/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 5 ms]
(4) HASKELL
(5) NumRed [SOUND, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) QDP
(9) TransformationProof [EQUIVALENT, 0 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 0 ms]
(12) QDP
(13) UsableRulesProof [EQUIVALENT, 0 ms]
(14) QDP
(15) QReductionProof [EQUIVALENT, 0 ms]
(16) QDP
(17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(18) YES


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
genericDrop :: Integral b => b  ->  [a]  ->  [a];
genericDrop 0 xs = xs;
genericDrop _ [] = [];
genericDrop n (_ : xs)  | n > 0 = genericDrop (n - 1) xs;
genericDrop _ _ = error [];

}
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;
genericDrop :: Integral a => a  ->  [b]  ->  [b];
genericDrop 0 xs = xs;
genericDrop xw [] = [];
genericDrop n (xx : xs)  | n > 0 = genericDrop (n - 1) xs;
genericDrop xy xz = error [];

}
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
"genericDrop 0 xs = xs;
genericDrop xw [] = [];
genericDrop n (xx : xs)|n > 0genericDrop (n - 1) xs;
genericDrop xy xz = error [];
"
is transformed to
"genericDrop zu xs = genericDrop5 zu xs;
genericDrop xw [] = genericDrop3 xw [];
genericDrop n (xx : xs) = genericDrop2 n (xx : xs);
genericDrop xy xz = genericDrop0 xy xz;
"
"genericDrop0 xy xz = error [];
"
"genericDrop1 n xx xs True = genericDrop (n - 1) xs;
genericDrop1 n xx xs False = genericDrop0 n (xx : xs);
"
"genericDrop2 n (xx : xs) = genericDrop1 n xx xs (n > 0);
genericDrop2 yv yw = genericDrop0 yv yw;
"
"genericDrop3 xw [] = [];
genericDrop3 yy yz = genericDrop2 yy yz;
"
"genericDrop4 True zu xs = xs;
genericDrop4 zv zw zx = genericDrop3 zw zx;
"
"genericDrop5 zu xs = genericDrop4 (zu == 0) zu xs;
genericDrop5 zy zz = genericDrop3 zy zz;
"

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

(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;
genericDrop :: Integral a => a  ->  [b]  ->  [b];
genericDrop zu xs = genericDrop5 zu xs;
genericDrop xw [] = genericDrop3 xw [];
genericDrop n (xx : xs) = genericDrop2 n (xx : xs);
genericDrop xy xz = genericDrop0 xy xz;

genericDrop0 xy xz = error [];

genericDrop1 n xx xs True = genericDrop (n - 1) xs;
genericDrop1 n xx xs False = genericDrop0 n (xx : xs);

genericDrop2 n (xx : xs) = genericDrop1 n xx xs (n > 0);
genericDrop2 yv yw = genericDrop0 yv yw;

genericDrop3 xw [] = [];
genericDrop3 yy yz = genericDrop2 yy yz;

genericDrop4 True zu xs = xs;
genericDrop4 zv zw zx = genericDrop3 zw zx;

genericDrop5 zu xs = genericDrop4 (zu == 0) zu xs;
genericDrop5 zy zz = genericDrop3 zy zz;

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

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

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

(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;
genericDrop :: Integral a => a  ->  [b]  ->  [b];
genericDrop zu xs = genericDrop5 zu xs;
genericDrop xw [] = genericDrop3 xw [];
genericDrop n (xx : xs) = genericDrop2 n (xx : xs);
genericDrop xy xz = genericDrop0 xy xz;

genericDrop0 xy xz = error [];

genericDrop1 n xx xs True = genericDrop (n - fromInt (Pos (Succ Zero))) xs;
genericDrop1 n xx xs False = genericDrop0 n (xx : xs);

genericDrop2 n (xx : xs) = genericDrop1 n xx xs (n > fromInt (Pos Zero));
genericDrop2 yv yw = genericDrop0 yv yw;

genericDrop3 xw [] = [];
genericDrop3 yy yz = genericDrop2 yy yz;

genericDrop4 True zu xs = xs;
genericDrop4 zv zw zx = genericDrop3 zw zx;

genericDrop5 zu xs = genericDrop4 (zu == fromInt (Pos Zero)) zu xs;
genericDrop5 zy zz = genericDrop3 zy zz;

}
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.genericDrop",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.genericDrop vuu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.genericDrop vuu3 vuu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="List.genericDrop5 vuu3 vuu4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="List.genericDrop4 (vuu3 == fromInt (Pos Zero)) vuu3 vuu4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="List.genericDrop4 (primEqInt vuu3 (fromInt (Pos Zero))) vuu3 vuu4",fontsize=16,color="burlywood",shape="box"];61[label="vuu3/Pos vuu30",fontsize=10,color="white",style="solid",shape="box"];7 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 8[label="",style="solid", color="burlywood", weight=3];
62[label="vuu3/Neg vuu30",fontsize=10,color="white",style="solid",shape="box"];7 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="List.genericDrop4 (primEqInt (Pos vuu30) (fromInt (Pos Zero))) (Pos vuu30) vuu4",fontsize=16,color="burlywood",shape="box"];63[label="vuu30/Succ vuu300",fontsize=10,color="white",style="solid",shape="box"];8 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 10[label="",style="solid", color="burlywood", weight=3];
64[label="vuu30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 11[label="",style="solid", color="burlywood", weight=3];
9[label="List.genericDrop4 (primEqInt (Neg vuu30) (fromInt (Pos Zero))) (Neg vuu30) vuu4",fontsize=16,color="burlywood",shape="box"];65[label="vuu30/Succ vuu300",fontsize=10,color="white",style="solid",shape="box"];9 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 12[label="",style="solid", color="burlywood", weight=3];
66[label="vuu30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 13[label="",style="solid", color="burlywood", weight=3];
10[label="List.genericDrop4 (primEqInt (Pos (Succ vuu300)) (fromInt (Pos Zero))) (Pos (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
11[label="List.genericDrop4 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (Pos Zero) vuu4",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3];
12[label="List.genericDrop4 (primEqInt (Neg (Succ vuu300)) (fromInt (Pos Zero))) (Neg (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
13[label="List.genericDrop4 (primEqInt (Neg Zero) (fromInt (Pos Zero))) (Neg Zero) vuu4",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
14[label="List.genericDrop4 (primEqInt (Pos (Succ vuu300)) (Pos Zero)) (Pos (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="List.genericDrop4 (primEqInt (Pos Zero) (Pos Zero)) (Pos Zero) vuu4",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="List.genericDrop4 (primEqInt (Neg (Succ vuu300)) (Pos Zero)) (Neg (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="List.genericDrop4 (primEqInt (Neg Zero) (Pos Zero)) (Neg Zero) vuu4",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="List.genericDrop4 False (Pos (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="List.genericDrop4 True (Pos Zero) vuu4",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="List.genericDrop4 False (Neg (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="List.genericDrop4 True (Neg Zero) vuu4",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="List.genericDrop3 (Pos (Succ vuu300)) vuu4",fontsize=16,color="burlywood",shape="box"];67[label="vuu4/vuu40 : vuu41",fontsize=10,color="white",style="solid",shape="box"];22 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 26[label="",style="solid", color="burlywood", weight=3];
68[label="vuu4/[]",fontsize=10,color="white",style="solid",shape="box"];22 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 27[label="",style="solid", color="burlywood", weight=3];
23[label="vuu4",fontsize=16,color="green",shape="box"];24[label="List.genericDrop3 (Neg (Succ vuu300)) vuu4",fontsize=16,color="burlywood",shape="box"];69[label="vuu4/vuu40 : vuu41",fontsize=10,color="white",style="solid",shape="box"];24 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 28[label="",style="solid", color="burlywood", weight=3];
70[label="vuu4/[]",fontsize=10,color="white",style="solid",shape="box"];24 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 29[label="",style="solid", color="burlywood", weight=3];
25[label="vuu4",fontsize=16,color="green",shape="box"];26[label="List.genericDrop3 (Pos (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="List.genericDrop3 (Pos (Succ vuu300)) []",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="List.genericDrop3 (Neg (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="List.genericDrop3 (Neg (Succ vuu300)) []",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="List.genericDrop2 (Pos (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="[]",fontsize=16,color="green",shape="box"];32[label="List.genericDrop2 (Neg (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
33[label="[]",fontsize=16,color="green",shape="box"];34[label="List.genericDrop1 (Pos (Succ vuu300)) vuu40 vuu41 (Pos (Succ vuu300) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="List.genericDrop1 (Neg (Succ vuu300)) vuu40 vuu41 (Neg (Succ vuu300) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="List.genericDrop1 (Pos (Succ vuu300)) vuu40 vuu41 (compare (Pos (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="List.genericDrop1 (Neg (Succ vuu300)) vuu40 vuu41 (compare (Neg (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="List.genericDrop1 (Pos (Succ vuu300)) vuu40 vuu41 (primCmpInt (Pos (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="List.genericDrop1 (Neg (Succ vuu300)) vuu40 vuu41 (primCmpInt (Neg (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40[label="List.genericDrop1 (Pos (Succ vuu300)) vuu40 vuu41 (primCmpInt (Pos (Succ vuu300)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];40 -> 42[label="",style="solid", color="black", weight=3];
41[label="List.genericDrop1 (Neg (Succ vuu300)) vuu40 vuu41 (primCmpInt (Neg (Succ vuu300)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
42[label="List.genericDrop1 (Pos (Succ vuu300)) vuu40 vuu41 (primCmpNat (Succ vuu300) Zero == GT)",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
43[label="List.genericDrop1 (Neg (Succ vuu300)) vuu40 vuu41 (LT == GT)",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
44[label="List.genericDrop1 (Pos (Succ vuu300)) vuu40 vuu41 (GT == GT)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
45[label="List.genericDrop1 (Neg (Succ vuu300)) vuu40 vuu41 False",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
46[label="List.genericDrop1 (Pos (Succ vuu300)) vuu40 vuu41 True",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
47[label="List.genericDrop0 (Neg (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
48 -> 4[label="",style="dashed", color="red", weight=0];
48[label="List.genericDrop (Pos (Succ vuu300) - fromInt (Pos (Succ Zero))) vuu41",fontsize=16,color="magenta"];48 -> 50[label="",style="dashed", color="magenta", weight=3];
48 -> 51[label="",style="dashed", color="magenta", weight=3];
49[label="error []",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3];
50[label="Pos (Succ vuu300) - fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];50 -> 53[label="",style="solid", color="black", weight=3];
51[label="vuu41",fontsize=16,color="green",shape="box"];52[label="error []",fontsize=16,color="red",shape="box"];53[label="primMinusInt (Pos (Succ vuu300)) (fromInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];53 -> 54[label="",style="solid", color="black", weight=3];
54[label="primMinusInt (Pos (Succ vuu300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];54 -> 55[label="",style="solid", color="black", weight=3];
55[label="primMinusNat (Succ vuu300) (Succ Zero)",fontsize=16,color="black",shape="box"];55 -> 56[label="",style="solid", color="black", weight=3];
56[label="primMinusNat vuu300 Zero",fontsize=16,color="burlywood",shape="box"];71[label="vuu300/Succ vuu3000",fontsize=10,color="white",style="solid",shape="box"];56 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 57[label="",style="solid", color="burlywood", weight=3];
72[label="vuu300/Zero",fontsize=10,color="white",style="solid",shape="box"];56 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 58[label="",style="solid", color="burlywood", weight=3];
57[label="primMinusNat (Succ vuu3000) Zero",fontsize=16,color="black",shape="box"];57 -> 59[label="",style="solid", color="black", weight=3];
58[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];58 -> 60[label="",style="solid", color="black", weight=3];
59[label="Pos (Succ vuu3000)",fontsize=16,color="green",shape="box"];60[label="Pos Zero",fontsize=16,color="green",shape="box"];}

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

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

   new_genericDrop(Pos(Succ(vuu300)), :(vuu40, vuu41), ba) -> new_genericDrop(new_primMinusNat(vuu300), vuu41, ba)

The TRS R consists of the following rules:

   new_primMinusNat(Succ(vuu3000)) -> Pos(Succ(vuu3000))
   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.
----------------------------------------

(9) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_genericDrop(Pos(Succ(vuu300)), :(vuu40, vuu41), ba) -> new_genericDrop(new_primMinusNat(vuu300), vuu41, ba) at position [0] we obtained the following new rules [LPAR04]:

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


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

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

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

The TRS R consists of the following rules:

   new_primMinusNat(Succ(vuu3000)) -> Pos(Succ(vuu3000))
   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.
----------------------------------------

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

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

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

The TRS R consists of the following rules:

   new_primMinusNat(Succ(vuu3000)) -> Pos(Succ(vuu3000))
   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) 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.
----------------------------------------

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

   new_genericDrop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericDrop(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.
----------------------------------------

(15) 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)


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

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

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

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


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

(18)
YES