Left Termination proof of /tmp/tmpQVWppV/avg.pl

Left Termination of the query pattern
avg(g,g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 87 ms)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇒, 13 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 PiDPToQDPProof (⇒, 21 ms)

↳8 QDP

↳9 MRRProof (⇔, 49 ms)

↳10 QDP

↳11 PisEmptyProof (⇔, 0 ms)

↳12 YES

(0) Obligation:

Clauses:

avg(s(X), Y, Z) :- avg(X, s(Y), Z).
avg(X, s(s(s(Y))), s(Z)) :- avg(s(X), Y, Z).
avg(0, 0, 0).
avg(0, s(0), 0).
avg(0, s(s(0)), s(0)).

Query: avg(g,g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: avg(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: avg(T7, s(T8), T10) | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T1, T2, T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 0 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 1 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 2 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 0\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 1 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 2 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: avg(T23, s(s(T24)), T26)\n\nT23 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 2 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: avg(s(T42), T43, T45)\n\nT42 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: avg(s(s(T61)), T62, T64)\n\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: avg(s(T71), T72, T74) | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\nX4 is free\nX5 is free\nX6 is free\nX64 is free\nX65 is free\nX66 is free\navg(T1, T2, T3) !=? avg(s(X4), X5, X6)\navg(T1, T2, T3) !=? avg(X64, s(s(s(X65))), s(X66))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 0 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 1 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 2 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 0\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 1 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 2 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: avg(T101, s(T102), T104)\n\nT101 is ground\nT102 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T101, s(s(s(T102))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 1\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 2 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: avg(s(s(T123)), T124, T126)\n\nT123 is ground\nT124 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T123, s(s(s(s(s(s(T124)))))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX4 is free\nX5 is free\nX6 is free\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true | avg(0, 0, T3), SCOPE: 1, CLAUSE: 3 | avg(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\nX4 is free\nX5 is free\nX6 is free\nX64 is free\nX65 is free\nX66 is free\navg(T1, T2, T3) !=? avg(s(X4), X5, X6)\navg(T1, T2, T3) !=? avg(X64, s(s(s(X65))), s(X66))\navg(T1, T2, T3) !=? avg(0, 0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: avg(0, 0, T3), SCOPE: 1, CLAUSE: 3 | avg(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: avg(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true | avg(0, s(0), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\nX4 is free\nX5 is free\nX6 is free\nX64 is free\nX65 is free\nX66 is free\navg(T1, T2, T3) !=? avg(s(X4), X5, X6)\navg(T1, T2, T3) !=? avg(X64, s(s(s(X65))), s(X66))\navg(T1, T2, T3) !=? avg(0, 0, 0)\navg(T1, T2, T3) !=? avg(0, s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: avg(0, s(0), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 62 [arrowhead = none ];
62 -> 2;

63 [label="EVAL with clause\navg(s(X4), X5, X6) :- avg(X4, s(X5), X6).\nand substitutionX4 -> T7,\nT1 -> s(T7),\nT2 -> T8,\nX5 -> T8,\nT3 -> T10,\nX6 -> T10,\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 63 [arrowhead = none ];
63 -> 3;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 64 [arrowhead = none ];
64 -> 4;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 65 [arrowhead = none ];
65 -> 5;

66 [label="EVAL with clause\navg(X64, s(s(s(X65))), s(X66)) :- avg(s(X64), X65, X66).\nand substitutionT1 -> T71,\nX64 -> T71,\nX65 -> T72,\nT2 -> s(s(s(T72))),\nX66 -> T74,\nT3 -> s(T74),\nT73 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 66 [arrowhead = none ];
66 -> 33;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 67 [arrowhead = none ];
67 -> 34;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 68 [arrowhead = none ];
68 -> 6;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 69 [arrowhead = none ];
69 -> 7;

70 [label="EVAL with clause\navg(s(X19), X20, X21) :- avg(X19, s(X20), X21).\nand substitutionX19 -> T23,\nT7 -> s(T23),\nT8 -> T24,\nX20 -> s(T24),\nT10 -> T26,\nX21 -> T26,\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 70 [arrowhead = none ];
70 -> 8;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 71 [arrowhead = none ];
71 -> 9;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 72 [arrowhead = none ];
72 -> 10;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 73 [arrowhead = none ];
73 -> 11;

74 [label="INSTANCE with matching:\nT1 -> T23\nT2 -> s(s(T24))\nT3 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
8 -> 74 [arrowhead = none , style=dashed];
74 -> 1[style=dashed];

75 [label="EVAL with clause\navg(X37, s(s(s(X38))), s(X39)) :- avg(s(X37), X38, X39).\nand substitutionT7 -> T42,\nX37 -> T42,\nX38 -> T43,\nT8 -> s(s(T43)),\nX39 -> T45,\nT10 -> s(T45),\nT44 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 75 [arrowhead = none ];
75 -> 12;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 76 [arrowhead = none ];
76 -> 13;

77 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 77 [arrowhead = none ];
77 -> 14;

78 [label="INSTANCE with matching:\nT1 -> s(T42)\nT2 -> T43\nT3 -> T45", fontsize=14, style = filled, fillcolor = lightgrey];
12 -> 78 [arrowhead = none , style=dashed];
78 -> 1[style=dashed];

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 79 [arrowhead = none ];
79 -> 15;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 80 [arrowhead = none ];
80 -> 16;

81 [label="EVAL with clause\navg(0, s(0), 0).\nand substitutionT7 -> 0,\nT8 -> 0,\nT10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 81 [arrowhead = none ];
81 -> 17;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 82 [arrowhead = none ];
82 -> 18;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 83 [arrowhead = none ];
83 -> 20;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 84 [arrowhead = none ];
84 -> 21;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 85 [arrowhead = none ];
85 -> 19;

86 [label="EVAL with clause\navg(0, s(s(0)), s(0)).\nand substitutionT7 -> 0,\nT8 -> s(0),\nT10 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 86 [arrowhead = none ];
86 -> 22;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 87 [arrowhead = none ];
87 -> 23;

88 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 88 [arrowhead = none ];
88 -> 25;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 89 [arrowhead = none ];
89 -> 24;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 90 [arrowhead = none ];
90 -> 26;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 91 [arrowhead = none ];
91 -> 27;

92 [label="EVAL with clause\navg(X55, s(s(s(X56))), s(X57)) :- avg(s(X55), X56, X57).\nand substitutionT7 -> T61,\nX55 -> s(T61),\nX56 -> T62,\nT8 -> s(s(s(T62))),\nX57 -> T64,\nT3 -> s(T64),\nT63 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 92 [arrowhead = none ];
92 -> 28;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 93 [arrowhead = none ];
93 -> 29;

94 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 94 [arrowhead = none ];
94 -> 30;

95 [label="INSTANCE with matching:\nT1 -> s(s(T61))\nT2 -> T62\nT3 -> T64", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 95 [arrowhead = none , style=dashed];
95 -> 1[style=dashed];

96 [label="BACKTRACK\nfor clause: avg(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 96 [arrowhead = none ];
96 -> 31;

97 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 97 [arrowhead = none ];
97 -> 32;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 98 [arrowhead = none ];
98 -> 35;

99 [label="EVAL with clause\navg(0, 0, 0).\nand substitutionT1 -> 0,\nT2 -> 0,\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 99 [arrowhead = none ];
99 -> 50;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 100 [arrowhead = none ];
100 -> 51;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 101 [arrowhead = none ];
101 -> 36;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 102 [arrowhead = none ];
102 -> 37;

103 [label="ONLY EVAL with clause\navg(s(X91), X92, X93) :- avg(X91, s(X92), X93).\nand substitutionT71 -> T101,\nX91 -> T101,\nT72 -> T102,\nX92 -> T102,\nT74 -> T104,\nX93 -> T104,\nT103 -> T104", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 103 [arrowhead = none ];
103 -> 38;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 104 [arrowhead = none ];
104 -> 39;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 105 [arrowhead = none ];
105 -> 40;

106 [label="INSTANCE with matching:\nT1 -> T101\nT2 -> s(T102)\nT3 -> T104", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 106 [arrowhead = none , style=dashed];
106 -> 1[style=dashed];

107 [label="EVAL with clause\navg(X112, s(s(s(X113))), s(X114)) :- avg(s(X112), X113, X114).\nand substitutionT71 -> T123,\nX112 -> s(T123),\nX113 -> T124,\nT72 -> s(s(s(T124))),\nX114 -> T126,\nT74 -> s(T126),\nT125 -> T126", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 107 [arrowhead = none ];
107 -> 41;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 108 [arrowhead = none ];
108 -> 42;

109 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 109 [arrowhead = none ];
109 -> 43;

110 [label="INSTANCE with matching:\nT1 -> s(s(T123))\nT2 -> T124\nT3 -> T126", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 110 [arrowhead = none , style=dashed];
110 -> 1[style=dashed];

111 [label="BACKTRACK\nfor clause: avg(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 111 [arrowhead = none ];
111 -> 44;

112 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 112 [arrowhead = none ];
112 -> 45;

113 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 113 [arrowhead = none ];
113 -> 46;

114 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 114 [arrowhead = none ];
114 -> 47;

115 [label="BACKTRACK\nfor clause: avg(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 115 [arrowhead = none ];
115 -> 48;

116 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 116 [arrowhead = none ];
116 -> 49;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 117 [arrowhead = none ];
117 -> 52;

118 [label="EVAL with clause\navg(0, s(0), 0).\nand substitutionT1 -> 0,\nT2 -> s(0),\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 118 [arrowhead = none ];
118 -> 55;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 119 [arrowhead = none ];
119 -> 56;

120 [label="BACKTRACK\nfor clause: avg(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 120 [arrowhead = none ];
120 -> 53;

121 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 121 [arrowhead = none ];
121 -> 54;

122 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 122 [arrowhead = none ];
122 -> 57;

123 [label="EVAL with clause\navg(0, s(s(0)), s(0)).\nand substitutionT1 -> 0,\nT2 -> s(s(0)),\nT3 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 123 [arrowhead = none ];
123 -> 59;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 124 [arrowhead = none ];
124 -> 60;

125 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 125 [arrowhead = none ];
125 -> 58;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 126 [arrowhead = none ];
126 -> 61;

}

(2) Obligation:

Triples:

avgA(s(s(X1)), X2, X3) :- avgA(X1, s(s(X2)), X3).
avgA(s(X1), s(s(X2)), s(X3)) :- avgA(s(X1), X2, X3).
avgA(s(X1), s(s(s(X2))), s(X3)) :- avgA(s(s(X1)), X2, X3).
avgA(X1, s(s(s(X2))), s(X3)) :- avgA(X1, s(X2), X3).
avgA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) :- avgA(s(s(X1)), X2, X3).

Clauses:

avgcA(s(s(X1)), X2, X3) :- avgcA(X1, s(s(X2)), X3).
avgcA(s(X1), s(s(X2)), s(X3)) :- avgcA(s(X1), X2, X3).
avgcA(s(0), 0, 0).
avgcA(s(0), s(0), s(0)).
avgcA(s(X1), s(s(s(X2))), s(X3)) :- avgcA(s(s(X1)), X2, X3).
avgcA(X1, s(s(s(X2))), s(X3)) :- avgcA(X1, s(X2), X3).
avgcA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) :- avgcA(s(s(X1)), X2, X3).
avgcA(0, 0, 0).
avgcA(0, s(0), 0).
avgcA(0, s(s(0)), s(0)).

Afs:

avgA(x1, x2, x3) = avgA(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
avgA_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

AVGA_IN_GGA(s(s(X1)), X2, X3) → U1_GGA(X1, X2, X3, avgA_in_gga(X1, s(s(X2)), X3))
AVGA_IN_GGA(s(s(X1)), X2, X3) → AVGA_IN_GGA(X1, s(s(X2)), X3)
AVGA_IN_GGA(s(X1), s(s(X2)), s(X3)) → U2_GGA(X1, X2, X3, avgA_in_gga(s(X1), X2, X3))
AVGA_IN_GGA(s(X1), s(s(X2)), s(X3)) → AVGA_IN_GGA(s(X1), X2, X3)
AVGA_IN_GGA(s(X1), s(s(s(X2))), s(X3)) → U3_GGA(X1, X2, X3, avgA_in_gga(s(s(X1)), X2, X3))
AVGA_IN_GGA(s(X1), s(s(s(X2))), s(X3)) → AVGA_IN_GGA(s(s(X1)), X2, X3)
AVGA_IN_GGA(X1, s(s(s(X2))), s(X3)) → U4_GGA(X1, X2, X3, avgA_in_gga(X1, s(X2), X3))
AVGA_IN_GGA(X1, s(s(s(X2))), s(X3)) → AVGA_IN_GGA(X1, s(X2), X3)
AVGA_IN_GGA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) → U5_GGA(X1, X2, X3, avgA_in_gga(s(s(X1)), X2, X3))
AVGA_IN_GGA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) → AVGA_IN_GGA(s(s(X1)), X2, X3)

R is empty.
The argument filtering Pi contains the following mapping:
avgA_in_gga(x1, x2, x3) = avgA_in_gga(x1, x2)
s(x1) = s(x1)
AVGA_IN_GGA(x1, x2, x3) = AVGA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)

We have to consider all (P,R,Pi)-chains

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

AVGA_IN_GGA(s(s(X1)), X2, X3) → U1_GGA(X1, X2, X3, avgA_in_gga(X1, s(s(X2)), X3))
AVGA_IN_GGA(s(s(X1)), X2, X3) → AVGA_IN_GGA(X1, s(s(X2)), X3)
AVGA_IN_GGA(s(X1), s(s(X2)), s(X3)) → U2_GGA(X1, X2, X3, avgA_in_gga(s(X1), X2, X3))
AVGA_IN_GGA(s(X1), s(s(X2)), s(X3)) → AVGA_IN_GGA(s(X1), X2, X3)
AVGA_IN_GGA(s(X1), s(s(s(X2))), s(X3)) → U3_GGA(X1, X2, X3, avgA_in_gga(s(s(X1)), X2, X3))
AVGA_IN_GGA(s(X1), s(s(s(X2))), s(X3)) → AVGA_IN_GGA(s(s(X1)), X2, X3)
AVGA_IN_GGA(X1, s(s(s(X2))), s(X3)) → U4_GGA(X1, X2, X3, avgA_in_gga(X1, s(X2), X3))
AVGA_IN_GGA(X1, s(s(s(X2))), s(X3)) → AVGA_IN_GGA(X1, s(X2), X3)
AVGA_IN_GGA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) → U5_GGA(X1, X2, X3, avgA_in_gga(s(s(X1)), X2, X3))
AVGA_IN_GGA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) → AVGA_IN_GGA(s(s(X1)), X2, X3)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

AVGA_IN_GGA(s(X1), s(s(X2)), s(X3)) → AVGA_IN_GGA(s(X1), X2, X3)
AVGA_IN_GGA(s(s(X1)), X2, X3) → AVGA_IN_GGA(X1, s(s(X2)), X3)
AVGA_IN_GGA(s(X1), s(s(s(X2))), s(X3)) → AVGA_IN_GGA(s(s(X1)), X2, X3)
AVGA_IN_GGA(X1, s(s(s(X2))), s(X3)) → AVGA_IN_GGA(X1, s(X2), X3)
AVGA_IN_GGA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) → AVGA_IN_GGA(s(s(X1)), X2, X3)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
AVGA_IN_GGA(x1, x2, x3) = AVGA_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(7) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AVGA_IN_GGA(s(X1), s(s(X2))) → AVGA_IN_GGA(s(X1), X2)
AVGA_IN_GGA(s(s(X1)), X2) → AVGA_IN_GGA(X1, s(s(X2)))
AVGA_IN_GGA(s(X1), s(s(s(X2)))) → AVGA_IN_GGA(s(s(X1)), X2)
AVGA_IN_GGA(X1, s(s(s(X2)))) → AVGA_IN_GGA(X1, s(X2))
AVGA_IN_GGA(X1, s(s(s(s(s(s(X2))))))) → AVGA_IN_GGA(s(s(X1)), X2)

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

(9) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

AVGA_IN_GGA(s(X1), s(s(X2))) → AVGA_IN_GGA(s(X1), X2)
AVGA_IN_GGA(s(s(X1)), X2) → AVGA_IN_GGA(X1, s(s(X2)))
AVGA_IN_GGA(s(X1), s(s(s(X2)))) → AVGA_IN_GGA(s(s(X1)), X2)
AVGA_IN_GGA(X1, s(s(s(X2)))) → AVGA_IN_GGA(X1, s(X2))
AVGA_IN_GGA(X1, s(s(s(s(s(s(X2))))))) → AVGA_IN_GGA(s(s(X1)), X2)

Used ordering: Knuth-Bendix order [KBO] with precedence:

s₁ > AVGA_{IN_GGA₂}

and weight map:

s_1=1
AVGA_IN_GGA_2=0

The variable weight is 1

(10) Obligation:

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

(11) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.