Left Termination of the query pattern
overlap(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(X1, []) :- ','(!, failure(a)).
member(X, Y) :- head(Y, X).
member(X, Y) :- ','(tail(Y, T), member(X, T)).
head([], X2).
head(.(H, X3), H).
tail([], []).
tail(.(X4, T), T).
failure(b).

Query: overlap(g,g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: overlap(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: overlap(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="B: ','(member(X9, T5), member(X9, T6))\n\nT5 is ground\nT6 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 1 | ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 2 | ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(','(!_2, failure(a)), member(X15, T6)) | ','(member(X9, []), member(X9, T6)), SCOPE: 2, CLAUSE: 2 | ','(member(X9, []), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nX9 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 2 | ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX9 is free\nX14 is free\nmember(X9, T5) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(failure(a), member(X15, T6))\n\nT6 is ground\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(failure(a), member(X15, T6)), SCOPE: 3, CLAUSE: 8\n\nT6 is ground\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground\nX9 is free\nX14 is free\nmember(X9, T5) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX9 is free\nX14 is free\nmember(X9, T5) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(head(T11, X36), member(X36, T6))\n\nT6 is ground\nT11 is ground\nX9 is free\nX14 is free\nX36 is free\nmember(X9, T11) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(head(T11, X36), member(X36, T6)), SCOPE: 4, CLAUSE: 4 | ','(head(T11, X36), member(X36, T6)), SCOPE: 4, CLAUSE: 5\n\nT6 is ground\nT11 is ground\nX9 is free\nX14 is free\nX36 is free\nmember(X9, T11) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(head(T11, X36), member(X36, T6)), SCOPE: 4, CLAUSE: 5\n\nT6 is ground\nT11 is ground\nX9 is free\nX14 is free\nX36 is free\nmember(X9, T11) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="C: member(T16, T6)\n\nT6 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: member(T16, T6), SCOPE: 5, CLAUSE: 1 | member(T16, T6), SCOPE: 5, CLAUSE: 2 | member(T16, T6), SCOPE: 5, CLAUSE: 3\n\nT6 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(!_5, failure(a)) | member(T21, []), SCOPE: 5, CLAUSE: 2 | member(T21, []), SCOPE: 5, CLAUSE: 3\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: member(T16, T6), SCOPE: 5, CLAUSE: 2 | member(T16, T6), SCOPE: 5, CLAUSE: 3\n\nT6 is ground\nT16 is ground\nX50 is free\nmember(T16, T6) !=? member(X50, [])", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: failure(a), SCOPE: 6, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: member(T16, T6), SCOPE: 5, CLAUSE: 2\n\nT6 is ground\nT16 is ground\nX50 is free\nmember(T16, T6) !=? member(X50, [])", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: member(T16, T6), SCOPE: 5, CLAUSE: 3\n\nT6 is ground\nT16 is ground\nX50 is free\nmember(T16, T6) !=? member(X50, [])", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: head(T31, T30)\n\nT30 is ground\nT31 is ground\nX50 is free\nmember(T30, T31) !=? member(X50, [])", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: head(T31, T30), SCOPE: 7, CLAUSE: 4 | head(T31, T30), SCOPE: 7, CLAUSE: 5\n\nT30 is ground\nT31 is ground\nX50 is free\nmember(T30, T31) !=? member(X50, [])", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: head(T31, T30), SCOPE: 7, CLAUSE: 5\n\nT30 is ground\nT31 is ground\nX50 is free\nmember(T30, T31) !=? member(X50, [])", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(tail(T45, X76), member(T44, X76))\n\nT44 is ground\nT45 is ground\nX50 is free\nX76 is free\nmember(T44, T45) !=? member(X50, [])", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(tail(T45, X76), member(T44, X76)), SCOPE: 8, CLAUSE: 6 | ','(tail(T45, X76), member(T44, X76)), SCOPE: 8, CLAUSE: 7\n\nT44 is ground\nT45 is ground\nX50 is free\nX76 is free\nmember(T44, T45) !=? member(X50, [])", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(tail(T45, X76), member(T44, X76)), SCOPE: 8, CLAUSE: 7\n\nT44 is ground\nT45 is ground\nX50 is free\nX76 is free\nmember(T44, T45) !=? member(X50, [])", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: member(T44, T52)\n\nT44 is ground\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: ','(','(tail(T56, X96), member(X97, X96)), member(X97, T6))\n\nT6 is ground\nT56 is ground\nX9 is free\nX14 is free\nX97 is free\nX96 is free\nmember(X9, T56) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: ','(','(tail(T56, X96), member(X97, X96)), member(X97, T6)), SCOPE: 9, CLAUSE: 6 | ','(','(tail(T56, X96), member(X97, X96)), member(X97, T6)), SCOPE: 9, CLAUSE: 7\n\nT6 is ground\nT56 is ground\nX9 is free\nX14 is free\nX97 is free\nX96 is free\nmember(X9, T56) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: ','(','(tail(T56, X96), member(X97, X96)), member(X97, T6)), SCOPE: 9, CLAUSE: 7\n\nT6 is ground\nT56 is ground\nX9 is free\nX14 is free\nX97 is free\nX96 is free\nmember(X9, T56) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: ','(member(X97, T62), member(X97, T6))\n\nT6 is ground\nT62 is ground\nX97 is free", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 116 [arrowhead = none ];
116 -> 4;

117 [label="ONLY EVAL with clause\noverlap(X7, X8) :- ','(member(X9, X7), member(X9, X8)).\nand substitutionT1 -> T5,\nX7 -> T5,\nT2 -> T6,\nX8 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 117 [arrowhead = none ];
117 -> 9;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 118 [arrowhead = none ];
118 -> 10;

119 [label="EVAL with clause\nmember(X14, []) :- ','(!_2, failure(a)).\nand substitutionX9 -> X15,\nX14 -> X15,\nT5 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 119 [arrowhead = none ];
119 -> 13;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 120 [arrowhead = none ];
120 -> 15;

121 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 121 [arrowhead = none ];
121 -> 16;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 122 [arrowhead = none ];
122 -> 24;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 123 [arrowhead = none ];
123 -> 25;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 124 [arrowhead = none ];
124 -> 22;

125 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 125 [arrowhead = none ];
125 -> 23;

126 [label="ONLY EVAL with clause\nmember(X34, X35) :- head(X35, X34).\nand substitutionX9 -> X36,\nX34 -> X36,\nT5 -> T11,\nX35 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 126 [arrowhead = none ];
126 -> 31;

127 [label="ONLY EVAL with clause\nmember(X94, X95) :- ','(tail(X95, X96), member(X94, X96)).\nand substitutionX9 -> X97,\nX94 -> X97,\nT5 -> T56,\nX95 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 127 [arrowhead = none ];
127 -> 108;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 128 [arrowhead = none ];
128 -> 33;

129 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (member(X9, T11), member(X14, []))", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 129 [arrowhead = none ];
129 -> 36;

130 [label="EVAL with clause\nhead(.(X45, X46), X45).\nand substitutionX45 -> T16,\nX46 -> T17,\nT11 -> .(T16, T17),\nX36 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 130 [arrowhead = none ];
130 -> 41;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 131 [arrowhead = none ];
131 -> 46;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 132 [arrowhead = none ];
132 -> 50;

133 [label="EVAL with clause\nmember(X50, []) :- ','(!_5, failure(a)).\nand substitutionT16 -> T21,\nX50 -> T21,\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 133 [arrowhead = none ];
133 -> 54;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 134 [arrowhead = none ];
134 -> 56;

135 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 135 [arrowhead = none ];
135 -> 57;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 136 [arrowhead = none ];
136 -> 62;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 137 [arrowhead = none ];
137 -> 64;

138 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 138 [arrowhead = none ];
138 -> 58;

139 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 139 [arrowhead = none ];
139 -> 59;

140 [label="ONLY EVAL with clause\nmember(X59, X60) :- head(X60, X59).\nand substitutionT16 -> T30,\nX59 -> T30,\nT6 -> T31,\nX60 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 140 [arrowhead = none ];
140 -> 74;

141 [label="ONLY EVAL with clause\nmember(X74, X75) :- ','(tail(X75, X76), member(X74, X76)).\nand substitutionT16 -> T44,\nX74 -> T44,\nT6 -> T45,\nX75 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 141 [arrowhead = none ];
141 -> 86;

142 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
74 -> 142 [arrowhead = none ];
142 -> 75;

143 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (member(T30, T31), member(X50, []))", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 143 [arrowhead = none ];
143 -> 76;

144 [label="EVAL with clause\nhead(.(X66, X67), X66).\nand substitutionX66 -> T37,\nX67 -> T38,\nT31 -> .(T37, T38),\nT30 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 144 [arrowhead = none ];
144 -> 82;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 145 [arrowhead = none ];
145 -> 84;

146 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 146 [arrowhead = none ];
146 -> 85;

147 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
86 -> 147 [arrowhead = none ];
147 -> 87;

148 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (member(T44, T45), member(X50, []))", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 148 [arrowhead = none ];
148 -> 88;

149 [label="EVAL with clause\ntail(.(X82, X83), X83).\nand substitutionX82 -> T51,\nX83 -> T52,\nT45 -> .(T51, T52),\nX76 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 149 [arrowhead = none ];
149 -> 90;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 150 [arrowhead = none ];
150 -> 91;

151 [label="INSTANCE with matching:\nT16 -> T44\nT6 -> T52", fontsize=14, style = filled, fillcolor = lightgrey];
90 -> 151 [arrowhead = none , style=dashed];
151 -> 41[style=dashed];

152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
108 -> 152 [arrowhead = none ];
152 -> 109;

153 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (member(X9, T56), member(X14, []))", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 153 [arrowhead = none ];
153 -> 110;

154 [label="EVAL with clause\ntail(.(X104, X105), X105).\nand substitutionX104 -> T61,\nX105 -> T62,\nT56 -> .(T61, T62),\nX96 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
110 -> 154 [arrowhead = none ];
154 -> 114;

155 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
110 -> 155 [arrowhead = none ];
155 -> 115;

156 [label="INSTANCE with matching:\nX9 -> X97\nT5 -> T62", fontsize=14, style = filled, fillcolor = lightgrey];
114 -> 156 [arrowhead = none , style=dashed];
156 -> 9[style=dashed];

}

(2) Obligation:

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

overlapA_in_gg(T5, T6) → U1_gg(T5, T6, pB_in_agg(X9, T5, T6))
pB_in_agg(T16, .(T16, T17), T6) → U3_agg(T16, T17, T6, memberC_in_gg(T16, T6))
memberC_in_gg(T37, .(T37, T38)) → memberC_out_gg(T37, .(T37, T38))
memberC_in_gg(T44, .(T51, T52)) → U2_gg(T44, T51, T52, memberC_in_gg(T44, T52))
U2_gg(T44, T51, T52, memberC_out_gg(T44, T52)) → memberC_out_gg(T44, .(T51, T52))
U3_agg(T16, T17, T6, memberC_out_gg(T16, T6)) → pB_out_agg(T16, .(T16, T17), T6)
pB_in_agg(X97, .(T61, T62), T6) → U4_agg(X97, T61, T62, T6, pB_in_agg(X97, T62, T6))
U4_agg(X97, T61, T62, T6, pB_out_agg(X97, T62, T6)) → pB_out_agg(X97, .(T61, T62), T6)
U1_gg(T5, T6, pB_out_agg(X9, T5, T6)) → overlapA_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
overlapA_in_gg(x1, x2)  =  overlapA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pB_in_agg(x1, x2, x3)  =  pB_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
pB_out_agg(x1, x2, x3)  =  pB_out_agg(x1, x2, x3)
U4_agg(x1, x2, x3, x4, x5)  =  U4_agg(x2, x3, x4, x5)
overlapA_out_gg(x1, x2)  =  overlapA_out_gg(x1, x2)

(3) 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:

OVERLAPA_IN_GG(T5, T6) → U1_GG(T5, T6, pB_in_agg(X9, T5, T6))
OVERLAPA_IN_GG(T5, T6) → PB_IN_AGG(X9, T5, T6)
PB_IN_AGG(T16, .(T16, T17), T6) → U3_AGG(T16, T17, T6, memberC_in_gg(T16, T6))
PB_IN_AGG(T16, .(T16, T17), T6) → MEMBERC_IN_GG(T16, T6)
MEMBERC_IN_GG(T44, .(T51, T52)) → U2_GG(T44, T51, T52, memberC_in_gg(T44, T52))
MEMBERC_IN_GG(T44, .(T51, T52)) → MEMBERC_IN_GG(T44, T52)
PB_IN_AGG(X97, .(T61, T62), T6) → U4_AGG(X97, T61, T62, T6, pB_in_agg(X97, T62, T6))
PB_IN_AGG(X97, .(T61, T62), T6) → PB_IN_AGG(X97, T62, T6)

The TRS R consists of the following rules:

overlapA_in_gg(T5, T6) → U1_gg(T5, T6, pB_in_agg(X9, T5, T6))
pB_in_agg(T16, .(T16, T17), T6) → U3_agg(T16, T17, T6, memberC_in_gg(T16, T6))
memberC_in_gg(T37, .(T37, T38)) → memberC_out_gg(T37, .(T37, T38))
memberC_in_gg(T44, .(T51, T52)) → U2_gg(T44, T51, T52, memberC_in_gg(T44, T52))
U2_gg(T44, T51, T52, memberC_out_gg(T44, T52)) → memberC_out_gg(T44, .(T51, T52))
U3_agg(T16, T17, T6, memberC_out_gg(T16, T6)) → pB_out_agg(T16, .(T16, T17), T6)
pB_in_agg(X97, .(T61, T62), T6) → U4_agg(X97, T61, T62, T6, pB_in_agg(X97, T62, T6))
U4_agg(X97, T61, T62, T6, pB_out_agg(X97, T62, T6)) → pB_out_agg(X97, .(T61, T62), T6)
U1_gg(T5, T6, pB_out_agg(X9, T5, T6)) → overlapA_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
overlapA_in_gg(x1, x2)  =  overlapA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pB_in_agg(x1, x2, x3)  =  pB_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
pB_out_agg(x1, x2, x3)  =  pB_out_agg(x1, x2, x3)
U4_agg(x1, x2, x3, x4, x5)  =  U4_agg(x2, x3, x4, x5)
overlapA_out_gg(x1, x2)  =  overlapA_out_gg(x1, x2)
OVERLAPA_IN_GG(x1, x2)  =  OVERLAPA_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
PB_IN_AGG(x1, x2, x3)  =  PB_IN_AGG(x2, x3)
U3_AGG(x1, x2, x3, x4)  =  U3_AGG(x1, x2, x3, x4)
MEMBERC_IN_GG(x1, x2)  =  MEMBERC_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x1, x2, x3, x4)
U4_AGG(x1, x2, x3, x4, x5)  =  U4_AGG(x2, x3, x4, x5)

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

(4) Obligation:

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

OVERLAPA_IN_GG(T5, T6) → U1_GG(T5, T6, pB_in_agg(X9, T5, T6))
OVERLAPA_IN_GG(T5, T6) → PB_IN_AGG(X9, T5, T6)
PB_IN_AGG(T16, .(T16, T17), T6) → U3_AGG(T16, T17, T6, memberC_in_gg(T16, T6))
PB_IN_AGG(T16, .(T16, T17), T6) → MEMBERC_IN_GG(T16, T6)
MEMBERC_IN_GG(T44, .(T51, T52)) → U2_GG(T44, T51, T52, memberC_in_gg(T44, T52))
MEMBERC_IN_GG(T44, .(T51, T52)) → MEMBERC_IN_GG(T44, T52)
PB_IN_AGG(X97, .(T61, T62), T6) → U4_AGG(X97, T61, T62, T6, pB_in_agg(X97, T62, T6))
PB_IN_AGG(X97, .(T61, T62), T6) → PB_IN_AGG(X97, T62, T6)

The TRS R consists of the following rules:

overlapA_in_gg(T5, T6) → U1_gg(T5, T6, pB_in_agg(X9, T5, T6))
pB_in_agg(T16, .(T16, T17), T6) → U3_agg(T16, T17, T6, memberC_in_gg(T16, T6))
memberC_in_gg(T37, .(T37, T38)) → memberC_out_gg(T37, .(T37, T38))
memberC_in_gg(T44, .(T51, T52)) → U2_gg(T44, T51, T52, memberC_in_gg(T44, T52))
U2_gg(T44, T51, T52, memberC_out_gg(T44, T52)) → memberC_out_gg(T44, .(T51, T52))
U3_agg(T16, T17, T6, memberC_out_gg(T16, T6)) → pB_out_agg(T16, .(T16, T17), T6)
pB_in_agg(X97, .(T61, T62), T6) → U4_agg(X97, T61, T62, T6, pB_in_agg(X97, T62, T6))
U4_agg(X97, T61, T62, T6, pB_out_agg(X97, T62, T6)) → pB_out_agg(X97, .(T61, T62), T6)
U1_gg(T5, T6, pB_out_agg(X9, T5, T6)) → overlapA_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
overlapA_in_gg(x1, x2)  =  overlapA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pB_in_agg(x1, x2, x3)  =  pB_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
pB_out_agg(x1, x2, x3)  =  pB_out_agg(x1, x2, x3)
U4_agg(x1, x2, x3, x4, x5)  =  U4_agg(x2, x3, x4, x5)
overlapA_out_gg(x1, x2)  =  overlapA_out_gg(x1, x2)
OVERLAPA_IN_GG(x1, x2)  =  OVERLAPA_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
PB_IN_AGG(x1, x2, x3)  =  PB_IN_AGG(x2, x3)
U3_AGG(x1, x2, x3, x4)  =  U3_AGG(x1, x2, x3, x4)
MEMBERC_IN_GG(x1, x2)  =  MEMBERC_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x1, x2, x3, x4)
U4_AGG(x1, x2, x3, x4, x5)  =  U4_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:

MEMBERC_IN_GG(T44, .(T51, T52)) → MEMBERC_IN_GG(T44, T52)

The TRS R consists of the following rules:

overlapA_in_gg(T5, T6) → U1_gg(T5, T6, pB_in_agg(X9, T5, T6))
pB_in_agg(T16, .(T16, T17), T6) → U3_agg(T16, T17, T6, memberC_in_gg(T16, T6))
memberC_in_gg(T37, .(T37, T38)) → memberC_out_gg(T37, .(T37, T38))
memberC_in_gg(T44, .(T51, T52)) → U2_gg(T44, T51, T52, memberC_in_gg(T44, T52))
U2_gg(T44, T51, T52, memberC_out_gg(T44, T52)) → memberC_out_gg(T44, .(T51, T52))
U3_agg(T16, T17, T6, memberC_out_gg(T16, T6)) → pB_out_agg(T16, .(T16, T17), T6)
pB_in_agg(X97, .(T61, T62), T6) → U4_agg(X97, T61, T62, T6, pB_in_agg(X97, T62, T6))
U4_agg(X97, T61, T62, T6, pB_out_agg(X97, T62, T6)) → pB_out_agg(X97, .(T61, T62), T6)
U1_gg(T5, T6, pB_out_agg(X9, T5, T6)) → overlapA_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
overlapA_in_gg(x1, x2)  =  overlapA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pB_in_agg(x1, x2, x3)  =  pB_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
pB_out_agg(x1, x2, x3)  =  pB_out_agg(x1, x2, x3)
U4_agg(x1, x2, x3, x4, x5)  =  U4_agg(x2, x3, x4, x5)
overlapA_out_gg(x1, x2)  =  overlapA_out_gg(x1, x2)
MEMBERC_IN_GG(x1, x2)  =  MEMBERC_IN_GG(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

MEMBERC_IN_GG(T44, .(T51, T52)) → MEMBERC_IN_GG(T44, T52)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

MEMBERC_IN_GG(T44, .(T51, T52)) → MEMBERC_IN_GG(T44, T52)

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

(12) 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:

• MEMBERC_IN_GG(T44, .(T51, T52)) → MEMBERC_IN_GG(T44, T52)
  The graph contains the following edges 1 >= 1, 2 > 2

(13) YES

(14) Obligation:

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

PB_IN_AGG(X97, .(T61, T62), T6) → PB_IN_AGG(X97, T62, T6)

The TRS R consists of the following rules:

overlapA_in_gg(T5, T6) → U1_gg(T5, T6, pB_in_agg(X9, T5, T6))
pB_in_agg(T16, .(T16, T17), T6) → U3_agg(T16, T17, T6, memberC_in_gg(T16, T6))
memberC_in_gg(T37, .(T37, T38)) → memberC_out_gg(T37, .(T37, T38))
memberC_in_gg(T44, .(T51, T52)) → U2_gg(T44, T51, T52, memberC_in_gg(T44, T52))
U2_gg(T44, T51, T52, memberC_out_gg(T44, T52)) → memberC_out_gg(T44, .(T51, T52))
U3_agg(T16, T17, T6, memberC_out_gg(T16, T6)) → pB_out_agg(T16, .(T16, T17), T6)
pB_in_agg(X97, .(T61, T62), T6) → U4_agg(X97, T61, T62, T6, pB_in_agg(X97, T62, T6))
U4_agg(X97, T61, T62, T6, pB_out_agg(X97, T62, T6)) → pB_out_agg(X97, .(T61, T62), T6)
U1_gg(T5, T6, pB_out_agg(X9, T5, T6)) → overlapA_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
overlapA_in_gg(x1, x2)  =  overlapA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pB_in_agg(x1, x2, x3)  =  pB_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
pB_out_agg(x1, x2, x3)  =  pB_out_agg(x1, x2, x3)
U4_agg(x1, x2, x3, x4, x5)  =  U4_agg(x2, x3, x4, x5)
overlapA_out_gg(x1, x2)  =  overlapA_out_gg(x1, x2)
PB_IN_AGG(x1, x2, x3)  =  PB_IN_AGG(x2, x3)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

PB_IN_AGG(X97, .(T61, T62), T6) → PB_IN_AGG(X97, T62, T6)

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

PB_IN_AGG(.(T61, T62), T6) → PB_IN_AGG(T62, T6)

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

(19) 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:

• PB_IN_AGG(.(T61, T62), T6) → PB_IN_AGG(T62, T6)
  The graph contains the following edges 1 > 1, 2 >= 2

(20) YES