Left Termination proof of /tmp/tmpVyQl94/average-ioi.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 88 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 12 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 56 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 PiDP

↳9 UsableRulesProof (⇔, 2 ms)

↳10 PiDP

↳11 PiDPToQDPProof (⇒, 0 ms)

↳12 QDP

↳13 QDPSizeChangeProof (⇔, 0 ms)

↳14 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,a,g)

(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\nT3 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\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | average(0, T2, 0), SCOPE: 1, CLAUSE: 1 | average(0, T2, 0), SCOPE: 1, CLAUSE: 2 | average(0, T2, 0), SCOPE: 1, CLAUSE: 3 | average(0, T2, 0), 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\nT3 is ground\naverage(T1, T2, T3) !=? average(0, 0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: average(0, T2, 0), SCOPE: 1, CLAUSE: 1 | average(0, T2, 0), SCOPE: 1, CLAUSE: 2 | average(0, T2, 0), SCOPE: 1, CLAUSE: 3 | average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | average(0, T2, 0), SCOPE: 1, CLAUSE: 2 | average(0, T2, 0), SCOPE: 1, CLAUSE: 3 | average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: average(0, T2, 0), SCOPE: 1, CLAUSE: 2 | average(0, T2, 0), SCOPE: 1, CLAUSE: 3 | average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\naverage(0, T2, 0) !=? average(0, s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: average(0, T2, 0), SCOPE: 1, CLAUSE: 2 | average(0, T2, 0), SCOPE: 1, CLAUSE: 3 | average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: average(0, T2, 0), SCOPE: 1, CLAUSE: 3 | average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: average(0, T2, 0), SCOPE: 1, CLAUSE: 3 | average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\naverage(0, T2, 0) !=? average(0, s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\naverage(0, T2, 0) !=? average(0, s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true | average(0, T2, 0), SCOPE: 1, CLAUSE: 2 | average(0, T2, 0), SCOPE: 1, CLAUSE: 3 | average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\naverage(0, T2, 0) !=? average(0, 0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: 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\naverage(T1, T2, T3) !=? average(0, 0, 0)\naverage(T1, T2, T3) !=? average(0, s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: average(0, T2, 0), SCOPE: 1, CLAUSE: 2 | average(0, T2, 0), SCOPE: 1, CLAUSE: 3 | average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\naverage(0, T2, 0) !=? average(0, 0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: average(0, T2, 0), SCOPE: 1, CLAUSE: 3 | average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\naverage(0, T2, 0) !=? average(0, 0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: average(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\naverage(0, T2, 0) !=? average(0, 0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true | average(0, T2, s(0)), SCOPE: 1, CLAUSE: 3 | average(0, T2, s(0)), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT3 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];
23 [label="23: average(0, T2, s(0)), SCOPE: 1, CLAUSE: 3 | average(0, T2, s(0)), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: average(0, T2, s(0)), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: average(s(0), T9, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: average(s(0), T9, 0), SCOPE: 2, CLAUSE: 0 | average(s(0), T9, 0), SCOPE: 2, CLAUSE: 1 | average(s(0), T9, 0), SCOPE: 2, CLAUSE: 2 | average(s(0), T9, 0), SCOPE: 2, CLAUSE: 3 | average(s(0), T9, 0), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: average(s(0), T9, 0), SCOPE: 2, CLAUSE: 1 | average(s(0), T9, 0), SCOPE: 2, CLAUSE: 2 | average(s(0), T9, 0), SCOPE: 2, CLAUSE: 3 | average(s(0), T9, 0), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: average(s(0), T9, 0), SCOPE: 2, CLAUSE: 2 | average(s(0), T9, 0), SCOPE: 2, CLAUSE: 3 | average(s(0), T9, 0), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: average(s(0), T9, 0), SCOPE: 2, CLAUSE: 3 | average(s(0), T9, 0), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: average(s(0), T9, 0), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: average(s(0), T9, 0), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: average(0, s(T17), 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: average(T22, s(T25), T24) | average(s(T22), T2, T24), SCOPE: 1, CLAUSE: 4\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT3 is ground\nX49 is free\nX50 is free\nX51 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(X49), X50, X51)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 0 | average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 1 | average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 2 | average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 3 | average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 4 | ?_3 | average(s(T22), T2, T24), SCOPE: 1, CLAUSE: 4\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 1 | average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 2 | average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 3 | average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 4 | ?_3 | average(s(T22), T2, T24), SCOPE: 1, CLAUSE: 4\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 1\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 2 | average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 3 | average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 4 | ?_3 | average(s(T22), T2, T24), SCOPE: 1, CLAUSE: 4\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 2\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 3 | average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 4 | ?_3 | average(s(T22), T2, T24), SCOPE: 1, CLAUSE: 4\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 3\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 4 | ?_3 | average(s(T22), T2, T24), SCOPE: 1, CLAUSE: 4\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: average(T38, s(s(T41)), T40)\n\nT38 is ground\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: average(T22, s(T25), T24), SCOPE: 3, CLAUSE: 4\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ?_3 | average(s(T22), T2, T24), SCOPE: 1, CLAUSE: 4\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: average(s(T54), T57, T56)\n\nT54 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: average(s(T22), T2, T24), SCOPE: 1, CLAUSE: 4\n\nT22 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: average(s(s(T64)), T67, T66)\n\nT64 is ground\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: average(s(T71), T74, T73)\n\nT71 is ground\nT73 is ground\nX49 is free\nX50 is free\nX51 is free\naverage(T71, T2, s(T73)) !=? average(0, s(s(0)), s(0))\naverage(T71, T2, s(T73)) !=? average(s(X49), X50, X51)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 0 | average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 1 | average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 2 | average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 3 | average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 4\n\nT71 is ground\nT73 is ground\nX49 is free\nX50 is free\nX51 is free\naverage(T71, T2, s(T73)) !=? average(0, s(s(0)), s(0))\naverage(T71, T2, s(T73)) !=? average(s(X49), X50, X51)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 1 | average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 2 | average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 3 | average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 4\n\nT71 is ground\nT73 is ground\nX49 is free\nX50 is free\nX51 is free\naverage(T71, T2, s(T73)) !=? average(0, s(s(0)), s(0))\naverage(T71, T2, s(T73)) !=? average(s(X49), X50, X51)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 2 | average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 3 | average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 4\n\nT71 is ground\nT73 is ground\nX49 is free\nX50 is free\nX51 is free\naverage(T71, T2, s(T73)) !=? average(0, s(s(0)), s(0))\naverage(T71, T2, s(T73)) !=? average(s(X49), X50, X51)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 3 | average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 4\n\nT71 is ground\nT73 is ground\nX49 is free\nX50 is free\nX51 is free\naverage(T71, T2, s(T73)) !=? average(0, s(s(0)), s(0))\naverage(T71, T2, s(T73)) !=? average(s(X49), X50, X51)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 3\n\nT71 is ground\nT73 is ground\nX49 is free\nX50 is free\nX51 is free\naverage(T71, T2, s(T73)) !=? average(0, s(s(0)), s(0))\naverage(T71, T2, s(T73)) !=? average(s(X49), X50, X51)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: average(s(T71), T74, T73), SCOPE: 4, CLAUSE: 4\n\nT71 is ground\nT73 is ground\nX49 is free\nX50 is free\nX51 is free\naverage(T71, T2, s(T73)) !=? average(0, s(s(0)), s(0))\naverage(T71, T2, s(T73)) !=? average(s(X49), X50, X51)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: average(T89, s(T92), T91)\n\nT89 is ground\nT91 is ground\nX49 is free\nX50 is free\nX51 is free\naverage(T89, T2, s(T91)) !=? average(0, s(s(0)), s(0))\naverage(T89, T2, s(T91)) !=? average(s(X49), X50, X51)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: average(s(s(T99)), T102, T101)\n\nT99 is ground\nT101 is ground\nX49 is free\nX50 is free\nX51 is free\naverage(T99, T2, s(s(T101))) !=? average(s(X49), X50, X51)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 71 [arrowhead = none ];
71 -> 2;

72 [label="EVAL with clause\naverage(0, 0, 0).\nand substitutionT1 -> 0,\nT2 -> 0,\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 72 [arrowhead = none ];
72 -> 3;

73 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 73 [arrowhead = none ];
73 -> 4;

74 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 74 [arrowhead = none ];
74 -> 5;

75 [label="EVAL with clause\naverage(0, s(0), 0).\nand substitutionT1 -> 0,\nT2 -> s(0),\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 75 [arrowhead = none ];
75 -> 15;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 76 [arrowhead = none ];
76 -> 16;

77 [label="EVAL with clause\naverage(0, s(0), 0).\nand substitutionT2 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 77 [arrowhead = none ];
77 -> 6;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 78 [arrowhead = none ];
78 -> 7;

79 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 79 [arrowhead = none ];
79 -> 8;

80 [label="BACKTRACK\nfor clause: average(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 80 [arrowhead = none ];
80 -> 12;

81 [label="BACKTRACK\nfor clause: average(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 81 [arrowhead = none ];
81 -> 9;

82 [label="BACKTRACK\nfor clause: average(s(X), Y, Z) :- average(X, s(Y), Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 82 [arrowhead = none ];
82 -> 10;

83 [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];
10 -> 83 [arrowhead = none ];
83 -> 11;

84 [label="BACKTRACK\nfor clause: average(s(X), Y, Z) :- average(X, s(Y), Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 84 [arrowhead = none ];
84 -> 13;

85 [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];
13 -> 85 [arrowhead = none ];
85 -> 14;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 86 [arrowhead = none ];
86 -> 17;

87 [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];
16 -> 87 [arrowhead = none ];
87 -> 21;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 88 [arrowhead = none ];
88 -> 22;

89 [label="BACKTRACK\nfor clause: average(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 89 [arrowhead = none ];
89 -> 18;

90 [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 -> 90 [arrowhead = none ];
90 -> 19;

91 [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 -> 91 [arrowhead = none ];
91 -> 20;

92 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 92 [arrowhead = none ];
92 -> 23;

93 [label="EVAL with clause\naverage(s(X49), X50, X51) :- average(X49, s(X50), X51).\nand substitutionX49 -> T22,\nT1 -> s(T22),\nT2 -> T25,\nX50 -> T25,\nT3 -> T24,\nX51 -> T24,\nT23 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 93 [arrowhead = none ];
93 -> 35;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 94 [arrowhead = none ];
94 -> 36;

95 [label="BACKTRACK\nfor clause: average(s(X), Y, Z) :- average(X, s(Y), Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 95 [arrowhead = none ];
95 -> 24;

96 [label="EVAL with clause\naverage(X25, s(s(s(X26))), s(X27)) :- average(s(X25), X26, X27).\nand substitutionX25 -> 0,\nX26 -> T9,\nT2 -> s(s(s(T9))),\nX27 -> 0,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 96 [arrowhead = none ];
96 -> 25;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 97 [arrowhead = none ];
97 -> 26;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 98 [arrowhead = none ];
98 -> 27;

99 [label="BACKTRACK\nfor clause: average(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 99 [arrowhead = none ];
99 -> 28;

100 [label="BACKTRACK\nfor clause: average(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 100 [arrowhead = none ];
100 -> 29;

101 [label="BACKTRACK\nfor clause: average(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 101 [arrowhead = none ];
101 -> 30;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 102 [arrowhead = none ];
102 -> 31;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 103 [arrowhead = none ];
103 -> 32;

104 [label="ONLY EVAL with clause\naverage(s(X40), X41, X42) :- average(X40, s(X41), X42).\nand substitutionX40 -> 0,\nT9 -> T17,\nX41 -> T17,\nX42 -> 0,\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 104 [arrowhead = none ];
104 -> 33;

105 [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];
32 -> 105 [arrowhead = none ];
105 -> 34;

106 [label="INSTANCE with matching:\nT1 -> 0\nT2 -> s(T17)\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 106 [arrowhead = none , style=dashed];
106 -> 1[style=dashed];

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 107 [arrowhead = none ];
107 -> 37;

108 [label="EVAL with clause\naverage(X94, s(s(s(X95))), s(X96)) :- average(s(X94), X95, X96).\nand substitutionT1 -> T71,\nX94 -> T71,\nX95 -> T74,\nT2 -> s(s(s(T74))),\nX96 -> T73,\nT3 -> s(T73),\nT72 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 108 [arrowhead = none ];
108 -> 60;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 109 [arrowhead = none ];
109 -> 61;

110 [label="BACKTRACK\nfor clause: average(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 110 [arrowhead = none ];
110 -> 38;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 111 [arrowhead = none ];
111 -> 39;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 112 [arrowhead = none ];
112 -> 40;

113 [label="EVAL with clause\naverage(0, s(0), 0).\nand substitutionT22 -> 0,\nT25 -> 0,\nT24 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 113 [arrowhead = none ];
113 -> 41;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 114 [arrowhead = none ];
114 -> 42;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 115 [arrowhead = none ];
115 -> 44;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 116 [arrowhead = none ];
116 -> 45;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 117 [arrowhead = none ];
117 -> 43;

118 [label="EVAL with clause\naverage(0, s(s(0)), s(0)).\nand substitutionT22 -> 0,\nT25 -> s(0),\nT24 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 118 [arrowhead = none ];
118 -> 46;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 119 [arrowhead = none ];
119 -> 47;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 120 [arrowhead = none ];
120 -> 49;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 121 [arrowhead = none ];
121 -> 50;

122 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 122 [arrowhead = none ];
122 -> 48;

123 [label="EVAL with clause\naverage(s(X64), X65, X66) :- average(X64, s(X65), X66).\nand substitutionX64 -> T38,\nT22 -> s(T38),\nT25 -> T41,\nX65 -> s(T41),\nT24 -> T40,\nX66 -> T40,\nT39 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 123 [arrowhead = none ];
123 -> 51;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 124 [arrowhead = none ];
124 -> 52;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 125 [arrowhead = none ];
125 -> 53;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 126 [arrowhead = none ];
126 -> 54;

127 [label="INSTANCE with matching:\nT1 -> T38\nT2 -> s(s(T41))\nT3 -> T40", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 127 [arrowhead = none , style=dashed];
127 -> 1[style=dashed];

128 [label="EVAL with clause\naverage(X79, s(s(s(X80))), s(X81)) :- average(s(X79), X80, X81).\nand substitutionT22 -> T54,\nX79 -> T54,\nX80 -> T57,\nT25 -> s(s(T57)),\nX81 -> T56,\nT24 -> s(T56),\nT55 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 128 [arrowhead = none ];
128 -> 55;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 129 [arrowhead = none ];
129 -> 56;

130 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 130 [arrowhead = none ];
130 -> 57;

131 [label="INSTANCE with matching:\nT1 -> s(T54)\nT2 -> T57\nT3 -> T56", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 131 [arrowhead = none , style=dashed];
131 -> 1[style=dashed];

132 [label="EVAL with clause\naverage(X88, s(s(s(X89))), s(X90)) :- average(s(X88), X89, X90).\nand substitutionT22 -> T64,\nX88 -> s(T64),\nX89 -> T67,\nT2 -> s(s(s(T67))),\nX90 -> T66,\nT24 -> s(T66),\nT65 -> T67", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 132 [arrowhead = none ];
132 -> 58;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 133 [arrowhead = none ];
133 -> 59;

134 [label="INSTANCE with matching:\nT1 -> s(s(T64))\nT2 -> T67\nT3 -> T66", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 134 [arrowhead = none , style=dashed];
134 -> 1[style=dashed];

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 135 [arrowhead = none ];
135 -> 62;

136 [label="BACKTRACK\nfor clause: average(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 136 [arrowhead = none ];
136 -> 63;

137 [label="BACKTRACK\nfor clause: average(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 137 [arrowhead = none ];
137 -> 64;

138 [label="BACKTRACK\nfor clause: average(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 138 [arrowhead = none ];
138 -> 65;

139 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 139 [arrowhead = none ];
139 -> 66;

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 140 [arrowhead = none ];
140 -> 67;

141 [label="ONLY EVAL with clause\naverage(s(X109), X110, X111) :- average(X109, s(X110), X111).\nand substitutionT71 -> T89,\nX109 -> T89,\nT74 -> T92,\nX110 -> T92,\nT73 -> T91,\nX111 -> T91,\nT90 -> T92", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 141 [arrowhead = none ];
141 -> 68;

142 [label="EVAL with clause\naverage(X118, s(s(s(X119))), s(X120)) :- average(s(X118), X119, X120).\nand substitutionT71 -> T99,\nX118 -> s(T99),\nX119 -> T102,\nT74 -> s(s(s(T102))),\nX120 -> T101,\nT73 -> s(T101),\nT100 -> T102", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 142 [arrowhead = none ];
142 -> 69;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 143 [arrowhead = none ];
143 -> 70;

144 [label="INSTANCE with matching:\nT1 -> T89\nT2 -> s(T92)\nT3 -> T91", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 144 [arrowhead = none , style=dashed];
144 -> 1[style=dashed];

145 [label="INSTANCE with matching:\nT1 -> s(s(T99))\nT2 -> T102\nT3 -> T101", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 145 [arrowhead = none , style=dashed];
145 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

averageA(0, 0, 0).
averageA(0, s(0), 0).
averageA(0, s(0), 0).
averageA(0, s(s(0)), s(0)).
averageA(0, s(s(s(T17))), s(0)) :- averageA(0, s(T17), 0).
averageA(s(0), 0, 0).
averageA(s(0), s(0), s(0)).
averageA(s(s(T38)), T41, T40) :- averageA(T38, s(s(T41)), T40).
averageA(s(T54), s(s(T57)), s(T56)) :- averageA(s(T54), T57, T56).
averageA(s(T64), s(s(s(T67))), s(T66)) :- averageA(s(s(T64)), T67, T66).
averageA(T89, s(s(s(T92))), s(T91)) :- averageA(T89, s(T92), T91).
averageA(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) :- averageA(s(s(T99)), T102, T101).

Query: averageA(g,a,g)

(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,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

averageA_in_gag(0, 0, 0) → averageA_out_gag(0, 0, 0)
averageA_in_gag(0, s(0), 0) → averageA_out_gag(0, s(0), 0)
averageA_in_gag(0, s(s(0)), s(0)) → averageA_out_gag(0, s(s(0)), s(0))
averageA_in_gag(0, s(s(s(T17))), s(0)) → U1_gag(T17, averageA_in_gag(0, s(T17), 0))
averageA_in_gag(s(0), 0, 0) → averageA_out_gag(s(0), 0, 0)
averageA_in_gag(s(0), s(0), s(0)) → averageA_out_gag(s(0), s(0), s(0))
averageA_in_gag(s(s(T38)), T41, T40) → U2_gag(T38, T41, T40, averageA_in_gag(T38, s(s(T41)), T40))
averageA_in_gag(s(T54), s(s(T57)), s(T56)) → U3_gag(T54, T57, T56, averageA_in_gag(s(T54), T57, T56))
averageA_in_gag(s(T64), s(s(s(T67))), s(T66)) → U4_gag(T64, T67, T66, averageA_in_gag(s(s(T64)), T67, T66))
averageA_in_gag(T89, s(s(s(T92))), s(T91)) → U5_gag(T89, T92, T91, averageA_in_gag(T89, s(T92), T91))
averageA_in_gag(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) → U6_gag(T99, T102, T101, averageA_in_gag(s(s(T99)), T102, T101))
U6_gag(T99, T102, T101, averageA_out_gag(s(s(T99)), T102, T101)) → averageA_out_gag(T99, s(s(s(s(s(s(T102)))))), s(s(T101)))
U5_gag(T89, T92, T91, averageA_out_gag(T89, s(T92), T91)) → averageA_out_gag(T89, s(s(s(T92))), s(T91))
U4_gag(T64, T67, T66, averageA_out_gag(s(s(T64)), T67, T66)) → averageA_out_gag(s(T64), s(s(s(T67))), s(T66))
U3_gag(T54, T57, T56, averageA_out_gag(s(T54), T57, T56)) → averageA_out_gag(s(T54), s(s(T57)), s(T56))
U2_gag(T38, T41, T40, averageA_out_gag(T38, s(s(T41)), T40)) → averageA_out_gag(s(s(T38)), T41, T40)
U1_gag(T17, averageA_out_gag(0, s(T17), 0)) → averageA_out_gag(0, s(s(s(T17))), s(0))

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_gag(0, 0, 0) → averageA_out_gag(0, 0, 0)
averageA_in_gag(0, s(0), 0) → averageA_out_gag(0, s(0), 0)
averageA_in_gag(0, s(s(0)), s(0)) → averageA_out_gag(0, s(s(0)), s(0))
averageA_in_gag(0, s(s(s(T17))), s(0)) → U1_gag(T17, averageA_in_gag(0, s(T17), 0))
averageA_in_gag(s(0), 0, 0) → averageA_out_gag(s(0), 0, 0)
averageA_in_gag(s(0), s(0), s(0)) → averageA_out_gag(s(0), s(0), s(0))
averageA_in_gag(s(s(T38)), T41, T40) → U2_gag(T38, T41, T40, averageA_in_gag(T38, s(s(T41)), T40))
averageA_in_gag(s(T54), s(s(T57)), s(T56)) → U3_gag(T54, T57, T56, averageA_in_gag(s(T54), T57, T56))
averageA_in_gag(s(T64), s(s(s(T67))), s(T66)) → U4_gag(T64, T67, T66, averageA_in_gag(s(s(T64)), T67, T66))
averageA_in_gag(T89, s(s(s(T92))), s(T91)) → U5_gag(T89, T92, T91, averageA_in_gag(T89, s(T92), T91))
averageA_in_gag(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) → U6_gag(T99, T102, T101, averageA_in_gag(s(s(T99)), T102, T101))
U6_gag(T99, T102, T101, averageA_out_gag(s(s(T99)), T102, T101)) → averageA_out_gag(T99, s(s(s(s(s(s(T102)))))), s(s(T101)))
U5_gag(T89, T92, T91, averageA_out_gag(T89, s(T92), T91)) → averageA_out_gag(T89, s(s(s(T92))), s(T91))
U4_gag(T64, T67, T66, averageA_out_gag(s(s(T64)), T67, T66)) → averageA_out_gag(s(T64), s(s(s(T67))), s(T66))
U3_gag(T54, T57, T56, averageA_out_gag(s(T54), T57, T56)) → averageA_out_gag(s(T54), s(s(T57)), s(T56))
U2_gag(T38, T41, T40, averageA_out_gag(T38, s(s(T41)), T40)) → averageA_out_gag(s(s(T38)), T41, T40)
U1_gag(T17, averageA_out_gag(0, s(T17), 0)) → averageA_out_gag(0, s(s(s(T17))), s(0))

(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_GAG(0, s(s(s(T17))), s(0)) → U1_GAG(T17, averageA_in_gag(0, s(T17), 0))
AVERAGEA_IN_GAG(0, s(s(s(T17))), s(0)) → AVERAGEA_IN_GAG(0, s(T17), 0)
AVERAGEA_IN_GAG(s(s(T38)), T41, T40) → U2_GAG(T38, T41, T40, averageA_in_gag(T38, s(s(T41)), T40))
AVERAGEA_IN_GAG(s(s(T38)), T41, T40) → AVERAGEA_IN_GAG(T38, s(s(T41)), T40)
AVERAGEA_IN_GAG(s(T54), s(s(T57)), s(T56)) → U3_GAG(T54, T57, T56, averageA_in_gag(s(T54), T57, T56))
AVERAGEA_IN_GAG(s(T54), s(s(T57)), s(T56)) → AVERAGEA_IN_GAG(s(T54), T57, T56)
AVERAGEA_IN_GAG(s(T64), s(s(s(T67))), s(T66)) → U4_GAG(T64, T67, T66, averageA_in_gag(s(s(T64)), T67, T66))
AVERAGEA_IN_GAG(s(T64), s(s(s(T67))), s(T66)) → AVERAGEA_IN_GAG(s(s(T64)), T67, T66)
AVERAGEA_IN_GAG(T89, s(s(s(T92))), s(T91)) → U5_GAG(T89, T92, T91, averageA_in_gag(T89, s(T92), T91))
AVERAGEA_IN_GAG(T89, s(s(s(T92))), s(T91)) → AVERAGEA_IN_GAG(T89, s(T92), T91)
AVERAGEA_IN_GAG(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) → U6_GAG(T99, T102, T101, averageA_in_gag(s(s(T99)), T102, T101))
AVERAGEA_IN_GAG(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) → AVERAGEA_IN_GAG(s(s(T99)), T102, T101)

The TRS R consists of the following rules:

averageA_in_gag(0, 0, 0) → averageA_out_gag(0, 0, 0)
averageA_in_gag(0, s(0), 0) → averageA_out_gag(0, s(0), 0)
averageA_in_gag(0, s(s(0)), s(0)) → averageA_out_gag(0, s(s(0)), s(0))
averageA_in_gag(0, s(s(s(T17))), s(0)) → U1_gag(T17, averageA_in_gag(0, s(T17), 0))
averageA_in_gag(s(0), 0, 0) → averageA_out_gag(s(0), 0, 0)
averageA_in_gag(s(0), s(0), s(0)) → averageA_out_gag(s(0), s(0), s(0))
averageA_in_gag(s(s(T38)), T41, T40) → U2_gag(T38, T41, T40, averageA_in_gag(T38, s(s(T41)), T40))
averageA_in_gag(s(T54), s(s(T57)), s(T56)) → U3_gag(T54, T57, T56, averageA_in_gag(s(T54), T57, T56))
averageA_in_gag(s(T64), s(s(s(T67))), s(T66)) → U4_gag(T64, T67, T66, averageA_in_gag(s(s(T64)), T67, T66))
averageA_in_gag(T89, s(s(s(T92))), s(T91)) → U5_gag(T89, T92, T91, averageA_in_gag(T89, s(T92), T91))
averageA_in_gag(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) → U6_gag(T99, T102, T101, averageA_in_gag(s(s(T99)), T102, T101))
U6_gag(T99, T102, T101, averageA_out_gag(s(s(T99)), T102, T101)) → averageA_out_gag(T99, s(s(s(s(s(s(T102)))))), s(s(T101)))
U5_gag(T89, T92, T91, averageA_out_gag(T89, s(T92), T91)) → averageA_out_gag(T89, s(s(s(T92))), s(T91))
U4_gag(T64, T67, T66, averageA_out_gag(s(s(T64)), T67, T66)) → averageA_out_gag(s(T64), s(s(s(T67))), s(T66))
U3_gag(T54, T57, T56, averageA_out_gag(s(T54), T57, T56)) → averageA_out_gag(s(T54), s(s(T57)), s(T56))
U2_gag(T38, T41, T40, averageA_out_gag(T38, s(s(T41)), T40)) → averageA_out_gag(s(s(T38)), T41, T40)
U1_gag(T17, averageA_out_gag(0, s(T17), 0)) → averageA_out_gag(0, s(s(s(T17))), s(0))

The argument filtering Pi contains the following mapping:
averageA_in_gag(x1, x2, x3) = averageA_in_gag(x1, x3)
0 = 0
averageA_out_gag(x1, x2, x3) = averageA_out_gag(x2)
s(x1) = s(x1)
U1_gag(x1, x2) = U1_gag(x2)
U2_gag(x1, x2, x3, x4) = U2_gag(x4)
U3_gag(x1, x2, x3, x4) = U3_gag(x4)
U4_gag(x1, x2, x3, x4) = U4_gag(x4)
U5_gag(x1, x2, x3, x4) = U5_gag(x4)
U6_gag(x1, x2, x3, x4) = U6_gag(x4)
AVERAGEA_IN_GAG(x1, x2, x3) = AVERAGEA_IN_GAG(x1, x3)
U1_GAG(x1, x2) = U1_GAG(x2)
U2_GAG(x1, x2, x3, x4) = U2_GAG(x4)
U3_GAG(x1, x2, x3, x4) = U3_GAG(x4)
U4_GAG(x1, x2, x3, x4) = U4_GAG(x4)
U5_GAG(x1, x2, x3, x4) = U5_GAG(x4)
U6_GAG(x1, x2, x3, x4) = U6_GAG(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_GAG(0, s(s(s(T17))), s(0)) → U1_GAG(T17, averageA_in_gag(0, s(T17), 0))
AVERAGEA_IN_GAG(0, s(s(s(T17))), s(0)) → AVERAGEA_IN_GAG(0, s(T17), 0)
AVERAGEA_IN_GAG(s(s(T38)), T41, T40) → U2_GAG(T38, T41, T40, averageA_in_gag(T38, s(s(T41)), T40))
AVERAGEA_IN_GAG(s(s(T38)), T41, T40) → AVERAGEA_IN_GAG(T38, s(s(T41)), T40)
AVERAGEA_IN_GAG(s(T54), s(s(T57)), s(T56)) → U3_GAG(T54, T57, T56, averageA_in_gag(s(T54), T57, T56))
AVERAGEA_IN_GAG(s(T54), s(s(T57)), s(T56)) → AVERAGEA_IN_GAG(s(T54), T57, T56)
AVERAGEA_IN_GAG(s(T64), s(s(s(T67))), s(T66)) → U4_GAG(T64, T67, T66, averageA_in_gag(s(s(T64)), T67, T66))
AVERAGEA_IN_GAG(s(T64), s(s(s(T67))), s(T66)) → AVERAGEA_IN_GAG(s(s(T64)), T67, T66)
AVERAGEA_IN_GAG(T89, s(s(s(T92))), s(T91)) → U5_GAG(T89, T92, T91, averageA_in_gag(T89, s(T92), T91))
AVERAGEA_IN_GAG(T89, s(s(s(T92))), s(T91)) → AVERAGEA_IN_GAG(T89, s(T92), T91)
AVERAGEA_IN_GAG(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) → U6_GAG(T99, T102, T101, averageA_in_gag(s(s(T99)), T102, T101))
AVERAGEA_IN_GAG(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) → AVERAGEA_IN_GAG(s(s(T99)), T102, T101)

The TRS R consists of the following rules:

averageA_in_gag(0, 0, 0) → averageA_out_gag(0, 0, 0)
averageA_in_gag(0, s(0), 0) → averageA_out_gag(0, s(0), 0)
averageA_in_gag(0, s(s(0)), s(0)) → averageA_out_gag(0, s(s(0)), s(0))
averageA_in_gag(0, s(s(s(T17))), s(0)) → U1_gag(T17, averageA_in_gag(0, s(T17), 0))
averageA_in_gag(s(0), 0, 0) → averageA_out_gag(s(0), 0, 0)
averageA_in_gag(s(0), s(0), s(0)) → averageA_out_gag(s(0), s(0), s(0))
averageA_in_gag(s(s(T38)), T41, T40) → U2_gag(T38, T41, T40, averageA_in_gag(T38, s(s(T41)), T40))
averageA_in_gag(s(T54), s(s(T57)), s(T56)) → U3_gag(T54, T57, T56, averageA_in_gag(s(T54), T57, T56))
averageA_in_gag(s(T64), s(s(s(T67))), s(T66)) → U4_gag(T64, T67, T66, averageA_in_gag(s(s(T64)), T67, T66))
averageA_in_gag(T89, s(s(s(T92))), s(T91)) → U5_gag(T89, T92, T91, averageA_in_gag(T89, s(T92), T91))
averageA_in_gag(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) → U6_gag(T99, T102, T101, averageA_in_gag(s(s(T99)), T102, T101))
U6_gag(T99, T102, T101, averageA_out_gag(s(s(T99)), T102, T101)) → averageA_out_gag(T99, s(s(s(s(s(s(T102)))))), s(s(T101)))
U5_gag(T89, T92, T91, averageA_out_gag(T89, s(T92), T91)) → averageA_out_gag(T89, s(s(s(T92))), s(T91))
U4_gag(T64, T67, T66, averageA_out_gag(s(s(T64)), T67, T66)) → averageA_out_gag(s(T64), s(s(s(T67))), s(T66))
U3_gag(T54, T57, T56, averageA_out_gag(s(T54), T57, T56)) → averageA_out_gag(s(T54), s(s(T57)), s(T56))
U2_gag(T38, T41, T40, averageA_out_gag(T38, s(s(T41)), T40)) → averageA_out_gag(s(s(T38)), T41, T40)
U1_gag(T17, averageA_out_gag(0, s(T17), 0)) → averageA_out_gag(0, s(s(s(T17))), s(0))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes.

(8) Obligation:

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

AVERAGEA_IN_GAG(s(T54), s(s(T57)), s(T56)) → AVERAGEA_IN_GAG(s(T54), T57, T56)
AVERAGEA_IN_GAG(s(s(T38)), T41, T40) → AVERAGEA_IN_GAG(T38, s(s(T41)), T40)
AVERAGEA_IN_GAG(s(T64), s(s(s(T67))), s(T66)) → AVERAGEA_IN_GAG(s(s(T64)), T67, T66)
AVERAGEA_IN_GAG(T89, s(s(s(T92))), s(T91)) → AVERAGEA_IN_GAG(T89, s(T92), T91)
AVERAGEA_IN_GAG(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) → AVERAGEA_IN_GAG(s(s(T99)), T102, T101)

The TRS R consists of the following rules:

averageA_in_gag(0, 0, 0) → averageA_out_gag(0, 0, 0)
averageA_in_gag(0, s(0), 0) → averageA_out_gag(0, s(0), 0)
averageA_in_gag(0, s(s(0)), s(0)) → averageA_out_gag(0, s(s(0)), s(0))
averageA_in_gag(0, s(s(s(T17))), s(0)) → U1_gag(T17, averageA_in_gag(0, s(T17), 0))
averageA_in_gag(s(0), 0, 0) → averageA_out_gag(s(0), 0, 0)
averageA_in_gag(s(0), s(0), s(0)) → averageA_out_gag(s(0), s(0), s(0))
averageA_in_gag(s(s(T38)), T41, T40) → U2_gag(T38, T41, T40, averageA_in_gag(T38, s(s(T41)), T40))
averageA_in_gag(s(T54), s(s(T57)), s(T56)) → U3_gag(T54, T57, T56, averageA_in_gag(s(T54), T57, T56))
averageA_in_gag(s(T64), s(s(s(T67))), s(T66)) → U4_gag(T64, T67, T66, averageA_in_gag(s(s(T64)), T67, T66))
averageA_in_gag(T89, s(s(s(T92))), s(T91)) → U5_gag(T89, T92, T91, averageA_in_gag(T89, s(T92), T91))
averageA_in_gag(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) → U6_gag(T99, T102, T101, averageA_in_gag(s(s(T99)), T102, T101))
U6_gag(T99, T102, T101, averageA_out_gag(s(s(T99)), T102, T101)) → averageA_out_gag(T99, s(s(s(s(s(s(T102)))))), s(s(T101)))
U5_gag(T89, T92, T91, averageA_out_gag(T89, s(T92), T91)) → averageA_out_gag(T89, s(s(s(T92))), s(T91))
U4_gag(T64, T67, T66, averageA_out_gag(s(s(T64)), T67, T66)) → averageA_out_gag(s(T64), s(s(s(T67))), s(T66))
U3_gag(T54, T57, T56, averageA_out_gag(s(T54), T57, T56)) → averageA_out_gag(s(T54), s(s(T57)), s(T56))
U2_gag(T38, T41, T40, averageA_out_gag(T38, s(s(T41)), T40)) → averageA_out_gag(s(s(T38)), T41, T40)
U1_gag(T17, averageA_out_gag(0, s(T17), 0)) → averageA_out_gag(0, s(s(s(T17))), s(0))

(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_GAG(s(T54), s(s(T57)), s(T56)) → AVERAGEA_IN_GAG(s(T54), T57, T56)
AVERAGEA_IN_GAG(s(s(T38)), T41, T40) → AVERAGEA_IN_GAG(T38, s(s(T41)), T40)
AVERAGEA_IN_GAG(s(T64), s(s(s(T67))), s(T66)) → AVERAGEA_IN_GAG(s(s(T64)), T67, T66)
AVERAGEA_IN_GAG(T89, s(s(s(T92))), s(T91)) → AVERAGEA_IN_GAG(T89, s(T92), T91)
AVERAGEA_IN_GAG(T99, s(s(s(s(s(s(T102)))))), s(s(T101))) → AVERAGEA_IN_GAG(s(s(T99)), T102, T101)

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

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_GAG(s(T54), s(T56)) → AVERAGEA_IN_GAG(s(T54), T56)
AVERAGEA_IN_GAG(s(s(T38)), T40) → AVERAGEA_IN_GAG(T38, T40)
AVERAGEA_IN_GAG(s(T64), s(T66)) → AVERAGEA_IN_GAG(s(s(T64)), T66)
AVERAGEA_IN_GAG(T89, s(T91)) → AVERAGEA_IN_GAG(T89, T91)
AVERAGEA_IN_GAG(T99, s(s(T101))) → AVERAGEA_IN_GAG(s(s(T99)), T101)

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

(13) 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:

AVERAGEA_IN_GAG(s(T54), s(T56)) → AVERAGEA_IN_GAG(s(T54), T56)
The graph contains the following edges 1 >= 1, 2 > 2
AVERAGEA_IN_GAG(s(s(T38)), T40) → AVERAGEA_IN_GAG(T38, T40)
The graph contains the following edges 1 > 1, 2 >= 2
AVERAGEA_IN_GAG(s(T64), s(T66)) → AVERAGEA_IN_GAG(s(s(T64)), T66)
The graph contains the following edges 2 > 2
AVERAGEA_IN_GAG(T89, s(T91)) → AVERAGEA_IN_GAG(T89, T91)
The graph contains the following edges 1 >= 1, 2 > 2
AVERAGEA_IN_GAG(T99, s(s(T101))) → AVERAGEA_IN_GAG(s(s(T99)), T101)
The graph contains the following edges 2 > 2

(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) QDPSizeChangeProof (EQUIVALENT transformation)

(14) YES