/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) AND
    (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (12) YES
    (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Main.WHNF a = WHNF a ;

dsEm :: (a  ->  b)  ->  a  ->  b;
dsEm f x = Main.seq x (f x);

enforceWHNF :: Main.WHNF b  ->  a  ->  a;
enforceWHNF (Main.WHNF x) y = y;

foldl' :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
foldl' f a Nil = a;
foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;

fromIntMyInt :: MyInt  ->  MyInt;
fromIntMyInt x = x;

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

psMyInt :: MyInt  ->  MyInt  ->  MyInt;
psMyInt = primPlusInt;

seq :: b  ->  a  ->  a;
seq x y = Main.enforceWHNF (Main.WHNF x) y;

sumMyInt :: List MyInt  ->  MyInt;
sumMyInt = foldl' psMyInt (fromIntMyInt (Main.Pos Main.Zero));

}

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

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Main.WHNF a = WHNF a ;

dsEm :: (a  ->  b)  ->  a  ->  b;
dsEm f x = Main.seq x (f x);

enforceWHNF :: Main.WHNF a  ->  b  ->  b;
enforceWHNF (Main.WHNF x) y = y;

foldl' :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
foldl' f a Nil = a;
foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;

fromIntMyInt :: MyInt  ->  MyInt;
fromIntMyInt x = x;

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

psMyInt :: MyInt  ->  MyInt  ->  MyInt;
psMyInt = primPlusInt;

seq :: a  ->  b  ->  b;
seq x y = Main.enforceWHNF (Main.WHNF x) y;

sumMyInt :: List MyInt  ->  MyInt;
sumMyInt = foldl' psMyInt (fromIntMyInt (Main.Pos Main.Zero));

}

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

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

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

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Main.WHNF a = WHNF a ;

dsEm :: (a  ->  b)  ->  a  ->  b;
dsEm f x = Main.seq x (f x);

enforceWHNF :: Main.WHNF b  ->  a  ->  a;
enforceWHNF (Main.WHNF x) y = y;

foldl' :: (a  ->  b  ->  a)  ->  a  ->  List b  ->  a;
foldl' f a Nil = a;
foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;

fromIntMyInt :: MyInt  ->  MyInt;
fromIntMyInt x = x;

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

psMyInt :: MyInt  ->  MyInt  ->  MyInt;
psMyInt = primPlusInt;

seq :: b  ->  a  ->  a;
seq x y = Main.enforceWHNF (Main.WHNF x) y;

sumMyInt :: List MyInt  ->  MyInt;
sumMyInt = foldl' psMyInt (fromIntMyInt (Main.Pos Main.Zero));

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="sumMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="sumMyInt vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl' psMyInt (fromIntMyInt (Pos Zero)) vx3",fontsize=16,color="burlywood",shape="box"];67[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 5[label="",style="solid", color="burlywood", weight=3];
68[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl' psMyInt (fromIntMyInt (Pos Zero)) (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl' psMyInt (fromIntMyInt (Pos Zero)) Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="dsEm (foldl' psMyInt) (psMyInt (fromIntMyInt (Pos Zero)) vx30) vx31",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="fromIntMyInt (Pos Zero)",fontsize=16,color="black",shape="triangle"];8 -> 10[label="",style="solid", color="black", weight=3];
9 -> 11[label="",style="dashed", color="red", weight=0];
9[label="seq (psMyInt (fromIntMyInt (Pos Zero)) vx30) (foldl' psMyInt (psMyInt (fromIntMyInt (Pos Zero)) vx30)) vx31",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
9 -> 13[label="",style="dashed", color="magenta", weight=3];
10[label="Pos Zero",fontsize=16,color="green",shape="box"];12 -> 8[label="",style="dashed", color="red", weight=0];
12[label="fromIntMyInt (Pos Zero)",fontsize=16,color="magenta"];13 -> 8[label="",style="dashed", color="red", weight=0];
13[label="fromIntMyInt (Pos Zero)",fontsize=16,color="magenta"];11[label="seq (psMyInt vx4 vx30) (foldl' psMyInt (psMyInt vx5 vx30)) vx31",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3];
14[label="enforceWHNF (WHNF (psMyInt vx4 vx30)) (foldl' psMyInt (psMyInt vx5 vx30)) vx31",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="foldl' psMyInt (psMyInt vx5 vx30) vx31",fontsize=16,color="burlywood",shape="box"];69[label="vx31/Cons vx310 vx311",fontsize=10,color="white",style="solid",shape="box"];15 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 16[label="",style="solid", color="burlywood", weight=3];
70[label="vx31/Nil",fontsize=10,color="white",style="solid",shape="box"];15 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 17[label="",style="solid", color="burlywood", weight=3];
16[label="foldl' psMyInt (psMyInt vx5 vx30) (Cons vx310 vx311)",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="foldl' psMyInt (psMyInt vx5 vx30) Nil",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="dsEm (foldl' psMyInt) (psMyInt (psMyInt vx5 vx30) vx310) vx311",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="psMyInt vx5 vx30",fontsize=16,color="black",shape="triangle"];19 -> 21[label="",style="solid", color="black", weight=3];
20 -> 11[label="",style="dashed", color="red", weight=0];
20[label="seq (psMyInt (psMyInt vx5 vx30) vx310) (foldl' psMyInt (psMyInt (psMyInt vx5 vx30) vx310)) vx311",fontsize=16,color="magenta"];20 -> 22[label="",style="dashed", color="magenta", weight=3];
20 -> 23[label="",style="dashed", color="magenta", weight=3];
20 -> 24[label="",style="dashed", color="magenta", weight=3];
20 -> 25[label="",style="dashed", color="magenta", weight=3];
21[label="primPlusInt vx5 vx30",fontsize=16,color="burlywood",shape="box"];71[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];21 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 26[label="",style="solid", color="burlywood", weight=3];
72[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];21 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 27[label="",style="solid", color="burlywood", weight=3];
22 -> 19[label="",style="dashed", color="red", weight=0];
22[label="psMyInt vx5 vx30",fontsize=16,color="magenta"];23[label="vx311",fontsize=16,color="green",shape="box"];24[label="vx310",fontsize=16,color="green",shape="box"];25 -> 19[label="",style="dashed", color="red", weight=0];
25[label="psMyInt vx5 vx30",fontsize=16,color="magenta"];26[label="primPlusInt (Pos vx50) vx30",fontsize=16,color="burlywood",shape="box"];73[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];26 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 28[label="",style="solid", color="burlywood", weight=3];
74[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];26 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 29[label="",style="solid", color="burlywood", weight=3];
27[label="primPlusInt (Neg vx50) vx30",fontsize=16,color="burlywood",shape="box"];75[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];27 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 30[label="",style="solid", color="burlywood", weight=3];
76[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];27 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 31[label="",style="solid", color="burlywood", weight=3];
28[label="primPlusInt (Pos vx50) (Pos vx300)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="primPlusInt (Pos vx50) (Neg vx300)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="primPlusInt (Neg vx50) (Pos vx300)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="primPlusInt (Neg vx50) (Neg vx300)",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="Pos (primPlusNat vx50 vx300)",fontsize=16,color="green",shape="box"];32 -> 36[label="",style="dashed", color="green", weight=3];
33[label="primMinusNat vx50 vx300",fontsize=16,color="burlywood",shape="triangle"];77[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];33 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 37[label="",style="solid", color="burlywood", weight=3];
78[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 38[label="",style="solid", color="burlywood", weight=3];
34 -> 33[label="",style="dashed", color="red", weight=0];
34[label="primMinusNat vx300 vx50",fontsize=16,color="magenta"];34 -> 39[label="",style="dashed", color="magenta", weight=3];
34 -> 40[label="",style="dashed", color="magenta", weight=3];
35[label="Neg (primPlusNat vx50 vx300)",fontsize=16,color="green",shape="box"];35 -> 41[label="",style="dashed", color="green", weight=3];
36[label="primPlusNat vx50 vx300",fontsize=16,color="burlywood",shape="triangle"];79[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];36 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 42[label="",style="solid", color="burlywood", weight=3];
80[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 43[label="",style="solid", color="burlywood", weight=3];
37[label="primMinusNat (Succ vx500) vx300",fontsize=16,color="burlywood",shape="box"];81[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];37 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 44[label="",style="solid", color="burlywood", weight=3];
82[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];37 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 45[label="",style="solid", color="burlywood", weight=3];
38[label="primMinusNat Zero vx300",fontsize=16,color="burlywood",shape="box"];83[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];38 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 46[label="",style="solid", color="burlywood", weight=3];
84[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];38 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 47[label="",style="solid", color="burlywood", weight=3];
39[label="vx50",fontsize=16,color="green",shape="box"];40[label="vx300",fontsize=16,color="green",shape="box"];41 -> 36[label="",style="dashed", color="red", weight=0];
41[label="primPlusNat vx50 vx300",fontsize=16,color="magenta"];41 -> 48[label="",style="dashed", color="magenta", weight=3];
41 -> 49[label="",style="dashed", color="magenta", weight=3];
42[label="primPlusNat (Succ vx500) vx300",fontsize=16,color="burlywood",shape="box"];85[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];42 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 50[label="",style="solid", color="burlywood", weight=3];
86[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 51[label="",style="solid", color="burlywood", weight=3];
43[label="primPlusNat Zero vx300",fontsize=16,color="burlywood",shape="box"];87[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];43 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 52[label="",style="solid", color="burlywood", weight=3];
88[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];43 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 53[label="",style="solid", color="burlywood", weight=3];
44[label="primMinusNat (Succ vx500) (Succ vx3000)",fontsize=16,color="black",shape="box"];44 -> 54[label="",style="solid", color="black", weight=3];
45[label="primMinusNat (Succ vx500) Zero",fontsize=16,color="black",shape="box"];45 -> 55[label="",style="solid", color="black", weight=3];
46[label="primMinusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];46 -> 56[label="",style="solid", color="black", weight=3];
47[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];47 -> 57[label="",style="solid", color="black", weight=3];
48[label="vx50",fontsize=16,color="green",shape="box"];49[label="vx300",fontsize=16,color="green",shape="box"];50[label="primPlusNat (Succ vx500) (Succ vx3000)",fontsize=16,color="black",shape="box"];50 -> 58[label="",style="solid", color="black", weight=3];
51[label="primPlusNat (Succ vx500) Zero",fontsize=16,color="black",shape="box"];51 -> 59[label="",style="solid", color="black", weight=3];
52[label="primPlusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];52 -> 60[label="",style="solid", color="black", weight=3];
53[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];53 -> 61[label="",style="solid", color="black", weight=3];
54 -> 33[label="",style="dashed", color="red", weight=0];
54[label="primMinusNat vx500 vx3000",fontsize=16,color="magenta"];54 -> 62[label="",style="dashed", color="magenta", weight=3];
54 -> 63[label="",style="dashed", color="magenta", weight=3];
55[label="Pos (Succ vx500)",fontsize=16,color="green",shape="box"];56[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];57[label="Pos Zero",fontsize=16,color="green",shape="box"];58[label="Succ (Succ (primPlusNat vx500 vx3000))",fontsize=16,color="green",shape="box"];58 -> 64[label="",style="dashed", color="green", weight=3];
59[label="Succ vx500",fontsize=16,color="green",shape="box"];60[label="Succ vx3000",fontsize=16,color="green",shape="box"];61[label="Zero",fontsize=16,color="green",shape="box"];62[label="vx3000",fontsize=16,color="green",shape="box"];63[label="vx500",fontsize=16,color="green",shape="box"];64 -> 36[label="",style="dashed", color="red", weight=0];
64[label="primPlusNat vx500 vx3000",fontsize=16,color="magenta"];64 -> 65[label="",style="dashed", color="magenta", weight=3];
64 -> 66[label="",style="dashed", color="magenta", weight=3];
65[label="vx500",fontsize=16,color="green",shape="box"];66[label="vx3000",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMinusNat(Main.Succ(vx500), Main.Succ(vx3000)) -> new_primMinusNat(vx500, vx3000)

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

(8) 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_primMinusNat(Main.Succ(vx500), Main.Succ(vx3000)) -> new_primMinusNat(vx500, vx3000)
The graph contains the following edges 1 > 1, 2 > 2


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

(9)
YES

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

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

   new_primPlusNat(Main.Succ(vx500), Main.Succ(vx3000)) -> new_primPlusNat(vx500, vx3000)

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

(11) 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_primPlusNat(Main.Succ(vx500), Main.Succ(vx3000)) -> new_primPlusNat(vx500, vx3000)
The graph contains the following edges 1 > 1, 2 > 2


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

(12)
YES

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

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

   new_seq(vx4, vx30, vx5, Cons(vx310, vx311)) -> new_seq(new_psMyInt(vx5, vx30), vx310, new_psMyInt(vx5, vx30), vx311)

The TRS R consists of the following rules:

   new_primMinusNat0(Main.Zero, Main.Zero) -> Main.Pos(Main.Zero)
   new_primMinusNat0(Main.Succ(vx500), Main.Succ(vx3000)) -> new_primMinusNat0(vx500, vx3000)
   new_primPlusNat0(Main.Succ(vx500), Main.Succ(vx3000)) -> Main.Succ(Main.Succ(new_primPlusNat0(vx500, vx3000)))
   new_primPlusNat0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNat0(Main.Zero, Main.Succ(vx3000)) -> Main.Neg(Main.Succ(vx3000))
   new_primPlusNat0(Main.Succ(vx500), Main.Zero) -> Main.Succ(vx500)
   new_primPlusNat0(Main.Zero, Main.Succ(vx3000)) -> Main.Succ(vx3000)
   new_psMyInt(Main.Pos(vx50), Main.Neg(vx300)) -> new_primMinusNat0(vx50, vx300)
   new_psMyInt(Main.Neg(vx50), Main.Pos(vx300)) -> new_primMinusNat0(vx300, vx50)
   new_psMyInt(Main.Pos(vx50), Main.Pos(vx300)) -> Main.Pos(new_primPlusNat0(vx50, vx300))
   new_psMyInt(Main.Neg(vx50), Main.Neg(vx300)) -> Main.Neg(new_primPlusNat0(vx50, vx300))
   new_primMinusNat0(Main.Succ(vx500), Main.Zero) -> Main.Pos(Main.Succ(vx500))

The set Q consists of the following terms:

   new_primMinusNat0(Main.Succ(x0), Main.Succ(x1))
   new_psMyInt(Main.Neg(x0), Main.Neg(x1))
   new_primMinusNat0(Main.Succ(x0), Main.Zero)
   new_primPlusNat0(Main.Zero, Main.Zero)
   new_psMyInt(Main.Pos(x0), Main.Neg(x1))
   new_psMyInt(Main.Neg(x0), Main.Pos(x1))
   new_primMinusNat0(Main.Zero, Main.Zero)
   new_primPlusNat0(Main.Zero, Main.Succ(x0))
   new_psMyInt(Main.Pos(x0), Main.Pos(x1))
   new_primPlusNat0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNat0(Main.Zero, Main.Succ(x0))
   new_primPlusNat0(Main.Succ(x0), Main.Zero)

We have to consider all minimal (P,Q,R)-chains.
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(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_seq(vx4, vx30, vx5, Cons(vx310, vx311)) -> new_seq(new_psMyInt(vx5, vx30), vx310, new_psMyInt(vx5, vx30), vx311)
The graph contains the following edges 4 > 2, 4 > 4


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(15)
YES