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


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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) AND
    (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) DependencyGraphProof [EQUIVALENT, 0 ms]
        (12) AND
            (13) QDP
                (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (15) YES
            (16) QDP
                (17) MRRProof [EQUIVALENT, 23 ms]
                (18) QDP
                (19) PisEmptyProof [EQUIVALENT, 0 ms]
                (20) YES


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data MyBool = MyTrue  | MyFalse ;

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

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

error :: a;
error = stop MyTrue;

primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primDivNatS Main.Zero Main.Zero = Main.error;
primDivNatS (Main.Succ x) Main.Zero = Main.error;
primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
primDivNatS Main.Zero (Main.Succ x) = Main.Zero;

primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
primDivNatS0 x y MyFalse = Main.Zero;

primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
primGEqNatS Main.Zero Main.Zero = MyTrue;

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

primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt vv vw = Main.error;

quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
quotMyInt = primQuotInt;

stop :: MyBool  ->  a;
stop MyFalse = stop MyFalse;

}

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

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data MyBool = MyTrue  | MyFalse ;

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

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

error :: a;
error = stop MyTrue;

primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primDivNatS Main.Zero Main.Zero = Main.error;
primDivNatS (Main.Succ x) Main.Zero = Main.error;
primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
primDivNatS Main.Zero (Main.Succ x) = Main.Zero;

primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
primDivNatS0 x y MyFalse = Main.Zero;

primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
primGEqNatS Main.Zero Main.Zero = MyTrue;

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

primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt vv vw = Main.error;

quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
quotMyInt = primQuotInt;

stop :: MyBool  ->  a;
stop MyFalse = stop MyFalse;

}

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

(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 MyBool = MyTrue  | MyFalse ;

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

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

error :: a;
error = stop MyTrue;

primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primDivNatS Main.Zero Main.Zero = Main.error;
primDivNatS (Main.Succ x) Main.Zero = Main.error;
primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
primDivNatS Main.Zero (Main.Succ x) = Main.Zero;

primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
primDivNatS0 x y MyFalse = Main.Zero;

primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
primGEqNatS Main.Zero Main.Zero = MyTrue;

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

primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt vv vw = Main.error;

quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
quotMyInt = primQuotInt;

stop :: MyBool  ->  a;
stop MyFalse = stop MyFalse;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="quotMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="quotMyInt vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="quotMyInt vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="primQuotInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];275[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 275[label="",style="solid", color="burlywood", weight=9];
275 -> 6[label="",style="solid", color="burlywood", weight=3];
276[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 276[label="",style="solid", color="burlywood", weight=9];
276 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="primQuotInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];277[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 277[label="",style="solid", color="burlywood", weight=9];
277 -> 8[label="",style="solid", color="burlywood", weight=3];
278[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 278[label="",style="solid", color="burlywood", weight=9];
278 -> 9[label="",style="solid", color="burlywood", weight=3];
7[label="primQuotInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];279[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 279[label="",style="solid", color="burlywood", weight=9];
279 -> 10[label="",style="solid", color="burlywood", weight=3];
280[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 280[label="",style="solid", color="burlywood", weight=9];
280 -> 11[label="",style="solid", color="burlywood", weight=3];
8[label="primQuotInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];281[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 281[label="",style="solid", color="burlywood", weight=9];
281 -> 12[label="",style="solid", color="burlywood", weight=3];
282[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 282[label="",style="solid", color="burlywood", weight=9];
282 -> 13[label="",style="solid", color="burlywood", weight=3];
9[label="primQuotInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];283[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 283[label="",style="solid", color="burlywood", weight=9];
283 -> 14[label="",style="solid", color="burlywood", weight=3];
284[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 284[label="",style="solid", color="burlywood", weight=9];
284 -> 15[label="",style="solid", color="burlywood", weight=3];
10[label="primQuotInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];285[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 285[label="",style="solid", color="burlywood", weight=9];
285 -> 16[label="",style="solid", color="burlywood", weight=3];
286[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 286[label="",style="solid", color="burlywood", weight=9];
286 -> 17[label="",style="solid", color="burlywood", weight=3];
11[label="primQuotInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];287[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 287[label="",style="solid", color="burlywood", weight=9];
287 -> 18[label="",style="solid", color="burlywood", weight=3];
288[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 288[label="",style="solid", color="burlywood", weight=9];
288 -> 19[label="",style="solid", color="burlywood", weight=3];
12[label="primQuotInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
13[label="primQuotInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="primQuotInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="primQuotInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
16[label="primQuotInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
17[label="primQuotInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
18[label="primQuotInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
19[label="primQuotInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
20[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3];
21[label="error",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3];
22[label="Neg (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3];
23 -> 21[label="",style="dashed", color="red", weight=0];
23[label="error",fontsize=16,color="magenta"];24[label="Neg (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];24 -> 31[label="",style="dashed", color="green", weight=3];
25 -> 21[label="",style="dashed", color="red", weight=0];
25[label="error",fontsize=16,color="magenta"];26[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];26 -> 32[label="",style="dashed", color="green", weight=3];
27 -> 21[label="",style="dashed", color="red", weight=0];
27[label="error",fontsize=16,color="magenta"];28[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];289[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 289[label="",style="solid", color="burlywood", weight=9];
289 -> 33[label="",style="solid", color="burlywood", weight=3];
290[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 290[label="",style="solid", color="burlywood", weight=9];
290 -> 34[label="",style="solid", color="burlywood", weight=3];
29[label="stop MyTrue",fontsize=16,color="black",shape="box"];29 -> 35[label="",style="solid", color="black", weight=3];
30 -> 28[label="",style="dashed", color="red", weight=0];
30[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];30 -> 36[label="",style="dashed", color="magenta", weight=3];
31 -> 28[label="",style="dashed", color="red", weight=0];
31[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 37[label="",style="dashed", color="magenta", weight=3];
32 -> 28[label="",style="dashed", color="red", weight=0];
32[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 38[label="",style="dashed", color="magenta", weight=3];
32 -> 39[label="",style="dashed", color="magenta", weight=3];
33[label="primDivNatS (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];33 -> 40[label="",style="solid", color="black", weight=3];
34[label="primDivNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 41[label="",style="solid", color="black", weight=3];
35[label="error []",fontsize=16,color="red",shape="box"];36[label="vz400",fontsize=16,color="green",shape="box"];37[label="vz30",fontsize=16,color="green",shape="box"];38[label="vz30",fontsize=16,color="green",shape="box"];39[label="vz400",fontsize=16,color="green",shape="box"];40[label="primDivNatS0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];291[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];40 -> 291[label="",style="solid", color="burlywood", weight=9];
291 -> 42[label="",style="solid", color="burlywood", weight=3];
292[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];40 -> 292[label="",style="solid", color="burlywood", weight=9];
292 -> 43[label="",style="solid", color="burlywood", weight=3];
41[label="Zero",fontsize=16,color="green",shape="box"];42[label="primDivNatS0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];293[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];42 -> 293[label="",style="solid", color="burlywood", weight=9];
293 -> 44[label="",style="solid", color="burlywood", weight=3];
294[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 294[label="",style="solid", color="burlywood", weight=9];
294 -> 45[label="",style="solid", color="burlywood", weight=3];
43[label="primDivNatS0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];295[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];43 -> 295[label="",style="solid", color="burlywood", weight=9];
295 -> 46[label="",style="solid", color="burlywood", weight=3];
296[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];43 -> 296[label="",style="solid", color="burlywood", weight=9];
296 -> 47[label="",style="solid", color="burlywood", weight=3];
44[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
45[label="primDivNatS0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
46[label="primDivNatS0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3];
47[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];47 -> 51[label="",style="solid", color="black", weight=3];
48 -> 212[label="",style="dashed", color="red", weight=0];
48[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];48 -> 213[label="",style="dashed", color="magenta", weight=3];
48 -> 214[label="",style="dashed", color="magenta", weight=3];
48 -> 215[label="",style="dashed", color="magenta", weight=3];
48 -> 216[label="",style="dashed", color="magenta", weight=3];
49[label="primDivNatS0 (Succ vz3000) Zero MyTrue",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3];
50[label="primDivNatS0 Zero (Succ vz4000) MyFalse",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3];
51[label="primDivNatS0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3];
213[label="vz3000",fontsize=16,color="green",shape="box"];214[label="vz3000",fontsize=16,color="green",shape="box"];215[label="vz4000",fontsize=16,color="green",shape="box"];216[label="vz4000",fontsize=16,color="green",shape="box"];212[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz23 vz24)",fontsize=16,color="burlywood",shape="triangle"];297[label="vz23/Succ vz230",fontsize=10,color="white",style="solid",shape="box"];212 -> 297[label="",style="solid", color="burlywood", weight=9];
297 -> 245[label="",style="solid", color="burlywood", weight=3];
298[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];212 -> 298[label="",style="solid", color="burlywood", weight=9];
298 -> 246[label="",style="solid", color="burlywood", weight=3];
54[label="Succ (primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];54 -> 61[label="",style="dashed", color="green", weight=3];
55[label="Zero",fontsize=16,color="green",shape="box"];56[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];56 -> 62[label="",style="dashed", color="green", weight=3];
245[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) vz24)",fontsize=16,color="burlywood",shape="box"];299[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];245 -> 299[label="",style="solid", color="burlywood", weight=9];
299 -> 247[label="",style="solid", color="burlywood", weight=3];
300[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];245 -> 300[label="",style="solid", color="burlywood", weight=9];
300 -> 248[label="",style="solid", color="burlywood", weight=3];
246[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero vz24)",fontsize=16,color="burlywood",shape="box"];301[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];246 -> 301[label="",style="solid", color="burlywood", weight=9];
301 -> 249[label="",style="solid", color="burlywood", weight=3];
302[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];246 -> 302[label="",style="solid", color="burlywood", weight=9];
302 -> 250[label="",style="solid", color="burlywood", weight=3];
61 -> 28[label="",style="dashed", color="red", weight=0];
61[label="primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];61 -> 67[label="",style="dashed", color="magenta", weight=3];
61 -> 68[label="",style="dashed", color="magenta", weight=3];
62 -> 28[label="",style="dashed", color="red", weight=0];
62[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];62 -> 69[label="",style="dashed", color="magenta", weight=3];
62 -> 70[label="",style="dashed", color="magenta", weight=3];
247[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) (Succ vz240))",fontsize=16,color="black",shape="box"];247 -> 251[label="",style="solid", color="black", weight=3];
248[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) Zero)",fontsize=16,color="black",shape="box"];248 -> 252[label="",style="solid", color="black", weight=3];
249[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero (Succ vz240))",fontsize=16,color="black",shape="box"];249 -> 253[label="",style="solid", color="black", weight=3];
250[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];250 -> 254[label="",style="solid", color="black", weight=3];
67[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="black",shape="triangle"];67 -> 76[label="",style="solid", color="black", weight=3];
68[label="Zero",fontsize=16,color="green",shape="box"];69[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];69 -> 77[label="",style="solid", color="black", weight=3];
70[label="Zero",fontsize=16,color="green",shape="box"];251 -> 212[label="",style="dashed", color="red", weight=0];
251[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz230 vz240)",fontsize=16,color="magenta"];251 -> 255[label="",style="dashed", color="magenta", weight=3];
251 -> 256[label="",style="dashed", color="magenta", weight=3];
252[label="primDivNatS0 (Succ vz21) (Succ vz22) MyTrue",fontsize=16,color="black",shape="triangle"];252 -> 257[label="",style="solid", color="black", weight=3];
253[label="primDivNatS0 (Succ vz21) (Succ vz22) MyFalse",fontsize=16,color="black",shape="box"];253 -> 258[label="",style="solid", color="black", weight=3];
254 -> 252[label="",style="dashed", color="red", weight=0];
254[label="primDivNatS0 (Succ vz21) (Succ vz22) MyTrue",fontsize=16,color="magenta"];76[label="Succ vz3000",fontsize=16,color="green",shape="box"];77[label="Zero",fontsize=16,color="green",shape="box"];255[label="vz230",fontsize=16,color="green",shape="box"];256[label="vz240",fontsize=16,color="green",shape="box"];257[label="Succ (primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22)))",fontsize=16,color="green",shape="box"];257 -> 259[label="",style="dashed", color="green", weight=3];
258[label="Zero",fontsize=16,color="green",shape="box"];259 -> 28[label="",style="dashed", color="red", weight=0];
259[label="primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22))",fontsize=16,color="magenta"];259 -> 260[label="",style="dashed", color="magenta", weight=3];
259 -> 261[label="",style="dashed", color="magenta", weight=3];
260[label="primMinusNatS (Succ vz21) (Succ vz22)",fontsize=16,color="black",shape="box"];260 -> 262[label="",style="solid", color="black", weight=3];
261[label="Succ vz22",fontsize=16,color="green",shape="box"];262[label="primMinusNatS vz21 vz22",fontsize=16,color="burlywood",shape="triangle"];303[label="vz21/Succ vz210",fontsize=10,color="white",style="solid",shape="box"];262 -> 303[label="",style="solid", color="burlywood", weight=9];
303 -> 263[label="",style="solid", color="burlywood", weight=3];
304[label="vz21/Zero",fontsize=10,color="white",style="solid",shape="box"];262 -> 304[label="",style="solid", color="burlywood", weight=9];
304 -> 264[label="",style="solid", color="burlywood", weight=3];
263[label="primMinusNatS (Succ vz210) vz22",fontsize=16,color="burlywood",shape="box"];305[label="vz22/Succ vz220",fontsize=10,color="white",style="solid",shape="box"];263 -> 305[label="",style="solid", color="burlywood", weight=9];
305 -> 265[label="",style="solid", color="burlywood", weight=3];
306[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];263 -> 306[label="",style="solid", color="burlywood", weight=9];
306 -> 266[label="",style="solid", color="burlywood", weight=3];
264[label="primMinusNatS Zero vz22",fontsize=16,color="burlywood",shape="box"];307[label="vz22/Succ vz220",fontsize=10,color="white",style="solid",shape="box"];264 -> 307[label="",style="solid", color="burlywood", weight=9];
307 -> 267[label="",style="solid", color="burlywood", weight=3];
308[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];264 -> 308[label="",style="solid", color="burlywood", weight=9];
308 -> 268[label="",style="solid", color="burlywood", weight=3];
265[label="primMinusNatS (Succ vz210) (Succ vz220)",fontsize=16,color="black",shape="box"];265 -> 269[label="",style="solid", color="black", weight=3];
266[label="primMinusNatS (Succ vz210) Zero",fontsize=16,color="black",shape="box"];266 -> 270[label="",style="solid", color="black", weight=3];
267[label="primMinusNatS Zero (Succ vz220)",fontsize=16,color="black",shape="box"];267 -> 271[label="",style="solid", color="black", weight=3];
268[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];268 -> 272[label="",style="solid", color="black", weight=3];
269 -> 262[label="",style="dashed", color="red", weight=0];
269[label="primMinusNatS vz210 vz220",fontsize=16,color="magenta"];269 -> 273[label="",style="dashed", color="magenta", weight=3];
269 -> 274[label="",style="dashed", color="magenta", weight=3];
270[label="Succ vz210",fontsize=16,color="green",shape="box"];271[label="Zero",fontsize=16,color="green",shape="box"];272[label="Zero",fontsize=16,color="green",shape="box"];273[label="vz210",fontsize=16,color="green",shape="box"];274[label="vz220",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

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

   new_primMinusNatS(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS(vz210, vz220)

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_primMinusNatS(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS(vz210, vz220)
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_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Main.Zero)
   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240)
   new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22))
   new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22)
   new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22))
   new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS2, Main.Zero)

The TRS R consists of the following rules:

   new_primMinusNatS0(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS0(vz210, vz220)
   new_primMinusNatS0(Main.Succ(vz210), Main.Zero) -> Main.Succ(vz210)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(vz220)) -> Main.Zero
   new_primMinusNatS1(vz3000) -> Main.Succ(vz3000)
   new_primMinusNatS2 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
   new_primMinusNatS1(x0)

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

(11) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
----------------------------------------

(12)
Complex Obligation (AND)

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

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

   new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22)
   new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22))
   new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240)
   new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22))

The TRS R consists of the following rules:

   new_primMinusNatS0(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS0(vz210, vz220)
   new_primMinusNatS0(Main.Succ(vz210), Main.Zero) -> Main.Succ(vz210)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(vz220)) -> Main.Zero
   new_primMinusNatS1(vz3000) -> Main.Succ(vz3000)
   new_primMinusNatS2 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
   new_primMinusNatS1(x0)

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

(14) QDPSizeChangeProof (EQUIVALENT)
We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 

Order:Polynomial interpretation [POLO]:

   POL(Main.Succ(x_1)) = 1 + x_1
   POL(Main.Zero) = 1
   POL(new_primMinusNatS0(x_1, x_2)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1


*new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4


*new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


*new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1, 2 >= 2


*new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1



We oriented the following set of usable rules [AAECC05,FROCOS05].

   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(vz220)) -> Main.Zero
   new_primMinusNatS0(Main.Succ(vz210), Main.Zero) -> Main.Succ(vz210)
   new_primMinusNatS0(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS0(vz210, vz220)

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

(15)
YES

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

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

   new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Main.Zero)

The TRS R consists of the following rules:

   new_primMinusNatS0(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS0(vz210, vz220)
   new_primMinusNatS0(Main.Succ(vz210), Main.Zero) -> Main.Succ(vz210)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(vz220)) -> Main.Zero
   new_primMinusNatS1(vz3000) -> Main.Succ(vz3000)
   new_primMinusNatS2 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
   new_primMinusNatS1(x0)

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

(17) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Main.Zero)

Strictly oriented rules of the TRS R:

   new_primMinusNatS0(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS0(vz210, vz220)
   new_primMinusNatS0(Main.Succ(vz210), Main.Zero) -> Main.Succ(vz210)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(vz220)) -> Main.Zero

Used ordering: Polynomial interpretation [POLO]:

   POL(Main.Succ(x_1)) = 1 + x_1
   POL(Main.Zero) = 2
   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
   POL(new_primMinusNatS0(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(new_primMinusNatS1(x_1)) = 1 + x_1
   POL(new_primMinusNatS2) = 2


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

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

   new_primMinusNatS1(vz3000) -> Main.Succ(vz3000)
   new_primMinusNatS2 -> Main.Zero

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0), Main.Zero)
   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
   new_primMinusNatS1(x0)

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

(19) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(20)
YES