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


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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) AND
    (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (12) YES
    (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES


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

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

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

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

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

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

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);

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));

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

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

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

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

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);

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));

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

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

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

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

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);

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));

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

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="msFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="msFloat vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="msFloat vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="primMinusFloat vx3 vx4",fontsize=16,color="burlywood",shape="box"];86[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];5 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 6[label="",style="solid", color="burlywood", weight=3];
6[label="primMinusFloat (Float vx30 vx31) vx4",fontsize=16,color="burlywood",shape="box"];87[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 7[label="",style="solid", color="burlywood", weight=3];
7[label="primMinusFloat (Float vx30 vx31) (Float vx40 vx41)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="Float (msMyInt vx30 vx40) (srMyInt vx31 vx41)",fontsize=16,color="green",shape="box"];8 -> 9[label="",style="dashed", color="green", weight=3];
8 -> 10[label="",style="dashed", color="green", weight=3];
9[label="msMyInt vx30 vx40",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="srMyInt vx31 vx41",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="primMinusInt vx30 vx40",fontsize=16,color="burlywood",shape="box"];88[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 13[label="",style="solid", color="burlywood", weight=3];
89[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="primMulInt vx31 vx41",fontsize=16,color="burlywood",shape="box"];90[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];12 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 15[label="",style="solid", color="burlywood", weight=3];
91[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];12 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 16[label="",style="solid", color="burlywood", weight=3];
13[label="primMinusInt (Pos vx300) vx40",fontsize=16,color="burlywood",shape="box"];92[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 17[label="",style="solid", color="burlywood", weight=3];
93[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 18[label="",style="solid", color="burlywood", weight=3];
14[label="primMinusInt (Neg vx300) vx40",fontsize=16,color="burlywood",shape="box"];94[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 19[label="",style="solid", color="burlywood", weight=3];
95[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 20[label="",style="solid", color="burlywood", weight=3];
15[label="primMulInt (Pos vx310) vx41",fontsize=16,color="burlywood",shape="box"];96[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];15 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 21[label="",style="solid", color="burlywood", weight=3];
97[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];15 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 22[label="",style="solid", color="burlywood", weight=3];
16[label="primMulInt (Neg vx310) vx41",fontsize=16,color="burlywood",shape="box"];98[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];16 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 23[label="",style="solid", color="burlywood", weight=3];
99[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];16 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 24[label="",style="solid", color="burlywood", weight=3];
17[label="primMinusInt (Pos vx300) (Pos vx400)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
18[label="primMinusInt (Pos vx300) (Neg vx400)",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
19[label="primMinusInt (Neg vx300) (Pos vx400)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
20[label="primMinusInt (Neg vx300) (Neg vx400)",fontsize=16,color="black",shape="box"];20 -> 28[label="",style="solid", color="black", weight=3];
21[label="primMulInt (Pos vx310) (Pos vx410)",fontsize=16,color="black",shape="box"];21 -> 29[label="",style="solid", color="black", weight=3];
22[label="primMulInt (Pos vx310) (Neg vx410)",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3];
23[label="primMulInt (Neg vx310) (Pos vx410)",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
24[label="primMulInt (Neg vx310) (Neg vx410)",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
25[label="primMinusNat vx300 vx400",fontsize=16,color="burlywood",shape="triangle"];100[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];25 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 33[label="",style="solid", color="burlywood", weight=3];
101[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 34[label="",style="solid", color="burlywood", weight=3];
26[label="Pos (primPlusNat vx300 vx400)",fontsize=16,color="green",shape="box"];26 -> 35[label="",style="dashed", color="green", weight=3];
27[label="Neg (primPlusNat vx300 vx400)",fontsize=16,color="green",shape="box"];27 -> 36[label="",style="dashed", color="green", weight=3];
28 -> 25[label="",style="dashed", color="red", weight=0];
28[label="primMinusNat vx400 vx300",fontsize=16,color="magenta"];28 -> 37[label="",style="dashed", color="magenta", weight=3];
28 -> 38[label="",style="dashed", color="magenta", weight=3];
29[label="Pos (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];29 -> 39[label="",style="dashed", color="green", weight=3];
30[label="Neg (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];30 -> 40[label="",style="dashed", color="green", weight=3];
31[label="Neg (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];31 -> 41[label="",style="dashed", color="green", weight=3];
32[label="Pos (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];32 -> 42[label="",style="dashed", color="green", weight=3];
33[label="primMinusNat (Succ vx3000) vx400",fontsize=16,color="burlywood",shape="box"];102[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];33 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 43[label="",style="solid", color="burlywood", weight=3];
103[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 44[label="",style="solid", color="burlywood", weight=3];
34[label="primMinusNat Zero vx400",fontsize=16,color="burlywood",shape="box"];104[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];34 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 45[label="",style="solid", color="burlywood", weight=3];
105[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];34 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 46[label="",style="solid", color="burlywood", weight=3];
35[label="primPlusNat vx300 vx400",fontsize=16,color="burlywood",shape="triangle"];106[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 47[label="",style="solid", color="burlywood", weight=3];
107[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 48[label="",style="solid", color="burlywood", weight=3];
36 -> 35[label="",style="dashed", color="red", weight=0];
36[label="primPlusNat vx300 vx400",fontsize=16,color="magenta"];36 -> 49[label="",style="dashed", color="magenta", weight=3];
36 -> 50[label="",style="dashed", color="magenta", weight=3];
37[label="vx400",fontsize=16,color="green",shape="box"];38[label="vx300",fontsize=16,color="green",shape="box"];39[label="primMulNat vx310 vx410",fontsize=16,color="burlywood",shape="triangle"];108[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];39 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 51[label="",style="solid", color="burlywood", weight=3];
109[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 52[label="",style="solid", color="burlywood", weight=3];
40 -> 39[label="",style="dashed", color="red", weight=0];
40[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];40 -> 53[label="",style="dashed", color="magenta", weight=3];
41 -> 39[label="",style="dashed", color="red", weight=0];
41[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];41 -> 54[label="",style="dashed", color="magenta", weight=3];
42 -> 39[label="",style="dashed", color="red", weight=0];
42[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];42 -> 55[label="",style="dashed", color="magenta", weight=3];
42 -> 56[label="",style="dashed", color="magenta", weight=3];
43[label="primMinusNat (Succ vx3000) (Succ vx4000)",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3];
44[label="primMinusNat (Succ vx3000) Zero",fontsize=16,color="black",shape="box"];44 -> 58[label="",style="solid", color="black", weight=3];
45[label="primMinusNat Zero (Succ vx4000)",fontsize=16,color="black",shape="box"];45 -> 59[label="",style="solid", color="black", weight=3];
46[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];46 -> 60[label="",style="solid", color="black", weight=3];
47[label="primPlusNat (Succ vx3000) vx400",fontsize=16,color="burlywood",shape="box"];110[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];47 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 61[label="",style="solid", color="burlywood", weight=3];
111[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 62[label="",style="solid", color="burlywood", weight=3];
48[label="primPlusNat Zero vx400",fontsize=16,color="burlywood",shape="box"];112[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];48 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 63[label="",style="solid", color="burlywood", weight=3];
113[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 113[label="",style="solid", color="burlywood", weight=9];
113 -> 64[label="",style="solid", color="burlywood", weight=3];
49[label="vx300",fontsize=16,color="green",shape="box"];50[label="vx400",fontsize=16,color="green",shape="box"];51[label="primMulNat (Succ vx3100) vx410",fontsize=16,color="burlywood",shape="box"];114[label="vx410/Succ vx4100",fontsize=10,color="white",style="solid",shape="box"];51 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 65[label="",style="solid", color="burlywood", weight=3];
115[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];51 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 66[label="",style="solid", color="burlywood", weight=3];
52[label="primMulNat Zero vx410",fontsize=16,color="burlywood",shape="box"];116[label="vx410/Succ vx4100",fontsize=10,color="white",style="solid",shape="box"];52 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 67[label="",style="solid", color="burlywood", weight=3];
117[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 117[label="",style="solid", color="burlywood", weight=9];
117 -> 68[label="",style="solid", color="burlywood", weight=3];
53[label="vx410",fontsize=16,color="green",shape="box"];54[label="vx310",fontsize=16,color="green",shape="box"];55[label="vx410",fontsize=16,color="green",shape="box"];56[label="vx310",fontsize=16,color="green",shape="box"];57 -> 25[label="",style="dashed", color="red", weight=0];
57[label="primMinusNat vx3000 vx4000",fontsize=16,color="magenta"];57 -> 69[label="",style="dashed", color="magenta", weight=3];
57 -> 70[label="",style="dashed", color="magenta", weight=3];
58[label="Pos (Succ vx3000)",fontsize=16,color="green",shape="box"];59[label="Neg (Succ vx4000)",fontsize=16,color="green",shape="box"];60[label="Pos Zero",fontsize=16,color="green",shape="box"];61[label="primPlusNat (Succ vx3000) (Succ vx4000)",fontsize=16,color="black",shape="box"];61 -> 71[label="",style="solid", color="black", weight=3];
62[label="primPlusNat (Succ vx3000) Zero",fontsize=16,color="black",shape="box"];62 -> 72[label="",style="solid", color="black", weight=3];
63[label="primPlusNat Zero (Succ vx4000)",fontsize=16,color="black",shape="box"];63 -> 73[label="",style="solid", color="black", weight=3];
64[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3];
65[label="primMulNat (Succ vx3100) (Succ vx4100)",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3];
66[label="primMulNat (Succ vx3100) Zero",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3];
67[label="primMulNat Zero (Succ vx4100)",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3];
68[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];68 -> 78[label="",style="solid", color="black", weight=3];
69[label="vx3000",fontsize=16,color="green",shape="box"];70[label="vx4000",fontsize=16,color="green",shape="box"];71[label="Succ (Succ (primPlusNat vx3000 vx4000))",fontsize=16,color="green",shape="box"];71 -> 79[label="",style="dashed", color="green", weight=3];
72[label="Succ vx3000",fontsize=16,color="green",shape="box"];73[label="Succ vx4000",fontsize=16,color="green",shape="box"];74[label="Zero",fontsize=16,color="green",shape="box"];75 -> 35[label="",style="dashed", color="red", weight=0];
75[label="primPlusNat (primMulNat vx3100 (Succ vx4100)) (Succ vx4100)",fontsize=16,color="magenta"];75 -> 80[label="",style="dashed", color="magenta", weight=3];
75 -> 81[label="",style="dashed", color="magenta", weight=3];
76[label="Zero",fontsize=16,color="green",shape="box"];77[label="Zero",fontsize=16,color="green",shape="box"];78[label="Zero",fontsize=16,color="green",shape="box"];79 -> 35[label="",style="dashed", color="red", weight=0];
79[label="primPlusNat vx3000 vx4000",fontsize=16,color="magenta"];79 -> 82[label="",style="dashed", color="magenta", weight=3];
79 -> 83[label="",style="dashed", color="magenta", weight=3];
80 -> 39[label="",style="dashed", color="red", weight=0];
80[label="primMulNat vx3100 (Succ vx4100)",fontsize=16,color="magenta"];80 -> 84[label="",style="dashed", color="magenta", weight=3];
80 -> 85[label="",style="dashed", color="magenta", weight=3];
81[label="Succ vx4100",fontsize=16,color="green",shape="box"];82[label="vx3000",fontsize=16,color="green",shape="box"];83[label="vx4000",fontsize=16,color="green",shape="box"];84[label="Succ vx4100",fontsize=16,color="green",shape="box"];85[label="vx3100",fontsize=16,color="green",shape="box"];}

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(6)
Complex Obligation (AND)

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

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

   new_primMulNat(Main.Succ(vx3100), Main.Succ(vx4100)) -> new_primMulNat(vx3100, Main.Succ(vx4100))

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_primMulNat(Main.Succ(vx3100), Main.Succ(vx4100)) -> new_primMulNat(vx3100, Main.Succ(vx4100))
The graph contains the following edges 1 > 1, 2 >= 2


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

(9)
YES

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

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

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

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

(11) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_primPlusNat(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_primPlusNat(vx3000, vx4000)
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(vx3000), Main.Succ(vx4000)) -> new_primMinusNat(vx3000, vx4000)

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


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

(15)
YES