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

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

↳3 PrologToPiTRSProof (⇐)

↳4 PiTRS

↳5 DependencyPairsProof (⇔)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 PiDP

↳9 UsableRulesProof (⇔)

↳10 PiDP

↳11 PiDPToQDPProof (⇐)

↳12 QDP

↳13 QDPSizeChangeProof (⇔)

↳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).

Queries:

average(g,a,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

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\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(X52), X53, X54)", 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(T67)), T70, T69)\n\nT67 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: average(s(T77), T80, T79)\n\nT77 is ground\nT79 is ground\naverage(T77, T2, s(T79)) !=? average(0, s(s(0)), s(0))\naverage(T77, T2, s(T79)) !=? average(s(X52), X53, X54)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 0 | average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 1 | average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 2 | average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 3 | average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 4\n\nT77 is ground\nT79 is ground\naverage(T77, T2, s(T79)) !=? average(0, s(s(0)), s(0))\naverage(T77, T2, s(T79)) !=? average(s(X52), X53, X54)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 1 | average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 2 | average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 3 | average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 4\n\nT77 is ground\nT79 is ground\naverage(T77, T2, s(T79)) !=? average(0, s(s(0)), s(0))\naverage(T77, T2, s(T79)) !=? average(s(X52), X53, X54)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 2 | average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 3 | average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 4\n\nT77 is ground\nT79 is ground\naverage(T77, T2, s(T79)) !=? average(0, s(s(0)), s(0))\naverage(T77, T2, s(T79)) !=? average(s(X52), X53, X54)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 3 | average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 4\n\nT77 is ground\nT79 is ground\naverage(T77, T2, s(T79)) !=? average(0, s(s(0)), s(0))\naverage(T77, T2, s(T79)) !=? average(s(X52), X53, X54)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 3\n\nT77 is ground\nT79 is ground\naverage(T77, T2, s(T79)) !=? average(0, s(s(0)), s(0))\naverage(T77, T2, s(T79)) !=? average(s(X52), X53, X54)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: average(s(T77), T80, T79), SCOPE: 4, CLAUSE: 4\n\nT77 is ground\nT79 is ground\naverage(T77, T2, s(T79)) !=? average(0, s(s(0)), s(0))\naverage(T77, T2, s(T79)) !=? average(s(X52), X53, X54)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: average(T95, s(T98), T97)\n\nT95 is ground\nT97 is ground\naverage(T95, T2, s(T97)) !=? average(0, s(s(0)), s(0))\naverage(T95, T2, s(T97)) !=? average(s(X52), X53, X54)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: average(s(s(T105)), T108, T107)\n\nT105 is ground\nT107 is ground\naverage(T105, T2, s(s(T107))) !=? average(s(X52), X53, X54)", 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 substitution\nT1 -> 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 = lightblue];
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 substitution\nT1 -> 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 = lightblue];
4 -> 76 [arrowhead = none ];
76 -> 16;

77 [label="EVAL with clause\naverage(0, s(0), 0).\nand substitution\nT2 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 77 [arrowhead = none ];
77 -> 6;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
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 = white];
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 = white];
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 = white];
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 = white];
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 = white];
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 substitution\nT1 -> 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 = lightblue];
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 = white];
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 = white];
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 = white];
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(X52), X53, X54) :- average(X52, s(X53), X54).\nand substitution\nX52 -> T22\nT1 -> s(T22)\nT2 -> T25\nX53 -> T25\nT3 -> T24\nX54 -> T24\nT23 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 93 [arrowhead = none ];
93 -> 35;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
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 substitution\nX25 -> 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 = lightblue];
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 = white];
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 = white];
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 = white];
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(X43), X44, X45) :- average(X43, s(X44), X45).\nand substitution\nX43 -> 0\nT9 -> T17\nX44 -> T17\nX45 -> 0\nT16 -> T17", fontsize=14, style = filled, fillcolor = white];
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 = white];
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(X115, s(s(s(X116))), s(X117)) :- average(s(X115), X116, X117).\nand substitution\nT1 -> T77\nX115 -> T77\nX116 -> T80\nT2 -> s(s(s(T80)))\nX117 -> T79\nT3 -> s(T79)\nT78 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 108 [arrowhead = none ];
108 -> 60;

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

110 [label="BACKTRACK\nfor clause: average(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
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 substitution\nT22 -> 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 = lightblue];
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 substitution\nT22 -> 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 = lightblue];
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(X70), X71, X72) :- average(X70, s(X71), X72).\nand substitution\nX70 -> T38\nT22 -> s(T38)\nT25 -> T41\nX71 -> s(T41)\nT24 -> T40\nX72 -> T40\nT39 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 123 [arrowhead = none ];
123 -> 51;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X88, s(s(s(X89))), s(X90)) :- average(s(X88), X89, X90).\nand substitution\nT22 -> T54\nX88 -> T54\nX89 -> T57\nT25 -> s(s(T57))\nX90 -> 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 = lightblue];
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(X103, s(s(s(X104))), s(X105)) :- average(s(X103), X104, X105).\nand substitution\nT22 -> T67\nX103 -> s(T67)\nX104 -> T70\nT2 -> s(s(s(T70)))\nX105 -> T69\nT24 -> s(T69)\nT68 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 132 [arrowhead = none ];
132 -> 58;

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

134 [label="INSTANCE with matching:\nT1 -> s(s(T67))\nT2 -> T70\nT3 -> T69", 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 = white];
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 = white];
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 = white];
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(X133), X134, X135) :- average(X133, s(X134), X135).\nand substitution\nT77 -> T95\nX133 -> T95\nT80 -> T98\nX134 -> T98\nT79 -> T97\nX135 -> T97\nT96 -> T98", fontsize=14, style = filled, fillcolor = white];
66 -> 141 [arrowhead = none ];
141 -> 68;

142 [label="EVAL with clause\naverage(X142, s(s(s(X143))), s(X144)) :- average(s(X142), X143, X144).\nand substitution\nT77 -> T105\nX142 -> s(T105)\nX143 -> T108\nT80 -> s(s(s(T108)))\nX144 -> T107\nT79 -> s(T107)\nT106 -> T108", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 142 [arrowhead = none ];
142 -> 69;

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

144 [label="INSTANCE with matching:\nT1 -> T95\nT2 -> s(T98)\nT3 -> T97", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 144 [arrowhead = none , style=dashed];
144 -> 1[style=dashed];

145 [label="INSTANCE with matching:\nT1 -> s(s(T105))\nT2 -> T108\nT3 -> T107", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 145 [arrowhead = none , style=dashed];
145 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

average1(0, 0, 0).
average1(0, s(0), 0).
average1(0, s(0), 0).
average1(0, s(s(0)), s(0)).
average1(0, s(s(s(T17))), s(0)) :- average1(0, s(T17), 0).
average1(s(0), 0, 0).
average1(s(0), s(0), s(0)).
average1(s(s(T38)), T41, T40) :- average1(T38, s(s(T41)), T40).
average1(s(T54), s(s(T57)), s(T56)) :- average1(s(T54), T57, T56).
average1(s(T67), s(s(s(T70))), s(T69)) :- average1(s(s(T67)), T70, T69).
average1(T95, s(s(s(T98))), s(T97)) :- average1(T95, s(T98), T97).
average1(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) :- average1(s(s(T105)), T108, T107).

Queries:

average1(g,a,g).

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

average1_in_gag(0, 0, 0) → average1_out_gag(0, 0, 0)
average1_in_gag(0, s(0), 0) → average1_out_gag(0, s(0), 0)
average1_in_gag(0, s(s(0)), s(0)) → average1_out_gag(0, s(s(0)), s(0))
average1_in_gag(0, s(s(s(T17))), s(0)) → U1_gag(T17, average1_in_gag(0, s(T17), 0))
average1_in_gag(s(0), 0, 0) → average1_out_gag(s(0), 0, 0)
average1_in_gag(s(0), s(0), s(0)) → average1_out_gag(s(0), s(0), s(0))
average1_in_gag(s(s(T38)), T41, T40) → U2_gag(T38, T41, T40, average1_in_gag(T38, s(s(T41)), T40))
average1_in_gag(s(T54), s(s(T57)), s(T56)) → U3_gag(T54, T57, T56, average1_in_gag(s(T54), T57, T56))
average1_in_gag(s(T67), s(s(s(T70))), s(T69)) → U4_gag(T67, T70, T69, average1_in_gag(s(s(T67)), T70, T69))
average1_in_gag(T95, s(s(s(T98))), s(T97)) → U5_gag(T95, T98, T97, average1_in_gag(T95, s(T98), T97))
average1_in_gag(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → U6_gag(T105, T108, T107, average1_in_gag(s(s(T105)), T108, T107))
U6_gag(T105, T108, T107, average1_out_gag(s(s(T105)), T108, T107)) → average1_out_gag(T105, s(s(s(s(s(s(T108)))))), s(s(T107)))
U5_gag(T95, T98, T97, average1_out_gag(T95, s(T98), T97)) → average1_out_gag(T95, s(s(s(T98))), s(T97))
U4_gag(T67, T70, T69, average1_out_gag(s(s(T67)), T70, T69)) → average1_out_gag(s(T67), s(s(s(T70))), s(T69))
U3_gag(T54, T57, T56, average1_out_gag(s(T54), T57, T56)) → average1_out_gag(s(T54), s(s(T57)), s(T56))
U2_gag(T38, T41, T40, average1_out_gag(T38, s(s(T41)), T40)) → average1_out_gag(s(s(T38)), T41, T40)
U1_gag(T17, average1_out_gag(0, s(T17), 0)) → average1_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:

average1_in_gag(0, 0, 0) → average1_out_gag(0, 0, 0)
average1_in_gag(0, s(0), 0) → average1_out_gag(0, s(0), 0)
average1_in_gag(0, s(s(0)), s(0)) → average1_out_gag(0, s(s(0)), s(0))
average1_in_gag(0, s(s(s(T17))), s(0)) → U1_gag(T17, average1_in_gag(0, s(T17), 0))
average1_in_gag(s(0), 0, 0) → average1_out_gag(s(0), 0, 0)
average1_in_gag(s(0), s(0), s(0)) → average1_out_gag(s(0), s(0), s(0))
average1_in_gag(s(s(T38)), T41, T40) → U2_gag(T38, T41, T40, average1_in_gag(T38, s(s(T41)), T40))
average1_in_gag(s(T54), s(s(T57)), s(T56)) → U3_gag(T54, T57, T56, average1_in_gag(s(T54), T57, T56))
average1_in_gag(s(T67), s(s(s(T70))), s(T69)) → U4_gag(T67, T70, T69, average1_in_gag(s(s(T67)), T70, T69))
average1_in_gag(T95, s(s(s(T98))), s(T97)) → U5_gag(T95, T98, T97, average1_in_gag(T95, s(T98), T97))
average1_in_gag(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → U6_gag(T105, T108, T107, average1_in_gag(s(s(T105)), T108, T107))
U6_gag(T105, T108, T107, average1_out_gag(s(s(T105)), T108, T107)) → average1_out_gag(T105, s(s(s(s(s(s(T108)))))), s(s(T107)))
U5_gag(T95, T98, T97, average1_out_gag(T95, s(T98), T97)) → average1_out_gag(T95, s(s(s(T98))), s(T97))
U4_gag(T67, T70, T69, average1_out_gag(s(s(T67)), T70, T69)) → average1_out_gag(s(T67), s(s(s(T70))), s(T69))
U3_gag(T54, T57, T56, average1_out_gag(s(T54), T57, T56)) → average1_out_gag(s(T54), s(s(T57)), s(T56))
U2_gag(T38, T41, T40, average1_out_gag(T38, s(s(T41)), T40)) → average1_out_gag(s(s(T38)), T41, T40)
U1_gag(T17, average1_out_gag(0, s(T17), 0)) → average1_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:

AVERAGE1_IN_GAG(0, s(s(s(T17))), s(0)) → U1_GAG(T17, average1_in_gag(0, s(T17), 0))
AVERAGE1_IN_GAG(0, s(s(s(T17))), s(0)) → AVERAGE1_IN_GAG(0, s(T17), 0)
AVERAGE1_IN_GAG(s(s(T38)), T41, T40) → U2_GAG(T38, T41, T40, average1_in_gag(T38, s(s(T41)), T40))
AVERAGE1_IN_GAG(s(s(T38)), T41, T40) → AVERAGE1_IN_GAG(T38, s(s(T41)), T40)
AVERAGE1_IN_GAG(s(T54), s(s(T57)), s(T56)) → U3_GAG(T54, T57, T56, average1_in_gag(s(T54), T57, T56))
AVERAGE1_IN_GAG(s(T54), s(s(T57)), s(T56)) → AVERAGE1_IN_GAG(s(T54), T57, T56)
AVERAGE1_IN_GAG(s(T67), s(s(s(T70))), s(T69)) → U4_GAG(T67, T70, T69, average1_in_gag(s(s(T67)), T70, T69))
AVERAGE1_IN_GAG(s(T67), s(s(s(T70))), s(T69)) → AVERAGE1_IN_GAG(s(s(T67)), T70, T69)
AVERAGE1_IN_GAG(T95, s(s(s(T98))), s(T97)) → U5_GAG(T95, T98, T97, average1_in_gag(T95, s(T98), T97))
AVERAGE1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVERAGE1_IN_GAG(T95, s(T98), T97)
AVERAGE1_IN_GAG(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → U6_GAG(T105, T108, T107, average1_in_gag(s(s(T105)), T108, T107))
AVERAGE1_IN_GAG(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → AVERAGE1_IN_GAG(s(s(T105)), T108, T107)

The TRS R consists of the following rules:

average1_in_gag(0, 0, 0) → average1_out_gag(0, 0, 0)
average1_in_gag(0, s(0), 0) → average1_out_gag(0, s(0), 0)
average1_in_gag(0, s(s(0)), s(0)) → average1_out_gag(0, s(s(0)), s(0))
average1_in_gag(0, s(s(s(T17))), s(0)) → U1_gag(T17, average1_in_gag(0, s(T17), 0))
average1_in_gag(s(0), 0, 0) → average1_out_gag(s(0), 0, 0)
average1_in_gag(s(0), s(0), s(0)) → average1_out_gag(s(0), s(0), s(0))
average1_in_gag(s(s(T38)), T41, T40) → U2_gag(T38, T41, T40, average1_in_gag(T38, s(s(T41)), T40))
average1_in_gag(s(T54), s(s(T57)), s(T56)) → U3_gag(T54, T57, T56, average1_in_gag(s(T54), T57, T56))
average1_in_gag(s(T67), s(s(s(T70))), s(T69)) → U4_gag(T67, T70, T69, average1_in_gag(s(s(T67)), T70, T69))
average1_in_gag(T95, s(s(s(T98))), s(T97)) → U5_gag(T95, T98, T97, average1_in_gag(T95, s(T98), T97))
average1_in_gag(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → U6_gag(T105, T108, T107, average1_in_gag(s(s(T105)), T108, T107))
U6_gag(T105, T108, T107, average1_out_gag(s(s(T105)), T108, T107)) → average1_out_gag(T105, s(s(s(s(s(s(T108)))))), s(s(T107)))
U5_gag(T95, T98, T97, average1_out_gag(T95, s(T98), T97)) → average1_out_gag(T95, s(s(s(T98))), s(T97))
U4_gag(T67, T70, T69, average1_out_gag(s(s(T67)), T70, T69)) → average1_out_gag(s(T67), s(s(s(T70))), s(T69))
U3_gag(T54, T57, T56, average1_out_gag(s(T54), T57, T56)) → average1_out_gag(s(T54), s(s(T57)), s(T56))
U2_gag(T38, T41, T40, average1_out_gag(T38, s(s(T41)), T40)) → average1_out_gag(s(s(T38)), T41, T40)
U1_gag(T17, average1_out_gag(0, s(T17), 0)) → average1_out_gag(0, s(s(s(T17))), s(0))

The argument filtering Pi contains the following mapping:
average1_in_gag(x1, x2, x3) = average1_in_gag(x1, x3)
0 = 0
average1_out_gag(x1, x2, x3) = average1_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)
AVERAGE1_IN_GAG(x1, x2, x3) = AVERAGE1_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:

AVERAGE1_IN_GAG(0, s(s(s(T17))), s(0)) → U1_GAG(T17, average1_in_gag(0, s(T17), 0))
AVERAGE1_IN_GAG(0, s(s(s(T17))), s(0)) → AVERAGE1_IN_GAG(0, s(T17), 0)
AVERAGE1_IN_GAG(s(s(T38)), T41, T40) → U2_GAG(T38, T41, T40, average1_in_gag(T38, s(s(T41)), T40))
AVERAGE1_IN_GAG(s(s(T38)), T41, T40) → AVERAGE1_IN_GAG(T38, s(s(T41)), T40)
AVERAGE1_IN_GAG(s(T54), s(s(T57)), s(T56)) → U3_GAG(T54, T57, T56, average1_in_gag(s(T54), T57, T56))
AVERAGE1_IN_GAG(s(T54), s(s(T57)), s(T56)) → AVERAGE1_IN_GAG(s(T54), T57, T56)
AVERAGE1_IN_GAG(s(T67), s(s(s(T70))), s(T69)) → U4_GAG(T67, T70, T69, average1_in_gag(s(s(T67)), T70, T69))
AVERAGE1_IN_GAG(s(T67), s(s(s(T70))), s(T69)) → AVERAGE1_IN_GAG(s(s(T67)), T70, T69)
AVERAGE1_IN_GAG(T95, s(s(s(T98))), s(T97)) → U5_GAG(T95, T98, T97, average1_in_gag(T95, s(T98), T97))
AVERAGE1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVERAGE1_IN_GAG(T95, s(T98), T97)
AVERAGE1_IN_GAG(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → U6_GAG(T105, T108, T107, average1_in_gag(s(s(T105)), T108, T107))
AVERAGE1_IN_GAG(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → AVERAGE1_IN_GAG(s(s(T105)), T108, T107)

The TRS R consists of the following rules:

average1_in_gag(0, 0, 0) → average1_out_gag(0, 0, 0)
average1_in_gag(0, s(0), 0) → average1_out_gag(0, s(0), 0)
average1_in_gag(0, s(s(0)), s(0)) → average1_out_gag(0, s(s(0)), s(0))
average1_in_gag(0, s(s(s(T17))), s(0)) → U1_gag(T17, average1_in_gag(0, s(T17), 0))
average1_in_gag(s(0), 0, 0) → average1_out_gag(s(0), 0, 0)
average1_in_gag(s(0), s(0), s(0)) → average1_out_gag(s(0), s(0), s(0))
average1_in_gag(s(s(T38)), T41, T40) → U2_gag(T38, T41, T40, average1_in_gag(T38, s(s(T41)), T40))
average1_in_gag(s(T54), s(s(T57)), s(T56)) → U3_gag(T54, T57, T56, average1_in_gag(s(T54), T57, T56))
average1_in_gag(s(T67), s(s(s(T70))), s(T69)) → U4_gag(T67, T70, T69, average1_in_gag(s(s(T67)), T70, T69))
average1_in_gag(T95, s(s(s(T98))), s(T97)) → U5_gag(T95, T98, T97, average1_in_gag(T95, s(T98), T97))
average1_in_gag(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → U6_gag(T105, T108, T107, average1_in_gag(s(s(T105)), T108, T107))
U6_gag(T105, T108, T107, average1_out_gag(s(s(T105)), T108, T107)) → average1_out_gag(T105, s(s(s(s(s(s(T108)))))), s(s(T107)))
U5_gag(T95, T98, T97, average1_out_gag(T95, s(T98), T97)) → average1_out_gag(T95, s(s(s(T98))), s(T97))
U4_gag(T67, T70, T69, average1_out_gag(s(s(T67)), T70, T69)) → average1_out_gag(s(T67), s(s(s(T70))), s(T69))
U3_gag(T54, T57, T56, average1_out_gag(s(T54), T57, T56)) → average1_out_gag(s(T54), s(s(T57)), s(T56))
U2_gag(T38, T41, T40, average1_out_gag(T38, s(s(T41)), T40)) → average1_out_gag(s(s(T38)), T41, T40)
U1_gag(T17, average1_out_gag(0, s(T17), 0)) → average1_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:

AVERAGE1_IN_GAG(s(T54), s(s(T57)), s(T56)) → AVERAGE1_IN_GAG(s(T54), T57, T56)
AVERAGE1_IN_GAG(s(s(T38)), T41, T40) → AVERAGE1_IN_GAG(T38, s(s(T41)), T40)
AVERAGE1_IN_GAG(s(T67), s(s(s(T70))), s(T69)) → AVERAGE1_IN_GAG(s(s(T67)), T70, T69)
AVERAGE1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVERAGE1_IN_GAG(T95, s(T98), T97)
AVERAGE1_IN_GAG(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → AVERAGE1_IN_GAG(s(s(T105)), T108, T107)

The TRS R consists of the following rules:

average1_in_gag(0, 0, 0) → average1_out_gag(0, 0, 0)
average1_in_gag(0, s(0), 0) → average1_out_gag(0, s(0), 0)
average1_in_gag(0, s(s(0)), s(0)) → average1_out_gag(0, s(s(0)), s(0))
average1_in_gag(0, s(s(s(T17))), s(0)) → U1_gag(T17, average1_in_gag(0, s(T17), 0))
average1_in_gag(s(0), 0, 0) → average1_out_gag(s(0), 0, 0)
average1_in_gag(s(0), s(0), s(0)) → average1_out_gag(s(0), s(0), s(0))
average1_in_gag(s(s(T38)), T41, T40) → U2_gag(T38, T41, T40, average1_in_gag(T38, s(s(T41)), T40))
average1_in_gag(s(T54), s(s(T57)), s(T56)) → U3_gag(T54, T57, T56, average1_in_gag(s(T54), T57, T56))
average1_in_gag(s(T67), s(s(s(T70))), s(T69)) → U4_gag(T67, T70, T69, average1_in_gag(s(s(T67)), T70, T69))
average1_in_gag(T95, s(s(s(T98))), s(T97)) → U5_gag(T95, T98, T97, average1_in_gag(T95, s(T98), T97))
average1_in_gag(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → U6_gag(T105, T108, T107, average1_in_gag(s(s(T105)), T108, T107))
U6_gag(T105, T108, T107, average1_out_gag(s(s(T105)), T108, T107)) → average1_out_gag(T105, s(s(s(s(s(s(T108)))))), s(s(T107)))
U5_gag(T95, T98, T97, average1_out_gag(T95, s(T98), T97)) → average1_out_gag(T95, s(s(s(T98))), s(T97))
U4_gag(T67, T70, T69, average1_out_gag(s(s(T67)), T70, T69)) → average1_out_gag(s(T67), s(s(s(T70))), s(T69))
U3_gag(T54, T57, T56, average1_out_gag(s(T54), T57, T56)) → average1_out_gag(s(T54), s(s(T57)), s(T56))
U2_gag(T38, T41, T40, average1_out_gag(T38, s(s(T41)), T40)) → average1_out_gag(s(s(T38)), T41, T40)
U1_gag(T17, average1_out_gag(0, s(T17), 0)) → average1_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:

AVERAGE1_IN_GAG(s(T54), s(s(T57)), s(T56)) → AVERAGE1_IN_GAG(s(T54), T57, T56)
AVERAGE1_IN_GAG(s(s(T38)), T41, T40) → AVERAGE1_IN_GAG(T38, s(s(T41)), T40)
AVERAGE1_IN_GAG(s(T67), s(s(s(T70))), s(T69)) → AVERAGE1_IN_GAG(s(s(T67)), T70, T69)
AVERAGE1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVERAGE1_IN_GAG(T95, s(T98), T97)
AVERAGE1_IN_GAG(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → AVERAGE1_IN_GAG(s(s(T105)), T108, T107)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
AVERAGE1_IN_GAG(x1, x2, x3) = AVERAGE1_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:

AVERAGE1_IN_GAG(s(T54), s(T56)) → AVERAGE1_IN_GAG(s(T54), T56)
AVERAGE1_IN_GAG(s(s(T38)), T40) → AVERAGE1_IN_GAG(T38, T40)
AVERAGE1_IN_GAG(s(T67), s(T69)) → AVERAGE1_IN_GAG(s(s(T67)), T69)
AVERAGE1_IN_GAG(T95, s(T97)) → AVERAGE1_IN_GAG(T95, T97)
AVERAGE1_IN_GAG(T105, s(s(T107))) → AVERAGE1_IN_GAG(s(s(T105)), T107)

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:

AVERAGE1_IN_GAG(s(T54), s(T56)) → AVERAGE1_IN_GAG(s(T54), T56)
The graph contains the following edges 1 >= 1, 2 > 2
AVERAGE1_IN_GAG(s(s(T38)), T40) → AVERAGE1_IN_GAG(T38, T40)
The graph contains the following edges 1 > 1, 2 >= 2
AVERAGE1_IN_GAG(s(T67), s(T69)) → AVERAGE1_IN_GAG(s(s(T67)), T69)
The graph contains the following edges 2 > 2
AVERAGE1_IN_GAG(T95, s(T97)) → AVERAGE1_IN_GAG(T95, T97)
The graph contains the following edges 1 >= 1, 2 > 2
AVERAGE1_IN_GAG(T105, s(s(T107))) → AVERAGE1_IN_GAG(s(s(T105)), T107)
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