Left Termination proof of /tmp/tmpeqzcww/avg.pl

Left Termination of the query pattern avg_in_3(g, g, a) 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 MRRProof (⇔)

↳14 QDP

↳15 QDPSizeChangeProof (⇔)

↳16 YES

(0) Obligation:

Clauses:

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

Queries:

avg(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: avg(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: avg(T7, s(T8), T10) | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\navg(T1, T2, T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 0 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 1 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 2 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 0\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 1 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 2 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: avg(T23, s(s(T24)), T26)\n\nT23 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 2 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: avg(s(T42), T43, T45)\n\nT42 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: avg(T7, s(T8), T10), SCOPE: 2, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: avg(s(s(T61)), T62, T64)\n\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: avg(s(T7), T8, T3), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: avg(s(T71), T72, T74) | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\navg(T1, T2, T3) !=? avg(s(X4), X5, X6)\navg(T1, T2, T3) !=? avg(X76, s(s(s(X77))), s(X78))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 0 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 1 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 2 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 0\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 1 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 2 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: avg(T95, s(T96), T98)\n\nT95 is ground\nT96 is ground\navg(T95, s(s(s(T96))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 1\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 2 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: avg(s(s(T114)), T115, T117)\n\nT114 is ground\nT115 is ground\navg(T114, s(s(s(s(s(s(T115)))))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: avg(s(T71), T72, T74), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ?_3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 2 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 3 | avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: avg(T71, s(s(s(T72))), T3), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT72 is ground\navg(T71, s(s(s(T72))), T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true | avg(0, 0, T3), SCOPE: 1, CLAUSE: 3 | avg(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\navg(T1, T2, T3) !=? avg(s(X4), X5, X6)\navg(T1, T2, T3) !=? avg(X76, s(s(s(X77))), s(X78))\navg(T1, T2, T3) !=? avg(0, 0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: avg(0, 0, T3), SCOPE: 1, CLAUSE: 3 | avg(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: avg(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true | avg(0, s(0), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\navg(T1, T2, T3) !=? avg(s(X4), X5, X6)\navg(T1, T2, T3) !=? avg(X76, s(s(s(X77))), s(X78))\navg(T1, T2, T3) !=? avg(0, 0, 0)\navg(T1, T2, T3) !=? avg(0, s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: avg(0, s(0), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 62 [arrowhead = none ];
62 -> 2;

63 [label="EVAL with clause\navg(s(X4), X5, X6) :- avg(X4, s(X5), X6).\nand substitution\nX4 -> T7\nT1 -> s(T7)\nT2 -> T8\nX5 -> T8\nT3 -> T10\nX6 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 63 [arrowhead = none ];
63 -> 3;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 64 [arrowhead = none ];
64 -> 4;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 65 [arrowhead = none ];
65 -> 5;

66 [label="EVAL with clause\navg(X76, s(s(s(X77))), s(X78)) :- avg(s(X76), X77, X78).\nand substitution\nT1 -> T71\nX76 -> T71\nX77 -> T72\nT2 -> s(s(s(T72)))\nX78 -> T74\nT3 -> s(T74)\nT73 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 66 [arrowhead = none ];
66 -> 33;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 67 [arrowhead = none ];
67 -> 34;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 68 [arrowhead = none ];
68 -> 6;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 69 [arrowhead = none ];
69 -> 7;

70 [label="EVAL with clause\navg(s(X22), X23, X24) :- avg(X22, s(X23), X24).\nand substitution\nX22 -> T23\nT7 -> s(T23)\nT8 -> T24\nX23 -> s(T24)\nT10 -> T26\nX24 -> T26\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 70 [arrowhead = none ];
70 -> 8;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 71 [arrowhead = none ];
71 -> 9;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 72 [arrowhead = none ];
72 -> 10;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 73 [arrowhead = none ];
73 -> 11;

74 [label="INSTANCE with matching:\nT1 -> T23\nT2 -> s(s(T24))\nT3 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
8 -> 74 [arrowhead = none , style=dashed];
74 -> 1[style=dashed];

75 [label="EVAL with clause\navg(X43, s(s(s(X44))), s(X45)) :- avg(s(X43), X44, X45).\nand substitution\nT7 -> T42\nX43 -> T42\nX44 -> T43\nT8 -> s(s(T43))\nX45 -> T45\nT10 -> s(T45)\nT44 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 75 [arrowhead = none ];
75 -> 12;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 76 [arrowhead = none ];
76 -> 13;

77 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
11 -> 77 [arrowhead = none ];
77 -> 14;

78 [label="INSTANCE with matching:\nT1 -> s(T42)\nT2 -> T43\nT3 -> T45", fontsize=14, style = filled, fillcolor = lightgrey];
12 -> 78 [arrowhead = none , style=dashed];
78 -> 1[style=dashed];

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 79 [arrowhead = none ];
79 -> 15;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 80 [arrowhead = none ];
80 -> 16;

81 [label="EVAL with clause\navg(0, s(0), 0).\nand substitution\nT7 -> 0\nT8 -> 0\nT10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 81 [arrowhead = none ];
81 -> 17;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 82 [arrowhead = none ];
82 -> 18;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 83 [arrowhead = none ];
83 -> 20;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 84 [arrowhead = none ];
84 -> 21;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 85 [arrowhead = none ];
85 -> 19;

86 [label="EVAL with clause\navg(0, s(s(0)), s(0)).\nand substitution\nT7 -> 0\nT8 -> s(0)\nT10 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 86 [arrowhead = none ];
86 -> 22;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 87 [arrowhead = none ];
87 -> 23;

88 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 88 [arrowhead = none ];
88 -> 25;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 89 [arrowhead = none ];
89 -> 24;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 90 [arrowhead = none ];
90 -> 26;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 91 [arrowhead = none ];
91 -> 27;

92 [label="EVAL with clause\navg(X64, s(s(s(X65))), s(X66)) :- avg(s(X64), X65, X66).\nand substitution\nT7 -> T61\nX64 -> s(T61)\nX65 -> T62\nT8 -> s(s(s(T62)))\nX66 -> T64\nT3 -> s(T64)\nT63 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 92 [arrowhead = none ];
92 -> 28;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 93 [arrowhead = none ];
93 -> 29;

94 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 94 [arrowhead = none ];
94 -> 30;

95 [label="INSTANCE with matching:\nT1 -> s(s(T61))\nT2 -> T62\nT3 -> T64", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 95 [arrowhead = none , style=dashed];
95 -> 1[style=dashed];

96 [label="BACKTRACK\nfor clause: avg(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
30 -> 96 [arrowhead = none ];
96 -> 31;

97 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
31 -> 97 [arrowhead = none ];
97 -> 32;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 98 [arrowhead = none ];
98 -> 35;

99 [label="EVAL with clause\navg(0, 0, 0).\nand substitution\nT1 -> 0\nT2 -> 0\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 99 [arrowhead = none ];
99 -> 50;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 100 [arrowhead = none ];
100 -> 51;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 101 [arrowhead = none ];
101 -> 36;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 102 [arrowhead = none ];
102 -> 37;

103 [label="ONLY EVAL with clause\navg(s(X100), X101, X102) :- avg(X100, s(X101), X102).\nand substitution\nT71 -> T95\nX100 -> T95\nT72 -> T96\nX101 -> T96\nT74 -> T98\nX102 -> T98\nT97 -> T98", fontsize=14, style = filled, fillcolor = white];
36 -> 103 [arrowhead = none ];
103 -> 38;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 104 [arrowhead = none ];
104 -> 39;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 105 [arrowhead = none ];
105 -> 40;

106 [label="INSTANCE with matching:\nT1 -> T95\nT2 -> s(T96)\nT3 -> T98", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 106 [arrowhead = none , style=dashed];
106 -> 1[style=dashed];

107 [label="EVAL with clause\navg(X118, s(s(s(X119))), s(X120)) :- avg(s(X118), X119, X120).\nand substitution\nT71 -> T114\nX118 -> s(T114)\nX119 -> T115\nT72 -> s(s(s(T115)))\nX120 -> T117\nT74 -> s(T117)\nT116 -> T117", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 107 [arrowhead = none ];
107 -> 41;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 108 [arrowhead = none ];
108 -> 42;

109 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
40 -> 109 [arrowhead = none ];
109 -> 43;

110 [label="INSTANCE with matching:\nT1 -> s(s(T114))\nT2 -> T115\nT3 -> T117", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 110 [arrowhead = none , style=dashed];
110 -> 1[style=dashed];

111 [label="BACKTRACK\nfor clause: avg(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
43 -> 111 [arrowhead = none ];
111 -> 44;

112 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
44 -> 112 [arrowhead = none ];
112 -> 45;

113 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 113 [arrowhead = none ];
113 -> 46;

114 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
46 -> 114 [arrowhead = none ];
114 -> 47;

115 [label="BACKTRACK\nfor clause: avg(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
47 -> 115 [arrowhead = none ];
115 -> 48;

116 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
48 -> 116 [arrowhead = none ];
116 -> 49;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 117 [arrowhead = none ];
117 -> 52;

118 [label="EVAL with clause\navg(0, s(0), 0).\nand substitution\nT1 -> 0\nT2 -> s(0)\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 118 [arrowhead = none ];
118 -> 55;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 119 [arrowhead = none ];
119 -> 56;

120 [label="BACKTRACK\nfor clause: avg(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
52 -> 120 [arrowhead = none ];
120 -> 53;

121 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
53 -> 121 [arrowhead = none ];
121 -> 54;

122 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 122 [arrowhead = none ];
122 -> 57;

123 [label="EVAL with clause\navg(0, s(s(0)), s(0)).\nand substitution\nT1 -> 0\nT2 -> s(s(0))\nT3 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 123 [arrowhead = none ];
123 -> 59;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 124 [arrowhead = none ];
124 -> 60;

125 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
57 -> 125 [arrowhead = none ];
125 -> 58;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 126 [arrowhead = none ];
126 -> 61;

}

(2) Obligation:

Clauses:

avg1(s(s(T23)), T24, T26) :- avg1(T23, s(s(T24)), T26).
avg1(s(T42), s(s(T43)), s(T45)) :- avg1(s(T42), T43, T45).
avg1(s(0), 0, 0).
avg1(s(0), s(0), s(0)).
avg1(s(T61), s(s(s(T62))), s(T64)) :- avg1(s(s(T61)), T62, T64).
avg1(T95, s(s(s(T96))), s(T98)) :- avg1(T95, s(T96), T98).
avg1(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) :- avg1(s(s(T114)), T115, T117).
avg1(0, 0, 0).
avg1(0, s(0), 0).
avg1(0, s(s(0)), s(0)).

Queries:

avg1(g,g,a).

(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:
avg1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

avg1_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
avg1_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
avg1_in_gga(s(0), 0, 0) → avg1_out_gga(s(0), 0, 0)
avg1_in_gga(s(0), s(0), s(0)) → avg1_out_gga(s(0), s(0), s(0))
avg1_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
avg1_in_gga(T95, s(s(s(T96))), s(T98)) → U4_gga(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
avg1_in_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_gga(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
avg1_in_gga(0, 0, 0) → avg1_out_gga(0, 0, 0)
avg1_in_gga(0, s(0), 0) → avg1_out_gga(0, s(0), 0)
avg1_in_gga(0, s(s(0)), s(0)) → avg1_out_gga(0, s(s(0)), s(0))
U5_gga(T114, T115, T117, avg1_out_gga(s(s(T114)), T115, T117)) → avg1_out_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117)))
U4_gga(T95, T96, T98, avg1_out_gga(T95, s(T96), T98)) → avg1_out_gga(T95, s(s(s(T96))), s(T98))
U3_gga(T61, T62, T64, avg1_out_gga(s(s(T61)), T62, T64)) → avg1_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avg1_out_gga(s(T42), T43, T45)) → avg1_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avg1_out_gga(T23, s(s(T24)), T26)) → avg1_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:

avg1_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
avg1_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
avg1_in_gga(s(0), 0, 0) → avg1_out_gga(s(0), 0, 0)
avg1_in_gga(s(0), s(0), s(0)) → avg1_out_gga(s(0), s(0), s(0))
avg1_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
avg1_in_gga(T95, s(s(s(T96))), s(T98)) → U4_gga(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
avg1_in_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_gga(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
avg1_in_gga(0, 0, 0) → avg1_out_gga(0, 0, 0)
avg1_in_gga(0, s(0), 0) → avg1_out_gga(0, s(0), 0)
avg1_in_gga(0, s(s(0)), s(0)) → avg1_out_gga(0, s(s(0)), s(0))
U5_gga(T114, T115, T117, avg1_out_gga(s(s(T114)), T115, T117)) → avg1_out_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117)))
U4_gga(T95, T96, T98, avg1_out_gga(T95, s(T96), T98)) → avg1_out_gga(T95, s(s(s(T96))), s(T98))
U3_gga(T61, T62, T64, avg1_out_gga(s(s(T61)), T62, T64)) → avg1_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avg1_out_gga(s(T42), T43, T45)) → avg1_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avg1_out_gga(T23, s(s(T24)), T26)) → avg1_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:

AVG1_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
AVG1_IN_GGA(s(s(T23)), T24, T26) → AVG1_IN_GGA(T23, s(s(T24)), T26)
AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → U2_GGA(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVG1_IN_GGA(s(T42), T43, T45)
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → U3_GGA(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVG1_IN_GGA(s(s(T61)), T62, T64)
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → U4_GGA(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → AVG1_IN_GGA(T95, s(T96), T98)
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_GGA(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → AVG1_IN_GGA(s(s(T114)), T115, T117)

The TRS R consists of the following rules:

avg1_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
avg1_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
avg1_in_gga(s(0), 0, 0) → avg1_out_gga(s(0), 0, 0)
avg1_in_gga(s(0), s(0), s(0)) → avg1_out_gga(s(0), s(0), s(0))
avg1_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
avg1_in_gga(T95, s(s(s(T96))), s(T98)) → U4_gga(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
avg1_in_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_gga(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
avg1_in_gga(0, 0, 0) → avg1_out_gga(0, 0, 0)
avg1_in_gga(0, s(0), 0) → avg1_out_gga(0, s(0), 0)
avg1_in_gga(0, s(s(0)), s(0)) → avg1_out_gga(0, s(s(0)), s(0))
U5_gga(T114, T115, T117, avg1_out_gga(s(s(T114)), T115, T117)) → avg1_out_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117)))
U4_gga(T95, T96, T98, avg1_out_gga(T95, s(T96), T98)) → avg1_out_gga(T95, s(s(s(T96))), s(T98))
U3_gga(T61, T62, T64, avg1_out_gga(s(s(T61)), T62, T64)) → avg1_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avg1_out_gga(s(T42), T43, T45)) → avg1_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avg1_out_gga(T23, s(s(T24)), T26)) → avg1_out_gga(s(s(T23)), T24, T26)

The argument filtering Pi contains the following mapping:
avg1_in_gga(x1, x2, x3) = avg1_in_gga(x1, x2)
s(x1) = s(x1)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
0 = 0
avg1_out_gga(x1, x2, x3) = avg1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4)
AVG1_IN_GGA(x1, x2, x3) = AVG1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)

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

(6) Obligation:

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

AVG1_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
AVG1_IN_GGA(s(s(T23)), T24, T26) → AVG1_IN_GGA(T23, s(s(T24)), T26)
AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → U2_GGA(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVG1_IN_GGA(s(T42), T43, T45)
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → U3_GGA(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVG1_IN_GGA(s(s(T61)), T62, T64)
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → U4_GGA(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → AVG1_IN_GGA(T95, s(T96), T98)
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_GGA(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → AVG1_IN_GGA(s(s(T114)), T115, T117)

The TRS R consists of the following rules:

avg1_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
avg1_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
avg1_in_gga(s(0), 0, 0) → avg1_out_gga(s(0), 0, 0)
avg1_in_gga(s(0), s(0), s(0)) → avg1_out_gga(s(0), s(0), s(0))
avg1_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
avg1_in_gga(T95, s(s(s(T96))), s(T98)) → U4_gga(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
avg1_in_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_gga(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
avg1_in_gga(0, 0, 0) → avg1_out_gga(0, 0, 0)
avg1_in_gga(0, s(0), 0) → avg1_out_gga(0, s(0), 0)
avg1_in_gga(0, s(s(0)), s(0)) → avg1_out_gga(0, s(s(0)), s(0))
U5_gga(T114, T115, T117, avg1_out_gga(s(s(T114)), T115, T117)) → avg1_out_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117)))
U4_gga(T95, T96, T98, avg1_out_gga(T95, s(T96), T98)) → avg1_out_gga(T95, s(s(s(T96))), s(T98))
U3_gga(T61, T62, T64, avg1_out_gga(s(s(T61)), T62, T64)) → avg1_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avg1_out_gga(s(T42), T43, T45)) → avg1_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avg1_out_gga(T23, s(s(T24)), T26)) → avg1_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:

AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVG1_IN_GGA(s(T42), T43, T45)
AVG1_IN_GGA(s(s(T23)), T24, T26) → AVG1_IN_GGA(T23, s(s(T24)), T26)
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVG1_IN_GGA(s(s(T61)), T62, T64)
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → AVG1_IN_GGA(T95, s(T96), T98)
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → AVG1_IN_GGA(s(s(T114)), T115, T117)

The TRS R consists of the following rules:

avg1_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
avg1_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
avg1_in_gga(s(0), 0, 0) → avg1_out_gga(s(0), 0, 0)
avg1_in_gga(s(0), s(0), s(0)) → avg1_out_gga(s(0), s(0), s(0))
avg1_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
avg1_in_gga(T95, s(s(s(T96))), s(T98)) → U4_gga(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
avg1_in_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_gga(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
avg1_in_gga(0, 0, 0) → avg1_out_gga(0, 0, 0)
avg1_in_gga(0, s(0), 0) → avg1_out_gga(0, s(0), 0)
avg1_in_gga(0, s(s(0)), s(0)) → avg1_out_gga(0, s(s(0)), s(0))
U5_gga(T114, T115, T117, avg1_out_gga(s(s(T114)), T115, T117)) → avg1_out_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117)))
U4_gga(T95, T96, T98, avg1_out_gga(T95, s(T96), T98)) → avg1_out_gga(T95, s(s(s(T96))), s(T98))
U3_gga(T61, T62, T64, avg1_out_gga(s(s(T61)), T62, T64)) → avg1_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avg1_out_gga(s(T42), T43, T45)) → avg1_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avg1_out_gga(T23, s(s(T24)), T26)) → avg1_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:

AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVG1_IN_GGA(s(T42), T43, T45)
AVG1_IN_GGA(s(s(T23)), T24, T26) → AVG1_IN_GGA(T23, s(s(T24)), T26)
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVG1_IN_GGA(s(s(T61)), T62, T64)
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → AVG1_IN_GGA(T95, s(T96), T98)
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → AVG1_IN_GGA(s(s(T114)), T115, T117)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
AVG1_IN_GGA(x1, x2, x3) = AVG1_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:

AVG1_IN_GGA(s(T42), s(s(T43))) → AVG1_IN_GGA(s(T42), T43)
AVG1_IN_GGA(s(s(T23)), T24) → AVG1_IN_GGA(T23, s(s(T24)))
AVG1_IN_GGA(s(T61), s(s(s(T62)))) → AVG1_IN_GGA(s(s(T61)), T62)
AVG1_IN_GGA(T95, s(s(s(T96)))) → AVG1_IN_GGA(T95, s(T96))
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115))))))) → AVG1_IN_GGA(s(s(T114)), T115)

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:

AVG1_IN_GGA(s(T42), s(s(T43))) → AVG1_IN_GGA(s(T42), T43)
AVG1_IN_GGA(s(T61), s(s(s(T62)))) → AVG1_IN_GGA(s(s(T61)), T62)
AVG1_IN_GGA(T95, s(s(s(T96)))) → AVG1_IN_GGA(T95, s(T96))
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115))))))) → AVG1_IN_GGA(s(s(T114)), T115)

Used ordering: Polynomial interpretation [POLO]:

POL(AVG1_IN_GGA(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 1 + x₁

(14) Obligation:

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

AVG1_IN_GGA(s(s(T23)), T24) → AVG1_IN_GGA(T23, s(s(T24)))

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

(15) 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:

AVG1_IN_GGA(s(s(T23)), T24) → AVG1_IN_GGA(T23, s(s(T24)))
The graph contains the following edges 1 > 1

(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) QDPSizeChangeProof (EQUIVALENT transformation)

(16) YES