Left Termination proof of /tmp/tmpnLvgMY/average.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 86 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 29 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 79 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 PiDP

↳9 UsableRulesProof (⇔, 0 ms)

↳10 PiDP

↳11 PiDPToQDPProof (⇒, 52 ms)

↳12 QDP

↳13 MRRProof (⇔, 38 ms)

↳14 QDP

↳15 PisEmptyProof (⇔, 0 ms)

↳16 YES

(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,g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

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\nX22 is free\nX23 is free\nX24 is free\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(T49)), T50, T52)\n\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: average(s(T56), T57, T59)\n\nT56 is ground\nT57 is ground\nX22 is free\nX23 is free\nX24 is free\naverage(T56, s(s(s(T57))), 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(T56), T57, T59), SCOPE: 3, CLAUSE: 0 | average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 1 | average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 2 | average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 3 | average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 4\n\nT56 is ground\nT57 is ground\nX22 is free\nX23 is free\nX24 is free\naverage(T56, s(s(s(T57))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 1 | average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 2 | average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 3 | average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 4\n\nT56 is ground\nT57 is ground\nX22 is free\nX23 is free\nX24 is free\naverage(T56, s(s(s(T57))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 2 | average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 3 | average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 4\n\nT56 is ground\nT57 is ground\nX22 is free\nX23 is free\nX24 is free\naverage(T56, s(s(s(T57))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 3 | average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 4\n\nT56 is ground\nT57 is ground\nX22 is free\nX23 is free\nX24 is free\naverage(T56, s(s(s(T57))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 3\n\nT56 is ground\nT57 is ground\nX22 is free\nX23 is free\nX24 is free\naverage(T56, s(s(s(T57))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: average(s(T56), T57, T59), SCOPE: 3, CLAUSE: 4\n\nT56 is ground\nT57 is ground\nX22 is free\nX23 is free\nX24 is free\naverage(T56, s(s(s(T57))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: average(T74, s(T75), T77)\n\nT74 is ground\nT75 is ground\nX22 is free\nX23 is free\nX24 is free\naverage(T74, s(s(s(T75))), T3) !=? average(s(X22), X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: average(s(s(T84)), T85, T87)\n\nT84 is ground\nT85 is ground\nX22 is free\nX23 is free\nX24 is free\naverage(T84, s(s(s(s(s(s(T85)))))), 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 substitutionT1 -> 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 = lightgreen];
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 substitutionT1 -> 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 = lightgreen];
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 = lightgreen];
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 = lightgreen];
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 = lightgreen];
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 = lightgreen];
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 substitutionT1 -> 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 = lightgreen];
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 = lightgreen];
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 = lightgreen];
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 = lightgreen];
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 substitutionX22 -> 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 = lightgreen];
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 = lightgreen];
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 = lightgreen];
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(X67, s(s(s(X68))), s(X69)) :- average(s(X67), X68, X69).\nand substitutionT1 -> T56,\nX67 -> T56,\nX68 -> T57,\nT2 -> s(s(s(T57))),\nX69 -> T59,\nT3 -> s(T59),\nT58 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 79 [arrowhead = none ];
79 -> 46;

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

81 [label="BACKTRACK\nfor clause: average(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionT7 -> 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 = lightgreen];
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 substitutionT7 -> 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 = lightgreen];
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(X37), X38, X39) :- average(X37, s(X38), X39).\nand substitutionX37 -> T23,\nT7 -> s(T23),\nT8 -> T24,\nX38 -> s(T24),\nT10 -> T26,\nX39 -> T26,\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 94 [arrowhead = none ];
94 -> 37;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X52, s(s(s(X53))), s(X54)) :- average(s(X52), X53, X54).\nand substitutionT7 -> T39,\nX52 -> T39,\nX53 -> T40,\nT8 -> s(s(T40)),\nX54 -> 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 = lightgreen];
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(X61, s(s(s(X62))), s(X63)) :- average(s(X61), X62, X63).\nand substitutionT7 -> T49,\nX61 -> s(T49),\nX62 -> T50,\nT8 -> s(s(s(T50))),\nX63 -> T52,\nT3 -> s(T52),\nT51 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 103 [arrowhead = none ];
103 -> 44;

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

105 [label="INSTANCE with matching:\nT1 -> s(s(T49))\nT2 -> T50\nT3 -> T52", 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 = lightgreen];
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 = lightgreen];
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 = lightgreen];
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(X82), X83, X84) :- average(X82, s(X83), X84).\nand substitutionT56 -> T74,\nX82 -> T74,\nT57 -> T75,\nX83 -> T75,\nT59 -> T77,\nX84 -> T77,\nT76 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 112 [arrowhead = none ];
112 -> 54;

113 [label="EVAL with clause\naverage(X91, s(s(s(X92))), s(X93)) :- average(s(X91), X92, X93).\nand substitutionT56 -> T84,\nX91 -> s(T84),\nX92 -> T85,\nT57 -> s(s(s(T85))),\nX93 -> T87,\nT59 -> s(T87),\nT86 -> T87", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 113 [arrowhead = none ];
113 -> 55;

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

115 [label="INSTANCE with matching:\nT1 -> T74\nT2 -> s(T75)\nT3 -> T77", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 115 [arrowhead = none , style=dashed];
115 -> 1[style=dashed];

116 [label="INSTANCE with matching:\nT1 -> s(s(T84))\nT2 -> T85\nT3 -> T87", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 116 [arrowhead = none , style=dashed];
116 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

averageA(0, 0, 0).
averageA(0, s(0), 0).
averageA(0, s(s(0)), s(0)).
averageA(s(0), 0, 0).
averageA(s(0), s(0), s(0)).
averageA(s(s(T23)), T24, T26) :- averageA(T23, s(s(T24)), T26).
averageA(s(T39), s(s(T40)), s(T42)) :- averageA(s(T39), T40, T42).
averageA(s(T49), s(s(s(T50))), s(T52)) :- averageA(s(s(T49)), T50, T52).
averageA(T74, s(s(s(T75))), s(T77)) :- averageA(T74, s(T75), T77).
averageA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) :- averageA(s(s(T84)), T85, T87).

Query: averageA(g,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
averageA_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

averageA_in_gga(0, 0, 0) → averageA_out_gga(0, 0, 0)
averageA_in_gga(0, s(0), 0) → averageA_out_gga(0, s(0), 0)
averageA_in_gga(0, s(s(0)), s(0)) → averageA_out_gga(0, s(s(0)), s(0))
averageA_in_gga(s(0), 0, 0) → averageA_out_gga(s(0), 0, 0)
averageA_in_gga(s(0), s(0), s(0)) → averageA_out_gga(s(0), s(0), s(0))
averageA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
averageA_in_gga(s(T39), s(s(T40)), s(T42)) → U2_gga(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
averageA_in_gga(s(T49), s(s(s(T50))), s(T52)) → U3_gga(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
averageA_in_gga(T74, s(s(s(T75))), s(T77)) → U4_gga(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
averageA_in_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_gga(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
U5_gga(T84, T85, T87, averageA_out_gga(s(s(T84)), T85, T87)) → averageA_out_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87)))
U4_gga(T74, T75, T77, averageA_out_gga(T74, s(T75), T77)) → averageA_out_gga(T74, s(s(s(T75))), s(T77))
U3_gga(T49, T50, T52, averageA_out_gga(s(s(T49)), T50, T52)) → averageA_out_gga(s(T49), s(s(s(T50))), s(T52))
U2_gga(T39, T40, T42, averageA_out_gga(s(T39), T40, T42)) → averageA_out_gga(s(T39), s(s(T40)), s(T42))
U1_gga(T23, T24, T26, averageA_out_gga(T23, s(s(T24)), T26)) → averageA_out_gga(s(s(T23)), T24, T26)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

averageA_in_gga(0, 0, 0) → averageA_out_gga(0, 0, 0)
averageA_in_gga(0, s(0), 0) → averageA_out_gga(0, s(0), 0)
averageA_in_gga(0, s(s(0)), s(0)) → averageA_out_gga(0, s(s(0)), s(0))
averageA_in_gga(s(0), 0, 0) → averageA_out_gga(s(0), 0, 0)
averageA_in_gga(s(0), s(0), s(0)) → averageA_out_gga(s(0), s(0), s(0))
averageA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
averageA_in_gga(s(T39), s(s(T40)), s(T42)) → U2_gga(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
averageA_in_gga(s(T49), s(s(s(T50))), s(T52)) → U3_gga(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
averageA_in_gga(T74, s(s(s(T75))), s(T77)) → U4_gga(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
averageA_in_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_gga(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
U5_gga(T84, T85, T87, averageA_out_gga(s(s(T84)), T85, T87)) → averageA_out_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87)))
U4_gga(T74, T75, T77, averageA_out_gga(T74, s(T75), T77)) → averageA_out_gga(T74, s(s(s(T75))), s(T77))
U3_gga(T49, T50, T52, averageA_out_gga(s(s(T49)), T50, T52)) → averageA_out_gga(s(T49), s(s(s(T50))), s(T52))
U2_gga(T39, T40, T42, averageA_out_gga(s(T39), T40, T42)) → averageA_out_gga(s(T39), s(s(T40)), s(T42))
U1_gga(T23, T24, T26, averageA_out_gga(T23, s(s(T24)), T26)) → averageA_out_gga(s(s(T23)), T24, T26)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → AVERAGEA_IN_GGA(T23, s(s(T24)), T26)
AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → U2_GGA(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → AVERAGEA_IN_GGA(s(T39), T40, T42)
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → U3_GGA(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → AVERAGEA_IN_GGA(s(s(T49)), T50, T52)
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → U4_GGA(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → AVERAGEA_IN_GGA(T74, s(T75), T77)
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_GGA(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → AVERAGEA_IN_GGA(s(s(T84)), T85, T87)

The TRS R consists of the following rules:

averageA_in_gga(0, 0, 0) → averageA_out_gga(0, 0, 0)
averageA_in_gga(0, s(0), 0) → averageA_out_gga(0, s(0), 0)
averageA_in_gga(0, s(s(0)), s(0)) → averageA_out_gga(0, s(s(0)), s(0))
averageA_in_gga(s(0), 0, 0) → averageA_out_gga(s(0), 0, 0)
averageA_in_gga(s(0), s(0), s(0)) → averageA_out_gga(s(0), s(0), s(0))
averageA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
averageA_in_gga(s(T39), s(s(T40)), s(T42)) → U2_gga(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
averageA_in_gga(s(T49), s(s(s(T50))), s(T52)) → U3_gga(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
averageA_in_gga(T74, s(s(s(T75))), s(T77)) → U4_gga(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
averageA_in_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_gga(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
U5_gga(T84, T85, T87, averageA_out_gga(s(s(T84)), T85, T87)) → averageA_out_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87)))
U4_gga(T74, T75, T77, averageA_out_gga(T74, s(T75), T77)) → averageA_out_gga(T74, s(s(s(T75))), s(T77))
U3_gga(T49, T50, T52, averageA_out_gga(s(s(T49)), T50, T52)) → averageA_out_gga(s(T49), s(s(s(T50))), s(T52))
U2_gga(T39, T40, T42, averageA_out_gga(s(T39), T40, T42)) → averageA_out_gga(s(T39), s(s(T40)), s(T42))
U1_gga(T23, T24, T26, averageA_out_gga(T23, s(s(T24)), T26)) → averageA_out_gga(s(s(T23)), T24, T26)

The argument filtering Pi contains the following mapping:
averageA_in_gga(x1, x2, x3) = averageA_in_gga(x1, x2)
0 = 0
averageA_out_gga(x1, x2, x3) = averageA_out_gga(x3)
s(x1) = s(x1)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
U5_gga(x1, x2, x3, x4) = U5_gga(x4)
AVERAGEA_IN_GGA(x1, x2, x3) = AVERAGEA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x4)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x4)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x4)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x4)

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

(6) Obligation:

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

AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → AVERAGEA_IN_GGA(T23, s(s(T24)), T26)
AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → U2_GGA(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → AVERAGEA_IN_GGA(s(T39), T40, T42)
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → U3_GGA(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → AVERAGEA_IN_GGA(s(s(T49)), T50, T52)
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → U4_GGA(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → AVERAGEA_IN_GGA(T74, s(T75), T77)
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_GGA(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → AVERAGEA_IN_GGA(s(s(T84)), T85, T87)

The TRS R consists of the following rules:

averageA_in_gga(0, 0, 0) → averageA_out_gga(0, 0, 0)
averageA_in_gga(0, s(0), 0) → averageA_out_gga(0, s(0), 0)
averageA_in_gga(0, s(s(0)), s(0)) → averageA_out_gga(0, s(s(0)), s(0))
averageA_in_gga(s(0), 0, 0) → averageA_out_gga(s(0), 0, 0)
averageA_in_gga(s(0), s(0), s(0)) → averageA_out_gga(s(0), s(0), s(0))
averageA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
averageA_in_gga(s(T39), s(s(T40)), s(T42)) → U2_gga(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
averageA_in_gga(s(T49), s(s(s(T50))), s(T52)) → U3_gga(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
averageA_in_gga(T74, s(s(s(T75))), s(T77)) → U4_gga(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
averageA_in_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_gga(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
U5_gga(T84, T85, T87, averageA_out_gga(s(s(T84)), T85, T87)) → averageA_out_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87)))
U4_gga(T74, T75, T77, averageA_out_gga(T74, s(T75), T77)) → averageA_out_gga(T74, s(s(s(T75))), s(T77))
U3_gga(T49, T50, T52, averageA_out_gga(s(s(T49)), T50, T52)) → averageA_out_gga(s(T49), s(s(s(T50))), s(T52))
U2_gga(T39, T40, T42, averageA_out_gga(s(T39), T40, T42)) → averageA_out_gga(s(T39), s(s(T40)), s(T42))
U1_gga(T23, T24, T26, averageA_out_gga(T23, s(s(T24)), T26)) → averageA_out_gga(s(s(T23)), T24, T26)

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → AVERAGEA_IN_GGA(s(T39), T40, T42)
AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → AVERAGEA_IN_GGA(T23, s(s(T24)), T26)
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → AVERAGEA_IN_GGA(s(s(T49)), T50, T52)
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → AVERAGEA_IN_GGA(T74, s(T75), T77)
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → AVERAGEA_IN_GGA(s(s(T84)), T85, T87)

The TRS R consists of the following rules:

averageA_in_gga(0, 0, 0) → averageA_out_gga(0, 0, 0)
averageA_in_gga(0, s(0), 0) → averageA_out_gga(0, s(0), 0)
averageA_in_gga(0, s(s(0)), s(0)) → averageA_out_gga(0, s(s(0)), s(0))
averageA_in_gga(s(0), 0, 0) → averageA_out_gga(s(0), 0, 0)
averageA_in_gga(s(0), s(0), s(0)) → averageA_out_gga(s(0), s(0), s(0))
averageA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
averageA_in_gga(s(T39), s(s(T40)), s(T42)) → U2_gga(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
averageA_in_gga(s(T49), s(s(s(T50))), s(T52)) → U3_gga(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
averageA_in_gga(T74, s(s(s(T75))), s(T77)) → U4_gga(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
averageA_in_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_gga(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
U5_gga(T84, T85, T87, averageA_out_gga(s(s(T84)), T85, T87)) → averageA_out_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87)))
U4_gga(T74, T75, T77, averageA_out_gga(T74, s(T75), T77)) → averageA_out_gga(T74, s(s(s(T75))), s(T77))
U3_gga(T49, T50, T52, averageA_out_gga(s(s(T49)), T50, T52)) → averageA_out_gga(s(T49), s(s(s(T50))), s(T52))
U2_gga(T39, T40, T42, averageA_out_gga(s(T39), T40, T42)) → averageA_out_gga(s(T39), s(s(T40)), s(T42))
U1_gga(T23, T24, T26, averageA_out_gga(T23, s(s(T24)), T26)) → averageA_out_gga(s(s(T23)), T24, T26)

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) Obligation:

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

AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → AVERAGEA_IN_GGA(s(T39), T40, T42)
AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → AVERAGEA_IN_GGA(T23, s(s(T24)), T26)
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → AVERAGEA_IN_GGA(s(s(T49)), T50, T52)
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → AVERAGEA_IN_GGA(T74, s(T75), T77)
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → AVERAGEA_IN_GGA(s(s(T84)), T85, T87)

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

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

AVERAGEA_IN_GGA(s(T39), s(s(T40))) → AVERAGEA_IN_GGA(s(T39), T40)
AVERAGEA_IN_GGA(s(s(T23)), T24) → AVERAGEA_IN_GGA(T23, s(s(T24)))
AVERAGEA_IN_GGA(s(T49), s(s(s(T50)))) → AVERAGEA_IN_GGA(s(s(T49)), T50)
AVERAGEA_IN_GGA(T74, s(s(s(T75)))) → AVERAGEA_IN_GGA(T74, s(T75))
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85))))))) → AVERAGEA_IN_GGA(s(s(T84)), T85)

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

(13) 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:

AVERAGEA_IN_GGA(s(T39), s(s(T40))) → AVERAGEA_IN_GGA(s(T39), T40)
AVERAGEA_IN_GGA(s(s(T23)), T24) → AVERAGEA_IN_GGA(T23, s(s(T24)))
AVERAGEA_IN_GGA(s(T49), s(s(s(T50)))) → AVERAGEA_IN_GGA(s(s(T49)), T50)
AVERAGEA_IN_GGA(T74, s(s(s(T75)))) → AVERAGEA_IN_GGA(T74, s(T75))
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85))))))) → AVERAGEA_IN_GGA(s(s(T84)), T85)

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

s₁ > AVERAGEA_{IN_GGA₂}

and weight map:

s_1=1
AVERAGEA_IN_GGA_2=0

The variable weight is 1

(14) Obligation:

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

(15) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) UsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) PiDPToQDPProof (SOUND transformation)

(12) Obligation:

(13) MRRProof (EQUIVALENT transformation)

(14) Obligation:

(15) PisEmptyProof (EQUIVALENT transformation)

(16) YES