/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) LetRed [EQUIVALENT, 11 ms]
(8) HASKELL
(9) NumRed [SOUND, 0 ms]
(10) HASKELL
(11) Narrow [SOUND, 0 ms]
(12) QDP
(13) TransformationProof [EQUIVALENT, 0 ms]
(14) QDP
(15) DependencyGraphProof [EQUIVALENT, 0 ms]
(16) QDP
(17) TransformationProof [EQUIVALENT, 0 ms]
(18) QDP
(19) DependencyGraphProof [EQUIVALENT, 0 ms]
(20) QDP
(21) UsableRulesProof [EQUIVALENT, 0 ms]
(22) QDP
(23) QReductionProof [EQUIVALENT, 0 ms]
(24) QDP
(25) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(26) YES


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\(_,xs'')->xs''"
is transformed to
"xs''0 (_,xs'') = xs'';
"
The following Lambda expression
"\(xs',_)->xs'"
is transformed to
"xs'0 (xs',_) = xs';
"

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

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

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"splitAt n xs|n <= 0([],xs);
splitAt vv [] = ([],[]);
splitAt n (x : xs) = (x : xs',xs'') where {
vu42  = splitAt (n - 1) xs;
;
xs'  = xs'0 vu42;
;
xs''  = xs''0 vu42;
;
xs''0 (vw,xs'') = xs'';
;
xs'0 (xs',vx) = xs';
} 
;
"
is transformed to
"splitAt n xs = splitAt3 n xs;
splitAt vv [] = splitAt1 vv [];
splitAt n (x : xs) = splitAt0 n (x : xs);
"
"splitAt0 n (x : xs) = (x : xs',xs'') where {
vu42  = splitAt (n - 1) xs;
;
xs'  = xs'0 vu42;
;
xs''  = xs''0 vu42;
;
xs''0 (vw,xs'') = xs'';
;
xs'0 (xs',vx) = xs';
} 
;
"
"splitAt1 vv [] = ([],[]);
splitAt1 ww wx = splitAt0 ww wx;
"
"splitAt2 n xs True = ([],xs);
splitAt2 n xs False = splitAt1 n xs;
"
"splitAt3 n xs = splitAt2 n xs (n <= 0);
splitAt3 wy wz = splitAt1 wy wz;
"
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

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

(6)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(7) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"(x : xs',xs'') where {
vu42  = splitAt (n - 1) xs;
;
xs'  = xs'0 vu42;
;
xs''  = xs''0 vu42;
;
xs''0 (vw,xs'') = xs'';
;
xs'0 (xs',vx) = xs';
} 
"
are unpacked to the following functions on top level
"splitAt0Xs''0 xu xv (vw,xs'') = xs'';
"
"splitAt0Xs'' xu xv = splitAt0Xs''0 xu xv (splitAt0Vu42 xu xv);
"
"splitAt0Vu42 xu xv = splitAt (xu - 1) xv;
"
"splitAt0Xs'0 xu xv (xs',vx) = xs';
"
"splitAt0Xs' xu xv = splitAt0Xs'0 xu xv (splitAt0Vu42 xu xv);
"

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

(8)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

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

(10)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(11) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="splitAt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="splitAt xw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="splitAt xw3 xw4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="splitAt3 xw3 xw4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="splitAt2 xw3 xw4 (xw3 <= Pos Zero)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="splitAt2 xw3 xw4 (compare xw3 (Pos Zero) /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="splitAt2 xw3 xw4 (not (compare xw3 (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="splitAt2 xw3 xw4 (not (primCmpInt xw3 (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];64[label="xw3/Pos xw30",fontsize=10,color="white",style="solid",shape="box"];9 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 10[label="",style="solid", color="burlywood", weight=3];
65[label="xw3/Neg xw30",fontsize=10,color="white",style="solid",shape="box"];9 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="splitAt2 (Pos xw30) xw4 (not (primCmpInt (Pos xw30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];66[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];10 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 12[label="",style="solid", color="burlywood", weight=3];
67[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 13[label="",style="solid", color="burlywood", weight=3];
11[label="splitAt2 (Neg xw30) xw4 (not (primCmpInt (Neg xw30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];68[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];11 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 14[label="",style="solid", color="burlywood", weight=3];
69[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 15[label="",style="solid", color="burlywood", weight=3];
12[label="splitAt2 (Pos (Succ xw300)) xw4 (not (primCmpInt (Pos (Succ xw300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
13[label="splitAt2 (Pos Zero) xw4 (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
14[label="splitAt2 (Neg (Succ xw300)) xw4 (not (primCmpInt (Neg (Succ xw300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="splitAt2 (Neg Zero) xw4 (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="splitAt2 (Pos (Succ xw300)) xw4 (not (primCmpNat (Succ xw300) Zero == GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="splitAt2 (Pos Zero) xw4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="splitAt2 (Neg (Succ xw300)) xw4 (not (LT == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="splitAt2 (Neg Zero) xw4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="splitAt2 (Pos (Succ xw300)) xw4 (not (GT == GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="splitAt2 (Pos Zero) xw4 (not False)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="splitAt2 (Neg (Succ xw300)) xw4 (not False)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="splitAt2 (Neg Zero) xw4 (not False)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="splitAt2 (Pos (Succ xw300)) xw4 (not True)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="splitAt2 (Pos Zero) xw4 True",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="splitAt2 (Neg (Succ xw300)) xw4 True",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="splitAt2 (Neg Zero) xw4 True",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="splitAt2 (Pos (Succ xw300)) xw4 False",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="([],xw4)",fontsize=16,color="green",shape="box"];30[label="([],xw4)",fontsize=16,color="green",shape="box"];31[label="([],xw4)",fontsize=16,color="green",shape="box"];32[label="splitAt1 (Pos (Succ xw300)) xw4",fontsize=16,color="burlywood",shape="box"];70[label="xw4/xw40 : xw41",fontsize=10,color="white",style="solid",shape="box"];32 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 33[label="",style="solid", color="burlywood", weight=3];
71[label="xw4/[]",fontsize=10,color="white",style="solid",shape="box"];32 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 34[label="",style="solid", color="burlywood", weight=3];
33[label="splitAt1 (Pos (Succ xw300)) (xw40 : xw41)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="splitAt1 (Pos (Succ xw300)) []",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="splitAt0 (Pos (Succ xw300)) (xw40 : xw41)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="([],[])",fontsize=16,color="green",shape="box"];37[label="(xw40 : splitAt0Xs' (Pos (Succ xw300)) xw41,splitAt0Xs'' (Pos (Succ xw300)) xw41)",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
37 -> 39[label="",style="dashed", color="green", weight=3];
38[label="splitAt0Xs' (Pos (Succ xw300)) xw41",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="splitAt0Xs'' (Pos (Succ xw300)) xw41",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40 -> 44[label="",style="dashed", color="red", weight=0];
40[label="splitAt0Xs'0 (Pos (Succ xw300)) xw41 (splitAt0Vu42 (Pos (Succ xw300)) xw41)",fontsize=16,color="magenta"];40 -> 45[label="",style="dashed", color="magenta", weight=3];
41 -> 49[label="",style="dashed", color="red", weight=0];
41[label="splitAt0Xs''0 (Pos (Succ xw300)) xw41 (splitAt0Vu42 (Pos (Succ xw300)) xw41)",fontsize=16,color="magenta"];41 -> 50[label="",style="dashed", color="magenta", weight=3];
45[label="splitAt0Vu42 (Pos (Succ xw300)) xw41",fontsize=16,color="black",shape="triangle"];45 -> 47[label="",style="solid", color="black", weight=3];
44[label="splitAt0Xs'0 (Pos (Succ xw300)) xw41 xw5",fontsize=16,color="burlywood",shape="triangle"];72[label="xw5/(xw50,xw51)",fontsize=10,color="white",style="solid",shape="box"];44 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 48[label="",style="solid", color="burlywood", weight=3];
50 -> 45[label="",style="dashed", color="red", weight=0];
50[label="splitAt0Vu42 (Pos (Succ xw300)) xw41",fontsize=16,color="magenta"];49[label="splitAt0Xs''0 (Pos (Succ xw300)) xw41 xw6",fontsize=16,color="burlywood",shape="triangle"];73[label="xw6/(xw60,xw61)",fontsize=10,color="white",style="solid",shape="box"];49 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 52[label="",style="solid", color="burlywood", weight=3];
47 -> 4[label="",style="dashed", color="red", weight=0];
47[label="splitAt (Pos (Succ xw300) - Pos (Succ Zero)) xw41",fontsize=16,color="magenta"];47 -> 53[label="",style="dashed", color="magenta", weight=3];
47 -> 54[label="",style="dashed", color="magenta", weight=3];
48[label="splitAt0Xs'0 (Pos (Succ xw300)) xw41 (xw50,xw51)",fontsize=16,color="black",shape="box"];48 -> 55[label="",style="solid", color="black", weight=3];
52[label="splitAt0Xs''0 (Pos (Succ xw300)) xw41 (xw60,xw61)",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="Pos (Succ xw300) - Pos (Succ Zero)",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="xw41",fontsize=16,color="green",shape="box"];55[label="xw50",fontsize=16,color="green",shape="box"];56[label="xw61",fontsize=16,color="green",shape="box"];57[label="primMinusInt (Pos (Succ xw300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];57 -> 58[label="",style="solid", color="black", weight=3];
58[label="primMinusNat (Succ xw300) (Succ Zero)",fontsize=16,color="black",shape="box"];58 -> 59[label="",style="solid", color="black", weight=3];
59[label="primMinusNat xw300 Zero",fontsize=16,color="burlywood",shape="box"];74[label="xw300/Succ xw3000",fontsize=10,color="white",style="solid",shape="box"];59 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 60[label="",style="solid", color="burlywood", weight=3];
75[label="xw300/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 61[label="",style="solid", color="burlywood", weight=3];
60[label="primMinusNat (Succ xw3000) Zero",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3];
61[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
62[label="Pos (Succ xw3000)",fontsize=16,color="green",shape="box"];63[label="Pos Zero",fontsize=16,color="green",shape="box"];}

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

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

   new_splitAt0Vu42(xw300, xw41, h) -> new_splitAt(new_primMinusNat(xw300), xw41, h)
   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt(new_primMinusNat(xw300), xw41, h)

The TRS R consists of the following rules:

   new_primMinusNat(Succ(xw3000)) -> Pos(Succ(xw3000))
   new_primMinusNat(Zero) -> Pos(Zero)

The set Q consists of the following terms:

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)

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

(13) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_splitAt0Vu42(xw300, xw41, h) -> new_splitAt(new_primMinusNat(xw300), xw41, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2),new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2))
   (new_splitAt0Vu42(Zero, y1, y2) -> new_splitAt(Pos(Zero), y1, y2),new_splitAt0Vu42(Zero, y1, y2) -> new_splitAt(Pos(Zero), y1, y2))


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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt(new_primMinusNat(xw300), xw41, h)
   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
   new_splitAt0Vu42(Zero, y1, y2) -> new_splitAt(Pos(Zero), y1, y2)

The TRS R consists of the following rules:

   new_primMinusNat(Succ(xw3000)) -> Pos(Succ(xw3000))
   new_primMinusNat(Zero) -> Pos(Zero)

The set Q consists of the following terms:

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)

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

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

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

   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt(new_primMinusNat(xw300), xw41, h)

The TRS R consists of the following rules:

   new_primMinusNat(Succ(xw3000)) -> Pos(Succ(xw3000))
   new_primMinusNat(Zero) -> Pos(Zero)

The set Q consists of the following terms:

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)

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

(17) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt(new_primMinusNat(xw300), xw41, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3),new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3))
   (new_splitAt(Pos(Succ(Zero)), :(y1, y2), y3) -> new_splitAt(Pos(Zero), y2, y3),new_splitAt(Pos(Succ(Zero)), :(y1, y2), y3) -> new_splitAt(Pos(Zero), y2, y3))


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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
   new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3)
   new_splitAt(Pos(Succ(Zero)), :(y1, y2), y3) -> new_splitAt(Pos(Zero), y2, y3)

The TRS R consists of the following rules:

   new_primMinusNat(Succ(xw3000)) -> Pos(Succ(xw3000))
   new_primMinusNat(Zero) -> Pos(Zero)

The set Q consists of the following terms:

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)

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

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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
   new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3)

The TRS R consists of the following rules:

   new_primMinusNat(Succ(xw3000)) -> Pos(Succ(xw3000))
   new_primMinusNat(Zero) -> Pos(Zero)

The set Q consists of the following terms:

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)

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

(21) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

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

   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
   new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3)

R is empty.
The set Q consists of the following terms:

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)

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

(23) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)


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(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
   new_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
   new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(25) 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_splitAt0Vu42(Succ(x0), y1, y2) -> new_splitAt(Pos(Succ(x0)), y1, y2)
The graph contains the following edges 2 >= 2, 3 >= 3


*new_splitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_splitAt(Pos(Succ(x0)), y2, y3)
The graph contains the following edges 2 > 2, 3 >= 3


*new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), h) -> new_splitAt0Vu42(xw300, xw41, h)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3


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(26)
YES