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


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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

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


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

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

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

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

primMulFloat :: Float  ->  Float  ->  Float;
primMulFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y1) (srMyInt x2 y2);

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

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

srFloat :: Float  ->  Float  ->  Float;
srFloat = primMulFloat;

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

}

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

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

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

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

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

primMulFloat :: Float  ->  Float  ->  Float;
primMulFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y1) (srMyInt x2 y2);

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

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

srFloat :: Float  ->  Float  ->  Float;
srFloat = primMulFloat;

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

}

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

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

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

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Float = Float MyInt MyInt ;

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

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

primMulFloat :: Float  ->  Float  ->  Float;
primMulFloat (Float x1 x2) (Float y1 y2) = Float (srMyInt x1 y1) (srMyInt x2 y2);

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

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

srFloat :: Float  ->  Float  ->  Float;
srFloat = primMulFloat;

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

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="srFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="srFloat vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="srFloat vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="primMulFloat vx3 vx4",fontsize=16,color="burlywood",shape="box"];64[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];5 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 6[label="",style="solid", color="burlywood", weight=3];
6[label="primMulFloat (Float vx30 vx31) vx4",fontsize=16,color="burlywood",shape="box"];65[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 7[label="",style="solid", color="burlywood", weight=3];
7[label="primMulFloat (Float vx30 vx31) (Float vx40 vx41)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="Float (srMyInt vx30 vx40) (srMyInt vx31 vx41)",fontsize=16,color="green",shape="box"];8 -> 9[label="",style="dashed", color="green", weight=3];
8 -> 10[label="",style="dashed", color="green", weight=3];
9[label="srMyInt vx30 vx40",fontsize=16,color="black",shape="triangle"];9 -> 11[label="",style="solid", color="black", weight=3];
10 -> 9[label="",style="dashed", color="red", weight=0];
10[label="srMyInt vx31 vx41",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
10 -> 13[label="",style="dashed", color="magenta", weight=3];
11[label="primMulInt vx30 vx40",fontsize=16,color="burlywood",shape="box"];66[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 14[label="",style="solid", color="burlywood", weight=3];
67[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 15[label="",style="solid", color="burlywood", weight=3];
12[label="vx31",fontsize=16,color="green",shape="box"];13[label="vx41",fontsize=16,color="green",shape="box"];14[label="primMulInt (Pos vx300) vx40",fontsize=16,color="burlywood",shape="box"];68[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 16[label="",style="solid", color="burlywood", weight=3];
69[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 17[label="",style="solid", color="burlywood", weight=3];
15[label="primMulInt (Neg vx300) vx40",fontsize=16,color="burlywood",shape="box"];70[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];15 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 18[label="",style="solid", color="burlywood", weight=3];
71[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];15 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 19[label="",style="solid", color="burlywood", weight=3];
16[label="primMulInt (Pos vx300) (Pos vx400)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="primMulInt (Pos vx300) (Neg vx400)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="primMulInt (Neg vx300) (Pos vx400)",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="primMulInt (Neg vx300) (Neg vx400)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="Pos (primMulNat vx300 vx400)",fontsize=16,color="green",shape="box"];20 -> 24[label="",style="dashed", color="green", weight=3];
21[label="Neg (primMulNat vx300 vx400)",fontsize=16,color="green",shape="box"];21 -> 25[label="",style="dashed", color="green", weight=3];
22[label="Neg (primMulNat vx300 vx400)",fontsize=16,color="green",shape="box"];22 -> 26[label="",style="dashed", color="green", weight=3];
23[label="Pos (primMulNat vx300 vx400)",fontsize=16,color="green",shape="box"];23 -> 27[label="",style="dashed", color="green", weight=3];
24[label="primMulNat vx300 vx400",fontsize=16,color="burlywood",shape="triangle"];72[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];24 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 28[label="",style="solid", color="burlywood", weight=3];
73[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 29[label="",style="solid", color="burlywood", weight=3];
25 -> 24[label="",style="dashed", color="red", weight=0];
25[label="primMulNat vx300 vx400",fontsize=16,color="magenta"];25 -> 30[label="",style="dashed", color="magenta", weight=3];
26 -> 24[label="",style="dashed", color="red", weight=0];
26[label="primMulNat vx300 vx400",fontsize=16,color="magenta"];26 -> 31[label="",style="dashed", color="magenta", weight=3];
27 -> 24[label="",style="dashed", color="red", weight=0];
27[label="primMulNat vx300 vx400",fontsize=16,color="magenta"];27 -> 32[label="",style="dashed", color="magenta", weight=3];
27 -> 33[label="",style="dashed", color="magenta", weight=3];
28[label="primMulNat (Succ vx3000) vx400",fontsize=16,color="burlywood",shape="box"];74[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 34[label="",style="solid", color="burlywood", weight=3];
75[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 35[label="",style="solid", color="burlywood", weight=3];
29[label="primMulNat Zero vx400",fontsize=16,color="burlywood",shape="box"];76[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 36[label="",style="solid", color="burlywood", weight=3];
77[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 37[label="",style="solid", color="burlywood", weight=3];
30[label="vx400",fontsize=16,color="green",shape="box"];31[label="vx300",fontsize=16,color="green",shape="box"];32[label="vx300",fontsize=16,color="green",shape="box"];33[label="vx400",fontsize=16,color="green",shape="box"];34[label="primMulNat (Succ vx3000) (Succ vx4000)",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
35[label="primMulNat (Succ vx3000) Zero",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
36[label="primMulNat Zero (Succ vx4000)",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
37[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38 -> 42[label="",style="dashed", color="red", weight=0];
38[label="primPlusNat (primMulNat vx3000 (Succ vx4000)) (Succ vx4000)",fontsize=16,color="magenta"];38 -> 43[label="",style="dashed", color="magenta", weight=3];
39[label="Zero",fontsize=16,color="green",shape="box"];40[label="Zero",fontsize=16,color="green",shape="box"];41[label="Zero",fontsize=16,color="green",shape="box"];43 -> 24[label="",style="dashed", color="red", weight=0];
43[label="primMulNat vx3000 (Succ vx4000)",fontsize=16,color="magenta"];43 -> 44[label="",style="dashed", color="magenta", weight=3];
43 -> 45[label="",style="dashed", color="magenta", weight=3];
42[label="primPlusNat vx5 (Succ vx4000)",fontsize=16,color="burlywood",shape="triangle"];78[label="vx5/Succ vx50",fontsize=10,color="white",style="solid",shape="box"];42 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 46[label="",style="solid", color="burlywood", weight=3];
79[label="vx5/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 47[label="",style="solid", color="burlywood", weight=3];
44[label="vx3000",fontsize=16,color="green",shape="box"];45[label="Succ vx4000",fontsize=16,color="green",shape="box"];46[label="primPlusNat (Succ vx50) (Succ vx4000)",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
47[label="primPlusNat Zero (Succ vx4000)",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
48[label="Succ (Succ (primPlusNat vx50 vx4000))",fontsize=16,color="green",shape="box"];48 -> 50[label="",style="dashed", color="green", weight=3];
49[label="Succ vx4000",fontsize=16,color="green",shape="box"];50[label="primPlusNat vx50 vx4000",fontsize=16,color="burlywood",shape="triangle"];80[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];50 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 51[label="",style="solid", color="burlywood", weight=3];
81[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 52[label="",style="solid", color="burlywood", weight=3];
51[label="primPlusNat (Succ vx500) vx4000",fontsize=16,color="burlywood",shape="box"];82[label="vx4000/Succ vx40000",fontsize=10,color="white",style="solid",shape="box"];51 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 53[label="",style="solid", color="burlywood", weight=3];
83[label="vx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];51 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 54[label="",style="solid", color="burlywood", weight=3];
52[label="primPlusNat Zero vx4000",fontsize=16,color="burlywood",shape="box"];84[label="vx4000/Succ vx40000",fontsize=10,color="white",style="solid",shape="box"];52 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 55[label="",style="solid", color="burlywood", weight=3];
85[label="vx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 56[label="",style="solid", color="burlywood", weight=3];
53[label="primPlusNat (Succ vx500) (Succ vx40000)",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="primPlusNat (Succ vx500) Zero",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
55[label="primPlusNat Zero (Succ vx40000)",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3];
56[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3];
57[label="Succ (Succ (primPlusNat vx500 vx40000))",fontsize=16,color="green",shape="box"];57 -> 61[label="",style="dashed", color="green", weight=3];
58[label="Succ vx500",fontsize=16,color="green",shape="box"];59[label="Succ vx40000",fontsize=16,color="green",shape="box"];60[label="Zero",fontsize=16,color="green",shape="box"];61 -> 50[label="",style="dashed", color="red", weight=0];
61[label="primPlusNat vx500 vx40000",fontsize=16,color="magenta"];61 -> 62[label="",style="dashed", color="magenta", weight=3];
61 -> 63[label="",style="dashed", color="magenta", weight=3];
62[label="vx500",fontsize=16,color="green",shape="box"];63[label="vx40000",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

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

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

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

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

From the DPs we obtained the following set of size-change graphs:
*new_primMulNat(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_primMulNat(vx3000, Main.Succ(vx4000))
The graph contains the following edges 1 > 1, 2 >= 2


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

(9)
YES

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

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

   new_primPlusNat(Main.Succ(vx500), Main.Succ(vx40000)) -> new_primPlusNat(vx500, vx40000)

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

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

From the DPs we obtained the following set of size-change graphs:
*new_primPlusNat(Main.Succ(vx500), Main.Succ(vx40000)) -> new_primPlusNat(vx500, vx40000)
The graph contains the following edges 1 > 1, 2 > 2


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

(12)
YES