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


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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


H-Termination with start terms of the given HASKELL could not be shown:

(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) MNOCProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) NonTerminationLoopProof [COMPLETE, 0 ms]
        (11) NO
    (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) YES
    (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES
(21) Narrow [COMPLETE, 0 ms]
(22) TRUE


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

(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 ;

enumFromThenMyInt :: MyInt  ->  MyInt  ->  List MyInt;
enumFromThenMyInt = numericEnumFromThen;

iterate :: (a  ->  a)  ->  a  ->  List a;
iterate f x = Cons x (iterate f (f x));

msMyInt :: MyInt  ->  MyInt  ->  MyInt;
msMyInt = primMinusInt;

numericEnumFromThen n m = iterate (psMyInt (msMyInt m n)) n;

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

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;

}

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

(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 ;

enumFromThenMyInt :: MyInt  ->  MyInt  ->  List MyInt;
enumFromThenMyInt = numericEnumFromThen;

iterate :: (a  ->  a)  ->  a  ->  List a;
iterate f x = Cons x (iterate f (f x));

msMyInt :: MyInt  ->  MyInt  ->  MyInt;
msMyInt = primMinusInt;

numericEnumFromThen n m = iterate (psMyInt (msMyInt m n)) n;

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

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;

}

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

(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 ;

enumFromThenMyInt :: MyInt  ->  MyInt  ->  List MyInt;
enumFromThenMyInt = numericEnumFromThen;

iterate :: (a  ->  a)  ->  a  ->  List a;
iterate f x = Cons x (iterate f (f x));

msMyInt :: MyInt  ->  MyInt  ->  MyInt;
msMyInt = primMinusInt;

numericEnumFromThen n m = iterate (psMyInt (msMyInt m n)) n;

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

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;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="enumFromThenMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="enumFromThenMyInt vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="enumFromThenMyInt vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="numericEnumFromThen vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="iterate (psMyInt (msMyInt vx4 vx3)) vx3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="Cons vx3 (iterate (psMyInt (msMyInt vx4 vx3)) (psMyInt (msMyInt vx4 vx3) vx3))",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="green", weight=3];
8 -> 13[label="",style="dashed", color="red", weight=0];
8[label="iterate (psMyInt (msMyInt vx4 vx3)) (psMyInt (msMyInt vx4 vx3) vx3)",fontsize=16,color="magenta"];8 -> 14[label="",style="dashed", color="magenta", weight=3];
14[label="vx3",fontsize=16,color="green",shape="box"];13[label="iterate (psMyInt (msMyInt vx4 vx3)) (psMyInt (msMyInt vx4 vx3) vx5)",fontsize=16,color="black",shape="triangle"];13 -> 16[label="",style="solid", color="black", weight=3];
16[label="Cons (psMyInt (msMyInt vx4 vx3) vx5) (iterate (psMyInt (msMyInt vx4 vx3)) (psMyInt (msMyInt vx4 vx3) (psMyInt (msMyInt vx4 vx3) vx5)))",fontsize=16,color="green",shape="box"];16 -> 17[label="",style="dashed", color="green", weight=3];
16 -> 18[label="",style="dashed", color="green", weight=3];
17[label="psMyInt (msMyInt vx4 vx3) vx5",fontsize=16,color="black",shape="triangle"];17 -> 19[label="",style="solid", color="black", weight=3];
18 -> 13[label="",style="dashed", color="red", weight=0];
18[label="iterate (psMyInt (msMyInt vx4 vx3)) (psMyInt (msMyInt vx4 vx3) (psMyInt (msMyInt vx4 vx3) vx5))",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
19[label="primPlusInt (msMyInt vx4 vx3) vx5",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20 -> 17[label="",style="dashed", color="red", weight=0];
20[label="psMyInt (msMyInt vx4 vx3) vx5",fontsize=16,color="magenta"];21[label="primPlusInt (primMinusInt vx4 vx3) vx5",fontsize=16,color="burlywood",shape="box"];192[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];21 -> 192[label="",style="solid", color="burlywood", weight=9];
192 -> 22[label="",style="solid", color="burlywood", weight=3];
193[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];21 -> 193[label="",style="solid", color="burlywood", weight=9];
193 -> 23[label="",style="solid", color="burlywood", weight=3];
22[label="primPlusInt (primMinusInt (Pos vx40) vx3) vx5",fontsize=16,color="burlywood",shape="box"];194[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];22 -> 194[label="",style="solid", color="burlywood", weight=9];
194 -> 24[label="",style="solid", color="burlywood", weight=3];
195[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];22 -> 195[label="",style="solid", color="burlywood", weight=9];
195 -> 25[label="",style="solid", color="burlywood", weight=3];
23[label="primPlusInt (primMinusInt (Neg vx40) vx3) vx5",fontsize=16,color="burlywood",shape="box"];196[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];23 -> 196[label="",style="solid", color="burlywood", weight=9];
196 -> 26[label="",style="solid", color="burlywood", weight=3];
197[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];23 -> 197[label="",style="solid", color="burlywood", weight=9];
197 -> 27[label="",style="solid", color="burlywood", weight=3];
24[label="primPlusInt (primMinusInt (Pos vx40) (Pos vx30)) vx5",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="primPlusInt (primMinusInt (Pos vx40) (Neg vx30)) vx5",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="primPlusInt (primMinusInt (Neg vx40) (Pos vx30)) vx5",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="primPlusInt (primMinusInt (Neg vx40) (Neg vx30)) vx5",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="primPlusInt (primMinusNat vx40 vx30) vx5",fontsize=16,color="burlywood",shape="triangle"];198[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];28 -> 198[label="",style="solid", color="burlywood", weight=9];
198 -> 32[label="",style="solid", color="burlywood", weight=3];
199[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 199[label="",style="solid", color="burlywood", weight=9];
199 -> 33[label="",style="solid", color="burlywood", weight=3];
29[label="primPlusInt (Pos (primPlusNat vx40 vx30)) vx5",fontsize=16,color="burlywood",shape="box"];200[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];29 -> 200[label="",style="solid", color="burlywood", weight=9];
200 -> 34[label="",style="solid", color="burlywood", weight=3];
201[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];29 -> 201[label="",style="solid", color="burlywood", weight=9];
201 -> 35[label="",style="solid", color="burlywood", weight=3];
30[label="primPlusInt (Neg (primPlusNat vx40 vx30)) vx5",fontsize=16,color="burlywood",shape="box"];202[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];30 -> 202[label="",style="solid", color="burlywood", weight=9];
202 -> 36[label="",style="solid", color="burlywood", weight=3];
203[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];30 -> 203[label="",style="solid", color="burlywood", weight=9];
203 -> 37[label="",style="solid", color="burlywood", weight=3];
31 -> 28[label="",style="dashed", color="red", weight=0];
31[label="primPlusInt (primMinusNat vx30 vx40) vx5",fontsize=16,color="magenta"];31 -> 38[label="",style="dashed", color="magenta", weight=3];
31 -> 39[label="",style="dashed", color="magenta", weight=3];
32[label="primPlusInt (primMinusNat (Succ vx400) vx30) vx5",fontsize=16,color="burlywood",shape="box"];204[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];32 -> 204[label="",style="solid", color="burlywood", weight=9];
204 -> 40[label="",style="solid", color="burlywood", weight=3];
205[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 205[label="",style="solid", color="burlywood", weight=9];
205 -> 41[label="",style="solid", color="burlywood", weight=3];
33[label="primPlusInt (primMinusNat Zero vx30) vx5",fontsize=16,color="burlywood",shape="box"];206[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];33 -> 206[label="",style="solid", color="burlywood", weight=9];
206 -> 42[label="",style="solid", color="burlywood", weight=3];
207[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 207[label="",style="solid", color="burlywood", weight=9];
207 -> 43[label="",style="solid", color="burlywood", weight=3];
34[label="primPlusInt (Pos (primPlusNat vx40 vx30)) (Pos vx50)",fontsize=16,color="black",shape="box"];34 -> 44[label="",style="solid", color="black", weight=3];
35[label="primPlusInt (Pos (primPlusNat vx40 vx30)) (Neg vx50)",fontsize=16,color="black",shape="box"];35 -> 45[label="",style="solid", color="black", weight=3];
36[label="primPlusInt (Neg (primPlusNat vx40 vx30)) (Pos vx50)",fontsize=16,color="black",shape="box"];36 -> 46[label="",style="solid", color="black", weight=3];
37[label="primPlusInt (Neg (primPlusNat vx40 vx30)) (Neg vx50)",fontsize=16,color="black",shape="box"];37 -> 47[label="",style="solid", color="black", weight=3];
38[label="vx30",fontsize=16,color="green",shape="box"];39[label="vx40",fontsize=16,color="green",shape="box"];40[label="primPlusInt (primMinusNat (Succ vx400) (Succ vx300)) vx5",fontsize=16,color="black",shape="box"];40 -> 48[label="",style="solid", color="black", weight=3];
41[label="primPlusInt (primMinusNat (Succ vx400) Zero) vx5",fontsize=16,color="black",shape="box"];41 -> 49[label="",style="solid", color="black", weight=3];
42[label="primPlusInt (primMinusNat Zero (Succ vx300)) vx5",fontsize=16,color="black",shape="box"];42 -> 50[label="",style="solid", color="black", weight=3];
43[label="primPlusInt (primMinusNat Zero Zero) vx5",fontsize=16,color="black",shape="box"];43 -> 51[label="",style="solid", color="black", weight=3];
44[label="Pos (primPlusNat (primPlusNat vx40 vx30) vx50)",fontsize=16,color="green",shape="box"];44 -> 52[label="",style="dashed", color="green", weight=3];
45[label="primMinusNat (primPlusNat vx40 vx30) vx50",fontsize=16,color="burlywood",shape="box"];208[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];45 -> 208[label="",style="solid", color="burlywood", weight=9];
208 -> 53[label="",style="solid", color="burlywood", weight=3];
209[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 209[label="",style="solid", color="burlywood", weight=9];
209 -> 54[label="",style="solid", color="burlywood", weight=3];
46[label="primMinusNat vx50 (primPlusNat vx40 vx30)",fontsize=16,color="burlywood",shape="box"];210[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];46 -> 210[label="",style="solid", color="burlywood", weight=9];
210 -> 55[label="",style="solid", color="burlywood", weight=3];
211[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 211[label="",style="solid", color="burlywood", weight=9];
211 -> 56[label="",style="solid", color="burlywood", weight=3];
47[label="Neg (primPlusNat (primPlusNat vx40 vx30) vx50)",fontsize=16,color="green",shape="box"];47 -> 57[label="",style="dashed", color="green", weight=3];
48 -> 28[label="",style="dashed", color="red", weight=0];
48[label="primPlusInt (primMinusNat vx400 vx300) vx5",fontsize=16,color="magenta"];48 -> 58[label="",style="dashed", color="magenta", weight=3];
48 -> 59[label="",style="dashed", color="magenta", weight=3];
49[label="primPlusInt (Pos (Succ vx400)) vx5",fontsize=16,color="burlywood",shape="box"];212[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];49 -> 212[label="",style="solid", color="burlywood", weight=9];
212 -> 60[label="",style="solid", color="burlywood", weight=3];
213[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];49 -> 213[label="",style="solid", color="burlywood", weight=9];
213 -> 61[label="",style="solid", color="burlywood", weight=3];
50[label="primPlusInt (Neg (Succ vx300)) vx5",fontsize=16,color="burlywood",shape="box"];214[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];50 -> 214[label="",style="solid", color="burlywood", weight=9];
214 -> 62[label="",style="solid", color="burlywood", weight=3];
215[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];50 -> 215[label="",style="solid", color="burlywood", weight=9];
215 -> 63[label="",style="solid", color="burlywood", weight=3];
51[label="primPlusInt (Pos Zero) vx5",fontsize=16,color="burlywood",shape="box"];216[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];51 -> 216[label="",style="solid", color="burlywood", weight=9];
216 -> 64[label="",style="solid", color="burlywood", weight=3];
217[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];51 -> 217[label="",style="solid", color="burlywood", weight=9];
217 -> 65[label="",style="solid", color="burlywood", weight=3];
52[label="primPlusNat (primPlusNat vx40 vx30) vx50",fontsize=16,color="burlywood",shape="triangle"];218[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];52 -> 218[label="",style="solid", color="burlywood", weight=9];
218 -> 66[label="",style="solid", color="burlywood", weight=3];
219[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 219[label="",style="solid", color="burlywood", weight=9];
219 -> 67[label="",style="solid", color="burlywood", weight=3];
53[label="primMinusNat (primPlusNat (Succ vx400) vx30) vx50",fontsize=16,color="burlywood",shape="box"];220[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];53 -> 220[label="",style="solid", color="burlywood", weight=9];
220 -> 68[label="",style="solid", color="burlywood", weight=3];
221[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 221[label="",style="solid", color="burlywood", weight=9];
221 -> 69[label="",style="solid", color="burlywood", weight=3];
54[label="primMinusNat (primPlusNat Zero vx30) vx50",fontsize=16,color="burlywood",shape="box"];222[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];54 -> 222[label="",style="solid", color="burlywood", weight=9];
222 -> 70[label="",style="solid", color="burlywood", weight=3];
223[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];54 -> 223[label="",style="solid", color="burlywood", weight=9];
223 -> 71[label="",style="solid", color="burlywood", weight=3];
55[label="primMinusNat (Succ vx500) (primPlusNat vx40 vx30)",fontsize=16,color="burlywood",shape="box"];224[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];55 -> 224[label="",style="solid", color="burlywood", weight=9];
224 -> 72[label="",style="solid", color="burlywood", weight=3];
225[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];55 -> 225[label="",style="solid", color="burlywood", weight=9];
225 -> 73[label="",style="solid", color="burlywood", weight=3];
56[label="primMinusNat Zero (primPlusNat vx40 vx30)",fontsize=16,color="burlywood",shape="box"];226[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];56 -> 226[label="",style="solid", color="burlywood", weight=9];
226 -> 74[label="",style="solid", color="burlywood", weight=3];
227[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];56 -> 227[label="",style="solid", color="burlywood", weight=9];
227 -> 75[label="",style="solid", color="burlywood", weight=3];
57 -> 52[label="",style="dashed", color="red", weight=0];
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172[label="primPlusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];172 -> 185[label="",style="solid", color="black", weight=3];
173[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];173 -> 186[label="",style="solid", color="black", weight=3];
174[label="vx300",fontsize=16,color="green",shape="box"];175[label="vx400",fontsize=16,color="green",shape="box"];176[label="vx300",fontsize=16,color="green",shape="box"];177[label="vx400",fontsize=16,color="green",shape="box"];178[label="vx500",fontsize=16,color="green",shape="box"];179 -> 126[label="",style="dashed", color="red", weight=0];
179[label="primMinusNat vx4000 vx5000",fontsize=16,color="magenta"];179 -> 187[label="",style="dashed", color="magenta", weight=3];
179 -> 188[label="",style="dashed", color="magenta", weight=3];
180[label="Pos (Succ vx4000)",fontsize=16,color="green",shape="box"];181[label="Neg (Succ vx5000)",fontsize=16,color="green",shape="box"];182[label="Pos Zero",fontsize=16,color="green",shape="box"];183[label="Succ (Succ (primPlusNat vx4000 vx3000))",fontsize=16,color="green",shape="box"];183 -> 189[label="",style="dashed", color="green", weight=3];
184[label="Succ vx4000",fontsize=16,color="green",shape="box"];185[label="Succ vx3000",fontsize=16,color="green",shape="box"];186[label="Zero",fontsize=16,color="green",shape="box"];187[label="vx4000",fontsize=16,color="green",shape="box"];188[label="vx5000",fontsize=16,color="green",shape="box"];189 -> 138[label="",style="dashed", color="red", weight=0];
189[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];189 -> 190[label="",style="dashed", color="magenta", weight=3];
189 -> 191[label="",style="dashed", color="magenta", weight=3];
190[label="vx3000",fontsize=16,color="green",shape="box"];191[label="vx4000",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

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

   new_iterate(vx4, vx3, vx5) -> new_iterate(vx4, vx3, new_psMyInt(vx4, vx3, vx5))

The TRS R consists of the following rules:

   new_primMinusNat0(vx400, Main.Succ(vx500)) -> new_primMinusNat1(vx400, vx500)
   new_primPlusInt0(Main.Zero, Main.Succ(vx300), Main.Pos(vx50)) -> new_primMinusNat3(vx50, vx300)
   new_primPlusNat1(Main.Succ(vx4000), Main.Zero) -> Main.Succ(vx4000)
   new_primPlusNat1(Main.Zero, Main.Succ(vx3000)) -> Main.Succ(vx3000)
   new_primMinusNat2(Main.Zero) -> Main.Pos(Main.Zero)
   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Zero)) -> new_primMinusNat2(Main.Zero)
   new_primMinusNat3(Main.Succ(vx500), vx300) -> new_primMinusNat1(vx500, vx300)
   new_psMyInt(Main.Neg(Main.Succ(vx400)), Main.Pos(Main.Succ(vx300)), Main.Pos(Main.Succ(vx500))) -> new_primMinusNat3(Main.Succ(vx500), Main.Succ(new_primPlusNat1(vx400, vx300)))
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Pos(vx50)) -> Main.Pos(new_primPlusNat3(vx50))
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Neg(vx50)) -> new_primMinusNat2(vx50)
   new_primPlusNat1(Main.Zero, Main.Zero) -> Main.Zero
   new_primPlusInt0(Main.Succ(vx400), Main.Zero, Main.Neg(vx50)) -> new_primMinusNat0(vx400, vx50)
   new_primPlusNat2(Main.Succ(vx400), Main.Succ(vx300), vx50) -> new_primPlusNat0(Main.Succ(new_primPlusNat1(vx400, vx300)), vx50)
   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Succ(vx500))) -> new_primMinusNat0(vx500, Main.Zero)
   new_psMyInt(Main.Pos(Main.Succ(vx400)), Main.Neg(Main.Zero), Main.Neg(vx50)) -> new_primMinusNat0(vx400, vx50)
   new_psMyInt(Main.Pos(Main.Zero), Main.Neg(Main.Succ(vx300)), Main.Neg(vx50)) -> new_primMinusNat0(vx300, vx50)
   new_primPlusNat1(Main.Succ(vx4000), Main.Succ(vx3000)) -> Main.Succ(Main.Succ(new_primPlusNat1(vx4000, vx3000)))
   new_psMyInt(Main.Neg(vx40), Main.Neg(vx30), vx5) -> new_primPlusInt0(vx30, vx40, vx5)
   new_primPlusNat3(Main.Succ(vx500)) -> Main.Succ(vx500)
   new_primMinusNat0(vx400, Main.Zero) -> Main.Pos(Main.Succ(vx400))
   new_psMyInt(Main.Neg(Main.Succ(vx400)), Main.Pos(Main.Zero), Main.Pos(Main.Succ(vx500))) -> new_primMinusNat3(Main.Succ(vx500), vx400)
   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Succ(vx300)), Main.Pos(Main.Succ(vx500))) -> new_primMinusNat3(Main.Succ(vx500), vx300)
   new_primPlusNat0(vx400, Main.Succ(vx500)) -> Main.Succ(Main.Succ(new_primPlusNat1(vx400, vx500)))
   new_primPlusNat3(Main.Zero) -> Main.Zero
   new_primPlusNat2(Main.Zero, Main.Zero, vx50) -> new_primPlusNat3(vx50)
   new_primMinusNat1(Main.Succ(vx4000), Main.Zero) -> Main.Pos(Main.Succ(vx4000))
   new_primPlusInt0(Main.Zero, Main.Succ(vx300), Main.Neg(vx50)) -> Main.Neg(new_primPlusNat0(vx300, vx50))
   new_psMyInt(Main.Neg(Main.Succ(vx400)), Main.Pos(Main.Zero), Main.Pos(Main.Zero)) -> new_primMinusNat3(Main.Zero, vx400)
   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Succ(vx300)), Main.Pos(Main.Zero)) -> new_primMinusNat3(Main.Zero, vx300)
   new_primPlusInt0(Main.Succ(vx400), Main.Succ(vx300), vx5) -> new_primPlusInt0(vx400, vx300, vx5)
   new_psMyInt(Main.Pos(vx40), Main.Neg(vx30), Main.Pos(vx50)) -> Main.Pos(new_primPlusNat2(vx40, vx30, vx50))
   new_primMinusNat1(Main.Zero, Main.Succ(vx5000)) -> Main.Neg(Main.Succ(vx5000))
   new_primMinusNat1(Main.Zero, Main.Zero) -> Main.Pos(Main.Zero)
   new_primMinusNat1(Main.Succ(vx4000), Main.Succ(vx5000)) -> new_primMinusNat1(vx4000, vx5000)
   new_primPlusNat2(Main.Succ(vx400), Main.Zero, vx50) -> new_primPlusNat0(vx400, vx50)
   new_primPlusNat2(Main.Zero, Main.Succ(vx300), vx50) -> new_primPlusNat0(vx300, vx50)
   new_primMinusNat3(Main.Zero, vx300) -> Main.Neg(Main.Succ(vx300))
   new_psMyInt(Main.Neg(vx40), Main.Pos(vx30), Main.Neg(vx50)) -> Main.Neg(new_primPlusNat2(vx40, vx30, vx50))
   new_primPlusNat0(vx400, Main.Zero) -> Main.Succ(vx400)
   new_psMyInt(Main.Neg(Main.Succ(vx400)), Main.Pos(Main.Succ(vx300)), Main.Pos(Main.Zero)) -> new_primMinusNat3(Main.Zero, Main.Succ(new_primPlusNat1(vx400, vx300)))
   new_primPlusInt0(Main.Succ(vx400), Main.Zero, Main.Pos(vx50)) -> Main.Pos(new_primPlusNat0(vx400, vx50))
   new_psMyInt(Main.Pos(vx40), Main.Pos(vx30), vx5) -> new_primPlusInt0(vx40, vx30, vx5)
   new_psMyInt(Main.Pos(Main.Zero), Main.Neg(Main.Zero), Main.Neg(vx50)) -> new_primMinusNat2(vx50)
   new_psMyInt(Main.Pos(Main.Succ(vx400)), Main.Neg(Main.Succ(vx300)), Main.Neg(vx50)) -> new_primMinusNat0(Main.Succ(new_primPlusNat1(vx400, vx300)), vx50)
   new_primMinusNat2(Main.Succ(vx500)) -> Main.Neg(Main.Succ(vx500))

The set Q consists of the following terms:

   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Succ(x0)), Main.Pos(Main.Zero))
   new_psMyInt(Main.Neg(Main.Succ(x0)), Main.Pos(Main.Succ(x1)), Main.Pos(Main.Succ(x2)))
   new_primPlusNat2(Main.Zero, Main.Succ(x0), x1)
   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Succ(x0)))
   new_primPlusNat3(Main.Succ(x0))
   new_primMinusNat1(Main.Succ(x0), Main.Zero)
   new_primPlusNat3(Main.Zero)
   new_primPlusInt0(Main.Zero, Main.Succ(x0), Main.Pos(x1))
   new_psMyInt(Main.Pos(Main.Succ(x0)), Main.Neg(Main.Zero), Main.Neg(x1))
   new_primPlusInt0(Main.Zero, Main.Succ(x0), Main.Neg(x1))
   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Succ(x0)), Main.Pos(Main.Succ(x1)))
   new_psMyInt(Main.Neg(Main.Succ(x0)), Main.Pos(Main.Zero), Main.Pos(Main.Succ(x1)))
   new_primPlusInt0(Main.Succ(x0), Main.Succ(x1), x2)
   new_psMyInt(Main.Neg(x0), Main.Neg(x1), x2)
   new_primPlusNat1(Main.Zero, Main.Zero)
   new_psMyInt(Main.Pos(x0), Main.Pos(x1), x2)
   new_psMyInt(Main.Neg(x0), Main.Pos(x1), Main.Neg(x2))
   new_psMyInt(Main.Pos(Main.Zero), Main.Neg(Main.Zero), Main.Neg(x0))
   new_primPlusInt0(Main.Succ(x0), Main.Zero, Main.Pos(x1))
   new_primMinusNat0(x0, Main.Succ(x1))
   new_psMyInt(Main.Pos(x0), Main.Neg(x1), Main.Pos(x2))
   new_primMinusNat3(Main.Zero, x0)
   new_primMinusNat2(Main.Succ(x0))
   new_primMinusNat3(Main.Succ(x0), x1)
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Neg(x0))
   new_primPlusNat0(x0, Main.Succ(x1))
   new_primPlusNat1(Main.Zero, Main.Succ(x0))
   new_primMinusNat2(Main.Zero)
   new_primMinusNat1(Main.Succ(x0), Main.Succ(x1))
   new_primPlusNat0(x0, Main.Zero)
   new_primPlusNat2(Main.Zero, Main.Zero, x0)
   new_psMyInt(Main.Pos(Main.Succ(x0)), Main.Neg(Main.Succ(x1)), Main.Neg(x2))
   new_primMinusNat1(Main.Zero, Main.Zero)
   new_primPlusNat1(Main.Succ(x0), Main.Succ(x1))
   new_psMyInt(Main.Neg(Main.Succ(x0)), Main.Pos(Main.Zero), Main.Pos(Main.Zero))
   new_primPlusNat1(Main.Succ(x0), Main.Zero)
   new_psMyInt(Main.Neg(Main.Succ(x0)), Main.Pos(Main.Succ(x1)), Main.Pos(Main.Zero))
   new_primPlusNat2(Main.Succ(x0), Main.Succ(x1), x2)
   new_primPlusNat2(Main.Succ(x0), Main.Zero, x1)
   new_primMinusNat0(x0, Main.Zero)
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Pos(x0))
   new_primMinusNat1(Main.Zero, Main.Succ(x0))
   new_psMyInt(Main.Pos(Main.Zero), Main.Neg(Main.Succ(x0)), Main.Neg(x1))
   new_primPlusInt0(Main.Succ(x0), Main.Zero, Main.Neg(x1))
   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Zero))

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

(8) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
----------------------------------------

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_iterate(vx4, vx3, vx5) -> new_iterate(vx4, vx3, new_psMyInt(vx4, vx3, vx5))

The TRS R consists of the following rules:

   new_primMinusNat0(vx400, Main.Succ(vx500)) -> new_primMinusNat1(vx400, vx500)
   new_primPlusInt0(Main.Zero, Main.Succ(vx300), Main.Pos(vx50)) -> new_primMinusNat3(vx50, vx300)
   new_primPlusNat1(Main.Succ(vx4000), Main.Zero) -> Main.Succ(vx4000)
   new_primPlusNat1(Main.Zero, Main.Succ(vx3000)) -> Main.Succ(vx3000)
   new_primMinusNat2(Main.Zero) -> Main.Pos(Main.Zero)
   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Zero)) -> new_primMinusNat2(Main.Zero)
   new_primMinusNat3(Main.Succ(vx500), vx300) -> new_primMinusNat1(vx500, vx300)
   new_psMyInt(Main.Neg(Main.Succ(vx400)), Main.Pos(Main.Succ(vx300)), Main.Pos(Main.Succ(vx500))) -> new_primMinusNat3(Main.Succ(vx500), Main.Succ(new_primPlusNat1(vx400, vx300)))
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Pos(vx50)) -> Main.Pos(new_primPlusNat3(vx50))
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Neg(vx50)) -> new_primMinusNat2(vx50)
   new_primPlusNat1(Main.Zero, Main.Zero) -> Main.Zero
   new_primPlusInt0(Main.Succ(vx400), Main.Zero, Main.Neg(vx50)) -> new_primMinusNat0(vx400, vx50)
   new_primPlusNat2(Main.Succ(vx400), Main.Succ(vx300), vx50) -> new_primPlusNat0(Main.Succ(new_primPlusNat1(vx400, vx300)), vx50)
   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Succ(vx500))) -> new_primMinusNat0(vx500, Main.Zero)
   new_psMyInt(Main.Pos(Main.Succ(vx400)), Main.Neg(Main.Zero), Main.Neg(vx50)) -> new_primMinusNat0(vx400, vx50)
   new_psMyInt(Main.Pos(Main.Zero), Main.Neg(Main.Succ(vx300)), Main.Neg(vx50)) -> new_primMinusNat0(vx300, vx50)
   new_primPlusNat1(Main.Succ(vx4000), Main.Succ(vx3000)) -> Main.Succ(Main.Succ(new_primPlusNat1(vx4000, vx3000)))
   new_psMyInt(Main.Neg(vx40), Main.Neg(vx30), vx5) -> new_primPlusInt0(vx30, vx40, vx5)
   new_primPlusNat3(Main.Succ(vx500)) -> Main.Succ(vx500)
   new_primMinusNat0(vx400, Main.Zero) -> Main.Pos(Main.Succ(vx400))
   new_psMyInt(Main.Neg(Main.Succ(vx400)), Main.Pos(Main.Zero), Main.Pos(Main.Succ(vx500))) -> new_primMinusNat3(Main.Succ(vx500), vx400)
   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Succ(vx300)), Main.Pos(Main.Succ(vx500))) -> new_primMinusNat3(Main.Succ(vx500), vx300)
   new_primPlusNat0(vx400, Main.Succ(vx500)) -> Main.Succ(Main.Succ(new_primPlusNat1(vx400, vx500)))
   new_primPlusNat3(Main.Zero) -> Main.Zero
   new_primPlusNat2(Main.Zero, Main.Zero, vx50) -> new_primPlusNat3(vx50)
   new_primMinusNat1(Main.Succ(vx4000), Main.Zero) -> Main.Pos(Main.Succ(vx4000))
   new_primPlusInt0(Main.Zero, Main.Succ(vx300), Main.Neg(vx50)) -> Main.Neg(new_primPlusNat0(vx300, vx50))
   new_psMyInt(Main.Neg(Main.Succ(vx400)), Main.Pos(Main.Zero), Main.Pos(Main.Zero)) -> new_primMinusNat3(Main.Zero, vx400)
   new_psMyInt(Main.Neg(Main.Zero), Main.Pos(Main.Succ(vx300)), Main.Pos(Main.Zero)) -> new_primMinusNat3(Main.Zero, vx300)
   new_primPlusInt0(Main.Succ(vx400), Main.Succ(vx300), vx5) -> new_primPlusInt0(vx400, vx300, vx5)
   new_psMyInt(Main.Pos(vx40), Main.Neg(vx30), Main.Pos(vx50)) -> Main.Pos(new_primPlusNat2(vx40, vx30, vx50))
   new_primMinusNat1(Main.Zero, Main.Succ(vx5000)) -> Main.Neg(Main.Succ(vx5000))
   new_primMinusNat1(Main.Zero, Main.Zero) -> Main.Pos(Main.Zero)
   new_primMinusNat1(Main.Succ(vx4000), Main.Succ(vx5000)) -> new_primMinusNat1(vx4000, vx5000)
   new_primPlusNat2(Main.Succ(vx400), Main.Zero, vx50) -> new_primPlusNat0(vx400, vx50)
   new_primPlusNat2(Main.Zero, Main.Succ(vx300), vx50) -> new_primPlusNat0(vx300, vx50)
   new_primMinusNat3(Main.Zero, vx300) -> Main.Neg(Main.Succ(vx300))
   new_psMyInt(Main.Neg(vx40), Main.Pos(vx30), Main.Neg(vx50)) -> Main.Neg(new_primPlusNat2(vx40, vx30, vx50))
   new_primPlusNat0(vx400, Main.Zero) -> Main.Succ(vx400)
   new_psMyInt(Main.Neg(Main.Succ(vx400)), Main.Pos(Main.Succ(vx300)), Main.Pos(Main.Zero)) -> new_primMinusNat3(Main.Zero, Main.Succ(new_primPlusNat1(vx400, vx300)))
   new_primPlusInt0(Main.Succ(vx400), Main.Zero, Main.Pos(vx50)) -> Main.Pos(new_primPlusNat0(vx400, vx50))
   new_psMyInt(Main.Pos(vx40), Main.Pos(vx30), vx5) -> new_primPlusInt0(vx40, vx30, vx5)
   new_psMyInt(Main.Pos(Main.Zero), Main.Neg(Main.Zero), Main.Neg(vx50)) -> new_primMinusNat2(vx50)
   new_psMyInt(Main.Pos(Main.Succ(vx400)), Main.Neg(Main.Succ(vx300)), Main.Neg(vx50)) -> new_primMinusNat0(Main.Succ(new_primPlusNat1(vx400, vx300)), vx50)
   new_primMinusNat2(Main.Succ(vx500)) -> Main.Neg(Main.Succ(vx500))

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

(10) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = new_iterate(vx4, vx3, vx5) evaluates to  t =new_iterate(vx4, vx3, new_psMyInt(vx4, vx3, vx5))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [vx5 / new_psMyInt(vx4, vx3, vx5)]
* Semiunifier: [ ]

--------------------------------------------------------------------------------
Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_iterate(vx4, vx3, vx5) to new_iterate(vx4, vx3, new_psMyInt(vx4, vx3, vx5)).




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

(11)
NO

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

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

   new_primPlusInt(Main.Succ(vx400), Main.Succ(vx300), vx5) -> new_primPlusInt(vx400, vx300, vx5)

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

(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_primPlusInt(Main.Succ(vx400), Main.Succ(vx300), vx5) -> new_primPlusInt(vx400, vx300, vx5)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3


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

(14)
YES

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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMinusNat(Main.Succ(vx4000), Main.Succ(vx5000)) -> new_primMinusNat(vx4000, vx5000)

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

(16) 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(vx4000), Main.Succ(vx5000)) -> new_primMinusNat(vx4000, vx5000)
The graph contains the following edges 1 > 1, 2 > 2


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

(17)
YES

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

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

   new_primPlusNat(Main.Succ(vx4000), Main.Succ(vx3000)) -> new_primPlusNat(vx4000, vx3000)

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

(19) 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(vx4000), Main.Succ(vx3000)) -> new_primPlusNat(vx4000, vx3000)
The graph contains the following edges 1 > 1, 2 > 2


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

(20)
YES

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

(21) Narrow (COMPLETE)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="enumFromThenMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="enumFromThenMyInt vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="enumFromThenMyInt vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="numericEnumFromThen vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="iterate (psMyInt (msMyInt vx4 vx3)) vx3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="Cons vx3 (iterate (psMyInt (msMyInt vx4 vx3)) (psMyInt (msMyInt vx4 vx3) vx3))",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="green", weight=3];
8 -> 13[label="",style="dashed", color="red", weight=0];
8[label="iterate (psMyInt (msMyInt vx4 vx3)) (psMyInt (msMyInt vx4 vx3) vx3)",fontsize=16,color="magenta"];8 -> 14[label="",style="dashed", color="magenta", weight=3];
14[label="vx3",fontsize=16,color="green",shape="box"];13[label="iterate (psMyInt (msMyInt vx4 vx3)) (psMyInt (msMyInt vx4 vx3) vx5)",fontsize=16,color="black",shape="triangle"];13 -> 16[label="",style="solid", color="black", weight=3];
16[label="Cons (psMyInt (msMyInt vx4 vx3) vx5) (iterate (psMyInt (msMyInt vx4 vx3)) (psMyInt (msMyInt vx4 vx3) (psMyInt (msMyInt vx4 vx3) vx5)))",fontsize=16,color="green",shape="box"];16 -> 17[label="",style="dashed", color="green", weight=3];
16 -> 18[label="",style="dashed", color="green", weight=3];
17[label="psMyInt (msMyInt vx4 vx3) vx5",fontsize=16,color="black",shape="triangle"];17 -> 19[label="",style="solid", color="black", weight=3];
18 -> 13[label="",style="dashed", color="red", weight=0];
18[label="iterate (psMyInt (msMyInt vx4 vx3)) (psMyInt (msMyInt vx4 vx3) (psMyInt (msMyInt vx4 vx3) vx5))",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
19[label="primPlusInt (msMyInt vx4 vx3) vx5",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20 -> 17[label="",style="dashed", color="red", weight=0];
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246 -> 124[label="",style="solid", color="burlywood", weight=3];
247[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];101 -> 247[label="",style="solid", color="burlywood", weight=9];
247 -> 125[label="",style="solid", color="burlywood", weight=3];
102[label="primMinusNat (Succ vx400) (Succ vx500)",fontsize=16,color="black",shape="box"];102 -> 126[label="",style="solid", color="black", weight=3];
103[label="primMinusNat (Succ vx400) Zero",fontsize=16,color="black",shape="box"];103 -> 127[label="",style="solid", color="black", weight=3];
104[label="primMinusNat (Succ vx500) (Succ vx300)",fontsize=16,color="black",shape="box"];104 -> 128[label="",style="solid", color="black", weight=3];
105[label="primMinusNat Zero (Succ vx300)",fontsize=16,color="black",shape="box"];105 -> 129[label="",style="solid", color="black", weight=3];
106 -> 101[label="",style="dashed", color="red", weight=0];
106[label="primPlusNat (Succ vx300) vx50",fontsize=16,color="magenta"];106 -> 130[label="",style="dashed", color="magenta", weight=3];
106 -> 131[label="",style="dashed", color="magenta", weight=3];
107[label="primPlusNat Zero vx50",fontsize=16,color="burlywood",shape="triangle"];248[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];107 -> 248[label="",style="solid", color="burlywood", weight=9];
248 -> 132[label="",style="solid", color="burlywood", weight=3];
249[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];107 -> 249[label="",style="solid", color="burlywood", weight=9];
249 -> 133[label="",style="solid", color="burlywood", weight=3];
108[label="primMinusNat Zero (Succ vx500)",fontsize=16,color="black",shape="box"];108 -> 134[label="",style="solid", color="black", weight=3];
109[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];109 -> 135[label="",style="solid", color="black", weight=3];
110 -> 101[label="",style="dashed", color="red", weight=0];
110[label="primPlusNat (Succ (Succ (primPlusNat vx400 vx300))) vx50",fontsize=16,color="magenta"];110 -> 136[label="",style="dashed", color="magenta", weight=3];
111 -> 101[label="",style="dashed", color="red", weight=0];
111[label="primPlusNat (Succ vx400) vx50",fontsize=16,color="magenta"];112 -> 101[label="",style="dashed", color="red", weight=0];
112[label="primPlusNat (Succ vx300) vx50",fontsize=16,color="magenta"];112 -> 137[label="",style="dashed", color="magenta", weight=3];
113 -> 107[label="",style="dashed", color="red", weight=0];
113[label="primPlusNat Zero vx50",fontsize=16,color="magenta"];114[label="Succ (primPlusNat vx400 vx300)",fontsize=16,color="green",shape="box"];114 -> 138[label="",style="dashed", color="green", weight=3];
115[label="vx300",fontsize=16,color="green",shape="box"];116 -> 81[label="",style="dashed", color="red", weight=0];
116[label="primMinusNat (Succ vx500) (Succ (Succ (primPlusNat vx400 vx300)))",fontsize=16,color="magenta"];116 -> 139[label="",style="dashed", color="magenta", weight=3];
116 -> 140[label="",style="dashed", color="magenta", weight=3];
117 -> 81[label="",style="dashed", color="red", weight=0];
117[label="primMinusNat (Succ vx500) (Succ vx400)",fontsize=16,color="magenta"];117 -> 141[label="",style="dashed", color="magenta", weight=3];
117 -> 142[label="",style="dashed", color="magenta", weight=3];
118 -> 81[label="",style="dashed", color="red", weight=0];
118[label="primMinusNat (Succ vx500) (Succ vx300)",fontsize=16,color="magenta"];118 -> 143[label="",style="dashed", color="magenta", weight=3];
119 -> 80[label="",style="dashed", color="red", weight=0];
119[label="primMinusNat (Succ vx500) Zero",fontsize=16,color="magenta"];119 -> 144[label="",style="dashed", color="magenta", weight=3];
119 -> 145[label="",style="dashed", color="magenta", weight=3];
120 -> 81[label="",style="dashed", color="red", weight=0];
120[label="primMinusNat Zero (Succ (Succ (primPlusNat vx400 vx300)))",fontsize=16,color="magenta"];120 -> 146[label="",style="dashed", color="magenta", weight=3];
120 -> 147[label="",style="dashed", color="magenta", weight=3];
121 -> 81[label="",style="dashed", color="red", weight=0];
121[label="primMinusNat Zero (Succ vx400)",fontsize=16,color="magenta"];121 -> 148[label="",style="dashed", color="magenta", weight=3];
121 -> 149[label="",style="dashed", color="magenta", weight=3];
122 -> 81[label="",style="dashed", color="red", weight=0];
122[label="primMinusNat Zero (Succ vx300)",fontsize=16,color="magenta"];122 -> 150[label="",style="dashed", color="magenta", weight=3];
123 -> 84[label="",style="dashed", color="red", weight=0];
123[label="primMinusNat Zero Zero",fontsize=16,color="magenta"];123 -> 151[label="",style="dashed", color="magenta", weight=3];
124[label="primPlusNat (Succ vx400) (Succ vx500)",fontsize=16,color="black",shape="box"];124 -> 152[label="",style="solid", color="black", weight=3];
125[label="primPlusNat (Succ vx400) Zero",fontsize=16,color="black",shape="box"];125 -> 153[label="",style="solid", color="black", weight=3];
126[label="primMinusNat vx400 vx500",fontsize=16,color="burlywood",shape="triangle"];250[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];126 -> 250[label="",style="solid", color="burlywood", weight=9];
250 -> 154[label="",style="solid", color="burlywood", weight=3];
251[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];126 -> 251[label="",style="solid", color="burlywood", weight=9];
251 -> 155[label="",style="solid", color="burlywood", weight=3];
127[label="Pos (Succ vx400)",fontsize=16,color="green",shape="box"];128 -> 126[label="",style="dashed", color="red", weight=0];
128[label="primMinusNat vx500 vx300",fontsize=16,color="magenta"];128 -> 156[label="",style="dashed", color="magenta", weight=3];
128 -> 157[label="",style="dashed", color="magenta", weight=3];
129[label="Neg (Succ vx300)",fontsize=16,color="green",shape="box"];130[label="vx300",fontsize=16,color="green",shape="box"];131[label="vx50",fontsize=16,color="green",shape="box"];132[label="primPlusNat Zero (Succ vx500)",fontsize=16,color="black",shape="box"];132 -> 158[label="",style="solid", color="black", weight=3];
133[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];133 -> 159[label="",style="solid", color="black", weight=3];
134[label="Neg (Succ vx500)",fontsize=16,color="green",shape="box"];135[label="Pos Zero",fontsize=16,color="green",shape="box"];136[label="Succ (primPlusNat vx400 vx300)",fontsize=16,color="green",shape="box"];136 -> 160[label="",style="dashed", color="green", weight=3];
137[label="vx300",fontsize=16,color="green",shape="box"];138[label="primPlusNat vx400 vx300",fontsize=16,color="burlywood",shape="triangle"];252[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];138 -> 252[label="",style="solid", color="burlywood", weight=9];
252 -> 161[label="",style="solid", color="burlywood", weight=3];
253[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];138 -> 253[label="",style="solid", color="burlywood", weight=9];
253 -> 162[label="",style="solid", color="burlywood", weight=3];
139[label="Succ (primPlusNat vx400 vx300)",fontsize=16,color="green",shape="box"];139 -> 163[label="",style="dashed", color="green", weight=3];
140[label="Succ vx500",fontsize=16,color="green",shape="box"];141[label="vx400",fontsize=16,color="green",shape="box"];142[label="Succ vx500",fontsize=16,color="green",shape="box"];143[label="Succ vx500",fontsize=16,color="green",shape="box"];144[label="Zero",fontsize=16,color="green",shape="box"];145[label="vx500",fontsize=16,color="green",shape="box"];146[label="Succ (primPlusNat vx400 vx300)",fontsize=16,color="green",shape="box"];146 -> 164[label="",style="dashed", color="green", weight=3];
147[label="Zero",fontsize=16,color="green",shape="box"];148[label="vx400",fontsize=16,color="green",shape="box"];149[label="Zero",fontsize=16,color="green",shape="box"];150[label="Zero",fontsize=16,color="green",shape="box"];151[label="Zero",fontsize=16,color="green",shape="box"];152[label="Succ (Succ (primPlusNat vx400 vx500))",fontsize=16,color="green",shape="box"];152 -> 165[label="",style="dashed", color="green", weight=3];
153[label="Succ vx400",fontsize=16,color="green",shape="box"];154[label="primMinusNat (Succ vx4000) vx500",fontsize=16,color="burlywood",shape="box"];254[label="vx500/Succ vx5000",fontsize=10,color="white",style="solid",shape="box"];154 -> 254[label="",style="solid", color="burlywood", weight=9];
254 -> 166[label="",style="solid", color="burlywood", weight=3];
255[label="vx500/Zero",fontsize=10,color="white",style="solid",shape="box"];154 -> 255[label="",style="solid", color="burlywood", weight=9];
255 -> 167[label="",style="solid", color="burlywood", weight=3];
155[label="primMinusNat Zero vx500",fontsize=16,color="burlywood",shape="box"];256[label="vx500/Succ vx5000",fontsize=10,color="white",style="solid",shape="box"];155 -> 256[label="",style="solid", color="burlywood", weight=9];
256 -> 168[label="",style="solid", color="burlywood", weight=3];
257[label="vx500/Zero",fontsize=10,color="white",style="solid",shape="box"];155 -> 257[label="",style="solid", color="burlywood", weight=9];
257 -> 169[label="",style="solid", color="burlywood", weight=3];
156[label="vx500",fontsize=16,color="green",shape="box"];157[label="vx300",fontsize=16,color="green",shape="box"];158[label="Succ vx500",fontsize=16,color="green",shape="box"];159[label="Zero",fontsize=16,color="green",shape="box"];160 -> 138[label="",style="dashed", color="red", weight=0];
160[label="primPlusNat vx400 vx300",fontsize=16,color="magenta"];161[label="primPlusNat (Succ vx4000) vx300",fontsize=16,color="burlywood",shape="box"];258[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];161 -> 258[label="",style="solid", color="burlywood", weight=9];
258 -> 170[label="",style="solid", color="burlywood", weight=3];
259[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];161 -> 259[label="",style="solid", color="burlywood", weight=9];
259 -> 171[label="",style="solid", color="burlywood", weight=3];
162[label="primPlusNat Zero vx300",fontsize=16,color="burlywood",shape="box"];260[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];162 -> 260[label="",style="solid", color="burlywood", weight=9];
260 -> 172[label="",style="solid", color="burlywood", weight=3];
261[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];162 -> 261[label="",style="solid", color="burlywood", weight=9];
261 -> 173[label="",style="solid", color="burlywood", weight=3];
163 -> 138[label="",style="dashed", color="red", weight=0];
163[label="primPlusNat vx400 vx300",fontsize=16,color="magenta"];163 -> 174[label="",style="dashed", color="magenta", weight=3];
163 -> 175[label="",style="dashed", color="magenta", weight=3];
164 -> 138[label="",style="dashed", color="red", weight=0];
164[label="primPlusNat vx400 vx300",fontsize=16,color="magenta"];164 -> 176[label="",style="dashed", color="magenta", weight=3];
164 -> 177[label="",style="dashed", color="magenta", weight=3];
165 -> 138[label="",style="dashed", color="red", weight=0];
165[label="primPlusNat vx400 vx500",fontsize=16,color="magenta"];165 -> 178[label="",style="dashed", color="magenta", weight=3];
166[label="primMinusNat (Succ vx4000) (Succ vx5000)",fontsize=16,color="black",shape="box"];166 -> 179[label="",style="solid", color="black", weight=3];
167[label="primMinusNat (Succ vx4000) Zero",fontsize=16,color="black",shape="box"];167 -> 180[label="",style="solid", color="black", weight=3];
168[label="primMinusNat Zero (Succ vx5000)",fontsize=16,color="black",shape="box"];168 -> 181[label="",style="solid", color="black", weight=3];
169[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];169 -> 182[label="",style="solid", color="black", weight=3];
170[label="primPlusNat (Succ vx4000) (Succ vx3000)",fontsize=16,color="black",shape="box"];170 -> 183[label="",style="solid", color="black", weight=3];
171[label="primPlusNat (Succ vx4000) Zero",fontsize=16,color="black",shape="box"];171 -> 184[label="",style="solid", color="black", weight=3];
172[label="primPlusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];172 -> 185[label="",style="solid", color="black", weight=3];
173[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];173 -> 186[label="",style="solid", color="black", weight=3];
174[label="vx300",fontsize=16,color="green",shape="box"];175[label="vx400",fontsize=16,color="green",shape="box"];176[label="vx300",fontsize=16,color="green",shape="box"];177[label="vx400",fontsize=16,color="green",shape="box"];178[label="vx500",fontsize=16,color="green",shape="box"];179 -> 126[label="",style="dashed", color="red", weight=0];
179[label="primMinusNat vx4000 vx5000",fontsize=16,color="magenta"];179 -> 187[label="",style="dashed", color="magenta", weight=3];
179 -> 188[label="",style="dashed", color="magenta", weight=3];
180[label="Pos (Succ vx4000)",fontsize=16,color="green",shape="box"];181[label="Neg (Succ vx5000)",fontsize=16,color="green",shape="box"];182[label="Pos Zero",fontsize=16,color="green",shape="box"];183[label="Succ (Succ (primPlusNat vx4000 vx3000))",fontsize=16,color="green",shape="box"];183 -> 189[label="",style="dashed", color="green", weight=3];
184[label="Succ vx4000",fontsize=16,color="green",shape="box"];185[label="Succ vx3000",fontsize=16,color="green",shape="box"];186[label="Zero",fontsize=16,color="green",shape="box"];187[label="vx4000",fontsize=16,color="green",shape="box"];188[label="vx5000",fontsize=16,color="green",shape="box"];189 -> 138[label="",style="dashed", color="red", weight=0];
189[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];189 -> 190[label="",style="dashed", color="magenta", weight=3];
189 -> 191[label="",style="dashed", color="magenta", weight=3];
190[label="vx3000",fontsize=16,color="green",shape="box"];191[label="vx4000",fontsize=16,color="green",shape="box"];}

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

(22)
TRUE