/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) QDP
(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(8) YES


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

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

data MyBool = MyTrue  | MyFalse ;

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

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

data Ordering = LT  | EQ  | GT ;

compareChar :: Main.Char  ->  Main.Char  ->  Ordering;
compareChar = primCmpChar;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = MyTrue;

ltChar :: Main.Char  ->  Main.Char  ->  MyBool;
ltChar x y = esEsOrdering (compareChar x y) LT;

primCmpChar :: Main.Char  ->  Main.Char  ->  Ordering;
primCmpChar (Main.Char x) (Main.Char y) = primCmpInt x y;

primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
primCmpInt (Main.Pos x) (Main.Neg y) = GT;
primCmpInt (Main.Neg x) (Main.Pos y) = LT;
primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;

primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
primCmpNat Main.Zero Main.Zero = EQ;
primCmpNat Main.Zero (Main.Succ y) = LT;
primCmpNat (Main.Succ x) Main.Zero = GT;
primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;

}

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

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

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

data MyBool = MyTrue  | MyFalse ;

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

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

data Ordering = LT  | EQ  | GT ;

compareChar :: Main.Char  ->  Main.Char  ->  Ordering;
compareChar = primCmpChar;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = MyTrue;

ltChar :: Main.Char  ->  Main.Char  ->  MyBool;
ltChar x y = esEsOrdering (compareChar x y) LT;

primCmpChar :: Main.Char  ->  Main.Char  ->  Ordering;
primCmpChar (Main.Char x) (Main.Char y) = primCmpInt x y;

primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
primCmpInt (Main.Pos x) (Main.Neg y) = GT;
primCmpInt (Main.Neg x) (Main.Pos y) = LT;
primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;

primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
primCmpNat Main.Zero Main.Zero = EQ;
primCmpNat Main.Zero (Main.Succ y) = LT;
primCmpNat (Main.Succ x) Main.Zero = GT;
primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;

}

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

(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 Main.Char = Char MyInt ;

data MyBool = MyTrue  | MyFalse ;

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

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

data Ordering = LT  | EQ  | GT ;

compareChar :: Main.Char  ->  Main.Char  ->  Ordering;
compareChar = primCmpChar;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = MyTrue;

ltChar :: Main.Char  ->  Main.Char  ->  MyBool;
ltChar x y = esEsOrdering (compareChar x y) LT;

primCmpChar :: Main.Char  ->  Main.Char  ->  Ordering;
primCmpChar (Main.Char x) (Main.Char y) = primCmpInt x y;

primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
primCmpInt (Main.Pos x) (Main.Neg y) = GT;
primCmpInt (Main.Neg x) (Main.Pos y) = LT;
primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;

primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
primCmpNat Main.Zero Main.Zero = EQ;
primCmpNat Main.Zero (Main.Succ y) = LT;
primCmpNat (Main.Succ x) Main.Zero = GT;
primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="ltChar",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="ltChar vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="ltChar vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="esEsOrdering (compareChar vx3 vx4) LT",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="esEsOrdering (primCmpChar vx3 vx4) LT",fontsize=16,color="burlywood",shape="box"];73[label="vx3/Char vx30",fontsize=10,color="white",style="solid",shape="box"];6 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 7[label="",style="solid", color="burlywood", weight=3];
7[label="esEsOrdering (primCmpChar (Char vx30) vx4) LT",fontsize=16,color="burlywood",shape="box"];74[label="vx4/Char vx40",fontsize=10,color="white",style="solid",shape="box"];7 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 8[label="",style="solid", color="burlywood", weight=3];
8[label="esEsOrdering (primCmpChar (Char vx30) (Char vx40)) LT",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="esEsOrdering (primCmpInt vx30 vx40) LT",fontsize=16,color="burlywood",shape="box"];75[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];9 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 10[label="",style="solid", color="burlywood", weight=3];
76[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];9 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="esEsOrdering (primCmpInt (Pos vx300) vx40) LT",fontsize=16,color="burlywood",shape="box"];77[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];10 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 12[label="",style="solid", color="burlywood", weight=3];
78[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 13[label="",style="solid", color="burlywood", weight=3];
11[label="esEsOrdering (primCmpInt (Neg vx300) vx40) LT",fontsize=16,color="burlywood",shape="box"];79[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];11 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 14[label="",style="solid", color="burlywood", weight=3];
80[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 15[label="",style="solid", color="burlywood", weight=3];
12[label="esEsOrdering (primCmpInt (Pos (Succ vx3000)) vx40) LT",fontsize=16,color="burlywood",shape="box"];81[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];12 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 16[label="",style="solid", color="burlywood", weight=3];
82[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];12 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 17[label="",style="solid", color="burlywood", weight=3];
13[label="esEsOrdering (primCmpInt (Pos Zero) vx40) LT",fontsize=16,color="burlywood",shape="box"];83[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 18[label="",style="solid", color="burlywood", weight=3];
84[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 19[label="",style="solid", color="burlywood", weight=3];
14[label="esEsOrdering (primCmpInt (Neg (Succ vx3000)) vx40) LT",fontsize=16,color="burlywood",shape="box"];85[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 20[label="",style="solid", color="burlywood", weight=3];
86[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 21[label="",style="solid", color="burlywood", weight=3];
15[label="esEsOrdering (primCmpInt (Neg Zero) vx40) LT",fontsize=16,color="burlywood",shape="box"];87[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];15 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 22[label="",style="solid", color="burlywood", weight=3];
88[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];15 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 23[label="",style="solid", color="burlywood", weight=3];
16[label="esEsOrdering (primCmpInt (Pos (Succ vx3000)) (Pos vx400)) LT",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
17[label="esEsOrdering (primCmpInt (Pos (Succ vx3000)) (Neg vx400)) LT",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
18[label="esEsOrdering (primCmpInt (Pos Zero) (Pos vx400)) LT",fontsize=16,color="burlywood",shape="box"];89[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];18 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 26[label="",style="solid", color="burlywood", weight=3];
90[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 27[label="",style="solid", color="burlywood", weight=3];
19[label="esEsOrdering (primCmpInt (Pos Zero) (Neg vx400)) LT",fontsize=16,color="burlywood",shape="box"];91[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];19 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 28[label="",style="solid", color="burlywood", weight=3];
92[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 29[label="",style="solid", color="burlywood", weight=3];
20[label="esEsOrdering (primCmpInt (Neg (Succ vx3000)) (Pos vx400)) LT",fontsize=16,color="black",shape="box"];20 -> 30[label="",style="solid", color="black", weight=3];
21[label="esEsOrdering (primCmpInt (Neg (Succ vx3000)) (Neg vx400)) LT",fontsize=16,color="black",shape="box"];21 -> 31[label="",style="solid", color="black", weight=3];
22[label="esEsOrdering (primCmpInt (Neg Zero) (Pos vx400)) LT",fontsize=16,color="burlywood",shape="box"];93[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];22 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 32[label="",style="solid", color="burlywood", weight=3];
94[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 33[label="",style="solid", color="burlywood", weight=3];
23[label="esEsOrdering (primCmpInt (Neg Zero) (Neg vx400)) LT",fontsize=16,color="burlywood",shape="box"];95[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];23 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 34[label="",style="solid", color="burlywood", weight=3];
96[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 35[label="",style="solid", color="burlywood", weight=3];
24[label="esEsOrdering (primCmpNat (Succ vx3000) vx400) LT",fontsize=16,color="burlywood",shape="triangle"];97[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];24 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 36[label="",style="solid", color="burlywood", weight=3];
98[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 37[label="",style="solid", color="burlywood", weight=3];
25[label="esEsOrdering GT LT",fontsize=16,color="black",shape="triangle"];25 -> 38[label="",style="solid", color="black", weight=3];
26[label="esEsOrdering (primCmpInt (Pos Zero) (Pos (Succ vx4000))) LT",fontsize=16,color="black",shape="box"];26 -> 39[label="",style="solid", color="black", weight=3];
27[label="esEsOrdering (primCmpInt (Pos Zero) (Pos Zero)) LT",fontsize=16,color="black",shape="box"];27 -> 40[label="",style="solid", color="black", weight=3];
28[label="esEsOrdering (primCmpInt (Pos Zero) (Neg (Succ vx4000))) LT",fontsize=16,color="black",shape="box"];28 -> 41[label="",style="solid", color="black", weight=3];
29[label="esEsOrdering (primCmpInt (Pos Zero) (Neg Zero)) LT",fontsize=16,color="black",shape="box"];29 -> 42[label="",style="solid", color="black", weight=3];
30[label="esEsOrdering LT LT",fontsize=16,color="black",shape="triangle"];30 -> 43[label="",style="solid", color="black", weight=3];
31[label="esEsOrdering (primCmpNat vx400 (Succ vx3000)) LT",fontsize=16,color="burlywood",shape="triangle"];99[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];31 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 44[label="",style="solid", color="burlywood", weight=3];
100[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];31 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 45[label="",style="solid", color="burlywood", weight=3];
32[label="esEsOrdering (primCmpInt (Neg Zero) (Pos (Succ vx4000))) LT",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
33[label="esEsOrdering (primCmpInt (Neg Zero) (Pos Zero)) LT",fontsize=16,color="black",shape="box"];33 -> 47[label="",style="solid", color="black", weight=3];
34[label="esEsOrdering (primCmpInt (Neg Zero) (Neg (Succ vx4000))) LT",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
35[label="esEsOrdering (primCmpInt (Neg Zero) (Neg Zero)) LT",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
36[label="esEsOrdering (primCmpNat (Succ vx3000) (Succ vx4000)) LT",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3];
37[label="esEsOrdering (primCmpNat (Succ vx3000) Zero) LT",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3];
38[label="MyFalse",fontsize=16,color="green",shape="box"];39 -> 31[label="",style="dashed", color="red", weight=0];
39[label="esEsOrdering (primCmpNat Zero (Succ vx4000)) LT",fontsize=16,color="magenta"];39 -> 52[label="",style="dashed", color="magenta", weight=3];
39 -> 53[label="",style="dashed", color="magenta", weight=3];
40[label="esEsOrdering EQ LT",fontsize=16,color="black",shape="triangle"];40 -> 54[label="",style="solid", color="black", weight=3];
41 -> 25[label="",style="dashed", color="red", weight=0];
41[label="esEsOrdering GT LT",fontsize=16,color="magenta"];42 -> 40[label="",style="dashed", color="red", weight=0];
42[label="esEsOrdering EQ LT",fontsize=16,color="magenta"];43[label="MyTrue",fontsize=16,color="green",shape="box"];44[label="esEsOrdering (primCmpNat (Succ vx4000) (Succ vx3000)) LT",fontsize=16,color="black",shape="box"];44 -> 55[label="",style="solid", color="black", weight=3];
45[label="esEsOrdering (primCmpNat Zero (Succ vx3000)) LT",fontsize=16,color="black",shape="box"];45 -> 56[label="",style="solid", color="black", weight=3];
46 -> 30[label="",style="dashed", color="red", weight=0];
46[label="esEsOrdering LT LT",fontsize=16,color="magenta"];47 -> 40[label="",style="dashed", color="red", weight=0];
47[label="esEsOrdering EQ LT",fontsize=16,color="magenta"];48 -> 24[label="",style="dashed", color="red", weight=0];
48[label="esEsOrdering (primCmpNat (Succ vx4000) Zero) LT",fontsize=16,color="magenta"];48 -> 57[label="",style="dashed", color="magenta", weight=3];
48 -> 58[label="",style="dashed", color="magenta", weight=3];
49 -> 40[label="",style="dashed", color="red", weight=0];
49[label="esEsOrdering EQ LT",fontsize=16,color="magenta"];50[label="esEsOrdering (primCmpNat vx3000 vx4000) LT",fontsize=16,color="burlywood",shape="triangle"];101[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];50 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 59[label="",style="solid", color="burlywood", weight=3];
102[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 60[label="",style="solid", color="burlywood", weight=3];
51 -> 25[label="",style="dashed", color="red", weight=0];
51[label="esEsOrdering GT LT",fontsize=16,color="magenta"];52[label="vx4000",fontsize=16,color="green",shape="box"];53[label="Zero",fontsize=16,color="green",shape="box"];54[label="MyFalse",fontsize=16,color="green",shape="box"];55 -> 50[label="",style="dashed", color="red", weight=0];
55[label="esEsOrdering (primCmpNat vx4000 vx3000) LT",fontsize=16,color="magenta"];55 -> 61[label="",style="dashed", color="magenta", weight=3];
55 -> 62[label="",style="dashed", color="magenta", weight=3];
56 -> 30[label="",style="dashed", color="red", weight=0];
56[label="esEsOrdering LT LT",fontsize=16,color="magenta"];57[label="Zero",fontsize=16,color="green",shape="box"];58[label="vx4000",fontsize=16,color="green",shape="box"];59[label="esEsOrdering (primCmpNat (Succ vx30000) vx4000) LT",fontsize=16,color="burlywood",shape="box"];103[label="vx4000/Succ vx40000",fontsize=10,color="white",style="solid",shape="box"];59 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 63[label="",style="solid", color="burlywood", weight=3];
104[label="vx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 64[label="",style="solid", color="burlywood", weight=3];
60[label="esEsOrdering (primCmpNat Zero vx4000) LT",fontsize=16,color="burlywood",shape="box"];105[label="vx4000/Succ vx40000",fontsize=10,color="white",style="solid",shape="box"];60 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 65[label="",style="solid", color="burlywood", weight=3];
106[label="vx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];60 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 66[label="",style="solid", color="burlywood", weight=3];
61[label="vx3000",fontsize=16,color="green",shape="box"];62[label="vx4000",fontsize=16,color="green",shape="box"];63[label="esEsOrdering (primCmpNat (Succ vx30000) (Succ vx40000)) LT",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
64[label="esEsOrdering (primCmpNat (Succ vx30000) Zero) LT",fontsize=16,color="black",shape="box"];64 -> 68[label="",style="solid", color="black", weight=3];
65[label="esEsOrdering (primCmpNat Zero (Succ vx40000)) LT",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
66[label="esEsOrdering (primCmpNat Zero Zero) LT",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
67 -> 50[label="",style="dashed", color="red", weight=0];
67[label="esEsOrdering (primCmpNat vx30000 vx40000) LT",fontsize=16,color="magenta"];67 -> 71[label="",style="dashed", color="magenta", weight=3];
67 -> 72[label="",style="dashed", color="magenta", weight=3];
68 -> 25[label="",style="dashed", color="red", weight=0];
68[label="esEsOrdering GT LT",fontsize=16,color="magenta"];69 -> 30[label="",style="dashed", color="red", weight=0];
69[label="esEsOrdering LT LT",fontsize=16,color="magenta"];70 -> 40[label="",style="dashed", color="red", weight=0];
70[label="esEsOrdering EQ LT",fontsize=16,color="magenta"];71[label="vx40000",fontsize=16,color="green",shape="box"];72[label="vx30000",fontsize=16,color="green",shape="box"];}

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

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

(7) 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_esEsOrdering(Main.Succ(vx30000), Main.Succ(vx40000)) -> new_esEsOrdering(vx30000, vx40000)
The graph contains the following edges 1 > 1, 2 > 2


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

(8)
YES