the following functions on top level
"nubByNubBy'2 wz (y : ys) xs = nubByNubBy'1 wz y ys xs (elem_by wz y xs);
"
"nubByNubBy'3 wz [] wu = [];
nubByNubBy'3 wz wx wy = nubByNubBy'2 wz wx wy;
"
"nubByNubBy'1 wz y ys xs True = nubByNubBy' wz ys xs;
nubByNubBy'1 wz y ys xs False = nubByNubBy'0 wz y ys xs otherwise;
"
"nubByNubBy' wz [] wu = nubByNubBy'3 wz [] wu;
nubByNubBy' wz (y : ys) xs = nubByNubBy'2 wz (y : ys) xs;
"
"nubByNubBy'0 wz y ys xs True = y : nubByNubBy' wz ys (y : xs);
"

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

(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;
elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
elem_by vy vz [] = False;
elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;

nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
nubBy eq l = nubByNubBy' eq l [];

nubByNubBy' wz [] wu = nubByNubBy'3 wz [] wu;
nubByNubBy' wz (y : ys) xs = nubByNubBy'2 wz (y : ys) xs;

nubByNubBy'0 wz y ys xs True = y : nubByNubBy' wz ys (y : xs);

nubByNubBy'1 wz y ys xs True = nubByNubBy' wz ys xs;
nubByNubBy'1 wz y ys xs False = nubByNubBy'0 wz y ys xs otherwise;

nubByNubBy'2 wz (y : ys) xs = nubByNubBy'1 wz y ys xs (elem_by wz y xs);

nubByNubBy'3 wz [] wu = [];
nubByNubBy'3 wz wx wy = nubByNubBy'2 wz wx wy;

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

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.nubBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.nubBy xu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.nubBy xu3 xu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="List.nubByNubBy' xu3 xu4 []",fontsize=16,color="burlywood",shape="box"];493[label="xu4/xu40 : xu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 493[label="",style="solid", color="burlywood", weight=9];
493 -> 6[label="",style="solid", color="burlywood", weight=3];
494[label="xu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 494[label="",style="solid", color="burlywood", weight=9];
494 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="List.nubByNubBy' xu3 (xu40 : xu41) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="List.nubByNubBy' xu3 [] []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="List.nubByNubBy'2 xu3 (xu40 : xu41) []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="List.nubByNubBy'3 xu3 [] []",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="List.nubByNubBy'1 xu3 xu40 xu41 [] (List.elem_by xu3 xu40 [])",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="[]",fontsize=16,color="green",shape="box"];12[label="List.nubByNubBy'1 xu3 xu40 xu41 [] False",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="List.nubByNubBy'0 xu3 xu40 xu41 [] otherwise",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="List.nubByNubBy'0 xu3 xu40 xu41 [] True",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="xu40 : List.nubByNubBy' xu3 xu41 (xu40 : [])",fontsize=16,color="green",shape="box"];15 -> 16[label="",style="dashed", color="green", weight=3];
16[label="List.nubByNubBy' xu3 xu41 (xu40 : [])",fontsize=16,color="burlywood",shape="triangle"];495[label="xu41/xu410 : xu411",fontsize=10,color="white",style="solid",shape="box"];16 -> 495[label="",style="solid", color="burlywood", weight=9];
495 -> 17[label="",style="solid", color="burlywood", weight=3];
496[label="xu41/[]",fontsize=10,color="white",style="solid",shape="box"];16 -> 496[label="",style="solid", color="burlywood", weight=9];
496 -> 18[label="",style="solid", color="burlywood", weight=3];
17[label="List.nubByNubBy' xu3 (xu410 : xu411) (xu40 : [])",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="List.nubByNubBy' xu3 [] (xu40 : [])",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="List.nubByNubBy'2 xu3 (xu410 : xu411) (xu40 : [])",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="List.nubByNubBy'3 xu3 [] (xu40 : [])",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
21[label="List.nubByNubBy'1 xu3 xu410 xu411 (xu40 : []) (List.elem_by xu3 xu410 (xu40 : []))",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="[]",fontsize=16,color="green",shape="box"];23 -> 452[label="",style="dashed", color="red", weight=0];
23[label="List.nubByNubBy'1 xu3 xu410 xu411 (xu40 : []) (xu3 xu40 xu410 || List.elem_by xu3 xu410 [])",fontsize=16,color="magenta"];23 -> 453[label="",style="dashed", color="magenta", weight=3];
23 -> 454[label="",style="dashed", color="magenta", weight=3];
23 -> 455[label="",style="dashed", color="magenta", weight=3];
23 -> 456[label="",style="dashed", color="magenta", weight=3];
23 -> 457[label="",style="dashed", color="magenta", weight=3];
23 -> 458[label="",style="dashed", color="magenta", weight=3];
23 -> 459[label="",style="dashed", color="magenta", weight=3];
453[label="xu40",fontsize=16,color="green",shape="box"];454[label="[]",fontsize=16,color="green",shape="box"];455[label="xu3",fontsize=16,color="green",shape="box"];456[label="[]",fontsize=16,color="green",shape="box"];457[label="xu3 xu40 xu410",fontsize=16,color="green",shape="box"];457 -> 461[label="",style="dashed", color="green", weight=3];
457 -> 462[label="",style="dashed", color="green", weight=3];
458[label="xu410",fontsize=16,color="green",shape="box"];459[label="xu411",fontsize=16,color="green",shape="box"];452[label="List.nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (xu39 || List.elem_by xu32 xu33 xu38)",fontsize=16,color="burlywood",shape="triangle"];497[label="xu39/False",fontsize=10,color="white",style="solid",shape="box"];452 -> 497[label="",style="solid", color="burlywood", weight=9];
497 -> 463[label="",style="solid", color="burlywood", weight=3];
498[label="xu39/True",fontsize=10,color="white",style="solid",shape="box"];452 -> 498[label="",style="solid", color="burlywood", weight=9];
498 -> 464[label="",style="solid", color="burlywood", weight=3];
461[label="xu40",fontsize=16,color="green",shape="box"];462[label="xu410",fontsize=16,color="green",shape="box"];463[label="List.nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (False || List.elem_by xu32 xu33 xu38)",fontsize=16,color="black",shape="box"];463 -> 467[label="",style="solid", color="black", weight=3];
464[label="List.nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (True || List.elem_by xu32 xu33 xu38)",fontsize=16,color="black",shape="box"];464 -> 468[label="",style="solid", color="black", weight=3];
467[label="List.nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (List.elem_by xu32 xu33 xu38)",fontsize=16,color="burlywood",shape="triangle"];499[label="xu38/xu380 : xu381",fontsize=10,color="white",style="solid",shape="box"];467 -> 499[label="",style="solid", color="burlywood", weight=9];
499 -> 469[label="",style="solid", color="burlywood", weight=3];
500[label="xu38/[]",fontsize=10,color="white",style="solid",shape="box"];467 -> 500[label="",style="solid", color="burlywood", weight=9];
500 -> 470[label="",style="solid", color="burlywood", weight=3];
468[label="List.nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) True",fontsize=16,color="black",shape="box"];468 -> 471[label="",style="solid", color="black", weight=3];
469[label="List.nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (List.elem_by xu32 xu33 (xu380 : xu381))",fontsize=16,color="black",shape="box"];469 -> 472[label="",style="solid", color="black", weight=3];
470[label="List.nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (List.elem_by xu32 xu33 [])",fontsize=16,color="black",shape="box"];470 -> 473[label="",style="solid", color="black", weight=3];
471[label="List.nubByNubBy' xu32 xu34 (xu35 : xu36)",fontsize=16,color="burlywood",shape="triangle"];501[label="xu34/xu340 : xu341",fontsize=10,color="white",style="solid",shape="box"];471 -> 501[label="",style="solid", color="burlywood", weight=9];
501 -> 474[label="",style="solid", color="burlywood", weight=3];
502[label="xu34/[]",fontsize=10,color="white",style="solid",shape="box"];471 -> 502[label="",style="solid", color="burlywood", weight=9];
502 -> 475[label="",style="solid", color="burlywood", weight=3];
472 -> 452[label="",style="dashed", color="red", weight=0];
472[label="List.nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (xu32 xu380 xu33 || List.elem_by xu32 xu33 xu381)",fontsize=16,color="magenta"];472 -> 476[label="",style="dashed", color="magenta", weight=3];
472 -> 477[label="",style="dashed", color="magenta", weight=3];
473[label="List.nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) False",fontsize=16,color="black",shape="box"];473 -> 478[label="",style="solid", color="black", weight=3];
474[label="List.nubByNubBy' xu32 (xu340 : xu341) (xu35 : xu36)",fontsize=16,color="black",shape="box"];474 -> 479[label="",style="solid", color="black", weight=3];
475[label="List.nubByNubBy' xu32 [] (xu35 : xu36)",fontsize=16,color="black",shape="box"];475 -> 480[label="",style="solid", color="black", weight=3];
476[label="xu381",fontsize=16,color="green",shape="box"];477[label="xu32 xu380 xu33",fontsize=16,color="green",shape="box"];477 -> 481[label="",style="dashed", color="green", weight=3];
477 -> 482[label="",style="dashed", color="green", weight=3];
478[label="List.nubByNubBy'0 xu32 xu33 xu34 (xu35 : xu36) otherwise",fontsize=16,color="black",shape="box"];478 -> 483[label="",style="solid", color="black", weight=3];
479[label="List.nubByNubBy'2 xu32 (xu340 : xu341) (xu35 : xu36)",fontsize=16,color="black",shape="box"];479 -> 484[label="",style="solid", color="black", weight=3];
480[label="List.nubByNubBy'3 xu32 [] (xu35 : xu36)",fontsize=16,color="black",shape="box"];480 -> 485[label="",style="solid", color="black", weight=3];
481[label="xu380",fontsize=16,color="green",shape="box"];482[label="xu33",fontsize=16,color="green",shape="box"];483[label="List.nubByNubBy'0 xu32 xu33 xu34 (xu35 : xu36) True",fontsize=16,color="black",shape="box"];483 -> 486[label="",style="solid", color="black", weight=3];
484 -> 467[label="",style="dashed", color="red", weight=0];
484[label="List.nubByNubBy'1 xu32 xu340 xu341 (xu35 : xu36) (List.elem_by xu32 xu340 (xu35 : xu36))",fontsize=16,color="magenta"];484 -> 487[label="",style="dashed", color="magenta", weight=3];
484 -> 488[label="",style="dashed", color="magenta", weight=3];
484 -> 489[label="",style="dashed", color="magenta", weight=3];
485[label="[]",fontsize=16,color="green",shape="box"];486[label="xu33 : List.nubByNubBy' xu32 xu34 (xu33 : xu35 : xu36)",fontsize=16,color="green",shape="box"];486 -> 490[label="",style="dashed", color="green", weight=3];
487[label="xu35 : xu36",fontsize=16,color="green",shape="box"];488[label="xu340",fontsize=16,color="green",shape="box"];489[label="xu341",fontsize=16,color="green",shape="box"];490 -> 471[label="",style="dashed", color="red", weight=0];
490[label="List.nubByNubBy' xu32 xu34 (xu33 : xu35 : xu36)",fontsize=16,color="magenta"];490 -> 491[label="",style="dashed", color="magenta", weight=3];
490 -> 492[label="",style="dashed", color="magenta", weight=3];
491[label="xu33",fontsize=16,color="green",shape="box"];492[label="xu35 : xu36",fontsize=16,color="green",shape="box"];}

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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_nubByNubBy'(xu32, :(xu340, xu341), xu35, xu36, ba) -> new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
   new_nubByNubBy'1(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) -> new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)
   new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, [], ba) -> new_nubByNubBy'(xu32, xu34, xu33, :(xu35, xu36), ba)
   new_nubByNubBy'10(xu32, xu33, :(xu340, xu341), xu35, xu36, xu38, ba) -> new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
   new_nubByNubBy'1(xu32, xu33, xu34, xu35, xu36, [], ba) -> new_nubByNubBy'(xu32, xu34, xu33, :(xu35, xu36), ba)
   new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) -> new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)

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

(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_nubByNubBy'1(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) -> new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)
   new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, [], ba) -> new_nubByNubBy'(xu32, xu34, xu33, :(xu35, xu36), ba)
   new_nubByNubBy'(xu32, :(xu340, xu341), xu35, xu36, ba) -> new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
   new_nubByNubBy'10(xu32, xu33, :(xu340, xu341), xu35, xu36, xu38, ba) -> new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
   new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) -> new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)

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

(11) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule new_nubByNubBy'1(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) -> new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba) we obtained the following new rules [LPAR04]:

   (new_nubByNubBy'1(z0, z1, z2, z3, z4, :(z3, z4), z5) -> new_nubByNubBy'10(z0, z1, z2, z3, z4, z4, z5),new_nubByNubBy'1(z0, z1, z2, z3, z4, :(z3, z4), z5) -> new_nubByNubBy'10(z0, z1, z2, z3, z4, z4, z5))


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

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

   new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, [], ba) -> new_nubByNubBy'(xu32, xu34, xu33, :(xu35, xu36), ba)
   new_nubByNubBy'(xu32, :(xu340, xu341), xu35, xu36, ba) -> new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
   new_nubByNubBy'10(xu32, xu33, :(xu340, xu341), xu35, xu36, xu38, ba) -> new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
   new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) -> new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)
   new_nubByNubBy'1(z0, z1, z2, z3, z4, :(z3, z4), z5) -> new_nubByNubBy'10(z0, z1, z2, z3, z4, z4, z5)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(13) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_nubByNubBy'(xu32, :(xu340, xu341), xu35, xu36, ba) -> new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4, 4 >= 5, 5 >= 7


*new_nubByNubBy'1(z0, z1, z2, z3, z4, :(z3, z4), z5) -> new_nubByNubBy'10(z0, z1, z2, z3, z4, z4, z5)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 6 > 4, 5 >= 5, 6 > 5, 5 >= 6, 6 > 6, 7 >= 7


*new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) -> new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 > 6, 7 >= 7


*new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, [], ba) -> new_nubByNubBy'(xu32, xu34, xu33, :(xu35, xu36), ba)
The graph contains the following edges 1 >= 1, 3 >= 2, 2 >= 3, 7 >= 5


*new_nubByNubBy'10(xu32, xu33, :(xu340, xu341), xu35, xu36, xu38, ba) -> new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5, 7 >= 7


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(14)
YES