/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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


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

(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 ;

data Ordering = LT  | EQ  | GT ;

data Ratio a = CnPc a a ;

compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
compareMyInt = primCmpInt;

esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
esEsMyInt = primEqInt;

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;

fromIntMyInt :: MyInt  ->  MyInt;
fromIntMyInt x = x;

gtMyInt :: MyInt  ->  MyInt  ->  MyBool;
gtMyInt x y = esEsOrdering (compareMyInt x y) GT;

otherwise :: MyBool;
otherwise = MyTrue;

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;

primEqInt :: MyInt  ->  MyInt  ->  MyBool;
primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt vv vw = MyFalse;

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

signumMyInt :: MyInt  ->  MyInt;
signumMyInt = signumReal;

signumRatio :: Ratio MyInt  ->  Ratio MyInt;
signumRatio (CnPc x y) = CnPc (signumMyInt x) (fromIntMyInt (Main.Pos (Main.Succ Main.Zero)));

signumReal x = signumReal3 x;

signumReal0 x MyTrue = fromIntMyInt (Main.Neg (Main.Succ Main.Zero));

signumReal1 x MyTrue = fromIntMyInt (Main.Pos (Main.Succ Main.Zero));
signumReal1 x MyFalse = signumReal0 x otherwise;

signumReal2 x MyTrue = fromIntMyInt (Main.Pos Main.Zero);
signumReal2 x MyFalse = signumReal1 x (gtMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

signumReal3 x = signumReal2 x (esEsMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

}

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

(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 ;

data Ordering = LT  | EQ  | GT ;

data Ratio a = CnPc a a ;

compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
compareMyInt = primCmpInt;

esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
esEsMyInt = primEqInt;

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;

fromIntMyInt :: MyInt  ->  MyInt;
fromIntMyInt x = x;

gtMyInt :: MyInt  ->  MyInt  ->  MyBool;
gtMyInt x y = esEsOrdering (compareMyInt x y) GT;

otherwise :: MyBool;
otherwise = MyTrue;

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;

primEqInt :: MyInt  ->  MyInt  ->  MyBool;
primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt vv vw = MyFalse;

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

signumMyInt :: MyInt  ->  MyInt;
signumMyInt = signumReal;

signumRatio :: Ratio MyInt  ->  Ratio MyInt;
signumRatio (CnPc x y) = CnPc (signumMyInt x) (fromIntMyInt (Main.Pos (Main.Succ Main.Zero)));

signumReal x = signumReal3 x;

signumReal0 x MyTrue = fromIntMyInt (Main.Neg (Main.Succ Main.Zero));

signumReal1 x MyTrue = fromIntMyInt (Main.Pos (Main.Succ Main.Zero));
signumReal1 x MyFalse = signumReal0 x otherwise;

signumReal2 x MyTrue = fromIntMyInt (Main.Pos Main.Zero);
signumReal2 x MyFalse = signumReal1 x (gtMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

signumReal3 x = signumReal2 x (esEsMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

}

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

(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 ;

data Ordering = LT  | EQ  | GT ;

data Ratio a = CnPc a a ;

compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
compareMyInt = primCmpInt;

esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
esEsMyInt = primEqInt;

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;

fromIntMyInt :: MyInt  ->  MyInt;
fromIntMyInt x = x;

gtMyInt :: MyInt  ->  MyInt  ->  MyBool;
gtMyInt x y = esEsOrdering (compareMyInt x y) GT;

otherwise :: MyBool;
otherwise = MyTrue;

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;

primEqInt :: MyInt  ->  MyInt  ->  MyBool;
primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt vv vw = MyFalse;

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

signumMyInt :: MyInt  ->  MyInt;
signumMyInt = signumReal;

signumRatio :: Ratio MyInt  ->  Ratio MyInt;
signumRatio (CnPc x y) = CnPc (signumMyInt x) (fromIntMyInt (Main.Pos (Main.Succ Main.Zero)));

signumReal x = signumReal3 x;

signumReal0 x MyTrue = fromIntMyInt (Main.Neg (Main.Succ Main.Zero));

signumReal1 x MyTrue = fromIntMyInt (Main.Pos (Main.Succ Main.Zero));
signumReal1 x MyFalse = signumReal0 x otherwise;

signumReal2 x MyTrue = fromIntMyInt (Main.Pos Main.Zero);
signumReal2 x MyFalse = signumReal1 x (gtMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

signumReal3 x = signumReal2 x (esEsMyInt x (fromIntMyInt (Main.Pos Main.Zero)));

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="signumRatio",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="signumRatio vz3",fontsize=16,color="burlywood",shape="triangle"];309[label="vz3/CnPc vz30 vz31",fontsize=10,color="white",style="solid",shape="box"];3 -> 309[label="",style="solid", color="burlywood", weight=9];
309 -> 4[label="",style="solid", color="burlywood", weight=3];
4[label="signumRatio (CnPc vz30 vz31)",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="CnPc (signumMyInt vz30) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
5 -> 7[label="",style="dashed", color="green", weight=3];
6[label="signumMyInt vz30",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="fromIntMyInt (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="signumReal vz30",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];10[label="signumReal3 vz30",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="signumReal2 vz30 (esEsMyInt vz30 (fromIntMyInt (Pos Zero)))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="signumReal2 vz30 (primEqInt vz30 (fromIntMyInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];310[label="vz30/Pos vz300",fontsize=10,color="white",style="solid",shape="box"];12 -> 310[label="",style="solid", color="burlywood", weight=9];
310 -> 13[label="",style="solid", color="burlywood", weight=3];
311[label="vz30/Neg vz300",fontsize=10,color="white",style="solid",shape="box"];12 -> 311[label="",style="solid", color="burlywood", weight=9];
311 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="signumReal2 (Pos vz300) (primEqInt (Pos vz300) (fromIntMyInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];312[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];13 -> 312[label="",style="solid", color="burlywood", weight=9];
312 -> 15[label="",style="solid", color="burlywood", weight=3];
313[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 313[label="",style="solid", color="burlywood", weight=9];
313 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="signumReal2 (Neg vz300) (primEqInt (Neg vz300) (fromIntMyInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];314[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];14 -> 314[label="",style="solid", color="burlywood", weight=9];
314 -> 17[label="",style="solid", color="burlywood", weight=3];
315[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 315[label="",style="solid", color="burlywood", weight=9];
315 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="signumReal2 (Pos (Succ vz3000)) (primEqInt (Pos (Succ vz3000)) (fromIntMyInt (Pos Zero)))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="signumReal2 (Pos Zero) (primEqInt (Pos Zero) (fromIntMyInt (Pos Zero)))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="signumReal2 (Neg (Succ vz3000)) (primEqInt (Neg (Succ vz3000)) (fromIntMyInt (Pos Zero)))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="signumReal2 (Neg Zero) (primEqInt (Neg Zero) (fromIntMyInt (Pos Zero)))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="signumReal2 (Pos (Succ vz3000)) (primEqInt (Pos (Succ vz3000)) (Pos Zero))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="signumReal2 (Pos Zero) (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="signumReal2 (Neg (Succ vz3000)) (primEqInt (Neg (Succ vz3000)) (Pos Zero))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="signumReal2 (Neg Zero) (primEqInt (Neg Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="signumReal2 (Pos (Succ vz3000)) MyFalse",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="signumReal2 (Pos Zero) MyTrue",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="signumReal2 (Neg (Succ vz3000)) MyFalse",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="signumReal2 (Neg Zero) MyTrue",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="signumReal1 (Pos (Succ vz3000)) (gtMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos Zero)))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="fromIntMyInt (Pos Zero)",fontsize=16,color="black",shape="triangle"];28 -> 32[label="",style="solid", color="black", weight=3];
29 -> 33[label="",style="dashed", color="red", weight=0];
29[label="signumReal1 (Neg (Succ vz3000)) (gtMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos Zero)))",fontsize=16,color="magenta"];29 -> 34[label="",style="dashed", color="magenta", weight=3];
30 -> 28[label="",style="dashed", color="red", weight=0];
30[label="fromIntMyInt (Pos Zero)",fontsize=16,color="magenta"];31 -> 35[label="",style="dashed", color="red", weight=0];
31[label="signumReal1 (Pos (Succ vz3000)) (esEsOrdering (compareMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos Zero))) GT)",fontsize=16,color="magenta"];31 -> 36[label="",style="dashed", color="magenta", weight=3];
32[label="Pos Zero",fontsize=16,color="green",shape="box"];34 -> 28[label="",style="dashed", color="red", weight=0];
34[label="fromIntMyInt (Pos Zero)",fontsize=16,color="magenta"];33[label="signumReal1 (Neg (Succ vz3000)) (gtMyInt (Neg (Succ vz3000)) vz4)",fontsize=16,color="black",shape="triangle"];33 -> 37[label="",style="solid", color="black", weight=3];
36 -> 28[label="",style="dashed", color="red", weight=0];
36[label="fromIntMyInt (Pos Zero)",fontsize=16,color="magenta"];35[label="signumReal1 (Pos (Succ vz3000)) (esEsOrdering (compareMyInt (Pos (Succ vz3000)) vz5) GT)",fontsize=16,color="black",shape="triangle"];35 -> 38[label="",style="solid", color="black", weight=3];
37[label="signumReal1 (Neg (Succ vz3000)) (esEsOrdering (compareMyInt (Neg (Succ vz3000)) vz4) GT)",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="signumReal1 (Pos (Succ vz3000)) (esEsOrdering (primCmpInt (Pos (Succ vz3000)) vz5) GT)",fontsize=16,color="burlywood",shape="box"];316[label="vz5/Pos vz50",fontsize=10,color="white",style="solid",shape="box"];38 -> 316[label="",style="solid", color="burlywood", weight=9];
316 -> 40[label="",style="solid", color="burlywood", weight=3];
317[label="vz5/Neg vz50",fontsize=10,color="white",style="solid",shape="box"];38 -> 317[label="",style="solid", color="burlywood", weight=9];
317 -> 41[label="",style="solid", color="burlywood", weight=3];
39[label="signumReal1 (Neg (Succ vz3000)) (esEsOrdering (primCmpInt (Neg (Succ vz3000)) vz4) GT)",fontsize=16,color="burlywood",shape="box"];318[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];39 -> 318[label="",style="solid", color="burlywood", weight=9];
318 -> 42[label="",style="solid", color="burlywood", weight=3];
319[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];39 -> 319[label="",style="solid", color="burlywood", weight=9];
319 -> 43[label="",style="solid", color="burlywood", weight=3];
40[label="signumReal1 (Pos (Succ vz3000)) (esEsOrdering (primCmpInt (Pos (Succ vz3000)) (Pos vz50)) GT)",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
41[label="signumReal1 (Pos (Succ vz3000)) (esEsOrdering (primCmpInt (Pos (Succ vz3000)) (Neg vz50)) GT)",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
42[label="signumReal1 (Neg (Succ vz3000)) (esEsOrdering (primCmpInt (Neg (Succ vz3000)) (Pos vz40)) GT)",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3];
43[label="signumReal1 (Neg (Succ vz3000)) (esEsOrdering (primCmpInt (Neg (Succ vz3000)) (Neg vz40)) GT)",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3];
44 -> 206[label="",style="dashed", color="red", weight=0];
44[label="signumReal1 (Pos (Succ vz3000)) (esEsOrdering (primCmpNat (Succ vz3000) vz50) GT)",fontsize=16,color="magenta"];44 -> 207[label="",style="dashed", color="magenta", weight=3];
44 -> 208[label="",style="dashed", color="magenta", weight=3];
44 -> 209[label="",style="dashed", color="magenta", weight=3];
45[label="signumReal1 (Pos (Succ vz3000)) (esEsOrdering GT GT)",fontsize=16,color="black",shape="triangle"];45 -> 50[label="",style="solid", color="black", weight=3];
46[label="signumReal1 (Neg (Succ vz3000)) (esEsOrdering LT GT)",fontsize=16,color="black",shape="triangle"];46 -> 51[label="",style="solid", color="black", weight=3];
47 -> 264[label="",style="dashed", color="red", weight=0];
47[label="signumReal1 (Neg (Succ vz3000)) (esEsOrdering (primCmpNat vz40 (Succ vz3000)) GT)",fontsize=16,color="magenta"];47 -> 265[label="",style="dashed", color="magenta", weight=3];
47 -> 266[label="",style="dashed", color="magenta", weight=3];
47 -> 267[label="",style="dashed", color="magenta", weight=3];
207[label="vz50",fontsize=16,color="green",shape="box"];208[label="Succ vz3000",fontsize=16,color="green",shape="box"];209[label="vz3000",fontsize=16,color="green",shape="box"];206[label="signumReal1 (Pos (Succ vz7)) (esEsOrdering (primCmpNat vz8 vz9) GT)",fontsize=16,color="burlywood",shape="triangle"];320[label="vz8/Succ vz80",fontsize=10,color="white",style="solid",shape="box"];206 -> 320[label="",style="solid", color="burlywood", weight=9];
320 -> 228[label="",style="solid", color="burlywood", weight=3];
321[label="vz8/Zero",fontsize=10,color="white",style="solid",shape="box"];206 -> 321[label="",style="solid", color="burlywood", weight=9];
321 -> 229[label="",style="solid", color="burlywood", weight=3];
50[label="signumReal1 (Pos (Succ vz3000)) MyTrue",fontsize=16,color="black",shape="box"];50 -> 56[label="",style="solid", color="black", weight=3];
51[label="signumReal1 (Neg (Succ vz3000)) MyFalse",fontsize=16,color="black",shape="triangle"];51 -> 57[label="",style="solid", color="black", weight=3];
265[label="vz40",fontsize=16,color="green",shape="box"];266[label="Succ vz3000",fontsize=16,color="green",shape="box"];267[label="vz3000",fontsize=16,color="green",shape="box"];264[label="signumReal1 (Neg (Succ vz13)) (esEsOrdering (primCmpNat vz14 vz15) GT)",fontsize=16,color="burlywood",shape="triangle"];322[label="vz14/Succ vz140",fontsize=10,color="white",style="solid",shape="box"];264 -> 322[label="",style="solid", color="burlywood", weight=9];
322 -> 289[label="",style="solid", color="burlywood", weight=3];
323[label="vz14/Zero",fontsize=10,color="white",style="solid",shape="box"];264 -> 323[label="",style="solid", color="burlywood", weight=9];
323 -> 290[label="",style="solid", color="burlywood", weight=3];
228[label="signumReal1 (Pos (Succ vz7)) (esEsOrdering (primCmpNat (Succ vz80) vz9) GT)",fontsize=16,color="burlywood",shape="box"];324[label="vz9/Succ vz90",fontsize=10,color="white",style="solid",shape="box"];228 -> 324[label="",style="solid", color="burlywood", weight=9];
324 -> 238[label="",style="solid", color="burlywood", weight=3];
325[label="vz9/Zero",fontsize=10,color="white",style="solid",shape="box"];228 -> 325[label="",style="solid", color="burlywood", weight=9];
325 -> 239[label="",style="solid", color="burlywood", weight=3];
229[label="signumReal1 (Pos (Succ vz7)) (esEsOrdering (primCmpNat Zero vz9) GT)",fontsize=16,color="burlywood",shape="box"];326[label="vz9/Succ vz90",fontsize=10,color="white",style="solid",shape="box"];229 -> 326[label="",style="solid", color="burlywood", weight=9];
326 -> 240[label="",style="solid", color="burlywood", weight=3];
327[label="vz9/Zero",fontsize=10,color="white",style="solid",shape="box"];229 -> 327[label="",style="solid", color="burlywood", weight=9];
327 -> 241[label="",style="solid", color="burlywood", weight=3];
56 -> 7[label="",style="dashed", color="red", weight=0];
56[label="fromIntMyInt (Pos (Succ Zero))",fontsize=16,color="magenta"];57[label="signumReal0 (Neg (Succ vz3000)) otherwise",fontsize=16,color="black",shape="box"];57 -> 62[label="",style="solid", color="black", weight=3];
289[label="signumReal1 (Neg (Succ vz13)) (esEsOrdering (primCmpNat (Succ vz140) vz15) GT)",fontsize=16,color="burlywood",shape="box"];328[label="vz15/Succ vz150",fontsize=10,color="white",style="solid",shape="box"];289 -> 328[label="",style="solid", color="burlywood", weight=9];
328 -> 292[label="",style="solid", color="burlywood", weight=3];
329[label="vz15/Zero",fontsize=10,color="white",style="solid",shape="box"];289 -> 329[label="",style="solid", color="burlywood", weight=9];
329 -> 293[label="",style="solid", color="burlywood", weight=3];
290[label="signumReal1 (Neg (Succ vz13)) (esEsOrdering (primCmpNat Zero vz15) GT)",fontsize=16,color="burlywood",shape="box"];330[label="vz15/Succ vz150",fontsize=10,color="white",style="solid",shape="box"];290 -> 330[label="",style="solid", color="burlywood", weight=9];
330 -> 294[label="",style="solid", color="burlywood", weight=3];
331[label="vz15/Zero",fontsize=10,color="white",style="solid",shape="box"];290 -> 331[label="",style="solid", color="burlywood", weight=9];
331 -> 295[label="",style="solid", color="burlywood", weight=3];
238[label="signumReal1 (Pos (Succ vz7)) (esEsOrdering (primCmpNat (Succ vz80) (Succ vz90)) GT)",fontsize=16,color="black",shape="box"];238 -> 249[label="",style="solid", color="black", weight=3];
239[label="signumReal1 (Pos (Succ vz7)) (esEsOrdering (primCmpNat (Succ vz80) Zero) GT)",fontsize=16,color="black",shape="box"];239 -> 250[label="",style="solid", color="black", weight=3];
240[label="signumReal1 (Pos (Succ vz7)) (esEsOrdering (primCmpNat Zero (Succ vz90)) GT)",fontsize=16,color="black",shape="box"];240 -> 251[label="",style="solid", color="black", weight=3];
241[label="signumReal1 (Pos (Succ vz7)) (esEsOrdering (primCmpNat Zero Zero) GT)",fontsize=16,color="black",shape="box"];241 -> 252[label="",style="solid", color="black", weight=3];
62[label="signumReal0 (Neg (Succ vz3000)) MyTrue",fontsize=16,color="black",shape="box"];62 -> 69[label="",style="solid", color="black", weight=3];
292[label="signumReal1 (Neg (Succ vz13)) (esEsOrdering (primCmpNat (Succ vz140) (Succ vz150)) GT)",fontsize=16,color="black",shape="box"];292 -> 297[label="",style="solid", color="black", weight=3];
293[label="signumReal1 (Neg (Succ vz13)) (esEsOrdering (primCmpNat (Succ vz140) Zero) GT)",fontsize=16,color="black",shape="box"];293 -> 298[label="",style="solid", color="black", weight=3];
294[label="signumReal1 (Neg (Succ vz13)) (esEsOrdering (primCmpNat Zero (Succ vz150)) GT)",fontsize=16,color="black",shape="box"];294 -> 299[label="",style="solid", color="black", weight=3];
295[label="signumReal1 (Neg (Succ vz13)) (esEsOrdering (primCmpNat Zero Zero) GT)",fontsize=16,color="black",shape="box"];295 -> 300[label="",style="solid", color="black", weight=3];
249 -> 206[label="",style="dashed", color="red", weight=0];
249[label="signumReal1 (Pos (Succ vz7)) (esEsOrdering (primCmpNat vz80 vz90) GT)",fontsize=16,color="magenta"];249 -> 259[label="",style="dashed", color="magenta", weight=3];
249 -> 260[label="",style="dashed", color="magenta", weight=3];
250 -> 45[label="",style="dashed", color="red", weight=0];
250[label="signumReal1 (Pos (Succ vz7)) (esEsOrdering GT GT)",fontsize=16,color="magenta"];250 -> 261[label="",style="dashed", color="magenta", weight=3];
251[label="signumReal1 (Pos (Succ vz7)) (esEsOrdering LT GT)",fontsize=16,color="black",shape="box"];251 -> 262[label="",style="solid", color="black", weight=3];
252[label="signumReal1 (Pos (Succ vz7)) (esEsOrdering EQ GT)",fontsize=16,color="black",shape="box"];252 -> 263[label="",style="solid", color="black", weight=3];
69[label="fromIntMyInt (Neg (Succ Zero))",fontsize=16,color="black",shape="triangle"];69 -> 78[label="",style="solid", color="black", weight=3];
297 -> 264[label="",style="dashed", color="red", weight=0];
297[label="signumReal1 (Neg (Succ vz13)) (esEsOrdering (primCmpNat vz140 vz150) GT)",fontsize=16,color="magenta"];297 -> 302[label="",style="dashed", color="magenta", weight=3];
297 -> 303[label="",style="dashed", color="magenta", weight=3];
298[label="signumReal1 (Neg (Succ vz13)) (esEsOrdering GT GT)",fontsize=16,color="black",shape="box"];298 -> 304[label="",style="solid", color="black", weight=3];
299 -> 46[label="",style="dashed", color="red", weight=0];
299[label="signumReal1 (Neg (Succ vz13)) (esEsOrdering LT GT)",fontsize=16,color="magenta"];299 -> 305[label="",style="dashed", color="magenta", weight=3];
300[label="signumReal1 (Neg (Succ vz13)) (esEsOrdering EQ GT)",fontsize=16,color="black",shape="box"];300 -> 306[label="",style="solid", color="black", weight=3];
259[label="vz90",fontsize=16,color="green",shape="box"];260[label="vz80",fontsize=16,color="green",shape="box"];261[label="vz7",fontsize=16,color="green",shape="box"];262[label="signumReal1 (Pos (Succ vz7)) MyFalse",fontsize=16,color="black",shape="triangle"];262 -> 291[label="",style="solid", color="black", weight=3];
263 -> 262[label="",style="dashed", color="red", weight=0];
263[label="signumReal1 (Pos (Succ vz7)) MyFalse",fontsize=16,color="magenta"];78[label="Neg (Succ Zero)",fontsize=16,color="green",shape="box"];302[label="vz140",fontsize=16,color="green",shape="box"];303[label="vz150",fontsize=16,color="green",shape="box"];304[label="signumReal1 (Neg (Succ vz13)) MyTrue",fontsize=16,color="black",shape="box"];304 -> 307[label="",style="solid", color="black", weight=3];
305[label="vz13",fontsize=16,color="green",shape="box"];306 -> 51[label="",style="dashed", color="red", weight=0];
306[label="signumReal1 (Neg (Succ vz13)) MyFalse",fontsize=16,color="magenta"];306 -> 308[label="",style="dashed", color="magenta", weight=3];
291[label="signumReal0 (Pos (Succ vz7)) otherwise",fontsize=16,color="black",shape="box"];291 -> 296[label="",style="solid", color="black", weight=3];
307 -> 7[label="",style="dashed", color="red", weight=0];
307[label="fromIntMyInt (Pos (Succ Zero))",fontsize=16,color="magenta"];308[label="vz13",fontsize=16,color="green",shape="box"];296[label="signumReal0 (Pos (Succ vz7)) MyTrue",fontsize=16,color="black",shape="box"];296 -> 301[label="",style="solid", color="black", weight=3];
301 -> 69[label="",style="dashed", color="red", weight=0];
301[label="fromIntMyInt (Neg (Succ Zero))",fontsize=16,color="magenta"];}

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

(6)
Complex Obligation (AND)

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

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

   new_signumReal1(vz13, Main.Succ(vz140), Main.Succ(vz150)) -> new_signumReal1(vz13, vz140, vz150)

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_signumReal1(vz13, Main.Succ(vz140), Main.Succ(vz150)) -> new_signumReal1(vz13, vz140, vz150)
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_signumReal10(vz7, Main.Succ(vz80), Main.Succ(vz90)) -> new_signumReal10(vz7, vz80, vz90)

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_signumReal10(vz7, Main.Succ(vz80), Main.Succ(vz90)) -> new_signumReal10(vz7, vz80, vz90)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3


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

(12)
YES