Left Termination of the query pattern overlap_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

overlap(Xs, Ys) :- ','(member(X, Xs), member(X, Ys)).
member(X, Y) :- ','(no(empty(Y)), head(Y, X)).
member(X, Y) :- ','(no(empty(Y)), ','(tail(Y, T), member(X, T))).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).

Queries:

overlap(g,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: overlap(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: overlap(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(member(X9, T5), member(X9, T6))\n\nT5 is ground\nT6 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 1 | ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 1\n\nT5 is ground\nT6 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(no(empty(T13)), ','(head(T13, X30), member(X30, T6)))\n\nT6 is ground\nT13 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(no(empty(T13)), ','(head(T13, X30), member(X30, T6))), SCOPE: 3, CLAUSE: 8 | ','(no(empty(T13)), ','(head(T13, X30), member(X30, T6))), SCOPE: 3, CLAUSE: 9\n\nT6 is ground\nT13 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(call(empty(T17)), ','(!_3, ','(failure(a), ','(head(T17, X30), member(X30, T6))))) | ','(no(empty(T17)), ','(head(T17, X30), member(X30, T6))), SCOPE: 3, CLAUSE: 9\n\nT6 is ground\nT17 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(empty(T17), ','(!_3, ','(failure(a), ','(head(T17, X30), member(X30, T6))))) | ?_4 | ','(no(empty(T17)), ','(head(T17, X30), member(X30, T6))), SCOPE: 3, CLAUSE: 9\n\nT6 is ground\nT17 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(empty(T17), ','(!_3, ','(failure(a), ','(head(T17, X30), member(X30, T6))))), SCOPE: 5, CLAUSE: 7 | ?_5 | ?_4 | ','(no(empty(T17)), ','(head(T17, X30), member(X30, T6))), SCOPE: 3, CLAUSE: 9\n\nT6 is ground\nT17 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(!_3, ','(failure(a), ','(head([], X30), member(X30, T6)))) | ?_5 | ?_4 | ','(no(empty([])), ','(head([], X30), member(X30, T6))), SCOPE: 3, CLAUSE: 9\n\nT6 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ?_5 | ?_4 | ','(no(empty(T17)), ','(head(T17, X30), member(X30, T6))), SCOPE: 3, CLAUSE: 9\n\nT6 is ground\nT17 is ground\nX30 is free; \nempty(T17) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(failure(a), ','(head([], X30), member(X30, T6)))\n\nT6 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(failure(a), ','(head([], X30), member(X30, T6))), SCOPE: 6, CLAUSE: 10\n\nT6 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ?_4 | ','(no(empty(T17)), ','(head(T17, X30), member(X30, T6))), SCOPE: 3, CLAUSE: 9\n\nT6 is ground\nT17 is ground\nX30 is free; \nempty(T17) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(no(empty(T17)), ','(head(T17, X30), member(X30, T6))), SCOPE: 3, CLAUSE: 9\n\nT6 is ground\nT17 is ground\nX30 is free; \nempty(T17) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(head(T22, X30), member(X30, T6))\n\nT6 is ground\nT22 is ground\nX30 is free; \nempty(T22) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T22, X30), member(X30, T6)), SCOPE: 7, CLAUSE: 3 | ','(head(T22, X30), member(X30, T6)), SCOPE: 7, CLAUSE: 4\n\nT6 is ground\nT22 is ground\nX30 is free; \nempty(T22) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(head(T22, X30), member(X30, T6)), SCOPE: 7, CLAUSE: 4\n\nT6 is ground\nT22 is ground\nX30 is free; \nempty(T22) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: member(T28, T6)\n\nT6 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: member(T28, T6), SCOPE: 8, CLAUSE: 1 | member(T28, T6), SCOPE: 8, CLAUSE: 2\n\nT6 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: member(T28, T6), SCOPE: 8, CLAUSE: 1\n\nT6 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: member(T28, T6), SCOPE: 8, CLAUSE: 2\n\nT6 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(no(empty(T41)), head(T41, T40))\n\nT40 is ground\nT41 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(no(empty(T41)), head(T41, T40)), SCOPE: 9, CLAUSE: 8 | ','(no(empty(T41)), head(T41, T40)), SCOPE: 9, CLAUSE: 9\n\nT40 is ground\nT41 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(call(empty(T44)), ','(!_9, ','(failure(a), head(T44, T40)))) | ','(no(empty(T44)), head(T44, T40)), SCOPE: 9, CLAUSE: 9\n\nT40 is ground\nT44 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(empty(T44), ','(!_9, ','(failure(a), head(T44, T40)))) | ?_10 | ','(no(empty(T44)), head(T44, T40)), SCOPE: 9, CLAUSE: 9\n\nT40 is ground\nT44 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(empty(T44), ','(!_9, ','(failure(a), head(T44, T40)))), SCOPE: 11, CLAUSE: 7 | ?_11 | ?_10 | ','(no(empty(T44)), head(T44, T40)), SCOPE: 9, CLAUSE: 9\n\nT40 is ground\nT44 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(!_9, ','(failure(a), head([], T40))) | ?_11 | ?_10 | ','(no(empty([])), head([], T40)), SCOPE: 9, CLAUSE: 9\n\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ?_11 | ?_10 | ','(no(empty(T44)), head(T44, T40)), SCOPE: 9, CLAUSE: 9\n\nT40 is ground\nT44 is ground\nempty(T44) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(failure(a), head([], T40))\n\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(failure(a), head([], T40)), SCOPE: 12, CLAUSE: 10\n\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ?_10 | ','(no(empty(T44)), head(T44, T40)), SCOPE: 9, CLAUSE: 9\n\nT40 is ground\nT44 is ground\nempty(T44) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(no(empty(T44)), head(T44, T40)), SCOPE: 9, CLAUSE: 9\n\nT40 is ground\nT44 is ground\nempty(T44) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: head(T47, T40)\n\nT40 is ground\nT47 is ground\nempty(T47) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: head(T47, T40), SCOPE: 13, CLAUSE: 3 | head(T47, T40), SCOPE: 13, CLAUSE: 4\n\nT40 is ground\nT47 is ground\nempty(T47) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: head(T47, T40), SCOPE: 13, CLAUSE: 4\n\nT40 is ground\nT47 is ground\nempty(T47) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(no(empty(T61)), ','(tail(T61, X96), member(T60, X96)))\n\nT60 is ground\nT61 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(no(empty(T61)), ','(tail(T61, X96), member(T60, X96))), SCOPE: 14, CLAUSE: 8 | ','(no(empty(T61)), ','(tail(T61, X96), member(T60, X96))), SCOPE: 14, CLAUSE: 9\n\nT60 is ground\nT61 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(call(empty(T65)), ','(!_14, ','(failure(a), ','(tail(T65, X96), member(T60, X96))))) | ','(no(empty(T65)), ','(tail(T65, X96), member(T60, X96))), SCOPE: 14, CLAUSE: 9\n\nT60 is ground\nT65 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(empty(T65), ','(!_14, ','(failure(a), ','(tail(T65, X96), member(T60, X96))))) | ?_15 | ','(no(empty(T65)), ','(tail(T65, X96), member(T60, X96))), SCOPE: 14, CLAUSE: 9\n\nT60 is ground\nT65 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(empty(T65), ','(!_14, ','(failure(a), ','(tail(T65, X96), member(T60, X96))))), SCOPE: 16, CLAUSE: 7 | ?_16 | ?_15 | ','(no(empty(T65)), ','(tail(T65, X96), member(T60, X96))), SCOPE: 14, CLAUSE: 9\n\nT60 is ground\nT65 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(!_14, ','(failure(a), ','(tail([], X96), member(T60, X96)))) | ?_16 | ?_15 | ','(no(empty([])), ','(tail([], X96), member(T60, X96))), SCOPE: 14, CLAUSE: 9\n\nT60 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ?_16 | ?_15 | ','(no(empty(T65)), ','(tail(T65, X96), member(T60, X96))), SCOPE: 14, CLAUSE: 9\n\nT60 is ground\nT65 is ground\nX96 is free; \nempty(T65) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(failure(a), ','(tail([], X96), member(T60, X96)))\n\nT60 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(failure(a), ','(tail([], X96), member(T60, X96))), SCOPE: 17, CLAUSE: 10\n\nT60 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ?_15 | ','(no(empty(T65)), ','(tail(T65, X96), member(T60, X96))), SCOPE: 14, CLAUSE: 9\n\nT60 is ground\nT65 is ground\nX96 is free; \nempty(T65) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(no(empty(T65)), ','(tail(T65, X96), member(T60, X96))), SCOPE: 14, CLAUSE: 9\n\nT60 is ground\nT65 is ground\nX96 is free; \nempty(T65) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(tail(T70, X96), member(T60, X96))\n\nT60 is ground\nT70 is ground\nX96 is free; \nempty(T70) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(tail(T70, X96), member(T60, X96)), SCOPE: 18, CLAUSE: 5 | ','(tail(T70, X96), member(T60, X96)), SCOPE: 18, CLAUSE: 6\n\nT60 is ground\nT70 is ground\nX96 is free; \nempty(T70) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(tail(T70, X96), member(T60, X96)), SCOPE: 18, CLAUSE: 6\n\nT60 is ground\nT70 is ground\nX96 is free; \nempty(T70) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: member(T60, T77)\n\nT60 is ground\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(no(empty(T82)), ','(tail(T82, X138), ','(member(X139, X138), member(X139, T6))))\n\nT6 is ground\nT82 is ground\nX139 is free; \nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(no(empty(T82)), ','(tail(T82, X138), ','(member(X139, X138), member(X139, T6)))), SCOPE: 19, CLAUSE: 8 | ','(no(empty(T82)), ','(tail(T82, X138), ','(member(X139, X138), member(X139, T6)))), SCOPE: 19, CLAUSE: 9\n\nT6 is ground\nT82 is ground\nX139 is free; \nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(call(empty(T85)), ','(!_19, ','(failure(a), ','(tail(T85, X138), ','(member(X139, X138), member(X139, T6)))))) | ','(no(empty(T85)), ','(tail(T85, X138), ','(member(X139, X138), member(X139, T6)))), SCOPE: 19, CLAUSE: 9\n\nT6 is ground\nT85 is ground\nX139 is free; \nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(empty(T85), ','(!_19, ','(failure(a), ','(tail(T85, X138), ','(member(X139, X138), member(X139, T6)))))) | ?_20 | ','(no(empty(T85)), ','(tail(T85, X138), ','(member(X139, X138), member(X139, T6)))), SCOPE: 19, CLAUSE: 9\n\nT6 is ground\nT85 is ground\nX139 is free; \nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(empty(T85), ','(!_19, ','(failure(a), ','(tail(T85, X138), ','(member(X139, X138), member(X139, T6)))))), SCOPE: 21, CLAUSE: 7 | ?_21 | ?_20 | ','(no(empty(T85)), ','(tail(T85, X138), ','(member(X139, X138), member(X139, T6)))), SCOPE: 19, CLAUSE: 9\n\nT6 is ground\nT85 is ground\nX139 is free; \nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(!_19, ','(failure(a), ','(tail([], X138), ','(member(X139, X138), member(X139, T6))))) | ?_21 | ?_20 | ','(no(empty([])), ','(tail([], X138), ','(member(X139, X138), member(X139, T6)))), SCOPE: 19, CLAUSE: 9\n\nT6 is ground\nX139 is free; \nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ?_21 | ?_20 | ','(no(empty(T85)), ','(tail(T85, X138), ','(member(X139, X138), member(X139, T6)))), SCOPE: 19, CLAUSE: 9\n\nT6 is ground\nT85 is ground\nX139 is free; \nX138 is free; \nempty(T85) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(failure(a), ','(tail([], X138), ','(member(X139, X138), member(X139, T6))))\n\nT6 is ground\nX139 is free; \nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(failure(a), ','(tail([], X138), ','(member(X139, X138), member(X139, T6)))), SCOPE: 22, CLAUSE: 10\n\nT6 is ground\nX139 is free; \nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ?_20 | ','(no(empty(T85)), ','(tail(T85, X138), ','(member(X139, X138), member(X139, T6)))), SCOPE: 19, CLAUSE: 9\n\nT6 is ground\nT85 is ground\nX139 is free; \nX138 is free; \nempty(T85) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(no(empty(T85)), ','(tail(T85, X138), ','(member(X139, X138), member(X139, T6)))), SCOPE: 19, CLAUSE: 9\n\nT6 is ground\nT85 is ground\nX139 is free; \nX138 is free; \nempty(T85) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(tail(T88, X138), ','(member(X139, X138), member(X139, T6)))\n\nT6 is ground\nT88 is ground\nX139 is free; \nX138 is free; \nempty(T88) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(tail(T88, X138), ','(member(X139, X138), member(X139, T6))), SCOPE: 23, CLAUSE: 5 | ','(tail(T88, X138), ','(member(X139, X138), member(X139, T6))), SCOPE: 23, CLAUSE: 6\n\nT6 is ground\nT88 is ground\nX139 is free; \nX138 is free; \nempty(T88) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(tail(T88, X138), ','(member(X139, X138), member(X139, T6))), SCOPE: 23, CLAUSE: 6\n\nT6 is ground\nT88 is ground\nX139 is free; \nX138 is free; \nempty(T88) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(member(X139, T94), member(X139, T6))\n\nT6 is ground\nT94 is ground\nX139 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 79 [arrowhead = none ];
79 -> 2;

80 [label="ONLY EVAL with clause\noverlap(X7, X8) :- ','(member(X9, X7), member(X9, X8)).\nand substitution\nT1 -> T5\nX7 -> T5\nT2 -> T6\nX8 -> T6", fontsize=14, style = filled, fillcolor = white];
2 -> 80 [arrowhead = none ];
80 -> 3;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 81 [arrowhead = none ];
81 -> 4;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 82 [arrowhead = none ];
82 -> 5;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 83 [arrowhead = none ];
83 -> 6;

84 [label="ONLY EVAL with clause\nmember(X28, X29) :- ','(no(empty(X29)), head(X29, X28)).\nand substitution\nX9 -> X30\nX28 -> X30\nT5 -> T13\nX29 -> T13", fontsize=14, style = filled, fillcolor = white];
5 -> 84 [arrowhead = none ];
84 -> 7;

85 [label="ONLY EVAL with clause\nmember(X136, X137) :- ','(no(empty(X137)), ','(tail(X137, X138), member(X136, X138))).\nand substitution\nX9 -> X139\nX136 -> X139\nT5 -> T82\nX137 -> T82", fontsize=14, style = filled, fillcolor = white];
6 -> 85 [arrowhead = none ];
85 -> 62;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 86 [arrowhead = none ];
86 -> 8;

87 [label="ONLY EVAL with clause\nno(X36) :- ','(call(X36), ','(!_3, failure(a))).\nand substitution\nT13 -> T17\nX36 -> empty(T17)", fontsize=14, style = filled, fillcolor = white];
8 -> 87 [arrowhead = none ];
87 -> 9;

88 [label="CALL", fontsize=14, style = filled, fillcolor = white];
9 -> 88 [arrowhead = none ];
88 -> 10;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 89 [arrowhead = none ];
89 -> 11;

90 [label="EVAL with clause\nempty([]).\nand substitution\nT17 -> []", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 90 [arrowhead = none ];
90 -> 12;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 91 [arrowhead = none ];
91 -> 13;

92 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 92 [arrowhead = none ];
92 -> 14;

93 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 93 [arrowhead = none ];
93 -> 17;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 94 [arrowhead = none ];
94 -> 15;

95 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
15 -> 95 [arrowhead = none ];
95 -> 16;

96 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 96 [arrowhead = none ];
96 -> 18;

97 [label="ONLY EVAL with clause\nno(X45).\nand substitution\nT17 -> T22\nX45 -> empty(T22)", fontsize=14, style = filled, fillcolor = white];
18 -> 97 [arrowhead = none ];
97 -> 19;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 98 [arrowhead = none ];
98 -> 20;

99 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T22), empty([]))", fontsize=14, style = filled, fillcolor = white];
20 -> 99 [arrowhead = none ];
99 -> 21;

100 [label="EVAL with clause\nhead(.(X55, X56), X55).\nand substitution\nX55 -> T28\nX56 -> T29\nT22 -> .(T28, T29)\nX30 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 100 [arrowhead = none ];
100 -> 22;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 101 [arrowhead = none ];
101 -> 23;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 102 [arrowhead = none ];
102 -> 24;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 103 [arrowhead = none ];
103 -> 25;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 104 [arrowhead = none ];
104 -> 26;

105 [label="ONLY EVAL with clause\nmember(X69, X70) :- ','(no(empty(X70)), head(X70, X69)).\nand substitution\nT28 -> T40\nX69 -> T40\nT6 -> T41\nX70 -> T41", fontsize=14, style = filled, fillcolor = white];
25 -> 105 [arrowhead = none ];
105 -> 27;

106 [label="ONLY EVAL with clause\nmember(X94, X95) :- ','(no(empty(X95)), ','(tail(X95, X96), member(X94, X96))).\nand substitution\nT28 -> T60\nX94 -> T60\nT6 -> T61\nX95 -> T61", fontsize=14, style = filled, fillcolor = white];
26 -> 106 [arrowhead = none ];
106 -> 45;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 107 [arrowhead = none ];
107 -> 28;

108 [label="ONLY EVAL with clause\nno(X73) :- ','(call(X73), ','(!_9, failure(a))).\nand substitution\nT41 -> T44\nX73 -> empty(T44)", fontsize=14, style = filled, fillcolor = white];
28 -> 108 [arrowhead = none ];
108 -> 29;

109 [label="CALL", fontsize=14, style = filled, fillcolor = white];
29 -> 109 [arrowhead = none ];
109 -> 30;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 110 [arrowhead = none ];
110 -> 31;

111 [label="EVAL with clause\nempty([]).\nand substitution\nT44 -> []", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 111 [arrowhead = none ];
111 -> 32;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 112 [arrowhead = none ];
112 -> 33;

113 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 113 [arrowhead = none ];
113 -> 34;

114 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 114 [arrowhead = none ];
114 -> 37;

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 115 [arrowhead = none ];
115 -> 35;

116 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
35 -> 116 [arrowhead = none ];
116 -> 36;

117 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 117 [arrowhead = none ];
117 -> 38;

118 [label="ONLY EVAL with clause\nno(X76).\nand substitution\nT44 -> T47\nX76 -> empty(T47)", fontsize=14, style = filled, fillcolor = white];
38 -> 118 [arrowhead = none ];
118 -> 39;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 119 [arrowhead = none ];
119 -> 40;

120 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T47), empty([]))", fontsize=14, style = filled, fillcolor = white];
40 -> 120 [arrowhead = none ];
120 -> 41;

121 [label="EVAL with clause\nhead(.(X82, X83), X82).\nand substitution\nX82 -> T53\nX83 -> T54\nT47 -> .(T53, T54)\nT40 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 121 [arrowhead = none ];
121 -> 42;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 122 [arrowhead = none ];
122 -> 43;

123 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 123 [arrowhead = none ];
123 -> 44;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 124 [arrowhead = none ];
124 -> 46;

125 [label="ONLY EVAL with clause\nno(X102) :- ','(call(X102), ','(!_14, failure(a))).\nand substitution\nT61 -> T65\nX102 -> empty(T65)", fontsize=14, style = filled, fillcolor = white];
46 -> 125 [arrowhead = none ];
125 -> 47;

126 [label="CALL", fontsize=14, style = filled, fillcolor = white];
47 -> 126 [arrowhead = none ];
126 -> 48;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 127 [arrowhead = none ];
127 -> 49;

128 [label="EVAL with clause\nempty([]).\nand substitution\nT65 -> []", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 128 [arrowhead = none ];
128 -> 50;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 129 [arrowhead = none ];
129 -> 51;

130 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 130 [arrowhead = none ];
130 -> 52;

131 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 131 [arrowhead = none ];
131 -> 55;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 132 [arrowhead = none ];
132 -> 53;

133 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
53 -> 133 [arrowhead = none ];
133 -> 54;

134 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 134 [arrowhead = none ];
134 -> 56;

135 [label="ONLY EVAL with clause\nno(X111).\nand substitution\nT65 -> T70\nX111 -> empty(T70)", fontsize=14, style = filled, fillcolor = white];
56 -> 135 [arrowhead = none ];
135 -> 57;

136 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 136 [arrowhead = none ];
136 -> 58;

137 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (empty(T70), empty([]))", fontsize=14, style = filled, fillcolor = white];
58 -> 137 [arrowhead = none ];
137 -> 59;

138 [label="EVAL with clause\ntail(.(X119, X120), X120).\nand substitution\nX119 -> T76\nX120 -> T77\nT70 -> .(T76, T77)\nX96 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 138 [arrowhead = none ];
138 -> 60;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 139 [arrowhead = none ];
139 -> 61;

140 [label="INSTANCE with matching:\nT28 -> T60\nT6 -> T77", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 140 [arrowhead = none , style=dashed];
140 -> 22[style=dashed];

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 141 [arrowhead = none ];
141 -> 63;

142 [label="ONLY EVAL with clause\nno(X146) :- ','(call(X146), ','(!_19, failure(a))).\nand substitution\nT82 -> T85\nX146 -> empty(T85)", fontsize=14, style = filled, fillcolor = white];
63 -> 142 [arrowhead = none ];
142 -> 64;

143 [label="CALL", fontsize=14, style = filled, fillcolor = white];
64 -> 143 [arrowhead = none ];
143 -> 65;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 144 [arrowhead = none ];
144 -> 66;

145 [label="EVAL with clause\nempty([]).\nand substitution\nT85 -> []", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 145 [arrowhead = none ];
145 -> 67;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 146 [arrowhead = none ];
146 -> 68;

147 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 147 [arrowhead = none ];
147 -> 69;

148 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 148 [arrowhead = none ];
148 -> 72;

149 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 149 [arrowhead = none ];
149 -> 70;

150 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
70 -> 150 [arrowhead = none ];
150 -> 71;

151 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 151 [arrowhead = none ];
151 -> 73;

152 [label="ONLY EVAL with clause\nno(X157).\nand substitution\nT85 -> T88\nX157 -> empty(T88)", fontsize=14, style = filled, fillcolor = white];
73 -> 152 [arrowhead = none ];
152 -> 74;

153 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
74 -> 153 [arrowhead = none ];
153 -> 75;

154 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (empty(T88), empty([]))", fontsize=14, style = filled, fillcolor = white];
75 -> 154 [arrowhead = none ];
154 -> 76;

155 [label="EVAL with clause\ntail(.(X166, X167), X167).\nand substitution\nX166 -> T93\nX167 -> T94\nT88 -> .(T93, T94)\nX138 -> T94", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 155 [arrowhead = none ];
155 -> 77;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 156 [arrowhead = none ];
156 -> 78;

157 [label="INSTANCE with matching:\nX9 -> X139\nT5 -> T94", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 157 [arrowhead = none , style=dashed];
157 -> 3[style=dashed];

}

(2) Obligation:

Triples:

member22(T60, .(T76, T77)) :- member22(T60, T77).
p3(T28, .(T28, T29), T6) :- member22(T28, T6).
p3(X139, .(T93, T94), T6) :- p3(X139, T94, T6).
overlap1(T5, T6) :- p3(X9, T5, T6).

Clauses:

memberc22(T53, .(T53, T54)).
memberc22(T60, .(T76, T77)) :- memberc22(T60, T77).
qc3(T28, .(T28, T29), T6) :- memberc22(T28, T6).
qc3(X139, .(T93, T94), T6) :- qc3(X139, T94, T6).

Afs:

overlap1(x1, x2)  =  overlap1(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
overlap1_in: (b,b)
p3_in: (f,b,b)
member22_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

OVERLAP1_IN_GG(T5, T6) → U4_GG(T5, T6, p3_in_agg(X9, T5, T6))
OVERLAP1_IN_GG(T5, T6) → P3_IN_AGG(X9, T5, T6)
P3_IN_AGG(T28, .(T28, T29), T6) → U2_AGG(T28, T29, T6, member22_in_gg(T28, T6))
P3_IN_AGG(T28, .(T28, T29), T6) → MEMBER22_IN_GG(T28, T6)
MEMBER22_IN_GG(T60, .(T76, T77)) → U1_GG(T60, T76, T77, member22_in_gg(T60, T77))
MEMBER22_IN_GG(T60, .(T76, T77)) → MEMBER22_IN_GG(T60, T77)
P3_IN_AGG(X139, .(T93, T94), T6) → U3_AGG(X139, T93, T94, T6, p3_in_agg(X139, T94, T6))
P3_IN_AGG(X139, .(T93, T94), T6) → P3_IN_AGG(X139, T94, T6)

R is empty.
The argument filtering Pi contains the following mapping:
p3_in_agg(x1, x2, x3)  =  p3_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
member22_in_gg(x1, x2)  =  member22_in_gg(x1, x2)
OVERLAP1_IN_GG(x1, x2)  =  OVERLAP1_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
P3_IN_AGG(x1, x2, x3)  =  P3_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4)  =  U2_AGG(x1, x2, x3, x4)
MEMBER22_IN_GG(x1, x2)  =  MEMBER22_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
U3_AGG(x1, x2, x3, x4, x5)  =  U3_AGG(x2, x3, x4, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

OVERLAP1_IN_GG(T5, T6) → U4_GG(T5, T6, p3_in_agg(X9, T5, T6))
OVERLAP1_IN_GG(T5, T6) → P3_IN_AGG(X9, T5, T6)
P3_IN_AGG(T28, .(T28, T29), T6) → U2_AGG(T28, T29, T6, member22_in_gg(T28, T6))
P3_IN_AGG(T28, .(T28, T29), T6) → MEMBER22_IN_GG(T28, T6)
MEMBER22_IN_GG(T60, .(T76, T77)) → U1_GG(T60, T76, T77, member22_in_gg(T60, T77))
MEMBER22_IN_GG(T60, .(T76, T77)) → MEMBER22_IN_GG(T60, T77)
P3_IN_AGG(X139, .(T93, T94), T6) → U3_AGG(X139, T93, T94, T6, p3_in_agg(X139, T94, T6))
P3_IN_AGG(X139, .(T93, T94), T6) → P3_IN_AGG(X139, T94, T6)

R is empty.
The argument filtering Pi contains the following mapping:
p3_in_agg(x1, x2, x3)  =  p3_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
member22_in_gg(x1, x2)  =  member22_in_gg(x1, x2)
OVERLAP1_IN_GG(x1, x2)  =  OVERLAP1_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
P3_IN_AGG(x1, x2, x3)  =  P3_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4)  =  U2_AGG(x1, x2, x3, x4)
MEMBER22_IN_GG(x1, x2)  =  MEMBER22_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
U3_AGG(x1, x2, x3, x4, x5)  =  U3_AGG(x2, x3, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

MEMBER22_IN_GG(T60, .(T76, T77)) → MEMBER22_IN_GG(T60, T77)

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

(8) PiDPToQDPProof (EQUIVALENT transformation)

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

(9) Obligation:

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

MEMBER22_IN_GG(T60, .(T76, T77)) → MEMBER22_IN_GG(T60, T77)

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

(10) 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:

• MEMBER22_IN_GG(T60, .(T76, T77)) → MEMBER22_IN_GG(T60, T77)
  The graph contains the following edges 1 >= 1, 2 > 2

(11) YES

(12) Obligation:

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

P3_IN_AGG(X139, .(T93, T94), T6) → P3_IN_AGG(X139, T94, T6)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
P3_IN_AGG(x1, x2, x3)  =  P3_IN_AGG(x2, x3)

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

(13) PiDPToQDPProof (SOUND transformation)

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

(14) Obligation:

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

P3_IN_AGG(.(T93, T94), T6) → P3_IN_AGG(T94, T6)

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:

• P3_IN_AGG(.(T93, T94), T6) → P3_IN_AGG(T94, T6)
  The graph contains the following edges 1 > 1, 2 >= 2

(16) YES