Left Termination of the query pattern average_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

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

Queries:

average(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: average(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: average(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | average(0, 0, T3), SCOPE: 1, CLAUSE: 1 | average(0, 0, T3), SCOPE: 1, CLAUSE: 2 | average(0, 0, T3), SCOPE: 1, CLAUSE: 3 | average(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: average(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\naverage(T1, T2, T3) !=? average(0, 0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: average(0, 0, T3), SCOPE: 1, CLAUSE: 1 | average(0, 0, T3), SCOPE: 1, CLAUSE: 2 | average(0, 0, T3), SCOPE: 1, CLAUSE: 3 | average(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: average(0, 0, T3), SCOPE: 1, CLAUSE: 2 | average(0, 0, T3), SCOPE: 1, CLAUSE: 3 | average(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: average(0, 0, T3), SCOPE: 1, CLAUSE: 3 | average(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: average(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | average(0, s(0), T3), SCOPE: 1, CLAUSE: 2 | average(0, s(0), T3), SCOPE: 1, CLAUSE: 3 | average(0, s(0), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: average(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\naverage(T1, T2, T3) !=? average(0, 0, 0)\naverage(T1, T2, T3) !=? average(0, s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: average(0, s(0), T3), SCOPE: 1, CLAUSE: 2 | average(0, s(0), T3), SCOPE: 1, CLAUSE: 3 | average(0, s(0), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: average(0, s(0), T3), SCOPE: 1, CLAUSE: 3 | average(0, s(0), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: average(0, s(0), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | average(0, s(s(0)), T3), SCOPE: 1, CLAUSE: 3 | average(0, s(s(0)), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\naverage(T1, T2, T3) !=? average(0, 0, 0)\naverage(T1, T2, T3) !=? average(0, s(0), 0)\naverage(T1, T2, T3) !=? average(0, s(s(0)), s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: average(0, s(s(0)), T3), SCOPE: 1, CLAUSE: 3 | average(0, s(s(0)), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: average(0, s(s(0)), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: average(T7, s(T8), T10) | average(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\naverage(T1, T2, T3) !=? average(0, 0, 0)\naverage(T1, T2, T3) !=? average(0, s(0), 0)\naverage(T1, T2, T3) !=? average(0, s(s(0)), s(0))\naverage(T1, T2, T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 0 | average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 1 | average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 2 | average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | average(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 1 | average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 2 | average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | average(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 2 | average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | average(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | average(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | average(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: average(T23, s(s(T24)), T26)\n\nT23 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: average(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ?_2 | average(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: average(s(T39), T40, T42)\n\nT39 is ground\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: average(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: average(s(s(T52)), T53, T55)\n\nT52 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: average(s(T62), T63, T65)\n\nT62 is ground\nT63 is ground\naverage(T62, s(s(s(T63))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 0 | average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 1 | average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 2 | average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 3 | average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 4\n\nT62 is ground\nT63 is ground\naverage(T62, s(s(s(T63))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 1 | average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 2 | average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 3 | average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 4\n\nT62 is ground\nT63 is ground\naverage(T62, s(s(s(T63))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 2 | average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 3 | average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 4\n\nT62 is ground\nT63 is ground\naverage(T62, s(s(s(T63))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 3 | average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 4\n\nT62 is ground\nT63 is ground\naverage(T62, s(s(s(T63))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 3\n\nT62 is ground\nT63 is ground\naverage(T62, s(s(s(T63))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: average(s(T62), T63, T65), SCOPE: 3, CLAUSE: 4\n\nT62 is ground\nT63 is ground\naverage(T62, s(s(s(T63))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: average(T80, s(T81), T83)\n\nT80 is ground\nT81 is ground\naverage(T80, s(s(s(T81))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: average(s(s(T90)), T91, T93)\n\nT90 is ground\nT91 is ground\naverage(T90, s(s(s(s(s(s(T91)))))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 57 [arrowhead = none ];
57 -> 2;

58 [label="EVAL with clause\naverage(0, 0, 0).\nand substitution\nT1 -> 0\nT2 -> 0\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 58 [arrowhead = none ];
58 -> 3;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 59 [arrowhead = none ];
59 -> 4;

60 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 60 [arrowhead = none ];
60 -> 5;

61 [label="EVAL with clause\naverage(0, s(0), 0).\nand substitution\nT1 -> 0\nT2 -> s(0)\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 61 [arrowhead = none ];
61 -> 10;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 62 [arrowhead = none ];
62 -> 11;

63 [label="BACKTRACK\nfor clause: average(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 63 [arrowhead = none ];
63 -> 6;

64 [label="BACKTRACK\nfor clause: average(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 64 [arrowhead = none ];
64 -> 7;

65 [label="BACKTRACK\nfor clause: average(s(X), Y, Z) :- average(X, s(Y), Z)because of non-unification", fontsize=14, style = filled, fillcolor = white];
7 -> 65 [arrowhead = none ];
65 -> 8;

66 [label="BACKTRACK\nfor clause: average(X, s(s(s(Y))), s(Z)) :- average(s(X), Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = white];
8 -> 66 [arrowhead = none ];
66 -> 9;

67 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 67 [arrowhead = none ];
67 -> 12;

68 [label="EVAL with clause\naverage(0, s(s(0)), s(0)).\nand substitution\nT1 -> 0\nT2 -> s(s(0))\nT3 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 68 [arrowhead = none ];
68 -> 16;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 69 [arrowhead = none ];
69 -> 17;

70 [label="BACKTRACK\nfor clause: average(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
12 -> 70 [arrowhead = none ];
70 -> 13;

71 [label="BACKTRACK\nfor clause: average(s(X), Y, Z) :- average(X, s(Y), Z)because of non-unification", fontsize=14, style = filled, fillcolor = white];
13 -> 71 [arrowhead = none ];
71 -> 14;

72 [label="BACKTRACK\nfor clause: average(X, s(s(s(Y))), s(Z)) :- average(s(X), Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = white];
14 -> 72 [arrowhead = none ];
72 -> 15;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 73 [arrowhead = none ];
73 -> 18;

74 [label="EVAL with clause\naverage(s(X22), X23, X24) :- average(X22, s(X23), X24).\nand substitution\nX22 -> T7\nT1 -> s(T7)\nT2 -> T8\nX23 -> T8\nT3 -> T10\nX24 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 74 [arrowhead = none ];
74 -> 21;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 75 [arrowhead = none ];
75 -> 22;

76 [label="BACKTRACK\nfor clause: average(s(X), Y, Z) :- average(X, s(Y), Z)because of non-unification", fontsize=14, style = filled, fillcolor = white];
18 -> 76 [arrowhead = none ];
76 -> 19;

77 [label="BACKTRACK\nfor clause: average(X, s(s(s(Y))), s(Z)) :- average(s(X), Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = white];
19 -> 77 [arrowhead = none ];
77 -> 20;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 78 [arrowhead = none ];
78 -> 23;

79 [label="EVAL with clause\naverage(X85, s(s(s(X86))), s(X87)) :- average(s(X85), X86, X87).\nand substitution\nT1 -> T62\nX85 -> T62\nX86 -> T63\nT2 -> s(s(s(T63)))\nX87 -> T65\nT3 -> s(T65)\nT64 -> T65", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 79 [arrowhead = none ];
79 -> 46;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 80 [arrowhead = none ];
80 -> 47;

81 [label="BACKTRACK\nfor clause: average(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 81 [arrowhead = none ];
81 -> 24;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 82 [arrowhead = none ];
82 -> 25;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 83 [arrowhead = none ];
83 -> 26;

84 [label="EVAL with clause\naverage(0, s(0), 0).\nand substitution\nT7 -> 0\nT8 -> 0\nT10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 84 [arrowhead = none ];
84 -> 27;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 85 [arrowhead = none ];
85 -> 28;

86 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 86 [arrowhead = none ];
86 -> 30;

87 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 87 [arrowhead = none ];
87 -> 31;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 88 [arrowhead = none ];
88 -> 29;

89 [label="EVAL with clause\naverage(0, s(s(0)), s(0)).\nand substitution\nT7 -> 0\nT8 -> s(0)\nT10 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 89 [arrowhead = none ];
89 -> 32;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 90 [arrowhead = none ];
90 -> 33;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 91 [arrowhead = none ];
91 -> 35;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 92 [arrowhead = none ];
92 -> 36;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 93 [arrowhead = none ];
93 -> 34;

94 [label="EVAL with clause\naverage(s(X40), X41, X42) :- average(X40, s(X41), X42).\nand substitution\nX40 -> T23\nT7 -> s(T23)\nT8 -> T24\nX41 -> s(T24)\nT10 -> T26\nX42 -> T26\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 94 [arrowhead = none ];
94 -> 37;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 95 [arrowhead = none ];
95 -> 38;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 96 [arrowhead = none ];
96 -> 39;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 97 [arrowhead = none ];
97 -> 40;

98 [label="INSTANCE with matching:\nT1 -> T23\nT2 -> s(s(T24))\nT3 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 98 [arrowhead = none , style=dashed];
98 -> 1[style=dashed];

99 [label="EVAL with clause\naverage(X58, s(s(s(X59))), s(X60)) :- average(s(X58), X59, X60).\nand substitution\nT7 -> T39\nX58 -> T39\nX59 -> T40\nT8 -> s(s(T40))\nX60 -> T42\nT10 -> s(T42)\nT41 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 99 [arrowhead = none ];
99 -> 41;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 100 [arrowhead = none ];
100 -> 42;

101 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 101 [arrowhead = none ];
101 -> 43;

102 [label="INSTANCE with matching:\nT1 -> s(T39)\nT2 -> T40\nT3 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 102 [arrowhead = none , style=dashed];
102 -> 1[style=dashed];

103 [label="EVAL with clause\naverage(X73, s(s(s(X74))), s(X75)) :- average(s(X73), X74, X75).\nand substitution\nT7 -> T52\nX73 -> s(T52)\nX74 -> T53\nT8 -> s(s(s(T53)))\nX75 -> T55\nT3 -> s(T55)\nT54 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 103 [arrowhead = none ];
103 -> 44;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 104 [arrowhead = none ];
104 -> 45;

105 [label="INSTANCE with matching:\nT1 -> s(s(T52))\nT2 -> T53\nT3 -> T55", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 105 [arrowhead = none , style=dashed];
105 -> 1[style=dashed];

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 106 [arrowhead = none ];
106 -> 48;

107 [label="BACKTRACK\nfor clause: average(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
48 -> 107 [arrowhead = none ];
107 -> 49;

108 [label="BACKTRACK\nfor clause: average(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
49 -> 108 [arrowhead = none ];
108 -> 50;

109 [label="BACKTRACK\nfor clause: average(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
50 -> 109 [arrowhead = none ];
109 -> 51;

110 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 110 [arrowhead = none ];
110 -> 52;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 111 [arrowhead = none ];
111 -> 53;

112 [label="ONLY EVAL with clause\naverage(s(X103), X104, X105) :- average(X103, s(X104), X105).\nand substitution\nT62 -> T80\nX103 -> T80\nT63 -> T81\nX104 -> T81\nT65 -> T83\nX105 -> T83\nT82 -> T83", fontsize=14, style = filled, fillcolor = white];
52 -> 112 [arrowhead = none ];
112 -> 54;

113 [label="EVAL with clause\naverage(X112, s(s(s(X113))), s(X114)) :- average(s(X112), X113, X114).\nand substitution\nT62 -> T90\nX112 -> s(T90)\nX113 -> T91\nT63 -> s(s(s(T91)))\nX114 -> T93\nT65 -> s(T93)\nT92 -> T93", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 113 [arrowhead = none ];
113 -> 55;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 114 [arrowhead = none ];
114 -> 56;

115 [label="INSTANCE with matching:\nT1 -> T80\nT2 -> s(T81)\nT3 -> T83", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 115 [arrowhead = none , style=dashed];
115 -> 1[style=dashed];

116 [label="INSTANCE with matching:\nT1 -> s(s(T90))\nT2 -> T91\nT3 -> T93", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 116 [arrowhead = none , style=dashed];
116 -> 1[style=dashed];

}

(2) Obligation:

Triples:

average1(s(s(T23)), T24, T26) :- average1(T23, s(s(T24)), T26).
average1(s(T39), s(s(T40)), s(T42)) :- average1(s(T39), T40, T42).
average1(s(T52), s(s(s(T53))), s(T55)) :- average1(s(s(T52)), T53, T55).
average1(T80, s(s(s(T81))), s(T83)) :- average1(T80, s(T81), T83).
average1(T90, s(s(s(s(s(s(T91)))))), s(s(T93))) :- average1(s(s(T90)), T91, T93).

Clauses:

averagec1(0, 0, 0).
averagec1(0, s(0), 0).
averagec1(0, s(s(0)), s(0)).
averagec1(s(0), 0, 0).
averagec1(s(0), s(0), s(0)).
averagec1(s(s(T23)), T24, T26) :- averagec1(T23, s(s(T24)), T26).
averagec1(s(T39), s(s(T40)), s(T42)) :- averagec1(s(T39), T40, T42).
averagec1(s(T52), s(s(s(T53))), s(T55)) :- averagec1(s(s(T52)), T53, T55).
averagec1(T80, s(s(s(T81))), s(T83)) :- averagec1(T80, s(T81), T83).
averagec1(T90, s(s(s(s(s(s(T91)))))), s(s(T93))) :- averagec1(s(s(T90)), T91, T93).

Afs:

average1(x1, x2, x3)  =  average1(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
average1_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

AVERAGE1_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, average1_in_gga(T23, s(s(T24)), T26))
AVERAGE1_IN_GGA(s(s(T23)), T24, T26) → AVERAGE1_IN_GGA(T23, s(s(T24)), T26)
AVERAGE1_IN_GGA(s(T39), s(s(T40)), s(T42)) → U2_GGA(T39, T40, T42, average1_in_gga(s(T39), T40, T42))
AVERAGE1_IN_GGA(s(T39), s(s(T40)), s(T42)) → AVERAGE1_IN_GGA(s(T39), T40, T42)
AVERAGE1_IN_GGA(s(T52), s(s(s(T53))), s(T55)) → U3_GGA(T52, T53, T55, average1_in_gga(s(s(T52)), T53, T55))
AVERAGE1_IN_GGA(s(T52), s(s(s(T53))), s(T55)) → AVERAGE1_IN_GGA(s(s(T52)), T53, T55)
AVERAGE1_IN_GGA(T80, s(s(s(T81))), s(T83)) → U4_GGA(T80, T81, T83, average1_in_gga(T80, s(T81), T83))
AVERAGE1_IN_GGA(T80, s(s(s(T81))), s(T83)) → AVERAGE1_IN_GGA(T80, s(T81), T83)
AVERAGE1_IN_GGA(T90, s(s(s(s(s(s(T91)))))), s(s(T93))) → U5_GGA(T90, T91, T93, average1_in_gga(s(s(T90)), T91, T93))
AVERAGE1_IN_GGA(T90, s(s(s(s(s(s(T91)))))), s(s(T93))) → AVERAGE1_IN_GGA(s(s(T90)), T91, T93)

R is empty.
The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
s(x1)  =  s(x1)
AVERAGE1_IN_GGA(x1, x2, x3)  =  AVERAGE1_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:

AVERAGE1_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, average1_in_gga(T23, s(s(T24)), T26))
AVERAGE1_IN_GGA(s(s(T23)), T24, T26) → AVERAGE1_IN_GGA(T23, s(s(T24)), T26)
AVERAGE1_IN_GGA(s(T39), s(s(T40)), s(T42)) → U2_GGA(T39, T40, T42, average1_in_gga(s(T39), T40, T42))
AVERAGE1_IN_GGA(s(T39), s(s(T40)), s(T42)) → AVERAGE1_IN_GGA(s(T39), T40, T42)
AVERAGE1_IN_GGA(s(T52), s(s(s(T53))), s(T55)) → U3_GGA(T52, T53, T55, average1_in_gga(s(s(T52)), T53, T55))
AVERAGE1_IN_GGA(s(T52), s(s(s(T53))), s(T55)) → AVERAGE1_IN_GGA(s(s(T52)), T53, T55)
AVERAGE1_IN_GGA(T80, s(s(s(T81))), s(T83)) → U4_GGA(T80, T81, T83, average1_in_gga(T80, s(T81), T83))
AVERAGE1_IN_GGA(T80, s(s(s(T81))), s(T83)) → AVERAGE1_IN_GGA(T80, s(T81), T83)
AVERAGE1_IN_GGA(T90, s(s(s(s(s(s(T91)))))), s(s(T93))) → U5_GGA(T90, T91, T93, average1_in_gga(s(s(T90)), T91, T93))
AVERAGE1_IN_GGA(T90, s(s(s(s(s(s(T91)))))), s(s(T93))) → AVERAGE1_IN_GGA(s(s(T90)), T91, T93)

R is empty.
The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
s(x1)  =  s(x1)
AVERAGE1_IN_GGA(x1, x2, x3)  =  AVERAGE1_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

(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:

AVERAGE1_IN_GGA(s(T39), s(s(T40)), s(T42)) → AVERAGE1_IN_GGA(s(T39), T40, T42)
AVERAGE1_IN_GGA(s(s(T23)), T24, T26) → AVERAGE1_IN_GGA(T23, s(s(T24)), T26)
AVERAGE1_IN_GGA(s(T52), s(s(s(T53))), s(T55)) → AVERAGE1_IN_GGA(s(s(T52)), T53, T55)
AVERAGE1_IN_GGA(T80, s(s(s(T81))), s(T83)) → AVERAGE1_IN_GGA(T80, s(T81), T83)
AVERAGE1_IN_GGA(T90, s(s(s(s(s(s(T91)))))), s(s(T93))) → AVERAGE1_IN_GGA(s(s(T90)), T91, T93)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
AVERAGE1_IN_GGA(x1, x2, x3)  =  AVERAGE1_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:

AVERAGE1_IN_GGA(s(T39), s(s(T40))) → AVERAGE1_IN_GGA(s(T39), T40)
AVERAGE1_IN_GGA(s(s(T23)), T24) → AVERAGE1_IN_GGA(T23, s(s(T24)))
AVERAGE1_IN_GGA(s(T52), s(s(s(T53)))) → AVERAGE1_IN_GGA(s(s(T52)), T53)
AVERAGE1_IN_GGA(T80, s(s(s(T81)))) → AVERAGE1_IN_GGA(T80, s(T81))
AVERAGE1_IN_GGA(T90, s(s(s(s(s(s(T91))))))) → AVERAGE1_IN_GGA(s(s(T90)), T91)

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:

AVERAGE1_IN_GGA(s(T39), s(s(T40))) → AVERAGE1_IN_GGA(s(T39), T40)
AVERAGE1_IN_GGA(s(T52), s(s(s(T53)))) → AVERAGE1_IN_GGA(s(s(T52)), T53)
AVERAGE1_IN_GGA(T80, s(s(s(T81)))) → AVERAGE1_IN_GGA(T80, s(T81))
AVERAGE1_IN_GGA(T90, s(s(s(s(s(s(T91))))))) → AVERAGE1_IN_GGA(s(s(T90)), T91)

Used ordering: Polynomial interpretation [POLO]:

POL(AVERAGE1_IN_GGA(x₁, x₂)) = x₁ + x₂   
POL(s(x₁)) = 1 + x₁   

(10) Obligation:

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

AVERAGE1_IN_GGA(s(s(T23)), T24) → AVERAGE1_IN_GGA(T23, s(s(T24)))

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

(11) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• AVERAGE1_IN_GGA(s(s(T23)), T24) → AVERAGE1_IN_GGA(T23, s(s(T24)))
  The graph contains the following edges 1 > 1

(12) YES