Left Termination of the query pattern
average(g,a,g)
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).

Query: average(g,a,g)

(1) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: average(T1, T2, T3)\n\nT1 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: 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\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: average(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: 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\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: average(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: 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\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: average(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: average(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: average(T16, s(T19), T18)\n\nT16 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: average(s(T26), T29, T28)\n\nT26 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 80 [arrowhead = none ];
80 -> 6;

81 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 81 [arrowhead = none ];
81 -> 7;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 82 [arrowhead = none ];
82 -> 8;

83 [label="EVAL with clause\naverage(0, 0, 0).\nand substitutionT1 -> 0,\nT2 -> 0,\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 83 [arrowhead = none ];
83 -> 9;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 84 [arrowhead = none ];
84 -> 12;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 85 [arrowhead = none ];
85 -> 19;

86 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 86 [arrowhead = none ];
86 -> 21;

87 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 87 [arrowhead = none ];
87 -> 15;

88 [label="EVAL with clause\naverage(0, s(0), 0).\nand substitutionT1 -> 0,\nT2 -> s(0),\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 88 [arrowhead = none ];
88 -> 28;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 89 [arrowhead = none ];
89 -> 29;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 90 [arrowhead = none ];
90 -> 33;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 91 [arrowhead = none ];
91 -> 34;

92 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 92 [arrowhead = none ];
92 -> 30;

93 [label="EVAL with clause\naverage(0, s(s(0)), s(0)).\nand substitutionT1 -> 0,\nT2 -> s(s(0)),\nT3 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 93 [arrowhead = none ];
93 -> 41;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 94 [arrowhead = none ];
94 -> 44;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 95 [arrowhead = none ];
95 -> 48;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 96 [arrowhead = none ];
96 -> 49;

97 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 97 [arrowhead = none ];
97 -> 45;

98 [label="EVAL with clause\naverage(s(X13), X14, X15) :- average(X13, s(X14), X15).\nand substitutionX13 -> T16,\nT1 -> s(T16),\nT2 -> T19,\nX14 -> T19,\nT3 -> T18,\nX15 -> T18,\nT17 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 98 [arrowhead = none ];
98 -> 58;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 99 [arrowhead = none ];
99 -> 60;

100 [label="EVAL with clause\naverage(X22, s(s(s(X23))), s(X24)) :- average(s(X22), X23, X24).\nand substitutionT1 -> T26,\nX22 -> T26,\nX23 -> T29,\nT2 -> s(s(s(T29))),\nX24 -> T28,\nT3 -> s(T28),\nT27 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 100 [arrowhead = none ];
100 -> 78;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 101 [arrowhead = none ];
101 -> 79;

102 [label="INSTANCE with matching:\nT1 -> T16\nT2 -> s(T19)\nT3 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 102 [arrowhead = none , style=dashed];
102 -> 2[style=dashed];

103 [label="INSTANCE with matching:\nT1 -> s(T26)\nT2 -> T29\nT3 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 103 [arrowhead = none , style=dashed];
103 -> 2[style=dashed];

}

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f2_in(0, 0) → f2_out1(0)
f2_in(0, 0) → f2_out1(s(0))
f2_in(0, s(0)) → f2_out1(s(s(0)))
f2_in(s(T16), T18) → U1(f2_in(T16, T18), s(T16), T18)
U1(f2_out1(s(T19)), s(T16), T18) → f2_out1(T19)
f2_in(T26, s(T28)) → U2(f2_in(s(T26), T28), T26, s(T28))
U2(f2_out1(T29), T26, s(T28)) → f2_out1(s(s(s(T29))))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
f2_in(x1, x2)  =  f2_in(x1, x2)
0  =  0
f2_out1(x1)  =  x1
s(x1)  =  s(x1)
U1(x1, x2, x3)  =  U1(x1, x2, x3)
U2(x1, x2, x3)  =  U2(x1, x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:

f2_in2 > 0 > s₁
f2_in2 > U1₃ > s₁
f2_in2 > U2₃ > s₁

Status:

f2_in2: [2,1]
0: multiset
s₁: multiset
U1₃: [1,2,3]
U2₃: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

f2_in(0, 0) → f2_out1(0)
f2_in(0, 0) → f2_out1(s(0))
f2_in(0, s(0)) → f2_out1(s(s(0)))
f2_in(s(T16), T18) → U1(f2_in(T16, T18), s(T16), T18)
U1(f2_out1(s(T19)), s(T16), T18) → f2_out1(T19)
f2_in(T26, s(T28)) → U2(f2_in(s(T26), T28), T26, s(T28))
U2(f2_out1(T29), T26, s(T28)) → f2_out1(s(s(s(T29))))

(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) YES