Left Termination proof of /tmp/tmpO2enP8/avg-bfb.pl

Left Termination of the query pattern avg_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:

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,a,g).

(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\nT3 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\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: avg(T7, s(T10), T9) | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT9 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\nT3 is ground\navg(T1, T2, T3) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 0 | avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 1 | avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 2 | avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 0\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 1 | avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 2 | avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: avg(T23, s(s(T26)), T25)\n\nT23 is ground\nT25 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(T10), T9), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 2 | avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: avg(s(T42), T45, T44)\n\nT42 is ground\nT44 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(T10), T9), SCOPE: 2, CLAUSE: 3 | avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: avg(T7, s(T10), T9), SCOPE: 2, CLAUSE: 4 | ?_2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT9 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(T10), T9), SCOPE: 2, CLAUSE: 4\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT9 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), T2, T9), SCOPE: 1, CLAUSE: 1 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 2 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: avg(s(s(T61)), T64, T63)\n\nT61 is ground\nT63 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), T2, T9), SCOPE: 1, CLAUSE: 3 | avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: avg(s(T7), T2, T9), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT9 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), T74, T73) | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 2 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? 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\nT3 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), T74, T73), SCOPE: 3, CLAUSE: 0 | avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 1 | avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 2 | avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 2 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 0\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 1 | avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 2 | avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 2 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: avg(T95, s(T98), T97)\n\nT95 is ground\nT97 is ground\navg(T95, T2, s(T97)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 1\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 2 | avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 2 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: avg(s(s(T114)), T117, T116)\n\nT114 is ground\nT116 is ground\navg(T114, T2, s(s(T116))) !=? 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), T74, T73), SCOPE: 3, CLAUSE: 3 | avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 2 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: avg(s(T71), T74, T73), SCOPE: 3, CLAUSE: 4 | ?_3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 2 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ?_3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 2 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 2 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 3 | avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: avg(T71, T2, s(T73)), SCOPE: 1, CLAUSE: 4\n\nT71 is ground\nT73 is ground\navg(T71, T2, s(T73)) !=? avg(s(X4), X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: true | avg(0, T2, 0), SCOPE: 1, CLAUSE: 3 | avg(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT3 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];
54 [label="54: avg(0, T2, 0), SCOPE: 1, CLAUSE: 3 | avg(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true | avg(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: avg(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\navg(0, T2, 0) !=? avg(0, s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: avg(0, T2, 0), 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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: true | avg(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\navg(0, T2, 0) !=? avg(0, 0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: avg(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT3 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];
62 [label="62: avg(0, T2, 0), SCOPE: 1, CLAUSE: 4\n\navg(0, T2, 0) !=? avg(0, 0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 67 [arrowhead = none ];
67 -> 2;

68 [label="EVAL with clause\navg(s(X4), X5, X6) :- avg(X4, s(X5), X6).\nand substitution\nX4 -> T7\nT1 -> s(T7)\nT2 -> T10\nX5 -> T10\nT3 -> T9\nX6 -> T9\nT8 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 68 [arrowhead = none ];
68 -> 3;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 69 [arrowhead = none ];
69 -> 4;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 70 [arrowhead = none ];
70 -> 5;

71 [label="EVAL with clause\navg(X76, s(s(s(X77))), s(X78)) :- avg(s(X76), X77, X78).\nand substitution\nT1 -> T71\nX76 -> T71\nX77 -> T74\nT2 -> s(s(s(T74)))\nX78 -> T73\nT3 -> s(T73)\nT72 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 71 [arrowhead = none ];
71 -> 33;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 72 [arrowhead = none ];
72 -> 34;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 73 [arrowhead = none ];
73 -> 6;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 74 [arrowhead = none ];
74 -> 7;

75 [label="EVAL with clause\navg(s(X22), X23, X24) :- avg(X22, s(X23), X24).\nand substitution\nX22 -> T23\nT7 -> s(T23)\nT10 -> T26\nX23 -> s(T26)\nT9 -> T25\nX24 -> T25\nT24 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 75 [arrowhead = none ];
75 -> 8;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 76 [arrowhead = none ];
76 -> 9;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 77 [arrowhead = none ];
77 -> 10;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 78 [arrowhead = none ];
78 -> 11;

79 [label="INSTANCE with matching:\nT1 -> T23\nT2 -> s(s(T26))\nT3 -> T25", fontsize=14, style = filled, fillcolor = lightgrey];
8 -> 79 [arrowhead = none , style=dashed];
79 -> 1[style=dashed];

80 [label="EVAL with clause\navg(X43, s(s(s(X44))), s(X45)) :- avg(s(X43), X44, X45).\nand substitution\nT7 -> T42\nX43 -> T42\nX44 -> T45\nT10 -> s(s(T45))\nX45 -> T44\nT9 -> s(T44)\nT43 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 80 [arrowhead = none ];
80 -> 12;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 81 [arrowhead = none ];
81 -> 13;

82 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
11 -> 82 [arrowhead = none ];
82 -> 14;

83 [label="INSTANCE with matching:\nT1 -> s(T42)\nT2 -> T45\nT3 -> T44", fontsize=14, style = filled, fillcolor = lightgrey];
12 -> 83 [arrowhead = none , style=dashed];
83 -> 1[style=dashed];

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 84 [arrowhead = none ];
84 -> 15;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 85 [arrowhead = none ];
85 -> 16;

86 [label="EVAL with clause\navg(0, s(0), 0).\nand substitution\nT7 -> 0\nT10 -> 0\nT9 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 86 [arrowhead = none ];
86 -> 17;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 87 [arrowhead = none ];
87 -> 18;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 88 [arrowhead = none ];
88 -> 20;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 89 [arrowhead = none ];
89 -> 21;

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 90 [arrowhead = none ];
90 -> 19;

91 [label="EVAL with clause\navg(0, s(s(0)), s(0)).\nand substitution\nT7 -> 0\nT10 -> s(0)\nT9 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 91 [arrowhead = none ];
91 -> 22;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 92 [arrowhead = none ];
92 -> 23;

93 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 93 [arrowhead = none ];
93 -> 25;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 94 [arrowhead = none ];
94 -> 24;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 95 [arrowhead = none ];
95 -> 26;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 96 [arrowhead = none ];
96 -> 27;

97 [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 -> T64\nT2 -> s(s(s(T64)))\nX66 -> T63\nT9 -> s(T63)\nT62 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 97 [arrowhead = none ];
97 -> 28;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 98 [arrowhead = none ];
98 -> 29;

99 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 99 [arrowhead = none ];
99 -> 30;

100 [label="INSTANCE with matching:\nT1 -> s(s(T61))\nT2 -> T64\nT3 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 100 [arrowhead = none , style=dashed];
100 -> 1[style=dashed];

101 [label="BACKTRACK\nfor clause: avg(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
30 -> 101 [arrowhead = none ];
101 -> 31;

102 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
31 -> 102 [arrowhead = none ];
102 -> 32;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 103 [arrowhead = none ];
103 -> 35;

104 [label="EVAL with clause\navg(0, 0, 0).\nand substitution\nT1 -> 0\nT2 -> 0\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 104 [arrowhead = none ];
104 -> 52;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 105 [arrowhead = none ];
105 -> 53;

106 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 106 [arrowhead = none ];
106 -> 36;

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

108 [label="ONLY EVAL with clause\navg(s(X100), X101, X102) :- avg(X100, s(X101), X102).\nand substitution\nT71 -> T95\nX100 -> T95\nT74 -> T98\nX101 -> T98\nT73 -> T97\nX102 -> T97\nT96 -> T98", fontsize=14, style = filled, fillcolor = white];
36 -> 108 [arrowhead = none ];
108 -> 38;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 109 [arrowhead = none ];
109 -> 39;

110 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 110 [arrowhead = none ];
110 -> 40;

111 [label="INSTANCE with matching:\nT1 -> T95\nT2 -> s(T98)\nT3 -> T97", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 111 [arrowhead = none , style=dashed];
111 -> 1[style=dashed];

112 [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 -> T117\nT74 -> s(s(s(T117)))\nX120 -> T116\nT73 -> s(T116)\nT115 -> T117", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 112 [arrowhead = none ];
112 -> 41;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 113 [arrowhead = none ];
113 -> 42;

114 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
40 -> 114 [arrowhead = none ];
114 -> 43;

115 [label="INSTANCE with matching:\nT1 -> s(s(T114))\nT2 -> T117\nT3 -> T116", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 115 [arrowhead = none , style=dashed];
115 -> 1[style=dashed];

116 [label="BACKTRACK\nfor clause: avg(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
43 -> 116 [arrowhead = none ];
116 -> 44;

117 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
44 -> 117 [arrowhead = none ];
117 -> 45;

118 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 118 [arrowhead = none ];
118 -> 46;

119 [label="BACKTRACK\nfor clause: avg(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
46 -> 119 [arrowhead = none ];
119 -> 47;

120 [label="BACKTRACK\nfor clause: avg(0, s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
47 -> 120 [arrowhead = none ];
120 -> 48;

121 [label="EVAL with clause\navg(0, s(s(0)), s(0)).\nand substitution\nT71 -> 0\nT2 -> s(s(0))\nT73 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 121 [arrowhead = none ];
121 -> 49;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 122 [arrowhead = none ];
122 -> 50;

123 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 123 [arrowhead = none ];
123 -> 51;

124 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 124 [arrowhead = none ];
124 -> 54;

125 [label="EVAL with clause\navg(0, s(0), 0).\nand substitution\nT1 -> 0\nT2 -> s(0)\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 125 [arrowhead = none ];
125 -> 60;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 126 [arrowhead = none ];
126 -> 61;

127 [label="EVAL with clause\navg(0, s(0), 0).\nand substitution\nT2 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 127 [arrowhead = none ];
127 -> 55;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 128 [arrowhead = none ];
128 -> 56;

129 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 129 [arrowhead = none ];
129 -> 57;

130 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
56 -> 130 [arrowhead = none ];
130 -> 59;

131 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
57 -> 131 [arrowhead = none ];
131 -> 58;

132 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 132 [arrowhead = none ];
132 -> 62;

133 [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];
61 -> 133 [arrowhead = none ];
133 -> 64;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 134 [arrowhead = none ];
134 -> 65;

135 [label="BACKTRACK\nfor clause: avg(0, s(s(0)), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
62 -> 135 [arrowhead = none ];
135 -> 63;

136 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 136 [arrowhead = none ];
136 -> 66;

}

(2) Obligation:

Clauses:

avg1(s(s(T23)), T26, T25) :- avg1(T23, s(s(T26)), T25).
avg1(s(T42), s(s(T45)), s(T44)) :- avg1(s(T42), T45, T44).
avg1(s(0), 0, 0).
avg1(s(0), s(0), s(0)).
avg1(s(T61), s(s(s(T64))), s(T63)) :- avg1(s(s(T61)), T64, T63).
avg1(T95, s(s(s(T98))), s(T97)) :- avg1(T95, s(T98), T97).
avg1(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) :- avg1(s(s(T114)), T117, T116).
avg1(0, s(s(0)), s(0)).
avg1(0, 0, 0).
avg1(0, s(0), 0).
avg1(0, s(0), 0).
avg1(0, s(s(0)), s(0)).

Queries:

avg1(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:
avg1_in: (b,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

avg1_in_gag(s(s(T23)), T26, T25) → U1_gag(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
avg1_in_gag(s(T42), s(s(T45)), s(T44)) → U2_gag(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
avg1_in_gag(s(0), 0, 0) → avg1_out_gag(s(0), 0, 0)
avg1_in_gag(s(0), s(0), s(0)) → avg1_out_gag(s(0), s(0), s(0))
avg1_in_gag(s(T61), s(s(s(T64))), s(T63)) → U3_gag(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
avg1_in_gag(T95, s(s(s(T98))), s(T97)) → U4_gag(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
avg1_in_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_gag(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
avg1_in_gag(0, s(s(0)), s(0)) → avg1_out_gag(0, s(s(0)), s(0))
avg1_in_gag(0, 0, 0) → avg1_out_gag(0, 0, 0)
avg1_in_gag(0, s(0), 0) → avg1_out_gag(0, s(0), 0)
U5_gag(T114, T117, T116, avg1_out_gag(s(s(T114)), T117, T116)) → avg1_out_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116)))
U4_gag(T95, T98, T97, avg1_out_gag(T95, s(T98), T97)) → avg1_out_gag(T95, s(s(s(T98))), s(T97))
U3_gag(T61, T64, T63, avg1_out_gag(s(s(T61)), T64, T63)) → avg1_out_gag(s(T61), s(s(s(T64))), s(T63))
U2_gag(T42, T45, T44, avg1_out_gag(s(T42), T45, T44)) → avg1_out_gag(s(T42), s(s(T45)), s(T44))
U1_gag(T23, T26, T25, avg1_out_gag(T23, s(s(T26)), T25)) → avg1_out_gag(s(s(T23)), T26, T25)

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_gag(s(s(T23)), T26, T25) → U1_gag(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
avg1_in_gag(s(T42), s(s(T45)), s(T44)) → U2_gag(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
avg1_in_gag(s(0), 0, 0) → avg1_out_gag(s(0), 0, 0)
avg1_in_gag(s(0), s(0), s(0)) → avg1_out_gag(s(0), s(0), s(0))
avg1_in_gag(s(T61), s(s(s(T64))), s(T63)) → U3_gag(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
avg1_in_gag(T95, s(s(s(T98))), s(T97)) → U4_gag(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
avg1_in_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_gag(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
avg1_in_gag(0, s(s(0)), s(0)) → avg1_out_gag(0, s(s(0)), s(0))
avg1_in_gag(0, 0, 0) → avg1_out_gag(0, 0, 0)
avg1_in_gag(0, s(0), 0) → avg1_out_gag(0, s(0), 0)
U5_gag(T114, T117, T116, avg1_out_gag(s(s(T114)), T117, T116)) → avg1_out_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116)))
U4_gag(T95, T98, T97, avg1_out_gag(T95, s(T98), T97)) → avg1_out_gag(T95, s(s(s(T98))), s(T97))
U3_gag(T61, T64, T63, avg1_out_gag(s(s(T61)), T64, T63)) → avg1_out_gag(s(T61), s(s(s(T64))), s(T63))
U2_gag(T42, T45, T44, avg1_out_gag(s(T42), T45, T44)) → avg1_out_gag(s(T42), s(s(T45)), s(T44))
U1_gag(T23, T26, T25, avg1_out_gag(T23, s(s(T26)), T25)) → avg1_out_gag(s(s(T23)), T26, T25)

(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_GAG(s(s(T23)), T26, T25) → U1_GAG(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
AVG1_IN_GAG(s(s(T23)), T26, T25) → AVG1_IN_GAG(T23, s(s(T26)), T25)
AVG1_IN_GAG(s(T42), s(s(T45)), s(T44)) → U2_GAG(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
AVG1_IN_GAG(s(T42), s(s(T45)), s(T44)) → AVG1_IN_GAG(s(T42), T45, T44)
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → U3_GAG(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → AVG1_IN_GAG(s(s(T61)), T64, T63)
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → U4_GAG(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVG1_IN_GAG(T95, s(T98), T97)
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_GAG(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T117, T116)

The TRS R consists of the following rules:

avg1_in_gag(s(s(T23)), T26, T25) → U1_gag(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
avg1_in_gag(s(T42), s(s(T45)), s(T44)) → U2_gag(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
avg1_in_gag(s(0), 0, 0) → avg1_out_gag(s(0), 0, 0)
avg1_in_gag(s(0), s(0), s(0)) → avg1_out_gag(s(0), s(0), s(0))
avg1_in_gag(s(T61), s(s(s(T64))), s(T63)) → U3_gag(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
avg1_in_gag(T95, s(s(s(T98))), s(T97)) → U4_gag(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
avg1_in_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_gag(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
avg1_in_gag(0, s(s(0)), s(0)) → avg1_out_gag(0, s(s(0)), s(0))
avg1_in_gag(0, 0, 0) → avg1_out_gag(0, 0, 0)
avg1_in_gag(0, s(0), 0) → avg1_out_gag(0, s(0), 0)
U5_gag(T114, T117, T116, avg1_out_gag(s(s(T114)), T117, T116)) → avg1_out_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116)))
U4_gag(T95, T98, T97, avg1_out_gag(T95, s(T98), T97)) → avg1_out_gag(T95, s(s(s(T98))), s(T97))
U3_gag(T61, T64, T63, avg1_out_gag(s(s(T61)), T64, T63)) → avg1_out_gag(s(T61), s(s(s(T64))), s(T63))
U2_gag(T42, T45, T44, avg1_out_gag(s(T42), T45, T44)) → avg1_out_gag(s(T42), s(s(T45)), s(T44))
U1_gag(T23, T26, T25, avg1_out_gag(T23, s(s(T26)), T25)) → avg1_out_gag(s(s(T23)), T26, T25)

The argument filtering Pi contains the following mapping:
avg1_in_gag(x1, x2, x3) = avg1_in_gag(x1, x3)
s(x1) = s(x1)
U1_gag(x1, x2, x3, x4) = U1_gag(x4)
U2_gag(x1, x2, x3, x4) = U2_gag(x4)
0 = 0
avg1_out_gag(x1, x2, x3) = avg1_out_gag(x2)
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)
AVG1_IN_GAG(x1, x2, x3) = AVG1_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4) = U1_GAG(x4)
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)

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

(6) Obligation:

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

AVG1_IN_GAG(s(s(T23)), T26, T25) → U1_GAG(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
AVG1_IN_GAG(s(s(T23)), T26, T25) → AVG1_IN_GAG(T23, s(s(T26)), T25)
AVG1_IN_GAG(s(T42), s(s(T45)), s(T44)) → U2_GAG(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
AVG1_IN_GAG(s(T42), s(s(T45)), s(T44)) → AVG1_IN_GAG(s(T42), T45, T44)
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → U3_GAG(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → AVG1_IN_GAG(s(s(T61)), T64, T63)
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → U4_GAG(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVG1_IN_GAG(T95, s(T98), T97)
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_GAG(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T117, T116)

The TRS R consists of the following rules:

avg1_in_gag(s(s(T23)), T26, T25) → U1_gag(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
avg1_in_gag(s(T42), s(s(T45)), s(T44)) → U2_gag(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
avg1_in_gag(s(0), 0, 0) → avg1_out_gag(s(0), 0, 0)
avg1_in_gag(s(0), s(0), s(0)) → avg1_out_gag(s(0), s(0), s(0))
avg1_in_gag(s(T61), s(s(s(T64))), s(T63)) → U3_gag(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
avg1_in_gag(T95, s(s(s(T98))), s(T97)) → U4_gag(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
avg1_in_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_gag(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
avg1_in_gag(0, s(s(0)), s(0)) → avg1_out_gag(0, s(s(0)), s(0))
avg1_in_gag(0, 0, 0) → avg1_out_gag(0, 0, 0)
avg1_in_gag(0, s(0), 0) → avg1_out_gag(0, s(0), 0)
U5_gag(T114, T117, T116, avg1_out_gag(s(s(T114)), T117, T116)) → avg1_out_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116)))
U4_gag(T95, T98, T97, avg1_out_gag(T95, s(T98), T97)) → avg1_out_gag(T95, s(s(s(T98))), s(T97))
U3_gag(T61, T64, T63, avg1_out_gag(s(s(T61)), T64, T63)) → avg1_out_gag(s(T61), s(s(s(T64))), s(T63))
U2_gag(T42, T45, T44, avg1_out_gag(s(T42), T45, T44)) → avg1_out_gag(s(T42), s(s(T45)), s(T44))
U1_gag(T23, T26, T25, avg1_out_gag(T23, s(s(T26)), T25)) → avg1_out_gag(s(s(T23)), T26, T25)

(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_GAG(s(T42), s(s(T45)), s(T44)) → AVG1_IN_GAG(s(T42), T45, T44)
AVG1_IN_GAG(s(s(T23)), T26, T25) → AVG1_IN_GAG(T23, s(s(T26)), T25)
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → AVG1_IN_GAG(s(s(T61)), T64, T63)
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVG1_IN_GAG(T95, s(T98), T97)
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T117, T116)

The TRS R consists of the following rules:

avg1_in_gag(s(s(T23)), T26, T25) → U1_gag(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
avg1_in_gag(s(T42), s(s(T45)), s(T44)) → U2_gag(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
avg1_in_gag(s(0), 0, 0) → avg1_out_gag(s(0), 0, 0)
avg1_in_gag(s(0), s(0), s(0)) → avg1_out_gag(s(0), s(0), s(0))
avg1_in_gag(s(T61), s(s(s(T64))), s(T63)) → U3_gag(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
avg1_in_gag(T95, s(s(s(T98))), s(T97)) → U4_gag(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
avg1_in_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_gag(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
avg1_in_gag(0, s(s(0)), s(0)) → avg1_out_gag(0, s(s(0)), s(0))
avg1_in_gag(0, 0, 0) → avg1_out_gag(0, 0, 0)
avg1_in_gag(0, s(0), 0) → avg1_out_gag(0, s(0), 0)
U5_gag(T114, T117, T116, avg1_out_gag(s(s(T114)), T117, T116)) → avg1_out_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116)))
U4_gag(T95, T98, T97, avg1_out_gag(T95, s(T98), T97)) → avg1_out_gag(T95, s(s(s(T98))), s(T97))
U3_gag(T61, T64, T63, avg1_out_gag(s(s(T61)), T64, T63)) → avg1_out_gag(s(T61), s(s(s(T64))), s(T63))
U2_gag(T42, T45, T44, avg1_out_gag(s(T42), T45, T44)) → avg1_out_gag(s(T42), s(s(T45)), s(T44))
U1_gag(T23, T26, T25, avg1_out_gag(T23, s(s(T26)), T25)) → avg1_out_gag(s(s(T23)), T26, T25)

(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_GAG(s(T42), s(s(T45)), s(T44)) → AVG1_IN_GAG(s(T42), T45, T44)
AVG1_IN_GAG(s(s(T23)), T26, T25) → AVG1_IN_GAG(T23, s(s(T26)), T25)
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → AVG1_IN_GAG(s(s(T61)), T64, T63)
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVG1_IN_GAG(T95, s(T98), T97)
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T117, T116)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
AVG1_IN_GAG(x1, x2, x3) = AVG1_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:

AVG1_IN_GAG(s(T42), s(T44)) → AVG1_IN_GAG(s(T42), T44)
AVG1_IN_GAG(s(s(T23)), T25) → AVG1_IN_GAG(T23, T25)
AVG1_IN_GAG(s(T61), s(T63)) → AVG1_IN_GAG(s(s(T61)), T63)
AVG1_IN_GAG(T95, s(T97)) → AVG1_IN_GAG(T95, T97)
AVG1_IN_GAG(T114, s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T116)

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:

AVG1_IN_GAG(s(T42), s(T44)) → AVG1_IN_GAG(s(T42), T44)
The graph contains the following edges 1 >= 1, 2 > 2
AVG1_IN_GAG(s(s(T23)), T25) → AVG1_IN_GAG(T23, T25)
The graph contains the following edges 1 > 1, 2 >= 2
AVG1_IN_GAG(s(T61), s(T63)) → AVG1_IN_GAG(s(s(T61)), T63)
The graph contains the following edges 2 > 2
AVG1_IN_GAG(T95, s(T97)) → AVG1_IN_GAG(T95, T97)
The graph contains the following edges 1 >= 1, 2 > 2
AVG1_IN_GAG(T114, s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T116)
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