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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(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;
}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="subtract",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="subtract vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="subtract vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="flip (-) vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="(-) vx4 vx3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="primMinusFloat vx4 vx3",fontsize=16,color="burlywood",shape="box"];94[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];7 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 8[label="",style="solid", color="burlywood", weight=3];
8[label="primMinusFloat (Float vx40 vx41) vx3",fontsize=16,color="burlywood",shape="box"];95[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];8 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 9[label="",style="solid", color="burlywood", weight=3];
9[label="primMinusFloat (Float vx40 vx41) (Float vx30 vx31)",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="Float (vx40 * vx31 - vx30 * vx41) (vx41 * vx31)",fontsize=16,color="green",shape="box"];10 -> 11[label="",style="dashed", color="green", weight=3];
10 -> 12[label="",style="dashed", color="green", weight=3];
11[label="vx40 * vx31 - vx30 * vx41",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="vx41 * vx31",fontsize=16,color="black",shape="triangle"];12 -> 14[label="",style="solid", color="black", weight=3];
13 -> 15[label="",style="dashed", color="red", weight=0];
13[label="primMinusInt (vx40 * vx31) (vx30 * vx41)",fontsize=16,color="magenta"];13 -> 16[label="",style="dashed", color="magenta", weight=3];
13 -> 17[label="",style="dashed", color="magenta", weight=3];
14[label="primMulInt vx41 vx31",fontsize=16,color="burlywood",shape="box"];96[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];14 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 18[label="",style="solid", color="burlywood", weight=3];
97[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];14 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 19[label="",style="solid", color="burlywood", weight=3];
16 -> 12[label="",style="dashed", color="red", weight=0];
16[label="vx40 * vx31",fontsize=16,color="magenta"];16 -> 20[label="",style="dashed", color="magenta", weight=3];
17 -> 12[label="",style="dashed", color="red", weight=0];
17[label="vx30 * vx41",fontsize=16,color="magenta"];17 -> 21[label="",style="dashed", color="magenta", weight=3];
17 -> 22[label="",style="dashed", color="magenta", weight=3];
15[label="primMinusInt vx6 vx5",fontsize=16,color="burlywood",shape="triangle"];98[label="vx6/Pos vx60",fontsize=10,color="white",style="solid",shape="box"];15 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 23[label="",style="solid", color="burlywood", weight=3];
99[label="vx6/Neg vx60",fontsize=10,color="white",style="solid",shape="box"];15 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 24[label="",style="solid", color="burlywood", weight=3];
18[label="primMulInt (Pos vx410) vx31",fontsize=16,color="burlywood",shape="box"];100[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];18 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 25[label="",style="solid", color="burlywood", weight=3];
101[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];18 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 26[label="",style="solid", color="burlywood", weight=3];
19[label="primMulInt (Neg vx410) vx31",fontsize=16,color="burlywood",shape="box"];102[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];19 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 27[label="",style="solid", color="burlywood", weight=3];
103[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];19 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 28[label="",style="solid", color="burlywood", weight=3];
20[label="vx40",fontsize=16,color="green",shape="box"];21[label="vx41",fontsize=16,color="green",shape="box"];22[label="vx30",fontsize=16,color="green",shape="box"];23[label="primMinusInt (Pos vx60) vx5",fontsize=16,color="burlywood",shape="box"];104[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];23 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 29[label="",style="solid", color="burlywood", weight=3];
105[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];23 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 30[label="",style="solid", color="burlywood", weight=3];
24[label="primMinusInt (Neg vx60) vx5",fontsize=16,color="burlywood",shape="box"];106[label="vx5/Pos vx50",fontsize=10,color="white",style="solid",shape="box"];24 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 31[label="",style="solid", color="burlywood", weight=3];
107[label="vx5/Neg vx50",fontsize=10,color="white",style="solid",shape="box"];24 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 32[label="",style="solid", color="burlywood", weight=3];
25[label="primMulInt (Pos vx410) (Pos vx310)",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
26[label="primMulInt (Pos vx410) (Neg vx310)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
27[label="primMulInt (Neg vx410) (Pos vx310)",fontsize=16,color="black",shape="box"];27 -> 35[label="",style="solid", color="black", weight=3];
28[label="primMulInt (Neg vx410) (Neg vx310)",fontsize=16,color="black",shape="box"];28 -> 36[label="",style="solid", color="black", weight=3];
29[label="primMinusInt (Pos vx60) (Pos vx50)",fontsize=16,color="black",shape="box"];29 -> 37[label="",style="solid", color="black", weight=3];
30[label="primMinusInt (Pos vx60) (Neg vx50)",fontsize=16,color="black",shape="box"];30 -> 38[label="",style="solid", color="black", weight=3];
31[label="primMinusInt (Neg vx60) (Pos vx50)",fontsize=16,color="black",shape="box"];31 -> 39[label="",style="solid", color="black", weight=3];
32[label="primMinusInt (Neg vx60) (Neg vx50)",fontsize=16,color="black",shape="box"];32 -> 40[label="",style="solid", color="black", weight=3];
33[label="Pos (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];33 -> 41[label="",style="dashed", color="green", weight=3];
34[label="Neg (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];34 -> 42[label="",style="dashed", color="green", weight=3];
35[label="Neg (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];35 -> 43[label="",style="dashed", color="green", weight=3];
36[label="Pos (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];36 -> 44[label="",style="dashed", color="green", weight=3];
37[label="primMinusNat vx60 vx50",fontsize=16,color="burlywood",shape="triangle"];108[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];37 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 45[label="",style="solid", color="burlywood", weight=3];
109[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];37 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 46[label="",style="solid", color="burlywood", weight=3];
38[label="Pos (primPlusNat vx60 vx50)",fontsize=16,color="green",shape="box"];38 -> 47[label="",style="dashed", color="green", weight=3];
39[label="Neg (primPlusNat vx60 vx50)",fontsize=16,color="green",shape="box"];39 -> 48[label="",style="dashed", color="green", weight=3];
40 -> 37[label="",style="dashed", color="red", weight=0];
40[label="primMinusNat vx50 vx60",fontsize=16,color="magenta"];40 -> 49[label="",style="dashed", color="magenta", weight=3];
40 -> 50[label="",style="dashed", color="magenta", weight=3];
41[label="primMulNat vx410 vx310",fontsize=16,color="burlywood",shape="triangle"];110[label="vx410/Succ vx4100",fontsize=10,color="white",style="solid",shape="box"];41 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 51[label="",style="solid", color="burlywood", weight=3];
111[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 52[label="",style="solid", color="burlywood", weight=3];
42 -> 41[label="",style="dashed", color="red", weight=0];
42[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];42 -> 53[label="",style="dashed", color="magenta", weight=3];
43 -> 41[label="",style="dashed", color="red", weight=0];
43[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];43 -> 54[label="",style="dashed", color="magenta", weight=3];
44 -> 41[label="",style="dashed", color="red", weight=0];
44[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];44 -> 55[label="",style="dashed", color="magenta", weight=3];
44 -> 56[label="",style="dashed", color="magenta", weight=3];
45[label="primMinusNat (Succ vx600) vx50",fontsize=16,color="burlywood",shape="box"];112[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];45 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 57[label="",style="solid", color="burlywood", weight=3];
113[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 113[label="",style="solid", color="burlywood", weight=9];
113 -> 58[label="",style="solid", color="burlywood", weight=3];
46[label="primMinusNat Zero vx50",fontsize=16,color="burlywood",shape="box"];114[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];46 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 59[label="",style="solid", color="burlywood", weight=3];
115[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 60[label="",style="solid", color="burlywood", weight=3];
47[label="primPlusNat vx60 vx50",fontsize=16,color="burlywood",shape="triangle"];116[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];47 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 61[label="",style="solid", color="burlywood", weight=3];
117[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 117[label="",style="solid", color="burlywood", weight=9];
117 -> 62[label="",style="solid", color="burlywood", weight=3];
48 -> 47[label="",style="dashed", color="red", weight=0];
48[label="primPlusNat vx60 vx50",fontsize=16,color="magenta"];48 -> 63[label="",style="dashed", color="magenta", weight=3];
48 -> 64[label="",style="dashed", color="magenta", weight=3];
49[label="vx50",fontsize=16,color="green",shape="box"];50[label="vx60",fontsize=16,color="green",shape="box"];51[label="primMulNat (Succ vx4100) vx310",fontsize=16,color="burlywood",shape="box"];118[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];51 -> 118[label="",style="solid", color="burlywood", weight=9];
118 -> 65[label="",style="solid", color="burlywood", weight=3];
119[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];51 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 66[label="",style="solid", color="burlywood", weight=3];
52[label="primMulNat Zero vx310",fontsize=16,color="burlywood",shape="box"];120[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];52 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 67[label="",style="solid", color="burlywood", weight=3];
121[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 68[label="",style="solid", color="burlywood", weight=3];
53[label="vx310",fontsize=16,color="green",shape="box"];54[label="vx410",fontsize=16,color="green",shape="box"];55[label="vx310",fontsize=16,color="green",shape="box"];56[label="vx410",fontsize=16,color="green",shape="box"];57[label="primMinusNat (Succ vx600) (Succ vx500)",fontsize=16,color="black",shape="box"];57 -> 69[label="",style="solid", color="black", weight=3];
58[label="primMinusNat (Succ vx600) Zero",fontsize=16,color="black",shape="box"];58 -> 70[label="",style="solid", color="black", weight=3];
59[label="primMinusNat Zero (Succ vx500)",fontsize=16,color="black",shape="box"];59 -> 71[label="",style="solid", color="black", weight=3];
60[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];60 -> 72[label="",style="solid", color="black", weight=3];
61[label="primPlusNat (Succ vx600) vx50",fontsize=16,color="burlywood",shape="box"];122[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];61 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 73[label="",style="solid", color="burlywood", weight=3];
123[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];61 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 74[label="",style="solid", color="burlywood", weight=3];
62[label="primPlusNat Zero vx50",fontsize=16,color="burlywood",shape="box"];124[label="vx50/Succ vx500",fontsize=10,color="white",style="solid",shape="box"];62 -> 124[label="",style="solid", color="burlywood", weight=9];
124 -> 75[label="",style="solid", color="burlywood", weight=3];
125[label="vx50/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 125[label="",style="solid", color="burlywood", weight=9];
125 -> 76[label="",style="solid", color="burlywood", weight=3];
63[label="vx60",fontsize=16,color="green",shape="box"];64[label="vx50",fontsize=16,color="green",shape="box"];65[label="primMulNat (Succ vx4100) (Succ vx3100)",fontsize=16,color="black",shape="box"];65 -> 77[label="",style="solid", color="black", weight=3];
66[label="primMulNat (Succ vx4100) Zero",fontsize=16,color="black",shape="box"];66 -> 78[label="",style="solid", color="black", weight=3];
67[label="primMulNat Zero (Succ vx3100)",fontsize=16,color="black",shape="box"];67 -> 79[label="",style="solid", color="black", weight=3];
68[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];68 -> 80[label="",style="solid", color="black", weight=3];
69 -> 37[label="",style="dashed", color="red", weight=0];
69[label="primMinusNat vx600 vx500",fontsize=16,color="magenta"];69 -> 81[label="",style="dashed", color="magenta", weight=3];
69 -> 82[label="",style="dashed", color="magenta", weight=3];
70[label="Pos (Succ vx600)",fontsize=16,color="green",shape="box"];71[label="Neg (Succ vx500)",fontsize=16,color="green",shape="box"];72[label="Pos Zero",fontsize=16,color="green",shape="box"];73[label="primPlusNat (Succ vx600) (Succ vx500)",fontsize=16,color="black",shape="box"];73 -> 83[label="",style="solid", color="black", weight=3];
74[label="primPlusNat (Succ vx600) Zero",fontsize=16,color="black",shape="box"];74 -> 84[label="",style="solid", color="black", weight=3];
75[label="primPlusNat Zero (Succ vx500)",fontsize=16,color="black",shape="box"];75 -> 85[label="",style="solid", color="black", weight=3];
76[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];76 -> 86[label="",style="solid", color="black", weight=3];
77 -> 47[label="",style="dashed", color="red", weight=0];
77[label="primPlusNat (primMulNat vx4100 (Succ vx3100)) (Succ vx3100)",fontsize=16,color="magenta"];77 -> 87[label="",style="dashed", color="magenta", weight=3];
77 -> 88[label="",style="dashed", color="magenta", weight=3];
78[label="Zero",fontsize=16,color="green",shape="box"];79[label="Zero",fontsize=16,color="green",shape="box"];80[label="Zero",fontsize=16,color="green",shape="box"];81[label="vx600",fontsize=16,color="green",shape="box"];82[label="vx500",fontsize=16,color="green",shape="box"];83[label="Succ (Succ (primPlusNat vx600 vx500))",fontsize=16,color="green",shape="box"];83 -> 89[label="",style="dashed", color="green", weight=3];
84[label="Succ vx600",fontsize=16,color="green",shape="box"];85[label="Succ vx500",fontsize=16,color="green",shape="box"];86[label="Zero",fontsize=16,color="green",shape="box"];87 -> 41[label="",style="dashed", color="red", weight=0];
87[label="primMulNat vx4100 (Succ vx3100)",fontsize=16,color="magenta"];87 -> 90[label="",style="dashed", color="magenta", weight=3];
87 -> 91[label="",style="dashed", color="magenta", weight=3];
88[label="Succ vx3100",fontsize=16,color="green",shape="box"];89 -> 47[label="",style="dashed", color="red", weight=0];
89[label="primPlusNat vx600 vx500",fontsize=16,color="magenta"];89 -> 92[label="",style="dashed", color="magenta", weight=3];
89 -> 93[label="",style="dashed", color="magenta", weight=3];
90[label="Succ vx3100",fontsize=16,color="green",shape="box"];91[label="vx4100",fontsize=16,color="green",shape="box"];92[label="vx600",fontsize=16,color="green",shape="box"];93[label="vx500",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(Succ(vx4100), Succ(vx3100)) -> new_primMulNat(vx4100, Succ(vx3100))

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(Succ(vx4100), Succ(vx3100)) -> new_primMulNat(vx4100, Succ(vx3100))
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(Succ(vx600), Succ(vx500)) -> new_primPlusNat(vx600, vx500)

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(Succ(vx600), Succ(vx500)) -> new_primPlusNat(vx600, vx500)
The graph contains the following edges 1 > 1, 2 > 2


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

(12)
YES

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

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

   new_primMinusNat(Succ(vx600), Succ(vx500)) -> new_primMinusNat(vx600, vx500)

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

(14) 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_primMinusNat(Succ(vx600), Succ(vx500)) -> new_primMinusNat(vx600, vx500)
The graph contains the following edges 1 > 1, 2 > 2


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

(15)
YES