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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog 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, T3), member(X9, T4))\n\nT3 is ground\nT4 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: member(X9, T3)\n\nT3 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: member(T5, T4)\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: member(X9, T3), SCOPE: 2, CLAUSE: 1 | member(X9, T3), SCOPE: 2, CLAUSE: 2\n\nT3 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: member(X9, T3), SCOPE: 2, CLAUSE: 1\n\nT3 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: member(X9, T3), SCOPE: 2, CLAUSE: 2\n\nT3 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(no(empty(T6)), head(T6, X16))\n\nT6 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(no(empty(T6)), head(T6, X16)), SCOPE: 3, CLAUSE: 8 | ','(no(empty(T6)), head(T6, X16)), SCOPE: 3, CLAUSE: 9 | ?_3\n\nT6 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(empty(T7), ','(!_3, ','(failure(a), head(T7, X16)))) | ','(no(empty(T7)), head(T7, X16)), SCOPE: 3, CLAUSE: 9 | ?_3\n\nT7 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(empty(T7), ','(!_3, ','(failure(a), head(T7, X16)))), SCOPE: 4, CLAUSE: 7 | ','(no(empty(T7)), head(T7, X16)), SCOPE: 3, CLAUSE: 9 | ?_3\n\nT7 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(!_3, ','(failure(a), head([], X16))) | ','(no(empty([])), head([], X16)), SCOPE: 3, CLAUSE: 9 | ?_3\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(no(empty(T7)), head(T7, X16)), SCOPE: 3, CLAUSE: 9\n\nT7 is ground\nX16 is free; \nempty(T7) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(failure(a), head([], X16))\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(failure(a), head([], X16)), SCOPE: 5, CLAUSE: 10\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: head(T8, X16)\n\nT8 is ground\nX16 is free; \nempty(T8) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: head(T8, X16), SCOPE: 6, CLAUSE: 3 | head(T8, X16), SCOPE: 6, CLAUSE: 4\n\nT8 is ground\nX16 is free; \nempty(T8) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: head(T8, X16), SCOPE: 6, CLAUSE: 4\n\nT8 is ground\nX16 is free; \nX22 is free; \nempty(T8) !=? empty([])\nhead(T8, X16) !=? head([], X22)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\nX16 is free\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\nX16 is free\nX22 is free; \nX25 is free; \nX26 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(no(empty(T11)), ','(tail(T11, X31), member(X32, X31)))\n\nT11 is ground\nX32 is free; \nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(no(empty(T11)), ','(tail(T11, X31), member(X32, X31))), SCOPE: 7, CLAUSE: 8 | ','(no(empty(T11)), ','(tail(T11, X31), member(X32, X31))), SCOPE: 7, CLAUSE: 9 | ?_7\n\nT11 is ground\nX32 is free; \nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(empty(T12), ','(!_7, ','(failure(a), ','(tail(T12, X31), member(X32, X31))))) | ','(no(empty(T12)), ','(tail(T12, X31), member(X32, X31))), SCOPE: 7, CLAUSE: 9 | ?_7\n\nT12 is ground\nX32 is free; \nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(empty(T12), ','(!_7, ','(failure(a), ','(tail(T12, X31), member(X32, X31))))), SCOPE: 8, CLAUSE: 7 | ','(no(empty(T12)), ','(tail(T12, X31), member(X32, X31))), SCOPE: 7, CLAUSE: 9 | ?_7\n\nT12 is ground\nX32 is free; \nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(!_7, ','(failure(a), ','(tail([], X31), member(X32, X31)))) | ','(no(empty([])), ','(tail([], X31), member(X32, X31))), SCOPE: 7, CLAUSE: 9 | ?_7\n\nX32 is free\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(no(empty(T12)), ','(tail(T12, X31), member(X32, X31))), SCOPE: 7, CLAUSE: 9\n\nT12 is ground\nX32 is free; \nX31 is free; \nempty(T12) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(failure(a), ','(tail([], X31), member(X32, X31)))\n\nX32 is free\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(failure(a), ','(tail([], X31), member(X32, X31))), SCOPE: 9, CLAUSE: 10\n\nX32 is free\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(tail(T13, X31), member(X32, X31))\n\nT13 is ground\nX32 is free; \nX31 is free; \nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(tail(T13, X31), member(X32, X31)), SCOPE: 10, CLAUSE: 5 | ','(tail(T13, X31), member(X32, X31)), SCOPE: 10, CLAUSE: 6\n\nT13 is ground\nX32 is free; \nX31 is free; \nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(tail(T13, X31), member(X32, X31)), SCOPE: 10, CLAUSE: 6\n\nT13 is ground\nX32 is free; \nX31 is free; \nempty(T13) !=? empty([])\ntail(T13, X31) !=? tail([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: member(X32, T15)\n\nT15 is ground\nX32 is free; \nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\nX31 is free\nX39 is free; \nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: member(T5, T4), SCOPE: 11, CLAUSE: 1 | member(T5, T4), SCOPE: 11, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: member(T5, T4), SCOPE: 11, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: member(T5, T4), SCOPE: 11, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(no(empty(T17)), head(T17, T18))\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(no(empty(T17)), head(T17, T18)), SCOPE: 12, CLAUSE: 8 | ','(no(empty(T17)), head(T17, T18)), SCOPE: 12, CLAUSE: 9 | ?_12\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(empty(T19), ','(!_12, ','(failure(a), head(T19, T18)))) | ','(no(empty(T19)), head(T19, T18)), SCOPE: 12, CLAUSE: 9 | ?_12\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(empty(T19), ','(!_12, ','(failure(a), head(T19, T18)))), SCOPE: 13, CLAUSE: 7 | ','(no(empty(T19)), head(T19, T18)), SCOPE: 12, CLAUSE: 9 | ?_12\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(!_12, ','(failure(a), head([], T18))) | ','(no(empty([])), head([], T18)), SCOPE: 12, CLAUSE: 9 | ?_12\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(no(empty(T19)), head(T19, T18)), SCOPE: 12, CLAUSE: 9\n\nT19 is ground\nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(failure(a), head([], T18))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(failure(a), head([], T18)), SCOPE: 14, CLAUSE: 10\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: head(T20, T18)\n\nT20 is ground\nempty(T20) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: head(T20, T18), SCOPE: 15, CLAUSE: 3 | head(T20, T18), SCOPE: 15, CLAUSE: 4\n\nT20 is ground\nempty(T20) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: head(T20, T18), SCOPE: 15, CLAUSE: 4\n\nT20 is ground\nX52 is free; \nempty(T20) !=? empty([])\nhead(T20, T18) !=? head([], X52)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\nX52 is free\nX55 is free; \nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(no(empty(T24)), ','(tail(T24, X61), member(T25, X61)))\n\nT24 is ground\nX61 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(no(empty(T24)), ','(tail(T24, X61), member(T25, X61))), SCOPE: 16, CLAUSE: 8 | ','(no(empty(T24)), ','(tail(T24, X61), member(T25, X61))), SCOPE: 16, CLAUSE: 9 | ?_16\n\nT24 is ground\nX61 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(empty(T26), ','(!_16, ','(failure(a), ','(tail(T26, X61), member(T25, X61))))) | ','(no(empty(T26)), ','(tail(T26, X61), member(T25, X61))), SCOPE: 16, CLAUSE: 9 | ?_16\n\nT26 is ground\nX61 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(empty(T26), ','(!_16, ','(failure(a), ','(tail(T26, X61), member(T25, X61))))), SCOPE: 17, CLAUSE: 7 | ','(no(empty(T26)), ','(tail(T26, X61), member(T25, X61))), SCOPE: 16, CLAUSE: 9 | ?_16\n\nT26 is ground\nX61 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(!_16, ','(failure(a), ','(tail([], X61), member(T25, X61)))) | ','(no(empty([])), ','(tail([], X61), member(T25, X61))), SCOPE: 16, CLAUSE: 9 | ?_16\n\nX61 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(no(empty(T26)), ','(tail(T26, X61), member(T25, X61))), SCOPE: 16, CLAUSE: 9\n\nT26 is ground\nX61 is free; \nempty(T26) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(failure(a), ','(tail([], X61), member(T25, X61)))\n\nX61 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(failure(a), ','(tail([], X61), member(T25, X61))), SCOPE: 18, CLAUSE: 10\n\nX61 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(tail(T27, X61), member(T25, X61))\n\nT27 is ground\nX61 is free; \nempty(T27) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(tail(T27, X61), member(T25, X61)), SCOPE: 19, CLAUSE: 5 | ','(tail(T27, X61), member(T25, X61)), SCOPE: 19, CLAUSE: 6\n\nT27 is ground\nX61 is free; \nempty(T27) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(tail(T27, X61), member(T25, X61)), SCOPE: 19, CLAUSE: 6\n\nT27 is ground\nX61 is free; \nempty(T27) !=? empty([])\ntail(T27, X61) !=? tail([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: member(T25, T29)\n\nT29 is ground\nX61 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\nX61 is free\nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 70 [arrowhead = none ];
70 -> 2;

71 [label="EVAL with clause\noverlap(X7, X8) :- ','(member(X9, X7), member(X9, X8)).\nand substitution\nT1 -> T3\nX7 -> T3\nT2 -> T4\nX8 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 71 [arrowhead = none ];
71 -> 3;

72 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
3 -> 72 [arrowhead = none ];
72 -> 4;

73 [label="SPLIT 2\nreplacements:\nX9 -> T5", fontsize=14, style = filled, fillcolor = violet];
3 -> 73 [arrowhead = none ];
73 -> 5;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
4 -> 74 [arrowhead = none ];
74 -> 6;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 75 [arrowhead = none ];
75 -> 38;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 76 [arrowhead = none ];
76 -> 7;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 77 [arrowhead = none ];
77 -> 8;

78 [label="EVAL with clause\nmember(X14, X15) :- ','(no(empty(X15)), head(X15, X14)).\nand substitution\nX9 -> X16\nX14 -> X16\nT3 -> T6\nX15 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 78 [arrowhead = none ];
78 -> 9;

79 [label="EVAL with clause\nmember(X29, X30) :- ','(no(empty(X30)), ','(tail(X30, X31), member(X29, X31))).\nand substitution\nX9 -> X32\nX29 -> X32\nT3 -> T11\nX30 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 79 [arrowhead = none ];
79 -> 24;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 80 [arrowhead = none ];
80 -> 10;

81 [label="EVAL with clause\nno(X18) :- ','(X18, ','(!, failure(a))).\nand substitution\nT6 -> T7\nX18 -> empty(T7)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 81 [arrowhead = none ];
81 -> 11;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 82 [arrowhead = none ];
82 -> 12;

83 [label="EVAL with clause\nempty([]).\nand substitution\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 83 [arrowhead = none ];
83 -> 13;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 84 [arrowhead = none ];
84 -> 14;

85 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 85 [arrowhead = none ];
85 -> 15;

86 [label="EVAL with clause\nno(X20).\nand substitution\nT7 -> T8\nX20 -> empty(T8)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 86 [arrowhead = none ];
86 -> 18;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 87 [arrowhead = none ];
87 -> 16;

88 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 88 [arrowhead = none ];
88 -> 17;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 89 [arrowhead = none ];
89 -> 19;

90 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T8), empty([]))", fontsize=14, style = filled, fillcolor = white];
19 -> 90 [arrowhead = none ];
90 -> 20;

91 [label="EVAL with clause\nhead(.(X25, X26), X25).\nand substitution\nX25 -> T9\nX26 -> T10\nT8 -> .(T9, T10)\nX16 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 91 [arrowhead = none ];
91 -> 21;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 92 [arrowhead = none ];
92 -> 22;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 93 [arrowhead = none ];
93 -> 23;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 94 [arrowhead = none ];
94 -> 25;

95 [label="EVAL with clause\nno(X34) :- ','(X34, ','(!, failure(a))).\nand substitution\nT11 -> T12\nX34 -> empty(T12)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 95 [arrowhead = none ];
95 -> 26;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 96 [arrowhead = none ];
96 -> 27;

97 [label="EVAL with clause\nempty([]).\nand substitution\nT12 -> []", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 97 [arrowhead = none ];
97 -> 28;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 98 [arrowhead = none ];
98 -> 29;

99 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 99 [arrowhead = none ];
99 -> 30;

100 [label="EVAL with clause\nno(X36).\nand substitution\nT12 -> T13\nX36 -> empty(T13)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 100 [arrowhead = none ];
100 -> 33;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 101 [arrowhead = none ];
101 -> 31;

102 [label="BACKTRACK\nfor clause: failure(b)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 -> 34;

104 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (empty(T13), empty([]))", fontsize=14, style = filled, fillcolor = white];
34 -> 104 [arrowhead = none ];
104 -> 35;

105 [label="EVAL with clause\ntail(.(X39, X40), X40).\nand substitution\nX39 -> T14\nX40 -> T15\nT13 -> .(T14, T15)\nX31 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 105 [arrowhead = none ];
105 -> 36;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 106 [arrowhead = none ];
106 -> 37;

107 [label="INSTANCE with matching:\nX9 -> X32\nT3 -> T15", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 107 [arrowhead = none , style=dashed];
107 -> 4[style=dashed];

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 108 [arrowhead = none ];
108 -> 39;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 109 [arrowhead = none ];
109 -> 40;

110 [label="EVAL with clause\nmember(X45, X46) :- ','(no(empty(X46)), head(X46, X45)).\nand substitution\nT5 -> T18\nX45 -> T18\nT4 -> T17\nX46 -> T17\nT16 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 110 [arrowhead = none ];
110 -> 41;

111 [label="EVAL with clause\nmember(X59, X60) :- ','(no(empty(X60)), ','(tail(X60, X61), member(X59, X61))).\nand substitution\nT5 -> T25\nX59 -> T25\nT4 -> T24\nX60 -> T24\nT23 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 111 [arrowhead = none ];
111 -> 56;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 112 [arrowhead = none ];
112 -> 42;

113 [label="EVAL with clause\nno(X48) :- ','(X48, ','(!, failure(a))).\nand substitution\nT17 -> T19\nX48 -> empty(T19)", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 113 [arrowhead = none ];
113 -> 43;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 114 [arrowhead = none ];
114 -> 44;

115 [label="EVAL with clause\nempty([]).\nand substitution\nT19 -> []", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 115 [arrowhead = none ];
115 -> 45;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 116 [arrowhead = none ];
116 -> 46;

117 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 117 [arrowhead = none ];
117 -> 47;

118 [label="EVAL with clause\nno(X50).\nand substitution\nT19 -> T20\nX50 -> empty(T20)", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 118 [arrowhead = none ];
118 -> 50;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 119 [arrowhead = none ];
119 -> 48;

120 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
48 -> 120 [arrowhead = none ];
120 -> 49;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 121 [arrowhead = none ];
121 -> 51;

122 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T20), empty([]))", fontsize=14, style = filled, fillcolor = white];
51 -> 122 [arrowhead = none ];
122 -> 52;

123 [label="EVAL with clause\nhead(.(X55, X56), X55).\nand substitution\nX55 -> T21\nX56 -> T22\nT20 -> .(T21, T22)\nT18 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 123 [arrowhead = none ];
123 -> 53;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 124 [arrowhead = none ];
124 -> 54;

125 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 125 [arrowhead = none ];
125 -> 55;

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 126 [arrowhead = none ];
126 -> 57;

127 [label="EVAL with clause\nno(X63) :- ','(X63, ','(!, failure(a))).\nand substitution\nT24 -> T26\nX63 -> empty(T26)", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 127 [arrowhead = none ];
127 -> 58;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 128 [arrowhead = none ];
128 -> 59;

129 [label="EVAL with clause\nempty([]).\nand substitution\nT26 -> []", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 129 [arrowhead = none ];
129 -> 60;

130 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 130 [arrowhead = none ];
130 -> 61;

131 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 131 [arrowhead = none ];
131 -> 62;

132 [label="EVAL with clause\nno(X65).\nand substitution\nT26 -> T27\nX65 -> empty(T27)", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 132 [arrowhead = none ];
132 -> 65;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 133 [arrowhead = none ];
133 -> 63;

134 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
63 -> 134 [arrowhead = none ];
134 -> 64;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 135 [arrowhead = none ];
135 -> 66;

136 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (empty(T27), empty([]))", fontsize=14, style = filled, fillcolor = white];
66 -> 136 [arrowhead = none ];
136 -> 67;

137 [label="EVAL with clause\ntail(.(X68, X69), X69).\nand substitution\nX68 -> T28\nX69 -> T29\nT27 -> .(T28, T29)\nX61 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 137 [arrowhead = none ];
137 -> 68;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 138 [arrowhead = none ];
138 -> 69;

139 [label="INSTANCE with matching:\nT5 -> T25\nT4 -> T29", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 139 [arrowhead = none , style=dashed];
139 -> 5[style=dashed];

}

(2) Obligation:

Clauses:

member4(T9, .(T9, T10)).
member4(X32, .(T14, T15)) :- member4(X32, T15).
member5(T21, .(T21, T22)).
member5(T25, .(T28, T29)) :- member5(T25, T29).
overlap1(T3, T4) :- member4(X9, T3).
overlap1(T3, .(T21, T22)) :- member4(T21, T3).
overlap1(T3, .(T28, T29)) :- ','(member4(T25, T3), member5(T25, T29)).

Queries:

overlap1(g,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:
overlap1_in: (b,b)
member4_in: (f,b) (b,b)
member5_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T9, .(T9, T10)) → member4_out_ag(T9, .(T9, T10))
member4_in_ag(X32, .(T14, T15)) → U1_ag(X32, T14, T15, member4_in_ag(X32, T15))
U1_ag(X32, T14, T15, member4_out_ag(X32, T15)) → member4_out_ag(X32, .(T14, T15))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T21, T22)) → U4_gg(T3, T21, T22, member4_in_gg(T21, T3))
member4_in_gg(T9, .(T9, T10)) → member4_out_gg(T9, .(T9, T10))
member4_in_gg(X32, .(T14, T15)) → U1_gg(X32, T14, T15, member4_in_gg(X32, T15))
U1_gg(X32, T14, T15, member4_out_gg(X32, T15)) → member4_out_gg(X32, .(T14, T15))
U4_gg(T3, T21, T22, member4_out_gg(T21, T3)) → overlap1_out_gg(T3, .(T21, T22))
overlap1_in_gg(T3, .(T28, T29)) → U5_gg(T3, T28, T29, member4_in_ag(T25, T3))
U5_gg(T3, T28, T29, member4_out_ag(T25, T3)) → U6_gg(T3, T28, T29, member5_in_gg(T25, T29))
member5_in_gg(T21, .(T21, T22)) → member5_out_gg(T21, .(T21, T22))
member5_in_gg(T25, .(T28, T29)) → U2_gg(T25, T28, T29, member5_in_gg(T25, T29))
U2_gg(T25, T28, T29, member5_out_gg(T25, T29)) → member5_out_gg(T25, .(T28, T29))
U6_gg(T3, T28, T29, member5_out_gg(T25, T29)) → overlap1_out_gg(T3, .(T28, T29))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T9, .(T9, T10)) → member4_out_ag(T9, .(T9, T10))
member4_in_ag(X32, .(T14, T15)) → U1_ag(X32, T14, T15, member4_in_ag(X32, T15))
U1_ag(X32, T14, T15, member4_out_ag(X32, T15)) → member4_out_ag(X32, .(T14, T15))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T21, T22)) → U4_gg(T3, T21, T22, member4_in_gg(T21, T3))
member4_in_gg(T9, .(T9, T10)) → member4_out_gg(T9, .(T9, T10))
member4_in_gg(X32, .(T14, T15)) → U1_gg(X32, T14, T15, member4_in_gg(X32, T15))
U1_gg(X32, T14, T15, member4_out_gg(X32, T15)) → member4_out_gg(X32, .(T14, T15))
U4_gg(T3, T21, T22, member4_out_gg(T21, T3)) → overlap1_out_gg(T3, .(T21, T22))
overlap1_in_gg(T3, .(T28, T29)) → U5_gg(T3, T28, T29, member4_in_ag(T25, T3))
U5_gg(T3, T28, T29, member4_out_ag(T25, T3)) → U6_gg(T3, T28, T29, member5_in_gg(T25, T29))
member5_in_gg(T21, .(T21, T22)) → member5_out_gg(T21, .(T21, T22))
member5_in_gg(T25, .(T28, T29)) → U2_gg(T25, T28, T29, member5_in_gg(T25, T29))
U2_gg(T25, T28, T29, member5_out_gg(T25, T29)) → member5_out_gg(T25, .(T28, T29))
U6_gg(T3, T28, T29, member5_out_gg(T25, T29)) → overlap1_out_gg(T3, .(T28, T29))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)

(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:

OVERLAP1_IN_GG(T3, T4) → U3_GG(T3, T4, member4_in_ag(X9, T3))
OVERLAP1_IN_GG(T3, T4) → MEMBER4_IN_AG(X9, T3)
MEMBER4_IN_AG(X32, .(T14, T15)) → U1_AG(X32, T14, T15, member4_in_ag(X32, T15))
MEMBER4_IN_AG(X32, .(T14, T15)) → MEMBER4_IN_AG(X32, T15)
OVERLAP1_IN_GG(T3, .(T21, T22)) → U4_GG(T3, T21, T22, member4_in_gg(T21, T3))
OVERLAP1_IN_GG(T3, .(T21, T22)) → MEMBER4_IN_GG(T21, T3)
MEMBER4_IN_GG(X32, .(T14, T15)) → U1_GG(X32, T14, T15, member4_in_gg(X32, T15))
MEMBER4_IN_GG(X32, .(T14, T15)) → MEMBER4_IN_GG(X32, T15)
OVERLAP1_IN_GG(T3, .(T28, T29)) → U5_GG(T3, T28, T29, member4_in_ag(T25, T3))
OVERLAP1_IN_GG(T3, .(T28, T29)) → MEMBER4_IN_AG(T25, T3)
U5_GG(T3, T28, T29, member4_out_ag(T25, T3)) → U6_GG(T3, T28, T29, member5_in_gg(T25, T29))
U5_GG(T3, T28, T29, member4_out_ag(T25, T3)) → MEMBER5_IN_GG(T25, T29)
MEMBER5_IN_GG(T25, .(T28, T29)) → U2_GG(T25, T28, T29, member5_in_gg(T25, T29))
MEMBER5_IN_GG(T25, .(T28, T29)) → MEMBER5_IN_GG(T25, T29)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T9, .(T9, T10)) → member4_out_ag(T9, .(T9, T10))
member4_in_ag(X32, .(T14, T15)) → U1_ag(X32, T14, T15, member4_in_ag(X32, T15))
U1_ag(X32, T14, T15, member4_out_ag(X32, T15)) → member4_out_ag(X32, .(T14, T15))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T21, T22)) → U4_gg(T3, T21, T22, member4_in_gg(T21, T3))
member4_in_gg(T9, .(T9, T10)) → member4_out_gg(T9, .(T9, T10))
member4_in_gg(X32, .(T14, T15)) → U1_gg(X32, T14, T15, member4_in_gg(X32, T15))
U1_gg(X32, T14, T15, member4_out_gg(X32, T15)) → member4_out_gg(X32, .(T14, T15))
U4_gg(T3, T21, T22, member4_out_gg(T21, T3)) → overlap1_out_gg(T3, .(T21, T22))
overlap1_in_gg(T3, .(T28, T29)) → U5_gg(T3, T28, T29, member4_in_ag(T25, T3))
U5_gg(T3, T28, T29, member4_out_ag(T25, T3)) → U6_gg(T3, T28, T29, member5_in_gg(T25, T29))
member5_in_gg(T21, .(T21, T22)) → member5_out_gg(T21, .(T21, T22))
member5_in_gg(T25, .(T28, T29)) → U2_gg(T25, T28, T29, member5_in_gg(T25, T29))
U2_gg(T25, T28, T29, member5_out_gg(T25, T29)) → member5_out_gg(T25, .(T28, T29))
U6_gg(T3, T28, T29, member5_out_gg(T25, T29)) → overlap1_out_gg(T3, .(T28, T29))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
OVERLAP1_IN_GG(x1, x2)  =  OVERLAP1_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
MEMBER4_IN_AG(x1, x2)  =  MEMBER4_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)
MEMBER4_IN_GG(x1, x2)  =  MEMBER4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x3, x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x4)
MEMBER5_IN_GG(x1, x2)  =  MEMBER5_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)

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

(6) Obligation:

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

OVERLAP1_IN_GG(T3, T4) → U3_GG(T3, T4, member4_in_ag(X9, T3))
OVERLAP1_IN_GG(T3, T4) → MEMBER4_IN_AG(X9, T3)
MEMBER4_IN_AG(X32, .(T14, T15)) → U1_AG(X32, T14, T15, member4_in_ag(X32, T15))
MEMBER4_IN_AG(X32, .(T14, T15)) → MEMBER4_IN_AG(X32, T15)
OVERLAP1_IN_GG(T3, .(T21, T22)) → U4_GG(T3, T21, T22, member4_in_gg(T21, T3))
OVERLAP1_IN_GG(T3, .(T21, T22)) → MEMBER4_IN_GG(T21, T3)
MEMBER4_IN_GG(X32, .(T14, T15)) → U1_GG(X32, T14, T15, member4_in_gg(X32, T15))
MEMBER4_IN_GG(X32, .(T14, T15)) → MEMBER4_IN_GG(X32, T15)
OVERLAP1_IN_GG(T3, .(T28, T29)) → U5_GG(T3, T28, T29, member4_in_ag(T25, T3))
OVERLAP1_IN_GG(T3, .(T28, T29)) → MEMBER4_IN_AG(T25, T3)
U5_GG(T3, T28, T29, member4_out_ag(T25, T3)) → U6_GG(T3, T28, T29, member5_in_gg(T25, T29))
U5_GG(T3, T28, T29, member4_out_ag(T25, T3)) → MEMBER5_IN_GG(T25, T29)
MEMBER5_IN_GG(T25, .(T28, T29)) → U2_GG(T25, T28, T29, member5_in_gg(T25, T29))
MEMBER5_IN_GG(T25, .(T28, T29)) → MEMBER5_IN_GG(T25, T29)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T9, .(T9, T10)) → member4_out_ag(T9, .(T9, T10))
member4_in_ag(X32, .(T14, T15)) → U1_ag(X32, T14, T15, member4_in_ag(X32, T15))
U1_ag(X32, T14, T15, member4_out_ag(X32, T15)) → member4_out_ag(X32, .(T14, T15))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T21, T22)) → U4_gg(T3, T21, T22, member4_in_gg(T21, T3))
member4_in_gg(T9, .(T9, T10)) → member4_out_gg(T9, .(T9, T10))
member4_in_gg(X32, .(T14, T15)) → U1_gg(X32, T14, T15, member4_in_gg(X32, T15))
U1_gg(X32, T14, T15, member4_out_gg(X32, T15)) → member4_out_gg(X32, .(T14, T15))
U4_gg(T3, T21, T22, member4_out_gg(T21, T3)) → overlap1_out_gg(T3, .(T21, T22))
overlap1_in_gg(T3, .(T28, T29)) → U5_gg(T3, T28, T29, member4_in_ag(T25, T3))
U5_gg(T3, T28, T29, member4_out_ag(T25, T3)) → U6_gg(T3, T28, T29, member5_in_gg(T25, T29))
member5_in_gg(T21, .(T21, T22)) → member5_out_gg(T21, .(T21, T22))
member5_in_gg(T25, .(T28, T29)) → U2_gg(T25, T28, T29, member5_in_gg(T25, T29))
U2_gg(T25, T28, T29, member5_out_gg(T25, T29)) → member5_out_gg(T25, .(T28, T29))
U6_gg(T3, T28, T29, member5_out_gg(T25, T29)) → overlap1_out_gg(T3, .(T28, T29))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
OVERLAP1_IN_GG(x1, x2)  =  OVERLAP1_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
MEMBER4_IN_AG(x1, x2)  =  MEMBER4_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)
MEMBER4_IN_GG(x1, x2)  =  MEMBER4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x3, x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x4)
MEMBER5_IN_GG(x1, x2)  =  MEMBER5_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

MEMBER5_IN_GG(T25, .(T28, T29)) → MEMBER5_IN_GG(T25, T29)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T9, .(T9, T10)) → member4_out_ag(T9, .(T9, T10))
member4_in_ag(X32, .(T14, T15)) → U1_ag(X32, T14, T15, member4_in_ag(X32, T15))
U1_ag(X32, T14, T15, member4_out_ag(X32, T15)) → member4_out_ag(X32, .(T14, T15))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T21, T22)) → U4_gg(T3, T21, T22, member4_in_gg(T21, T3))
member4_in_gg(T9, .(T9, T10)) → member4_out_gg(T9, .(T9, T10))
member4_in_gg(X32, .(T14, T15)) → U1_gg(X32, T14, T15, member4_in_gg(X32, T15))
U1_gg(X32, T14, T15, member4_out_gg(X32, T15)) → member4_out_gg(X32, .(T14, T15))
U4_gg(T3, T21, T22, member4_out_gg(T21, T3)) → overlap1_out_gg(T3, .(T21, T22))
overlap1_in_gg(T3, .(T28, T29)) → U5_gg(T3, T28, T29, member4_in_ag(T25, T3))
U5_gg(T3, T28, T29, member4_out_ag(T25, T3)) → U6_gg(T3, T28, T29, member5_in_gg(T25, T29))
member5_in_gg(T21, .(T21, T22)) → member5_out_gg(T21, .(T21, T22))
member5_in_gg(T25, .(T28, T29)) → U2_gg(T25, T28, T29, member5_in_gg(T25, T29))
U2_gg(T25, T28, T29, member5_out_gg(T25, T29)) → member5_out_gg(T25, .(T28, T29))
U6_gg(T3, T28, T29, member5_out_gg(T25, T29)) → overlap1_out_gg(T3, .(T28, T29))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
MEMBER5_IN_GG(x1, x2)  =  MEMBER5_IN_GG(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(11) Obligation:

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

MEMBER5_IN_GG(T25, .(T28, T29)) → MEMBER5_IN_GG(T25, T29)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

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

MEMBER5_IN_GG(T25, .(T28, T29)) → MEMBER5_IN_GG(T25, T29)

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

(14) 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:

• MEMBER5_IN_GG(T25, .(T28, T29)) → MEMBER5_IN_GG(T25, T29)
  The graph contains the following edges 1 >= 1, 2 > 2

(15) TRUE

(16) Obligation:

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

MEMBER4_IN_GG(X32, .(T14, T15)) → MEMBER4_IN_GG(X32, T15)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T9, .(T9, T10)) → member4_out_ag(T9, .(T9, T10))
member4_in_ag(X32, .(T14, T15)) → U1_ag(X32, T14, T15, member4_in_ag(X32, T15))
U1_ag(X32, T14, T15, member4_out_ag(X32, T15)) → member4_out_ag(X32, .(T14, T15))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T21, T22)) → U4_gg(T3, T21, T22, member4_in_gg(T21, T3))
member4_in_gg(T9, .(T9, T10)) → member4_out_gg(T9, .(T9, T10))
member4_in_gg(X32, .(T14, T15)) → U1_gg(X32, T14, T15, member4_in_gg(X32, T15))
U1_gg(X32, T14, T15, member4_out_gg(X32, T15)) → member4_out_gg(X32, .(T14, T15))
U4_gg(T3, T21, T22, member4_out_gg(T21, T3)) → overlap1_out_gg(T3, .(T21, T22))
overlap1_in_gg(T3, .(T28, T29)) → U5_gg(T3, T28, T29, member4_in_ag(T25, T3))
U5_gg(T3, T28, T29, member4_out_ag(T25, T3)) → U6_gg(T3, T28, T29, member5_in_gg(T25, T29))
member5_in_gg(T21, .(T21, T22)) → member5_out_gg(T21, .(T21, T22))
member5_in_gg(T25, .(T28, T29)) → U2_gg(T25, T28, T29, member5_in_gg(T25, T29))
U2_gg(T25, T28, T29, member5_out_gg(T25, T29)) → member5_out_gg(T25, .(T28, T29))
U6_gg(T3, T28, T29, member5_out_gg(T25, T29)) → overlap1_out_gg(T3, .(T28, T29))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
MEMBER4_IN_GG(x1, x2)  =  MEMBER4_IN_GG(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) Obligation:

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

MEMBER4_IN_GG(X32, .(T14, T15)) → MEMBER4_IN_GG(X32, T15)

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

(19) PiDPToQDPProof (EQUIVALENT transformation)

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

(20) Obligation:

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

MEMBER4_IN_GG(X32, .(T14, T15)) → MEMBER4_IN_GG(X32, T15)

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

(21) 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:

• MEMBER4_IN_GG(X32, .(T14, T15)) → MEMBER4_IN_GG(X32, T15)
  The graph contains the following edges 1 >= 1, 2 > 2

(22) TRUE

(23) Obligation:

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

MEMBER4_IN_AG(X32, .(T14, T15)) → MEMBER4_IN_AG(X32, T15)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T9, .(T9, T10)) → member4_out_ag(T9, .(T9, T10))
member4_in_ag(X32, .(T14, T15)) → U1_ag(X32, T14, T15, member4_in_ag(X32, T15))
U1_ag(X32, T14, T15, member4_out_ag(X32, T15)) → member4_out_ag(X32, .(T14, T15))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T21, T22)) → U4_gg(T3, T21, T22, member4_in_gg(T21, T3))
member4_in_gg(T9, .(T9, T10)) → member4_out_gg(T9, .(T9, T10))
member4_in_gg(X32, .(T14, T15)) → U1_gg(X32, T14, T15, member4_in_gg(X32, T15))
U1_gg(X32, T14, T15, member4_out_gg(X32, T15)) → member4_out_gg(X32, .(T14, T15))
U4_gg(T3, T21, T22, member4_out_gg(T21, T3)) → overlap1_out_gg(T3, .(T21, T22))
overlap1_in_gg(T3, .(T28, T29)) → U5_gg(T3, T28, T29, member4_in_ag(T25, T3))
U5_gg(T3, T28, T29, member4_out_ag(T25, T3)) → U6_gg(T3, T28, T29, member5_in_gg(T25, T29))
member5_in_gg(T21, .(T21, T22)) → member5_out_gg(T21, .(T21, T22))
member5_in_gg(T25, .(T28, T29)) → U2_gg(T25, T28, T29, member5_in_gg(T25, T29))
U2_gg(T25, T28, T29, member5_out_gg(T25, T29)) → member5_out_gg(T25, .(T28, T29))
U6_gg(T3, T28, T29, member5_out_gg(T25, T29)) → overlap1_out_gg(T3, .(T28, T29))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
member4_in_ag(x1, x2)  =  member4_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
member4_out_ag(x1, x2)  =  member4_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
member4_in_gg(x1, x2)  =  member4_in_gg(x1, x2)
member4_out_gg(x1, x2)  =  member4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
member5_in_gg(x1, x2)  =  member5_in_gg(x1, x2)
member5_out_gg(x1, x2)  =  member5_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
MEMBER4_IN_AG(x1, x2)  =  MEMBER4_IN_AG(x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(25) Obligation:

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

MEMBER4_IN_AG(X32, .(T14, T15)) → MEMBER4_IN_AG(X32, T15)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
MEMBER4_IN_AG(x1, x2)  =  MEMBER4_IN_AG(x2)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

MEMBER4_IN_AG(.(T14, T15)) → MEMBER4_IN_AG(T15)

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

(28) 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:

• MEMBER4_IN_AG(.(T14, T15)) → MEMBER4_IN_AG(T15)
  The graph contains the following edges 1 > 1

(29) TRUE