/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (12) YES
    (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) QDP
        (20) MNOCProof [EQUIVALENT, 0 ms]
        (21) QDP
        (22) NonTerminationLoopProof [COMPLETE, 0 ms]
        (23) NO
(24) Narrow [COMPLETE, 0 ms]
(25) TRUE


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Float = Float MyInt MyInt ;

data List a = Cons a (List a)  | Nil ;

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

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

enumFromThenFloat :: Float  ->  Float  ->  List Float;
enumFromThenFloat = numericEnumFromThen;

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

msFloat :: Float  ->  Float  ->  Float;
msFloat = primMinusFloat;

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

numericEnumFromThen n m = iterate (psFloat (msFloat m n)) n;

primMinusFloat :: Float  ->  Float  ->  Float;
primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);

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;

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

primPlusFloat :: Float  ->  Float  ->  Float;
primPlusFloat (Float x1 x2) (Float y1 y2) = Float (psMyInt x1 y1) (srMyInt x2 y2);

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));

psFloat :: Float  ->  Float  ->  Float;
psFloat = primPlusFloat;

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

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

}

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

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Float = Float MyInt MyInt ;

data List a = Cons a (List a)  | Nil ;

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

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

enumFromThenFloat :: Float  ->  Float  ->  List Float;
enumFromThenFloat = numericEnumFromThen;

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

msFloat :: Float  ->  Float  ->  Float;
msFloat = primMinusFloat;

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

numericEnumFromThen n m = iterate (psFloat (msFloat m n)) n;

primMinusFloat :: Float  ->  Float  ->  Float;
primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);

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;

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

primPlusFloat :: Float  ->  Float  ->  Float;
primPlusFloat (Float x1 x2) (Float y1 y2) = Float (psMyInt x1 y1) (srMyInt x2 y2);

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));

psFloat :: Float  ->  Float  ->  Float;
psFloat = primPlusFloat;

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

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

}

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

(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 Float = Float MyInt MyInt ;

data List a = Cons a (List a)  | Nil ;

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

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

enumFromThenFloat :: Float  ->  Float  ->  List Float;
enumFromThenFloat = numericEnumFromThen;

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

msFloat :: Float  ->  Float  ->  Float;
msFloat = primMinusFloat;

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

numericEnumFromThen n m = iterate (psFloat (msFloat m n)) n;

primMinusFloat :: Float  ->  Float  ->  Float;
primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);

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;

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

primPlusFloat :: Float  ->  Float  ->  Float;
primPlusFloat (Float x1 x2) (Float y1 y2) = Float (psMyInt x1 y1) (srMyInt x2 y2);

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));

psFloat :: Float  ->  Float  ->  Float;
psFloat = primPlusFloat;

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

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="enumFromThenFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="enumFromThenFloat vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="enumFromThenFloat 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 (psFloat (msFloat vx4 vx3)) vx3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="Cons vx3 (iterate (psFloat (msFloat vx4 vx3)) (psFloat (msFloat 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 (psFloat (msFloat vx4 vx3)) (psFloat (msFloat 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 (psFloat (msFloat vx4 vx3)) (psFloat (msFloat vx4 vx3) vx5)",fontsize=16,color="black",shape="triangle"];13 -> 16[label="",style="solid", color="black", weight=3];
16[label="Cons (psFloat (msFloat vx4 vx3) vx5) (iterate (psFloat (msFloat vx4 vx3)) (psFloat (msFloat vx4 vx3) (psFloat (msFloat 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="psFloat (msFloat 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 (psFloat (msFloat vx4 vx3)) (psFloat (msFloat vx4 vx3) (psFloat (msFloat vx4 vx3) vx5))",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
19[label="primPlusFloat (msFloat 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="psFloat (msFloat vx4 vx3) vx5",fontsize=16,color="magenta"];21[label="primPlusFloat (primMinusFloat vx4 vx3) vx5",fontsize=16,color="burlywood",shape="box"];286[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];21 -> 286[label="",style="solid", color="burlywood", weight=9];
286 -> 22[label="",style="solid", color="burlywood", weight=3];
22[label="primPlusFloat (primMinusFloat (Float vx40 vx41) vx3) vx5",fontsize=16,color="burlywood",shape="box"];287[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];22 -> 287[label="",style="solid", color="burlywood", weight=9];
287 -> 23[label="",style="solid", color="burlywood", weight=3];
23[label="primPlusFloat (primMinusFloat (Float vx40 vx41) (Float vx30 vx31)) vx5",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
24[label="primPlusFloat (Float (msMyInt vx40 vx30) (srMyInt vx41 vx31)) vx5",fontsize=16,color="burlywood",shape="box"];288[label="vx5/Float vx50 vx51",fontsize=10,color="white",style="solid",shape="box"];24 -> 288[label="",style="solid", color="burlywood", weight=9];
288 -> 25[label="",style="solid", color="burlywood", weight=3];
25[label="primPlusFloat (Float (msMyInt vx40 vx30) (srMyInt vx41 vx31)) (Float vx50 vx51)",fontsize=16,color="black",shape="box"];25 -> 26[label="",style="solid", color="black", weight=3];
26[label="Float (psMyInt (msMyInt vx40 vx30) vx50) (srMyInt (srMyInt vx41 vx31) vx51)",fontsize=16,color="green",shape="box"];26 -> 27[label="",style="dashed", color="green", weight=3];
26 -> 28[label="",style="dashed", color="green", weight=3];
27[label="psMyInt (msMyInt vx40 vx30) vx50",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="srMyInt (srMyInt vx41 vx31) vx51",fontsize=16,color="black",shape="box"];28 -> 30[label="",style="solid", color="black", weight=3];
29[label="primPlusInt (msMyInt vx40 vx30) vx50",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="primMulInt (srMyInt vx41 vx31) vx51",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
31[label="primPlusInt (primMinusInt vx40 vx30) vx50",fontsize=16,color="burlywood",shape="box"];289[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];31 -> 289[label="",style="solid", color="burlywood", weight=9];
289 -> 33[label="",style="solid", color="burlywood", weight=3];
290[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];31 -> 290[label="",style="solid", color="burlywood", weight=9];
290 -> 34[label="",style="solid", color="burlywood", weight=3];
32[label="primMulInt (primMulInt vx41 vx31) vx51",fontsize=16,color="burlywood",shape="box"];291[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];32 -> 291[label="",style="solid", color="burlywood", weight=9];
291 -> 35[label="",style="solid", color="burlywood", weight=3];
292[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];32 -> 292[label="",style="solid", color="burlywood", weight=9];
292 -> 36[label="",style="solid", color="burlywood", weight=3];
33[label="primPlusInt (primMinusInt (Pos vx400) vx30) vx50",fontsize=16,color="burlywood",shape="box"];293[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];33 -> 293[label="",style="solid", color="burlywood", weight=9];
293 -> 37[label="",style="solid", color="burlywood", weight=3];
294[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];33 -> 294[label="",style="solid", color="burlywood", weight=9];
294 -> 38[label="",style="solid", color="burlywood", weight=3];
34[label="primPlusInt (primMinusInt (Neg vx400) vx30) vx50",fontsize=16,color="burlywood",shape="box"];295[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];34 -> 295[label="",style="solid", color="burlywood", weight=9];
295 -> 39[label="",style="solid", color="burlywood", weight=3];
296[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];34 -> 296[label="",style="solid", color="burlywood", weight=9];
296 -> 40[label="",style="solid", color="burlywood", weight=3];
35[label="primMulInt (primMulInt (Pos vx410) vx31) vx51",fontsize=16,color="burlywood",shape="box"];297[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];35 -> 297[label="",style="solid", color="burlywood", weight=9];
297 -> 41[label="",style="solid", color="burlywood", weight=3];
298[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];35 -> 298[label="",style="solid", color="burlywood", weight=9];
298 -> 42[label="",style="solid", color="burlywood", weight=3];
36[label="primMulInt (primMulInt (Neg vx410) vx31) vx51",fontsize=16,color="burlywood",shape="box"];299[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];36 -> 299[label="",style="solid", color="burlywood", weight=9];
299 -> 43[label="",style="solid", color="burlywood", weight=3];
300[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];36 -> 300[label="",style="solid", color="burlywood", weight=9];
300 -> 44[label="",style="solid", color="burlywood", weight=3];
37[label="primPlusInt (primMinusInt (Pos vx400) (Pos vx300)) vx50",fontsize=16,color="black",shape="box"];37 -> 45[label="",style="solid", color="black", weight=3];
38[label="primPlusInt (primMinusInt (Pos vx400) (Neg vx300)) vx50",fontsize=16,color="black",shape="box"];38 -> 46[label="",style="solid", color="black", weight=3];
39[label="primPlusInt (primMinusInt (Neg vx400) (Pos vx300)) vx50",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3];
40[label="primPlusInt (primMinusInt (Neg vx400) (Neg vx300)) vx50",fontsize=16,color="black",shape="box"];40 -> 48[label="",style="solid", color="black", weight=3];
41[label="primMulInt (primMulInt (Pos vx410) (Pos vx310)) vx51",fontsize=16,color="black",shape="box"];41 -> 49[label="",style="solid", color="black", weight=3];
42[label="primMulInt (primMulInt (Pos vx410) (Neg vx310)) vx51",fontsize=16,color="black",shape="box"];42 -> 50[label="",style="solid", color="black", weight=3];
43[label="primMulInt (primMulInt (Neg vx410) (Pos vx310)) vx51",fontsize=16,color="black",shape="box"];43 -> 51[label="",style="solid", color="black", weight=3];
44[label="primMulInt (primMulInt (Neg vx410) (Neg vx310)) vx51",fontsize=16,color="black",shape="box"];44 -> 52[label="",style="solid", color="black", weight=3];
45[label="primPlusInt (primMinusNat vx400 vx300) vx50",fontsize=16,color="burlywood",shape="triangle"];301[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];45 -> 301[label="",style="solid", color="burlywood", weight=9];
301 -> 53[label="",style="solid", color="burlywood", weight=3];
302[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 302[label="",style="solid", color="burlywood", weight=9];
302 -> 54[label="",style="solid", color="burlywood", weight=3];
46[label="primPlusInt (Pos (primPlusNat vx400 vx300)) vx50",fontsize=16,color="burlywood",shape="box"];303[label="vx50/Pos vx500",fontsize=10,color="white",style="solid",shape="box"];46 -> 303[label="",style="solid", color="burlywood", weight=9];
303 -> 55[label="",style="solid", color="burlywood", weight=3];
304[label="vx50/Neg vx500",fontsize=10,color="white",style="solid",shape="box"];46 -> 304[label="",style="solid", color="burlywood", weight=9];
304 -> 56[label="",style="solid", color="burlywood", weight=3];
47[label="primPlusInt (Neg (primPlusNat vx400 vx300)) vx50",fontsize=16,color="burlywood",shape="box"];305[label="vx50/Pos vx500",fontsize=10,color="white",style="solid",shape="box"];47 -> 305[label="",style="solid", color="burlywood", weight=9];
305 -> 57[label="",style="solid", color="burlywood", weight=3];
306[label="vx50/Neg vx500",fontsize=10,color="white",style="solid",shape="box"];47 -> 306[label="",style="solid", color="burlywood", weight=9];
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134[label="primMinusNat (Succ vx3000) vx500",fontsize=16,color="magenta"];134 -> 162[label="",style="dashed", color="magenta", weight=3];
135 -> 127[label="",style="dashed", color="red", weight=0];
135[label="primMinusNat Zero vx500",fontsize=16,color="magenta"];136[label="primMinusNat (Succ vx5000) (primPlusNat (Succ vx4000) (Succ vx3000))",fontsize=16,color="black",shape="box"];136 -> 163[label="",style="solid", color="black", weight=3];
137[label="primMinusNat (Succ vx5000) (primPlusNat (Succ vx4000) Zero)",fontsize=16,color="black",shape="box"];137 -> 164[label="",style="solid", color="black", weight=3];
138[label="primMinusNat (Succ vx5000) (primPlusNat Zero (Succ vx3000))",fontsize=16,color="black",shape="box"];138 -> 165[label="",style="solid", color="black", weight=3];
139[label="primMinusNat (Succ vx5000) (primPlusNat Zero Zero)",fontsize=16,color="black",shape="box"];139 -> 166[label="",style="solid", color="black", weight=3];
140[label="primMinusNat Zero (primPlusNat (Succ vx4000) (Succ vx3000))",fontsize=16,color="black",shape="box"];140 -> 167[label="",style="solid", color="black", weight=3];
141[label="primMinusNat Zero (primPlusNat (Succ vx4000) Zero)",fontsize=16,color="black",shape="box"];141 -> 168[label="",style="solid", color="black", weight=3];
142[label="primMinusNat Zero (primPlusNat Zero (Succ vx3000))",fontsize=16,color="black",shape="box"];142 -> 169[label="",style="solid", color="black", weight=3];
143[label="primMinusNat Zero (primPlusNat Zero Zero)",fontsize=16,color="black",shape="box"];143 -> 170[label="",style="solid", color="black", weight=3];
250[label="vx310",fontsize=16,color="green",shape="box"];251[label="vx410",fontsize=16,color="green",shape="box"];252[label="primMulNat (Succ vx60) vx510",fontsize=16,color="burlywood",shape="box"];355[label="vx510/Succ vx5100",fontsize=10,color="white",style="solid",shape="box"];252 -> 355[label="",style="solid", color="burlywood", weight=9];
355 -> 269[label="",style="solid", color="burlywood", weight=3];
356[label="vx510/Zero",fontsize=10,color="white",style="solid",shape="box"];252 -> 356[label="",style="solid", color="burlywood", weight=9];
356 -> 270[label="",style="solid", color="burlywood", weight=3];
253[label="primMulNat Zero vx510",fontsize=16,color="burlywood",shape="box"];357[label="vx510/Succ vx5100",fontsize=10,color="white",style="solid",shape="box"];253 -> 357[label="",style="solid", color="burlywood", weight=9];
357 -> 271[label="",style="solid", color="burlywood", weight=3];
358[label="vx510/Zero",fontsize=10,color="white",style="solid",shape="box"];253 -> 358[label="",style="solid", color="burlywood", weight=9];
358 -> 272[label="",style="solid", color="burlywood", weight=3];
254[label="vx310",fontsize=16,color="green",shape="box"];255[label="vx410",fontsize=16,color="green",shape="box"];256[label="vx310",fontsize=16,color="green",shape="box"];257[label="vx410",fontsize=16,color="green",shape="box"];258[label="vx310",fontsize=16,color="green",shape="box"];259[label="vx410",fontsize=16,color="green",shape="box"];148[label="primPlusNat (Succ vx4000) vx500",fontsize=16,color="burlywood",shape="triangle"];359[label="vx500/Succ vx5000",fontsize=10,color="white",style="solid",shape="box"];148 -> 359[label="",style="solid", color="burlywood", weight=9];
359 -> 175[label="",style="solid", color="burlywood", weight=3];
360[label="vx500/Zero",fontsize=10,color="white",style="solid",shape="box"];148 -> 360[label="",style="solid", color="burlywood", weight=9];
360 -> 176[label="",style="solid", color="burlywood", weight=3];
149[label="primMinusNat (Succ vx4000) (Succ vx5000)",fontsize=16,color="black",shape="box"];149 -> 177[label="",style="solid", color="black", weight=3];
150[label="primMinusNat (Succ vx4000) Zero",fontsize=16,color="black",shape="box"];150 -> 178[label="",style="solid", color="black", weight=3];
151[label="primMinusNat (Succ vx5000) (Succ vx3000)",fontsize=16,color="black",shape="box"];151 -> 179[label="",style="solid", color="black", weight=3];
152[label="primMinusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];152 -> 180[label="",style="solid", color="black", weight=3];
153 -> 148[label="",style="dashed", color="red", weight=0];
153[label="primPlusNat (Succ vx3000) vx500",fontsize=16,color="magenta"];153 -> 181[label="",style="dashed", color="magenta", weight=3];
153 -> 182[label="",style="dashed", color="magenta", weight=3];
154[label="primPlusNat Zero vx500",fontsize=16,color="burlywood",shape="triangle"];361[label="vx500/Succ vx5000",fontsize=10,color="white",style="solid",shape="box"];154 -> 361[label="",style="solid", color="burlywood", weight=9];
361 -> 183[label="",style="solid", color="burlywood", weight=3];
362[label="vx500/Zero",fontsize=10,color="white",style="solid",shape="box"];154 -> 362[label="",style="solid", color="burlywood", weight=9];
362 -> 184[label="",style="solid", color="burlywood", weight=3];
155[label="primMinusNat Zero (Succ vx5000)",fontsize=16,color="black",shape="box"];155 -> 185[label="",style="solid", color="black", weight=3];
156[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];156 -> 186[label="",style="solid", color="black", weight=3];
157 -> 148[label="",style="dashed", color="red", weight=0];
157[label="primPlusNat (Succ (Succ (primPlusNat vx4000 vx3000))) vx500",fontsize=16,color="magenta"];157 -> 187[label="",style="dashed", color="magenta", weight=3];
158 -> 148[label="",style="dashed", color="red", weight=0];
158[label="primPlusNat (Succ vx4000) vx500",fontsize=16,color="magenta"];159 -> 148[label="",style="dashed", color="red", weight=0];
159[label="primPlusNat (Succ vx3000) vx500",fontsize=16,color="magenta"];159 -> 188[label="",style="dashed", color="magenta", weight=3];
160 -> 154[label="",style="dashed", color="red", weight=0];
160[label="primPlusNat Zero vx500",fontsize=16,color="magenta"];161[label="Succ (primPlusNat vx4000 vx3000)",fontsize=16,color="green",shape="box"];161 -> 189[label="",style="dashed", color="green", weight=3];
162[label="vx3000",fontsize=16,color="green",shape="box"];163 -> 124[label="",style="dashed", color="red", weight=0];
163[label="primMinusNat (Succ vx5000) (Succ (Succ (primPlusNat vx4000 vx3000)))",fontsize=16,color="magenta"];163 -> 190[label="",style="dashed", color="magenta", weight=3];
163 -> 191[label="",style="dashed", color="magenta", weight=3];
164 -> 124[label="",style="dashed", color="red", weight=0];
164[label="primMinusNat (Succ vx5000) (Succ vx4000)",fontsize=16,color="magenta"];164 -> 192[label="",style="dashed", color="magenta", weight=3];
164 -> 193[label="",style="dashed", color="magenta", weight=3];
165 -> 124[label="",style="dashed", color="red", weight=0];
165[label="primMinusNat (Succ vx5000) (Succ vx3000)",fontsize=16,color="magenta"];165 -> 194[label="",style="dashed", color="magenta", weight=3];
166 -> 123[label="",style="dashed", color="red", weight=0];
166[label="primMinusNat (Succ vx5000) Zero",fontsize=16,color="magenta"];166 -> 195[label="",style="dashed", color="magenta", weight=3];
166 -> 196[label="",style="dashed", color="magenta", weight=3];
167 -> 124[label="",style="dashed", color="red", weight=0];
167[label="primMinusNat Zero (Succ (Succ (primPlusNat vx4000 vx3000)))",fontsize=16,color="magenta"];167 -> 197[label="",style="dashed", color="magenta", weight=3];
167 -> 198[label="",style="dashed", color="magenta", weight=3];
168 -> 124[label="",style="dashed", color="red", weight=0];
168[label="primMinusNat Zero (Succ vx4000)",fontsize=16,color="magenta"];168 -> 199[label="",style="dashed", color="magenta", weight=3];
168 -> 200[label="",style="dashed", color="magenta", weight=3];
169 -> 124[label="",style="dashed", color="red", weight=0];
169[label="primMinusNat Zero (Succ vx3000)",fontsize=16,color="magenta"];169 -> 201[label="",style="dashed", color="magenta", weight=3];
170 -> 127[label="",style="dashed", color="red", weight=0];
170[label="primMinusNat Zero Zero",fontsize=16,color="magenta"];170 -> 202[label="",style="dashed", color="magenta", weight=3];
269[label="primMulNat (Succ vx60) (Succ vx5100)",fontsize=16,color="black",shape="box"];269 -> 276[label="",style="solid", color="black", weight=3];
270[label="primMulNat (Succ vx60) Zero",fontsize=16,color="black",shape="box"];270 -> 277[label="",style="solid", color="black", weight=3];
271[label="primMulNat Zero (Succ vx5100)",fontsize=16,color="black",shape="box"];271 -> 278[label="",style="solid", color="black", weight=3];
272[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];272 -> 279[label="",style="solid", color="black", weight=3];
175[label="primPlusNat (Succ vx4000) (Succ vx5000)",fontsize=16,color="black",shape="box"];175 -> 207[label="",style="solid", color="black", weight=3];
176[label="primPlusNat (Succ vx4000) Zero",fontsize=16,color="black",shape="box"];176 -> 208[label="",style="solid", color="black", weight=3];
177[label="primMinusNat vx4000 vx5000",fontsize=16,color="burlywood",shape="triangle"];363[label="vx4000/Succ vx40000",fontsize=10,color="white",style="solid",shape="box"];177 -> 363[label="",style="solid", color="burlywood", weight=9];
363 -> 209[label="",style="solid", color="burlywood", weight=3];
364[label="vx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];177 -> 364[label="",style="solid", color="burlywood", weight=9];
364 -> 210[label="",style="solid", color="burlywood", weight=3];
178[label="Pos (Succ vx4000)",fontsize=16,color="green",shape="box"];179 -> 177[label="",style="dashed", color="red", weight=0];
179[label="primMinusNat vx5000 vx3000",fontsize=16,color="magenta"];179 -> 211[label="",style="dashed", color="magenta", weight=3];
179 -> 212[label="",style="dashed", color="magenta", weight=3];
180[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];181[label="vx3000",fontsize=16,color="green",shape="box"];182[label="vx500",fontsize=16,color="green",shape="box"];183[label="primPlusNat Zero (Succ vx5000)",fontsize=16,color="black",shape="box"];183 -> 213[label="",style="solid", color="black", weight=3];
184[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];184 -> 214[label="",style="solid", color="black", weight=3];
185[label="Neg (Succ vx5000)",fontsize=16,color="green",shape="box"];186[label="Pos Zero",fontsize=16,color="green",shape="box"];187[label="Succ (primPlusNat vx4000 vx3000)",fontsize=16,color="green",shape="box"];187 -> 215[label="",style="dashed", color="green", weight=3];
188[label="vx3000",fontsize=16,color="green",shape="box"];189[label="primPlusNat vx4000 vx3000",fontsize=16,color="burlywood",shape="triangle"];365[label="vx4000/Succ vx40000",fontsize=10,color="white",style="solid",shape="box"];189 -> 365[label="",style="solid", color="burlywood", weight=9];
365 -> 216[label="",style="solid", color="burlywood", weight=3];
366[label="vx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];189 -> 366[label="",style="solid", color="burlywood", weight=9];
366 -> 217[label="",style="solid", color="burlywood", weight=3];
190[label="Succ vx5000",fontsize=16,color="green",shape="box"];191[label="Succ (primPlusNat vx4000 vx3000)",fontsize=16,color="green",shape="box"];191 -> 218[label="",style="dashed", color="green", weight=3];
192[label="Succ vx5000",fontsize=16,color="green",shape="box"];193[label="vx4000",fontsize=16,color="green",shape="box"];194[label="Succ vx5000",fontsize=16,color="green",shape="box"];195[label="Zero",fontsize=16,color="green",shape="box"];196[label="vx5000",fontsize=16,color="green",shape="box"];197[label="Zero",fontsize=16,color="green",shape="box"];198[label="Succ (primPlusNat vx4000 vx3000)",fontsize=16,color="green",shape="box"];198 -> 219[label="",style="dashed", color="green", weight=3];
199[label="Zero",fontsize=16,color="green",shape="box"];200[label="vx4000",fontsize=16,color="green",shape="box"];201[label="Zero",fontsize=16,color="green",shape="box"];202[label="Zero",fontsize=16,color="green",shape="box"];276 -> 189[label="",style="dashed", color="red", weight=0];
276[label="primPlusNat (primMulNat vx60 (Succ vx5100)) (Succ vx5100)",fontsize=16,color="magenta"];276 -> 282[label="",style="dashed", color="magenta", weight=3];
276 -> 283[label="",style="dashed", color="magenta", weight=3];
277[label="Zero",fontsize=16,color="green",shape="box"];278[label="Zero",fontsize=16,color="green",shape="box"];279[label="Zero",fontsize=16,color="green",shape="box"];207[label="Succ (Succ (primPlusNat vx4000 vx5000))",fontsize=16,color="green",shape="box"];207 -> 224[label="",style="dashed", color="green", weight=3];
208[label="Succ vx4000",fontsize=16,color="green",shape="box"];209[label="primMinusNat (Succ vx40000) vx5000",fontsize=16,color="burlywood",shape="box"];367[label="vx5000/Succ vx50000",fontsize=10,color="white",style="solid",shape="box"];209 -> 367[label="",style="solid", color="burlywood", weight=9];
367 -> 225[label="",style="solid", color="burlywood", weight=3];
368[label="vx5000/Zero",fontsize=10,color="white",style="solid",shape="box"];209 -> 368[label="",style="solid", color="burlywood", weight=9];
368 -> 226[label="",style="solid", color="burlywood", weight=3];
210[label="primMinusNat Zero vx5000",fontsize=16,color="burlywood",shape="box"];369[label="vx5000/Succ vx50000",fontsize=10,color="white",style="solid",shape="box"];210 -> 369[label="",style="solid", color="burlywood", weight=9];
369 -> 227[label="",style="solid", color="burlywood", weight=3];
370[label="vx5000/Zero",fontsize=10,color="white",style="solid",shape="box"];210 -> 370[label="",style="solid", color="burlywood", weight=9];
370 -> 228[label="",style="solid", color="burlywood", weight=3];
211[label="vx3000",fontsize=16,color="green",shape="box"];212[label="vx5000",fontsize=16,color="green",shape="box"];213[label="Succ vx5000",fontsize=16,color="green",shape="box"];214[label="Zero",fontsize=16,color="green",shape="box"];215 -> 189[label="",style="dashed", color="red", weight=0];
215[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];216[label="primPlusNat (Succ vx40000) vx3000",fontsize=16,color="burlywood",shape="box"];371[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];216 -> 371[label="",style="solid", color="burlywood", weight=9];
371 -> 229[label="",style="solid", color="burlywood", weight=3];
372[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];216 -> 372[label="",style="solid", color="burlywood", weight=9];
372 -> 230[label="",style="solid", color="burlywood", weight=3];
217[label="primPlusNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];373[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];217 -> 373[label="",style="solid", color="burlywood", weight=9];
373 -> 231[label="",style="solid", color="burlywood", weight=3];
374[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];217 -> 374[label="",style="solid", color="burlywood", weight=9];
374 -> 232[label="",style="solid", color="burlywood", weight=3];
218 -> 189[label="",style="dashed", color="red", weight=0];
218[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];218 -> 233[label="",style="dashed", color="magenta", weight=3];
218 -> 234[label="",style="dashed", color="magenta", weight=3];
219 -> 189[label="",style="dashed", color="red", weight=0];
219[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];219 -> 235[label="",style="dashed", color="magenta", weight=3];
219 -> 236[label="",style="dashed", color="magenta", weight=3];
282 -> 237[label="",style="dashed", color="red", weight=0];
282[label="primMulNat vx60 (Succ vx5100)",fontsize=16,color="magenta"];282 -> 284[label="",style="dashed", color="magenta", weight=3];
282 -> 285[label="",style="dashed", color="magenta", weight=3];
283[label="Succ vx5100",fontsize=16,color="green",shape="box"];224 -> 189[label="",style="dashed", color="red", weight=0];
224[label="primPlusNat vx4000 vx5000",fontsize=16,color="magenta"];224 -> 260[label="",style="dashed", color="magenta", weight=3];
225[label="primMinusNat (Succ vx40000) (Succ vx50000)",fontsize=16,color="black",shape="box"];225 -> 261[label="",style="solid", color="black", weight=3];
226[label="primMinusNat (Succ vx40000) Zero",fontsize=16,color="black",shape="box"];226 -> 262[label="",style="solid", color="black", weight=3];
227[label="primMinusNat Zero (Succ vx50000)",fontsize=16,color="black",shape="box"];227 -> 263[label="",style="solid", color="black", weight=3];
228[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];228 -> 264[label="",style="solid", color="black", weight=3];
229[label="primPlusNat (Succ vx40000) (Succ vx30000)",fontsize=16,color="black",shape="box"];229 -> 265[label="",style="solid", color="black", weight=3];
230[label="primPlusNat (Succ vx40000) Zero",fontsize=16,color="black",shape="box"];230 -> 266[label="",style="solid", color="black", weight=3];
231[label="primPlusNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];231 -> 267[label="",style="solid", color="black", weight=3];
232[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];232 -> 268[label="",style="solid", color="black", weight=3];
233[label="vx4000",fontsize=16,color="green",shape="box"];234[label="vx3000",fontsize=16,color="green",shape="box"];235[label="vx4000",fontsize=16,color="green",shape="box"];236[label="vx3000",fontsize=16,color="green",shape="box"];284[label="Succ vx5100",fontsize=16,color="green",shape="box"];285[label="vx60",fontsize=16,color="green",shape="box"];260[label="vx5000",fontsize=16,color="green",shape="box"];261 -> 177[label="",style="dashed", color="red", weight=0];
261[label="primMinusNat vx40000 vx50000",fontsize=16,color="magenta"];261 -> 273[label="",style="dashed", color="magenta", weight=3];
261 -> 274[label="",style="dashed", color="magenta", weight=3];
262[label="Pos (Succ vx40000)",fontsize=16,color="green",shape="box"];263[label="Neg (Succ vx50000)",fontsize=16,color="green",shape="box"];264[label="Pos Zero",fontsize=16,color="green",shape="box"];265[label="Succ (Succ (primPlusNat vx40000 vx30000))",fontsize=16,color="green",shape="box"];265 -> 275[label="",style="dashed", color="green", weight=3];
266[label="Succ vx40000",fontsize=16,color="green",shape="box"];267[label="Succ vx30000",fontsize=16,color="green",shape="box"];268[label="Zero",fontsize=16,color="green",shape="box"];273[label="vx50000",fontsize=16,color="green",shape="box"];274[label="vx40000",fontsize=16,color="green",shape="box"];275 -> 189[label="",style="dashed", color="red", weight=0];
275[label="primPlusNat vx40000 vx30000",fontsize=16,color="magenta"];275 -> 280[label="",style="dashed", color="magenta", weight=3];
275 -> 281[label="",style="dashed", color="magenta", weight=3];
280[label="vx40000",fontsize=16,color="green",shape="box"];281[label="vx30000",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

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

   new_primPlusInt(Main.Succ(vx4000), Main.Succ(vx3000), vx50) -> new_primPlusInt(vx4000, vx3000, vx50)

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_primPlusInt(Main.Succ(vx4000), Main.Succ(vx3000), vx50) -> new_primPlusInt(vx4000, vx3000, vx50)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3


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

(9)
YES

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

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

   new_primMulNat(Main.Succ(vx60), Main.Succ(vx5100)) -> new_primMulNat(vx60, Main.Succ(vx5100))

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_primMulNat(Main.Succ(vx60), Main.Succ(vx5100)) -> new_primMulNat(vx60, Main.Succ(vx5100))
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_primMinusNat(Main.Succ(vx40000), Main.Succ(vx50000)) -> new_primMinusNat(vx40000, vx50000)

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

(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_primMinusNat(Main.Succ(vx40000), Main.Succ(vx50000)) -> new_primMinusNat(vx40000, vx50000)
The graph contains the following edges 1 > 1, 2 > 2


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

(15)
YES

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

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

   new_primPlusNat(Main.Succ(vx40000), Main.Succ(vx30000)) -> new_primPlusNat(vx40000, vx30000)

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_primPlusNat(Main.Succ(vx40000), Main.Succ(vx30000)) -> new_primPlusNat(vx40000, vx30000)
The graph contains the following edges 1 > 1, 2 > 2


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

(18)
YES

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

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   new_primPlusNat1(vx4000, Main.Zero) -> Main.Succ(vx4000)
   new_primMulInt0(vx410, vx310, Main.Pos(vx510)) -> Main.Pos(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
   new_primPlusInt1(Main.Pos(vx400), Main.Pos(vx300), vx50) -> new_primPlusInt0(vx400, vx300, vx50)
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Pos(vx500)) -> Main.Pos(new_primPlusNat2(vx500))
   new_primPlusInt1(Main.Neg(vx400), Main.Neg(vx300), vx50) -> new_primPlusInt0(vx300, vx400, vx50)
   new_primPlusNat0(Main.Succ(vx40000), Main.Succ(vx30000)) -> Main.Succ(Main.Succ(new_primPlusNat0(vx40000, vx30000)))
   new_primPlusInt1(Main.Pos(Main.Zero), Main.Neg(Main.Zero), Main.Neg(vx500)) -> new_primMinusNat3(vx500)
   new_primMinusNat0(Main.Zero, vx3000) -> Main.Neg(Main.Succ(vx3000))
   new_primPlusInt0(Main.Zero, Main.Succ(vx3000), Main.Pos(vx500)) -> new_primMinusNat0(vx500, vx3000)
   new_primMulInt(Main.Neg(vx410), Main.Neg(vx310), vx51) -> new_primMulInt0(vx410, vx310, vx51)
   new_primPlusInt1(Main.Pos(vx400), Main.Neg(vx300), Main.Pos(vx500)) -> Main.Pos(new_primPlusNat3(vx400, vx300, vx500))
   new_primMulInt(Main.Pos(vx410), Main.Pos(vx310), vx51) -> new_primMulInt0(vx410, vx310, vx51)
   new_primMinusNat3(Main.Succ(vx5000)) -> Main.Neg(Main.Succ(vx5000))
   new_psFloat(Float(vx40, vx41), Float(vx30, vx31), Float(vx50, vx51)) -> Float(new_primPlusInt1(vx40, vx30, vx50), new_primMulInt(vx41, vx31, vx51))
   new_primMulNat0(Main.Succ(vx60), Main.Succ(vx5100)) -> new_primPlusNat0(new_primMulNat0(vx60, Main.Succ(vx5100)), Main.Succ(vx5100))
   new_primPlusNat2(Main.Zero) -> Main.Zero
   new_primPlusInt0(Main.Succ(vx4000), Main.Succ(vx3000), vx50) -> new_primPlusInt0(vx4000, vx3000, vx50)
   new_primMinusNat1(Main.Zero, Main.Succ(vx50000)) -> Main.Neg(Main.Succ(vx50000))
   new_primMinusNat1(Main.Zero, Main.Zero) -> Main.Pos(Main.Zero)
   new_primPlusNat0(Main.Succ(vx40000), Main.Zero) -> Main.Succ(vx40000)
   new_primPlusNat0(Main.Zero, Main.Succ(vx30000)) -> Main.Succ(vx30000)
   new_primPlusNat1(vx4000, Main.Succ(vx5000)) -> Main.Succ(Main.Succ(new_primPlusNat0(vx4000, vx5000)))
   new_primMulInt0(vx410, vx310, Main.Neg(vx510)) -> Main.Neg(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
   new_primPlusInt0(Main.Zero, Main.Succ(vx3000), Main.Neg(vx500)) -> Main.Neg(new_primPlusNat1(vx3000, vx500))
   new_primPlusNat3(Main.Zero, Main.Zero, vx500) -> new_primPlusNat2(vx500)
   new_primPlusInt1(Main.Neg(Main.Succ(vx4000)), Main.Pos(Main.Zero), Main.Pos(Main.Succ(vx5000))) -> new_primMinusNat0(Main.Succ(vx5000), vx4000)
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Succ(vx3000)), Main.Pos(Main.Succ(vx5000))) -> new_primMinusNat0(Main.Succ(vx5000), vx3000)
   new_primPlusNat3(Main.Succ(vx4000), Main.Zero, vx500) -> new_primPlusNat1(vx4000, vx500)
   new_primPlusNat3(Main.Zero, Main.Succ(vx3000), vx500) -> new_primPlusNat1(vx3000, vx500)
   new_primPlusNat3(Main.Succ(vx4000), Main.Succ(vx3000), vx500) -> new_primPlusNat1(Main.Succ(new_primPlusNat0(vx4000, vx3000)), vx500)
   new_primPlusInt1(Main.Neg(Main.Succ(vx4000)), Main.Pos(Main.Succ(vx3000)), Main.Pos(Main.Succ(vx5000))) -> new_primMinusNat0(Main.Succ(vx5000), Main.Succ(new_primPlusNat0(vx4000, vx3000)))
   new_primPlusInt1(Main.Neg(Main.Succ(vx4000)), Main.Pos(Main.Succ(vx3000)), Main.Pos(Main.Zero)) -> new_primMinusNat0(Main.Zero, Main.Succ(new_primPlusNat0(vx4000, vx3000)))
   new_primPlusInt0(Main.Succ(vx4000), Main.Zero, Main.Pos(vx500)) -> Main.Pos(new_primPlusNat1(vx4000, vx500))
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Zero)) -> new_primMinusNat3(Main.Zero)
   new_primMulNat0(Main.Succ(vx60), Main.Zero) -> Main.Zero
   new_primMulNat0(Main.Zero, Main.Succ(vx5100)) -> Main.Zero
   new_primPlusInt0(Main.Succ(vx4000), Main.Zero, Main.Neg(vx500)) -> new_primMinusNat2(vx4000, vx500)
   new_primPlusInt1(Main.Neg(vx400), Main.Pos(vx300), Main.Neg(vx500)) -> Main.Neg(new_primPlusNat3(vx400, vx300, vx500))
   new_primMulInt(Main.Pos(vx410), Main.Neg(vx310), vx51) -> new_primMulInt1(vx410, vx310, vx51)
   new_primMulInt(Main.Neg(vx410), Main.Pos(vx310), vx51) -> new_primMulInt1(vx410, vx310, vx51)
   new_primMinusNat2(vx4000, Main.Zero) -> Main.Pos(Main.Succ(vx4000))
   new_primMinusNat2(vx4000, Main.Succ(vx5000)) -> new_primMinusNat1(vx4000, vx5000)
   new_primMinusNat1(Main.Succ(vx40000), Main.Zero) -> Main.Pos(Main.Succ(vx40000))
   new_primMinusNat3(Main.Zero) -> Main.Pos(Main.Zero)
   new_primPlusInt1(Main.Neg(Main.Succ(vx4000)), Main.Pos(Main.Zero), Main.Pos(Main.Zero)) -> new_primMinusNat0(Main.Zero, vx4000)
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Succ(vx3000)), Main.Pos(Main.Zero)) -> new_primMinusNat0(Main.Zero, vx3000)
   new_primMinusNat1(Main.Succ(vx40000), Main.Succ(vx50000)) -> new_primMinusNat1(vx40000, vx50000)
   new_primMulInt1(vx410, vx310, Main.Neg(vx510)) -> Main.Pos(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
   new_primPlusInt1(Main.Pos(Main.Succ(vx4000)), Main.Neg(Main.Succ(vx3000)), Main.Neg(vx500)) -> new_primMinusNat2(Main.Succ(new_primPlusNat0(vx4000, vx3000)), vx500)
   new_primMulNat0(Main.Zero, Main.Zero) -> Main.Zero
   new_primPlusNat0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNat0(Main.Succ(vx5000), vx3000) -> new_primMinusNat1(vx5000, vx3000)
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Neg(vx500)) -> new_primMinusNat3(vx500)
   new_primPlusInt1(Main.Pos(Main.Succ(vx4000)), Main.Neg(Main.Zero), Main.Neg(vx500)) -> new_primMinusNat2(vx4000, vx500)
   new_primPlusInt1(Main.Pos(Main.Zero), Main.Neg(Main.Succ(vx3000)), Main.Neg(vx500)) -> new_primMinusNat2(vx3000, vx500)
   new_primMulInt1(vx410, vx310, Main.Pos(vx510)) -> Main.Neg(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Succ(vx5000))) -> new_primMinusNat2(vx5000, Main.Zero)
   new_primPlusNat2(Main.Succ(vx5000)) -> Main.Succ(vx5000)

The set Q consists of the following terms:

   new_primPlusInt1(Main.Neg(Main.Succ(x0)), Main.Pos(Main.Succ(x1)), Main.Pos(Main.Succ(x2)))
   new_primMulNat0(Main.Succ(x0), Main.Zero)
   new_primMinusNat2(x0, Main.Zero)
   new_primPlusNat3(Main.Succ(x0), Main.Succ(x1), x2)
   new_primMulInt(Main.Neg(x0), Main.Neg(x1), x2)
   new_primPlusInt0(Main.Succ(x0), Main.Zero, Main.Neg(x1))
   new_primMulInt(Main.Pos(x0), Main.Neg(x1), x2)
   new_primMulInt(Main.Neg(x0), Main.Pos(x1), x2)
   new_primMinusNat1(Main.Zero, Main.Succ(x0))
   new_primPlusInt0(Main.Zero, Main.Succ(x0), Main.Neg(x1))
   new_primMulInt(Main.Pos(x0), Main.Pos(x1), x2)
   new_primPlusInt0(Main.Succ(x0), Main.Zero, Main.Pos(x1))
   new_primMinusNat0(Main.Zero, x0)
   new_primPlusNat0(Main.Succ(x0), Main.Succ(x1))
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Pos(x0))
   new_primMulNat0(Main.Zero, Main.Succ(x0))
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Succ(x0)), Main.Pos(Main.Zero))
   new_primPlusInt0(Main.Succ(x0), Main.Succ(x1), x2)
   new_primPlusNat0(Main.Zero, Main.Zero)
   new_psFloat(Float(x0, x1), Float(x2, x3), Float(x4, x5))
   new_primPlusNat1(x0, Main.Succ(x1))
   new_primMulInt0(x0, x1, Main.Pos(x2))
   new_primPlusNat3(Main.Zero, Main.Zero, x0)
   new_primMinusNat1(Main.Zero, Main.Zero)
   new_primMinusNat2(x0, Main.Succ(x1))
   new_primPlusInt1(Main.Neg(x0), Main.Pos(x1), Main.Neg(x2))
   new_primMinusNat3(Main.Succ(x0))
   new_primMulNat0(Main.Succ(x0), Main.Succ(x1))
   new_primPlusInt1(Main.Pos(Main.Succ(x0)), Main.Neg(Main.Zero), Main.Neg(x1))
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Zero))
   new_primPlusInt1(Main.Neg(Main.Succ(x0)), Main.Pos(Main.Zero), Main.Pos(Main.Succ(x1)))
   new_primPlusNat1(x0, Main.Zero)
   new_primPlusInt1(Main.Pos(Main.Succ(x0)), Main.Neg(Main.Succ(x1)), Main.Neg(x2))
   new_primPlusInt1(Main.Neg(x0), Main.Neg(x1), x2)
   new_primPlusInt1(Main.Pos(x0), Main.Neg(x1), Main.Pos(x2))
   new_primPlusNat2(Main.Zero)
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Succ(x0)), Main.Pos(Main.Succ(x1)))
   new_primMinusNat3(Main.Zero)
   new_primMinusNat1(Main.Succ(x0), Main.Zero)
   new_primMulInt0(x0, x1, Main.Neg(x2))
   new_primPlusInt1(Main.Pos(Main.Zero), Main.Neg(Main.Succ(x0)), Main.Neg(x1))
   new_primPlusNat3(Main.Zero, Main.Succ(x0), x1)
   new_primPlusInt1(Main.Neg(Main.Succ(x0)), Main.Pos(Main.Succ(x1)), Main.Pos(Main.Zero))
   new_primPlusNat2(Main.Succ(x0))
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Neg(x0))
   new_primPlusInt1(Main.Neg(Main.Succ(x0)), Main.Pos(Main.Zero), Main.Pos(Main.Zero))
   new_primMulInt1(x0, x1, Main.Neg(x2))
   new_primPlusNat0(Main.Zero, Main.Succ(x0))
   new_primPlusInt1(Main.Pos(x0), Main.Pos(x1), x2)
   new_primMulInt1(x0, x1, Main.Pos(x2))
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Succ(x0)))
   new_primPlusInt1(Main.Pos(Main.Zero), Main.Neg(Main.Zero), Main.Neg(x0))
   new_primPlusNat3(Main.Succ(x0), Main.Zero, x1)
   new_primPlusNat0(Main.Succ(x0), Main.Zero)
   new_primPlusInt0(Main.Zero, Main.Succ(x0), Main.Pos(x1))
   new_primMinusNat0(Main.Succ(x0), x1)
   new_primMulNat0(Main.Zero, Main.Zero)
   new_primMinusNat1(Main.Succ(x0), Main.Succ(x1))

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

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

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   new_primPlusNat1(vx4000, Main.Zero) -> Main.Succ(vx4000)
   new_primMulInt0(vx410, vx310, Main.Pos(vx510)) -> Main.Pos(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
   new_primPlusInt1(Main.Pos(vx400), Main.Pos(vx300), vx50) -> new_primPlusInt0(vx400, vx300, vx50)
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Pos(vx500)) -> Main.Pos(new_primPlusNat2(vx500))
   new_primPlusInt1(Main.Neg(vx400), Main.Neg(vx300), vx50) -> new_primPlusInt0(vx300, vx400, vx50)
   new_primPlusNat0(Main.Succ(vx40000), Main.Succ(vx30000)) -> Main.Succ(Main.Succ(new_primPlusNat0(vx40000, vx30000)))
   new_primPlusInt1(Main.Pos(Main.Zero), Main.Neg(Main.Zero), Main.Neg(vx500)) -> new_primMinusNat3(vx500)
   new_primMinusNat0(Main.Zero, vx3000) -> Main.Neg(Main.Succ(vx3000))
   new_primPlusInt0(Main.Zero, Main.Succ(vx3000), Main.Pos(vx500)) -> new_primMinusNat0(vx500, vx3000)
   new_primMulInt(Main.Neg(vx410), Main.Neg(vx310), vx51) -> new_primMulInt0(vx410, vx310, vx51)
   new_primPlusInt1(Main.Pos(vx400), Main.Neg(vx300), Main.Pos(vx500)) -> Main.Pos(new_primPlusNat3(vx400, vx300, vx500))
   new_primMulInt(Main.Pos(vx410), Main.Pos(vx310), vx51) -> new_primMulInt0(vx410, vx310, vx51)
   new_primMinusNat3(Main.Succ(vx5000)) -> Main.Neg(Main.Succ(vx5000))
   new_psFloat(Float(vx40, vx41), Float(vx30, vx31), Float(vx50, vx51)) -> Float(new_primPlusInt1(vx40, vx30, vx50), new_primMulInt(vx41, vx31, vx51))
   new_primMulNat0(Main.Succ(vx60), Main.Succ(vx5100)) -> new_primPlusNat0(new_primMulNat0(vx60, Main.Succ(vx5100)), Main.Succ(vx5100))
   new_primPlusNat2(Main.Zero) -> Main.Zero
   new_primPlusInt0(Main.Succ(vx4000), Main.Succ(vx3000), vx50) -> new_primPlusInt0(vx4000, vx3000, vx50)
   new_primMinusNat1(Main.Zero, Main.Succ(vx50000)) -> Main.Neg(Main.Succ(vx50000))
   new_primMinusNat1(Main.Zero, Main.Zero) -> Main.Pos(Main.Zero)
   new_primPlusNat0(Main.Succ(vx40000), Main.Zero) -> Main.Succ(vx40000)
   new_primPlusNat0(Main.Zero, Main.Succ(vx30000)) -> Main.Succ(vx30000)
   new_primPlusNat1(vx4000, Main.Succ(vx5000)) -> Main.Succ(Main.Succ(new_primPlusNat0(vx4000, vx5000)))
   new_primMulInt0(vx410, vx310, Main.Neg(vx510)) -> Main.Neg(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
   new_primPlusInt0(Main.Zero, Main.Succ(vx3000), Main.Neg(vx500)) -> Main.Neg(new_primPlusNat1(vx3000, vx500))
   new_primPlusNat3(Main.Zero, Main.Zero, vx500) -> new_primPlusNat2(vx500)
   new_primPlusInt1(Main.Neg(Main.Succ(vx4000)), Main.Pos(Main.Zero), Main.Pos(Main.Succ(vx5000))) -> new_primMinusNat0(Main.Succ(vx5000), vx4000)
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Succ(vx3000)), Main.Pos(Main.Succ(vx5000))) -> new_primMinusNat0(Main.Succ(vx5000), vx3000)
   new_primPlusNat3(Main.Succ(vx4000), Main.Zero, vx500) -> new_primPlusNat1(vx4000, vx500)
   new_primPlusNat3(Main.Zero, Main.Succ(vx3000), vx500) -> new_primPlusNat1(vx3000, vx500)
   new_primPlusNat3(Main.Succ(vx4000), Main.Succ(vx3000), vx500) -> new_primPlusNat1(Main.Succ(new_primPlusNat0(vx4000, vx3000)), vx500)
   new_primPlusInt1(Main.Neg(Main.Succ(vx4000)), Main.Pos(Main.Succ(vx3000)), Main.Pos(Main.Succ(vx5000))) -> new_primMinusNat0(Main.Succ(vx5000), Main.Succ(new_primPlusNat0(vx4000, vx3000)))
   new_primPlusInt1(Main.Neg(Main.Succ(vx4000)), Main.Pos(Main.Succ(vx3000)), Main.Pos(Main.Zero)) -> new_primMinusNat0(Main.Zero, Main.Succ(new_primPlusNat0(vx4000, vx3000)))
   new_primPlusInt0(Main.Succ(vx4000), Main.Zero, Main.Pos(vx500)) -> Main.Pos(new_primPlusNat1(vx4000, vx500))
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Zero)) -> new_primMinusNat3(Main.Zero)
   new_primMulNat0(Main.Succ(vx60), Main.Zero) -> Main.Zero
   new_primMulNat0(Main.Zero, Main.Succ(vx5100)) -> Main.Zero
   new_primPlusInt0(Main.Succ(vx4000), Main.Zero, Main.Neg(vx500)) -> new_primMinusNat2(vx4000, vx500)
   new_primPlusInt1(Main.Neg(vx400), Main.Pos(vx300), Main.Neg(vx500)) -> Main.Neg(new_primPlusNat3(vx400, vx300, vx500))
   new_primMulInt(Main.Pos(vx410), Main.Neg(vx310), vx51) -> new_primMulInt1(vx410, vx310, vx51)
   new_primMulInt(Main.Neg(vx410), Main.Pos(vx310), vx51) -> new_primMulInt1(vx410, vx310, vx51)
   new_primMinusNat2(vx4000, Main.Zero) -> Main.Pos(Main.Succ(vx4000))
   new_primMinusNat2(vx4000, Main.Succ(vx5000)) -> new_primMinusNat1(vx4000, vx5000)
   new_primMinusNat1(Main.Succ(vx40000), Main.Zero) -> Main.Pos(Main.Succ(vx40000))
   new_primMinusNat3(Main.Zero) -> Main.Pos(Main.Zero)
   new_primPlusInt1(Main.Neg(Main.Succ(vx4000)), Main.Pos(Main.Zero), Main.Pos(Main.Zero)) -> new_primMinusNat0(Main.Zero, vx4000)
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Succ(vx3000)), Main.Pos(Main.Zero)) -> new_primMinusNat0(Main.Zero, vx3000)
   new_primMinusNat1(Main.Succ(vx40000), Main.Succ(vx50000)) -> new_primMinusNat1(vx40000, vx50000)
   new_primMulInt1(vx410, vx310, Main.Neg(vx510)) -> Main.Pos(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
   new_primPlusInt1(Main.Pos(Main.Succ(vx4000)), Main.Neg(Main.Succ(vx3000)), Main.Neg(vx500)) -> new_primMinusNat2(Main.Succ(new_primPlusNat0(vx4000, vx3000)), vx500)
   new_primMulNat0(Main.Zero, Main.Zero) -> Main.Zero
   new_primPlusNat0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNat0(Main.Succ(vx5000), vx3000) -> new_primMinusNat1(vx5000, vx3000)
   new_primPlusInt0(Main.Zero, Main.Zero, Main.Neg(vx500)) -> new_primMinusNat3(vx500)
   new_primPlusInt1(Main.Pos(Main.Succ(vx4000)), Main.Neg(Main.Zero), Main.Neg(vx500)) -> new_primMinusNat2(vx4000, vx500)
   new_primPlusInt1(Main.Pos(Main.Zero), Main.Neg(Main.Succ(vx3000)), Main.Neg(vx500)) -> new_primMinusNat2(vx3000, vx500)
   new_primMulInt1(vx410, vx310, Main.Pos(vx510)) -> Main.Neg(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
   new_primPlusInt1(Main.Neg(Main.Zero), Main.Pos(Main.Zero), Main.Pos(Main.Succ(vx5000))) -> new_primMinusNat2(vx5000, Main.Zero)
   new_primPlusNat2(Main.Succ(vx5000)) -> Main.Succ(vx5000)

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

(22) 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_psFloat(vx4, vx3, vx5))

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




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

(23)
NO

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

(24) Narrow (COMPLETE)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="enumFromThenFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="enumFromThenFloat vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="enumFromThenFloat 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 (psFloat (msFloat vx4 vx3)) vx3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="Cons vx3 (iterate (psFloat (msFloat vx4 vx3)) (psFloat (msFloat 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 (psFloat (msFloat vx4 vx3)) (psFloat (msFloat 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 (psFloat (msFloat vx4 vx3)) (psFloat (msFloat vx4 vx3) vx5)",fontsize=16,color="black",shape="triangle"];13 -> 16[label="",style="solid", color="black", weight=3];
16[label="Cons (psFloat (msFloat vx4 vx3) vx5) (iterate (psFloat (msFloat vx4 vx3)) (psFloat (msFloat vx4 vx3) (psFloat (msFloat 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="psFloat (msFloat 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 (psFloat (msFloat vx4 vx3)) (psFloat (msFloat vx4 vx3) (psFloat (msFloat vx4 vx3) vx5))",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
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370[label="vx5000/Zero",fontsize=10,color="white",style="solid",shape="box"];210 -> 370[label="",style="solid", color="burlywood", weight=9];
370 -> 228[label="",style="solid", color="burlywood", weight=3];
211[label="vx3000",fontsize=16,color="green",shape="box"];212[label="vx5000",fontsize=16,color="green",shape="box"];213[label="Succ vx5000",fontsize=16,color="green",shape="box"];214[label="Zero",fontsize=16,color="green",shape="box"];215 -> 189[label="",style="dashed", color="red", weight=0];
215[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];216[label="primPlusNat (Succ vx40000) vx3000",fontsize=16,color="burlywood",shape="box"];371[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];216 -> 371[label="",style="solid", color="burlywood", weight=9];
371 -> 229[label="",style="solid", color="burlywood", weight=3];
372[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];216 -> 372[label="",style="solid", color="burlywood", weight=9];
372 -> 230[label="",style="solid", color="burlywood", weight=3];
217[label="primPlusNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];373[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];217 -> 373[label="",style="solid", color="burlywood", weight=9];
373 -> 231[label="",style="solid", color="burlywood", weight=3];
374[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];217 -> 374[label="",style="solid", color="burlywood", weight=9];
374 -> 232[label="",style="solid", color="burlywood", weight=3];
218 -> 189[label="",style="dashed", color="red", weight=0];
218[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];218 -> 233[label="",style="dashed", color="magenta", weight=3];
218 -> 234[label="",style="dashed", color="magenta", weight=3];
219 -> 189[label="",style="dashed", color="red", weight=0];
219[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];219 -> 235[label="",style="dashed", color="magenta", weight=3];
219 -> 236[label="",style="dashed", color="magenta", weight=3];
282 -> 237[label="",style="dashed", color="red", weight=0];
282[label="primMulNat vx60 (Succ vx5100)",fontsize=16,color="magenta"];282 -> 284[label="",style="dashed", color="magenta", weight=3];
282 -> 285[label="",style="dashed", color="magenta", weight=3];
283[label="Succ vx5100",fontsize=16,color="green",shape="box"];224 -> 189[label="",style="dashed", color="red", weight=0];
224[label="primPlusNat vx4000 vx5000",fontsize=16,color="magenta"];224 -> 260[label="",style="dashed", color="magenta", weight=3];
225[label="primMinusNat (Succ vx40000) (Succ vx50000)",fontsize=16,color="black",shape="box"];225 -> 261[label="",style="solid", color="black", weight=3];
226[label="primMinusNat (Succ vx40000) Zero",fontsize=16,color="black",shape="box"];226 -> 262[label="",style="solid", color="black", weight=3];
227[label="primMinusNat Zero (Succ vx50000)",fontsize=16,color="black",shape="box"];227 -> 263[label="",style="solid", color="black", weight=3];
228[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];228 -> 264[label="",style="solid", color="black", weight=3];
229[label="primPlusNat (Succ vx40000) (Succ vx30000)",fontsize=16,color="black",shape="box"];229 -> 265[label="",style="solid", color="black", weight=3];
230[label="primPlusNat (Succ vx40000) Zero",fontsize=16,color="black",shape="box"];230 -> 266[label="",style="solid", color="black", weight=3];
231[label="primPlusNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];231 -> 267[label="",style="solid", color="black", weight=3];
232[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];232 -> 268[label="",style="solid", color="black", weight=3];
233[label="vx4000",fontsize=16,color="green",shape="box"];234[label="vx3000",fontsize=16,color="green",shape="box"];235[label="vx4000",fontsize=16,color="green",shape="box"];236[label="vx3000",fontsize=16,color="green",shape="box"];284[label="Succ vx5100",fontsize=16,color="green",shape="box"];285[label="vx60",fontsize=16,color="green",shape="box"];260[label="vx5000",fontsize=16,color="green",shape="box"];261 -> 177[label="",style="dashed", color="red", weight=0];
261[label="primMinusNat vx40000 vx50000",fontsize=16,color="magenta"];261 -> 273[label="",style="dashed", color="magenta", weight=3];
261 -> 274[label="",style="dashed", color="magenta", weight=3];
262[label="Pos (Succ vx40000)",fontsize=16,color="green",shape="box"];263[label="Neg (Succ vx50000)",fontsize=16,color="green",shape="box"];264[label="Pos Zero",fontsize=16,color="green",shape="box"];265[label="Succ (Succ (primPlusNat vx40000 vx30000))",fontsize=16,color="green",shape="box"];265 -> 275[label="",style="dashed", color="green", weight=3];
266[label="Succ vx40000",fontsize=16,color="green",shape="box"];267[label="Succ vx30000",fontsize=16,color="green",shape="box"];268[label="Zero",fontsize=16,color="green",shape="box"];273[label="vx50000",fontsize=16,color="green",shape="box"];274[label="vx40000",fontsize=16,color="green",shape="box"];275 -> 189[label="",style="dashed", color="red", weight=0];
275[label="primPlusNat vx40000 vx30000",fontsize=16,color="magenta"];275 -> 280[label="",style="dashed", color="magenta", weight=3];
275 -> 281[label="",style="dashed", color="magenta", weight=3];
280[label="vx40000",fontsize=16,color="green",shape="box"];281[label="vx30000",fontsize=16,color="green",shape="box"];}

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

(25)
TRUE