/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 MyBool = MyTrue  | MyFalse ;

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

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

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

fsEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
fsEsMyInt x y = not (esEsMyInt x y);

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

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;

}

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

(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 ;

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

fsEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
fsEsMyInt x y = not (esEsMyInt x y);

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

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;

}

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

(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 ;

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

fsEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
fsEsMyInt x y = not (esEsMyInt x y);

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

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;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="fsEsMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="fsEsMyInt vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="fsEsMyInt vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="not (esEsMyInt vz3 vz4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="not (primEqInt vz3 vz4)",fontsize=16,color="burlywood",shape="box"];63[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];6 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 7[label="",style="solid", color="burlywood", weight=3];
64[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];6 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="not (primEqInt (Pos vz30) vz4)",fontsize=16,color="burlywood",shape="box"];65[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];7 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 9[label="",style="solid", color="burlywood", weight=3];
66[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];7 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="not (primEqInt (Neg vz30) vz4)",fontsize=16,color="burlywood",shape="box"];67[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];8 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 11[label="",style="solid", color="burlywood", weight=3];
68[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 12[label="",style="solid", color="burlywood", weight=3];
9[label="not (primEqInt (Pos (Succ vz300)) vz4)",fontsize=16,color="burlywood",shape="box"];69[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];9 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 13[label="",style="solid", color="burlywood", weight=3];
70[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];9 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 14[label="",style="solid", color="burlywood", weight=3];
10[label="not (primEqInt (Pos Zero) vz4)",fontsize=16,color="burlywood",shape="box"];71[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];10 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 15[label="",style="solid", color="burlywood", weight=3];
72[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];10 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 16[label="",style="solid", color="burlywood", weight=3];
11[label="not (primEqInt (Neg (Succ vz300)) vz4)",fontsize=16,color="burlywood",shape="box"];73[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];11 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 17[label="",style="solid", color="burlywood", weight=3];
74[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];11 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 18[label="",style="solid", color="burlywood", weight=3];
12[label="not (primEqInt (Neg Zero) vz4)",fontsize=16,color="burlywood",shape="box"];75[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];12 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 19[label="",style="solid", color="burlywood", weight=3];
76[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];12 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 20[label="",style="solid", color="burlywood", weight=3];
13[label="not (primEqInt (Pos (Succ vz300)) (Pos vz40))",fontsize=16,color="burlywood",shape="box"];77[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];13 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 21[label="",style="solid", color="burlywood", weight=3];
78[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 22[label="",style="solid", color="burlywood", weight=3];
14[label="not (primEqInt (Pos (Succ vz300)) (Neg vz40))",fontsize=16,color="black",shape="box"];14 -> 23[label="",style="solid", color="black", weight=3];
15[label="not (primEqInt (Pos Zero) (Pos vz40))",fontsize=16,color="burlywood",shape="box"];79[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];15 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 24[label="",style="solid", color="burlywood", weight=3];
80[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 25[label="",style="solid", color="burlywood", weight=3];
16[label="not (primEqInt (Pos Zero) (Neg vz40))",fontsize=16,color="burlywood",shape="box"];81[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];16 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 26[label="",style="solid", color="burlywood", weight=3];
82[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 27[label="",style="solid", color="burlywood", weight=3];
17[label="not (primEqInt (Neg (Succ vz300)) (Pos vz40))",fontsize=16,color="black",shape="box"];17 -> 28[label="",style="solid", color="black", weight=3];
18[label="not (primEqInt (Neg (Succ vz300)) (Neg vz40))",fontsize=16,color="burlywood",shape="box"];83[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];18 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 29[label="",style="solid", color="burlywood", weight=3];
84[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 30[label="",style="solid", color="burlywood", weight=3];
19[label="not (primEqInt (Neg Zero) (Pos vz40))",fontsize=16,color="burlywood",shape="box"];85[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];19 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 31[label="",style="solid", color="burlywood", weight=3];
86[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 32[label="",style="solid", color="burlywood", weight=3];
20[label="not (primEqInt (Neg Zero) (Neg vz40))",fontsize=16,color="burlywood",shape="box"];87[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];20 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 33[label="",style="solid", color="burlywood", weight=3];
88[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 34[label="",style="solid", color="burlywood", weight=3];
21[label="not (primEqInt (Pos (Succ vz300)) (Pos (Succ vz400)))",fontsize=16,color="black",shape="box"];21 -> 35[label="",style="solid", color="black", weight=3];
22[label="not (primEqInt (Pos (Succ vz300)) (Pos Zero))",fontsize=16,color="black",shape="box"];22 -> 36[label="",style="solid", color="black", weight=3];
23[label="not MyFalse",fontsize=16,color="black",shape="triangle"];23 -> 37[label="",style="solid", color="black", weight=3];
24[label="not (primEqInt (Pos Zero) (Pos (Succ vz400)))",fontsize=16,color="black",shape="box"];24 -> 38[label="",style="solid", color="black", weight=3];
25[label="not (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];25 -> 39[label="",style="solid", color="black", weight=3];
26[label="not (primEqInt (Pos Zero) (Neg (Succ vz400)))",fontsize=16,color="black",shape="box"];26 -> 40[label="",style="solid", color="black", weight=3];
27[label="not (primEqInt (Pos Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];27 -> 41[label="",style="solid", color="black", weight=3];
28 -> 23[label="",style="dashed", color="red", weight=0];
28[label="not MyFalse",fontsize=16,color="magenta"];29[label="not (primEqInt (Neg (Succ vz300)) (Neg (Succ vz400)))",fontsize=16,color="black",shape="box"];29 -> 42[label="",style="solid", color="black", weight=3];
30[label="not (primEqInt (Neg (Succ vz300)) (Neg Zero))",fontsize=16,color="black",shape="box"];30 -> 43[label="",style="solid", color="black", weight=3];
31[label="not (primEqInt (Neg Zero) (Pos (Succ vz400)))",fontsize=16,color="black",shape="box"];31 -> 44[label="",style="solid", color="black", weight=3];
32[label="not (primEqInt (Neg Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];32 -> 45[label="",style="solid", color="black", weight=3];
33[label="not (primEqInt (Neg Zero) (Neg (Succ vz400)))",fontsize=16,color="black",shape="box"];33 -> 46[label="",style="solid", color="black", weight=3];
34[label="not (primEqInt (Neg Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];34 -> 47[label="",style="solid", color="black", weight=3];
35[label="not (primEqNat vz300 vz400)",fontsize=16,color="burlywood",shape="triangle"];89[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 48[label="",style="solid", color="burlywood", weight=3];
90[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 49[label="",style="solid", color="burlywood", weight=3];
36 -> 23[label="",style="dashed", color="red", weight=0];
36[label="not MyFalse",fontsize=16,color="magenta"];37[label="MyTrue",fontsize=16,color="green",shape="box"];38 -> 23[label="",style="dashed", color="red", weight=0];
38[label="not MyFalse",fontsize=16,color="magenta"];39[label="not MyTrue",fontsize=16,color="black",shape="triangle"];39 -> 50[label="",style="solid", color="black", weight=3];
40 -> 23[label="",style="dashed", color="red", weight=0];
40[label="not MyFalse",fontsize=16,color="magenta"];41 -> 39[label="",style="dashed", color="red", weight=0];
41[label="not MyTrue",fontsize=16,color="magenta"];42 -> 35[label="",style="dashed", color="red", weight=0];
42[label="not (primEqNat vz300 vz400)",fontsize=16,color="magenta"];42 -> 51[label="",style="dashed", color="magenta", weight=3];
42 -> 52[label="",style="dashed", color="magenta", weight=3];
43 -> 23[label="",style="dashed", color="red", weight=0];
43[label="not MyFalse",fontsize=16,color="magenta"];44 -> 23[label="",style="dashed", color="red", weight=0];
44[label="not MyFalse",fontsize=16,color="magenta"];45 -> 39[label="",style="dashed", color="red", weight=0];
45[label="not MyTrue",fontsize=16,color="magenta"];46 -> 23[label="",style="dashed", color="red", weight=0];
46[label="not MyFalse",fontsize=16,color="magenta"];47 -> 39[label="",style="dashed", color="red", weight=0];
47[label="not MyTrue",fontsize=16,color="magenta"];48[label="not (primEqNat (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];91[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];48 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 53[label="",style="solid", color="burlywood", weight=3];
92[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 54[label="",style="solid", color="burlywood", weight=3];
49[label="not (primEqNat Zero vz400)",fontsize=16,color="burlywood",shape="box"];93[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];49 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 55[label="",style="solid", color="burlywood", weight=3];
94[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 56[label="",style="solid", color="burlywood", weight=3];
50[label="MyFalse",fontsize=16,color="green",shape="box"];51[label="vz300",fontsize=16,color="green",shape="box"];52[label="vz400",fontsize=16,color="green",shape="box"];53[label="not (primEqNat (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="not (primEqNat (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
55[label="not (primEqNat Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3];
56[label="not (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3];
57 -> 35[label="",style="dashed", color="red", weight=0];
57[label="not (primEqNat vz3000 vz4000)",fontsize=16,color="magenta"];57 -> 61[label="",style="dashed", color="magenta", weight=3];
57 -> 62[label="",style="dashed", color="magenta", weight=3];
58 -> 23[label="",style="dashed", color="red", weight=0];
58[label="not MyFalse",fontsize=16,color="magenta"];59 -> 23[label="",style="dashed", color="red", weight=0];
59[label="not MyFalse",fontsize=16,color="magenta"];60 -> 39[label="",style="dashed", color="red", weight=0];
60[label="not MyTrue",fontsize=16,color="magenta"];61[label="vz3000",fontsize=16,color="green",shape="box"];62[label="vz4000",fontsize=16,color="green",shape="box"];}

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

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

   new_not(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_not(vz3000, vz4000)

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_not(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_not(vz3000, vz4000)
The graph contains the following edges 1 > 1, 2 > 2


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

(8)
YES