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

(0) Obligation:

Clauses:

reach(X, Y, Edges, Not_Visited) :- member(.(X, .(Y, [])), Edges).
reach(X, Z, Edges, Not_Visited) :- ','(member(.(X, .(Y, [])), Edges), ','(member(Y, Not_Visited), ','(delete(Y, Not_Visited, V1), reach(Y, Z, Edges, V1)))).
member(H, .(H, L)).
member(X, .(H, L)) :- member(X, L).
delete(X, .(X, Y), Y).
delete(X, .(H, T1), .(H, T2)) :- delete(X, T1, T2).

Query: reach(g,g,g,g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: reach(T3, T4, T5, T6)\n\nT3 is ground\nT4 is ground\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: reach(T3, T4, T5, T6), SCOPE: 1, CLAUSE: 0 | reach(T3, T4, T5, T6), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nT4 is ground\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: member(.(T11, .(T12, [])), T13) | reach(T11, T12, T13, T14), SCOPE: 1, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: member(.(T11, .(T12, [])), T13), SCOPE: 2, CLAUSE: 2 | member(.(T11, .(T12, [])), T13), SCOPE: 2, CLAUSE: 3 | ?_2 | reach(T11, T12, T13, T14), SCOPE: 1, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: member(.(T11, .(T12, [])), T13), SCOPE: 2, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: member(.(T11, .(T12, [])), T13), SCOPE: 2, CLAUSE: 3 | ?_2 | reach(T11, T12, T13, T14), SCOPE: 1, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: member(.(T11, .(T12, [])), T13), SCOPE: 2, CLAUSE: 3\n\nT11 is ground\nT12 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ?_2 | reach(T11, T12, T13, T14), SCOPE: 1, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: member(.(T46, .(T47, [])), T49)\n\nT46 is ground\nT47 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: member(.(T46, .(T47, [])), T49), SCOPE: 3, CLAUSE: 2 | member(.(T46, .(T47, [])), T49), SCOPE: 3, CLAUSE: 3\n\nT46 is ground\nT47 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: member(.(T46, .(T47, [])), T49), SCOPE: 3, CLAUSE: 2\n\nT46 is ground\nT47 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: member(.(T46, .(T47, [])), T49), SCOPE: 3, CLAUSE: 3\n\nT46 is ground\nT47 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: member(.(T79, .(T80, [])), T82)\n\nT79 is ground\nT80 is ground\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: reach(T11, T12, T13, T14), SCOPE: 1, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(member(.(T103, .(X81, [])), T105), ','(member(X81, T106), ','(delete(X81, T106, X82), reach(X81, T104, T105, X82))))\n\nT103 is ground\nT104 is ground\nT105 is ground\nT106 is ground\nX81 is free\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: member(.(T103, .(X81, [])), T105)\n\nT103 is ground\nT105 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(member(T113, T106), ','(delete(T113, T106, X82), reach(T113, T104, T105, X82)))\n\nT104 is ground\nT105 is ground\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: member(.(T103, .(X81, [])), T105), SCOPE: 4, CLAUSE: 2 | member(.(T103, .(X81, [])), T105), SCOPE: 4, CLAUSE: 3\n\nT103 is ground\nT105 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: member(.(T103, .(X81, [])), T105), SCOPE: 4, CLAUSE: 2\n\nT103 is ground\nT105 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: member(.(T103, .(X81, [])), T105), SCOPE: 4, CLAUSE: 3\n\nT103 is ground\nT105 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: member(.(T141, .(X121, [])), T143)\n\nT141 is ground\nT143 is ground\nX121 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: member(T113, T106)\n\nT106 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(delete(T113, T106, X82), reach(T113, T104, T105, X82))\n\nT104 is ground\nT105 is ground\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: member(T113, T106), SCOPE: 5, CLAUSE: 2 | member(T113, T106), SCOPE: 5, CLAUSE: 3\n\nT106 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: member(T113, T106), SCOPE: 5, CLAUSE: 2\n\nT106 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: member(T113, T106), SCOPE: 5, CLAUSE: 3\n\nT106 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: member(T174, T176)\n\nT174 is ground\nT176 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: delete(T113, T106, X82)\n\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: reach(T113, T104, T105, T185)\n\nT104 is ground\nT105 is ground\nT113 is ground\nT185 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: delete(T113, T106, X82), SCOPE: 6, CLAUSE: 4 | delete(T113, T106, X82), SCOPE: 6, CLAUSE: 5\n\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: delete(T113, T106, X82), SCOPE: 6, CLAUSE: 4\n\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: delete(T113, T106, X82), SCOPE: 6, CLAUSE: 5\n\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: delete(T206, T208, X191)\n\nT206 is ground\nT208 is ground\nX191 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 54 [arrowhead = none ];
54 -> 2;

55 [label="ONLY EVAL with clause\nreach(X5, X6, X7, X8) :- member(.(X5, .(X6, [])), X7).\nand substitutionT3 -> T11,\nX5 -> T11,\nT4 -> T12,\nX6 -> T12,\nT5 -> T13,\nX7 -> T13,\nT6 -> T14,\nX8 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 55 [arrowhead = none ];
55 -> 3;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 56 [arrowhead = none ];
56 -> 4;

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 57 [arrowhead = none ];
57 -> 5;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 58 [arrowhead = none ];
58 -> 6;

59 [label="EVAL with clause\nmember(X17, .(X17, X18)).\nand substitutionT11 -> T27,\nT12 -> T28,\nX17 -> .(T27, .(T28, [])),\nX18 -> T29,\nT13 -> .(.(T27, .(T28, [])), T29)", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 59 [arrowhead = none ];
59 -> 7;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 60 [arrowhead = none ];
60 -> 8;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 61 [arrowhead = none ];
61 -> 10;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 62 [arrowhead = none ];
62 -> 11;

63 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 63 [arrowhead = none ];
63 -> 9;

64 [label="EVAL with clause\nmember(X31, .(X32, X33)) :- member(X31, X33).\nand substitutionT11 -> T46,\nT12 -> T47,\nX31 -> .(T46, .(T47, [])),\nX32 -> T48,\nX33 -> T49,\nT13 -> .(T48, T49)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 64 [arrowhead = none ];
64 -> 12;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 65 [arrowhead = none ];
65 -> 13;

66 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 66 [arrowhead = none ];
66 -> 22;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 67 [arrowhead = none ];
67 -> 14;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 68 [arrowhead = none ];
68 -> 15;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 69 [arrowhead = none ];
69 -> 16;

70 [label="EVAL with clause\nmember(X46, .(X46, X47)).\nand substitutionT46 -> T68,\nT47 -> T69,\nX46 -> .(T68, .(T69, [])),\nX47 -> T70,\nT49 -> .(.(T68, .(T69, [])), T70)", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 70 [arrowhead = none ];
70 -> 17;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 71 [arrowhead = none ];
71 -> 18;

72 [label="EVAL with clause\nmember(X54, .(X55, X56)) :- member(X54, X56).\nand substitutionT46 -> T79,\nT47 -> T80,\nX54 -> .(T79, .(T80, [])),\nX55 -> T81,\nX56 -> T82,\nT49 -> .(T81, T82)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 72 [arrowhead = none ];
72 -> 20;

73 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 73 [arrowhead = none ];
73 -> 21;

74 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 74 [arrowhead = none ];
74 -> 19;

75 [label="INSTANCE with matching:\nT46 -> T79\nT47 -> T80\nT49 -> T82", fontsize=14, style = filled, fillcolor = lightgrey];
20 -> 75 [arrowhead = none , style=dashed];
75 -> 12[style=dashed];

76 [label="ONLY EVAL with clause\nreach(X77, X78, X79, X80) :- ','(member(.(X77, .(X81, [])), X79), ','(member(X81, X80), ','(delete(X81, X80, X82), reach(X81, X78, X79, X82)))).\nand substitutionT11 -> T103,\nX77 -> T103,\nT12 -> T104,\nX78 -> T104,\nT13 -> T105,\nX79 -> T105,\nT14 -> T106,\nX80 -> T106", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 76 [arrowhead = none ];
76 -> 23;

77 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
23 -> 77 [arrowhead = none ];
77 -> 24;

78 [label="SPLIT 2\nnew knowledge:\nT103 is ground\nT113 is ground\nT105 is ground\nreplacements:X81 -> T113", fontsize=14, style = filled, fillcolor = violet];
23 -> 78 [arrowhead = none ];
78 -> 25;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 79 [arrowhead = none ];
79 -> 26;

80 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 80 [arrowhead = none ];
80 -> 34;

81 [label="SPLIT 2\nnew knowledge:\nT113 is ground\nT106 is ground", fontsize=14, style = filled, fillcolor = violet];
25 -> 81 [arrowhead = none ];
81 -> 35;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 82 [arrowhead = none ];
82 -> 27;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 83 [arrowhead = none ];
83 -> 28;

84 [label="EVAL with clause\nmember(X107, .(X107, X108)).\nand substitutionT103 -> T132,\nX81 -> T133,\nX107 -> .(T132, .(T133, [])),\nX109 -> T133,\nX108 -> T134,\nT105 -> .(.(T132, .(T133, [])), T134)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 84 [arrowhead = none ];
84 -> 29;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 85 [arrowhead = none ];
85 -> 30;

86 [label="EVAL with clause\nmember(X118, .(X119, X120)) :- member(X118, X120).\nand substitutionT103 -> T141,\nX81 -> X121,\nX118 -> .(T141, .(X121, [])),\nX119 -> T142,\nX120 -> T143,\nT105 -> .(T142, T143)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 86 [arrowhead = none ];
86 -> 32;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 87 [arrowhead = none ];
87 -> 33;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 88 [arrowhead = none ];
88 -> 31;

89 [label="INSTANCE with matching:\nT103 -> T141\nX81 -> X121\nT105 -> T143", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 89 [arrowhead = none , style=dashed];
89 -> 24[style=dashed];

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 90 [arrowhead = none ];
90 -> 36;

91 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
35 -> 91 [arrowhead = none ];
91 -> 44;

92 [label="SPLIT 2\nnew knowledge:\nT113 is ground\nT106 is ground\nT185 is ground\nreplacements:X82 -> T185", fontsize=14, style = filled, fillcolor = violet];
35 -> 92 [arrowhead = none ];
92 -> 45;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 93 [arrowhead = none ];
93 -> 37;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 94 [arrowhead = none ];
94 -> 38;

95 [label="EVAL with clause\nmember(X144, .(X144, X145)).\nand substitutionT113 -> T166,\nX144 -> T166,\nX145 -> T167,\nT106 -> .(T166, T167)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 95 [arrowhead = none ];
95 -> 39;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 96 [arrowhead = none ];
96 -> 40;

97 [label="EVAL with clause\nmember(X152, .(X153, X154)) :- member(X152, X154).\nand substitutionT113 -> T174,\nX152 -> T174,\nX153 -> T175,\nX154 -> T176,\nT106 -> .(T175, T176)", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 97 [arrowhead = none ];
97 -> 42;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 98 [arrowhead = none ];
98 -> 43;

99 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 99 [arrowhead = none ];
99 -> 41;

100 [label="INSTANCE with matching:\nT113 -> T174\nT106 -> T176", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 100 [arrowhead = none , style=dashed];
100 -> 34[style=dashed];

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 101 [arrowhead = none ];
101 -> 46;

102 [label="INSTANCE with matching:\nT3 -> T113\nT4 -> T104\nT5 -> T105\nT6 -> T185", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 102 [arrowhead = none , style=dashed];
102 -> 1[style=dashed];

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 103 [arrowhead = none ];
103 -> 47;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 104 [arrowhead = none ];
104 -> 48;

105 [label="EVAL with clause\ndelete(X175, .(X175, X176), X176).\nand substitutionT113 -> T198,\nX175 -> T198,\nX176 -> T199,\nT106 -> .(T198, T199),\nX82 -> T199", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 105 [arrowhead = none ];
105 -> 49;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 106 [arrowhead = none ];
106 -> 50;

107 [label="EVAL with clause\ndelete(X187, .(X188, X189), .(X188, X190)) :- delete(X187, X189, X190).\nand substitutionT113 -> T206,\nX187 -> T206,\nX188 -> T207,\nX189 -> T208,\nT106 -> .(T207, T208),\nX190 -> X191,\nX82 -> .(T207, X191)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 107 [arrowhead = none ];
107 -> 52;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 108 [arrowhead = none ];
108 -> 53;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 109 [arrowhead = none ];
109 -> 51;

110 [label="INSTANCE with matching:\nT113 -> T206\nT106 -> T208\nX82 -> X191", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 110 [arrowhead = none , style=dashed];
110 -> 44[style=dashed];

}

(2) Obligation:

Clauses:

memberA(T68, T69, .(.(T68, .(T69, [])), T70)).
memberA(T79, T80, .(T81, T82)) :- memberA(T79, T80, T82).
memberB(T132, T133, .(.(T132, .(T133, [])), T134)).
memberB(T141, X121, .(T142, T143)) :- memberB(T141, X121, T143).
memberC(T166, .(T166, T167)).
memberC(T174, .(T175, T176)) :- memberC(T174, T176).
deleteD(T198, .(T198, T199), T199).
deleteD(T206, .(T207, T208), .(T207, X191)) :- deleteD(T206, T208, X191).
reachE(T27, T28, .(.(T27, .(T28, [])), T29), T14).
reachE(T46, T47, .(T48, T49), T14) :- memberA(T46, T47, T49).
reachE(T103, T104, T105, T106) :- memberB(T103, X81, T105).
reachE(T103, T104, T105, T106) :- ','(memberB(T103, T113, T105), memberC(T113, T106)).
reachE(T103, T104, T105, T106) :- ','(memberB(T103, T113, T105), ','(memberC(T113, T106), deleteD(T113, T106, X82))).
reachE(T103, T104, T105, T106) :- ','(memberB(T103, T113, T105), ','(memberC(T113, T106), ','(deleteD(T113, T106, T185), reachE(T113, T104, T105, T185)))).

Query: reachE(g,g,g,g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
reachE_in: (b,b,b,b)
memberA_in: (b,b,b)
memberB_in: (b,f,b)
memberC_in: (b,b)
deleteD_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

reachE_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachE_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachE_in_gggg(T46, T47, .(T48, T49), T14) → U5_gggg(T46, T47, T48, T49, T14, memberA_in_ggg(T46, T47, T49))
memberA_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberA_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberA_in_ggg(T79, T80, .(T81, T82)) → U1_ggg(T79, T80, T81, T82, memberA_in_ggg(T79, T80, T82))
U1_ggg(T79, T80, T81, T82, memberA_out_ggg(T79, T80, T82)) → memberA_out_ggg(T79, T80, .(T81, T82))
U5_gggg(T46, T47, T48, T49, T14, memberA_out_ggg(T46, T47, T49)) → reachE_out_gggg(T46, T47, .(T48, T49), T14)
reachE_in_gggg(T103, T104, T105, T106) → U6_gggg(T103, T104, T105, T106, memberB_in_gag(T103, X81, T105))
memberB_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberB_in_gag(T141, X121, .(T142, T143)) → U2_gag(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
U2_gag(T141, X121, T142, T143, memberB_out_gag(T141, X121, T143)) → memberB_out_gag(T141, X121, .(T142, T143))
U6_gggg(T103, T104, T105, T106, memberB_out_gag(T103, X81, T105)) → reachE_out_gggg(T103, T104, T105, T106)
reachE_in_gggg(T103, T104, T105, T106) → U7_gggg(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_gggg(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_gggg(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg(T166, .(T166, T167))
memberC_in_gg(T174, .(T175, T176)) → U3_gg(T174, T175, T176, memberC_in_gg(T174, T176))
U3_gg(T174, T175, T176, memberC_out_gg(T174, T176)) → memberC_out_gg(T174, .(T175, T176))
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U9_gggg(T103, T104, T105, T106, deleteD_in_gga(T113, T106, X82))
deleteD_in_gga(T198, .(T198, T199), T199) → deleteD_out_gga(T198, .(T198, T199), T199)
deleteD_in_gga(T206, .(T207, T208), .(T207, X191)) → U4_gga(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
U4_gga(T206, T207, T208, X191, deleteD_out_gga(T206, T208, X191)) → deleteD_out_gga(T206, .(T207, T208), .(T207, X191))
U9_gggg(T103, T104, T105, T106, deleteD_out_gga(T113, T106, X82)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_gggg(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_gggg(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → U11_gggg(T103, T104, T105, T106, reachE_in_gggg(T113, T104, T105, T185))
U11_gggg(T103, T104, T105, T106, reachE_out_gggg(T113, T104, T105, T185)) → reachE_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachE_in_gggg(x1, x2, x3, x4)  =  reachE_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachE_out_gggg(x1, x2, x3, x4)  =  reachE_out_gggg
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x6)
memberA_in_ggg(x1, x2, x3)  =  memberA_in_ggg(x1, x2, x3)
memberA_out_ggg(x1, x2, x3)  =  memberA_out_ggg
U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
U6_gggg(x1, x2, x3, x4, x5)  =  U6_gggg(x5)
memberB_in_gag(x1, x2, x3)  =  memberB_in_gag(x1, x3)
memberB_out_gag(x1, x2, x3)  =  memberB_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U7_gggg(x1, x2, x3, x4, x5)  =  U7_gggg(x2, x3, x4, x5)
U8_gggg(x1, x2, x3, x4, x5, x6)  =  U8_gggg(x2, x3, x4, x5, x6)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U9_gggg(x1, x2, x3, x4, x5)  =  U9_gggg(x5)
deleteD_in_gga(x1, x2, x3)  =  deleteD_in_gga(x1, x2)
deleteD_out_gga(x1, x2, x3)  =  deleteD_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
U10_gggg(x1, x2, x3, x4, x5, x6)  =  U10_gggg(x2, x3, x5, x6)
U11_gggg(x1, x2, x3, x4, x5)  =  U11_gggg(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

reachE_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachE_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachE_in_gggg(T46, T47, .(T48, T49), T14) → U5_gggg(T46, T47, T48, T49, T14, memberA_in_ggg(T46, T47, T49))
memberA_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberA_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberA_in_ggg(T79, T80, .(T81, T82)) → U1_ggg(T79, T80, T81, T82, memberA_in_ggg(T79, T80, T82))
U1_ggg(T79, T80, T81, T82, memberA_out_ggg(T79, T80, T82)) → memberA_out_ggg(T79, T80, .(T81, T82))
U5_gggg(T46, T47, T48, T49, T14, memberA_out_ggg(T46, T47, T49)) → reachE_out_gggg(T46, T47, .(T48, T49), T14)
reachE_in_gggg(T103, T104, T105, T106) → U6_gggg(T103, T104, T105, T106, memberB_in_gag(T103, X81, T105))
memberB_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberB_in_gag(T141, X121, .(T142, T143)) → U2_gag(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
U2_gag(T141, X121, T142, T143, memberB_out_gag(T141, X121, T143)) → memberB_out_gag(T141, X121, .(T142, T143))
U6_gggg(T103, T104, T105, T106, memberB_out_gag(T103, X81, T105)) → reachE_out_gggg(T103, T104, T105, T106)
reachE_in_gggg(T103, T104, T105, T106) → U7_gggg(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_gggg(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_gggg(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg(T166, .(T166, T167))
memberC_in_gg(T174, .(T175, T176)) → U3_gg(T174, T175, T176, memberC_in_gg(T174, T176))
U3_gg(T174, T175, T176, memberC_out_gg(T174, T176)) → memberC_out_gg(T174, .(T175, T176))
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U9_gggg(T103, T104, T105, T106, deleteD_in_gga(T113, T106, X82))
deleteD_in_gga(T198, .(T198, T199), T199) → deleteD_out_gga(T198, .(T198, T199), T199)
deleteD_in_gga(T206, .(T207, T208), .(T207, X191)) → U4_gga(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
U4_gga(T206, T207, T208, X191, deleteD_out_gga(T206, T208, X191)) → deleteD_out_gga(T206, .(T207, T208), .(T207, X191))
U9_gggg(T103, T104, T105, T106, deleteD_out_gga(T113, T106, X82)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_gggg(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_gggg(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → U11_gggg(T103, T104, T105, T106, reachE_in_gggg(T113, T104, T105, T185))
U11_gggg(T103, T104, T105, T106, reachE_out_gggg(T113, T104, T105, T185)) → reachE_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachE_in_gggg(x1, x2, x3, x4)  =  reachE_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachE_out_gggg(x1, x2, x3, x4)  =  reachE_out_gggg
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x6)
memberA_in_ggg(x1, x2, x3)  =  memberA_in_ggg(x1, x2, x3)
memberA_out_ggg(x1, x2, x3)  =  memberA_out_ggg
U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
U6_gggg(x1, x2, x3, x4, x5)  =  U6_gggg(x5)
memberB_in_gag(x1, x2, x3)  =  memberB_in_gag(x1, x3)
memberB_out_gag(x1, x2, x3)  =  memberB_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U7_gggg(x1, x2, x3, x4, x5)  =  U7_gggg(x2, x3, x4, x5)
U8_gggg(x1, x2, x3, x4, x5, x6)  =  U8_gggg(x2, x3, x4, x5, x6)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U9_gggg(x1, x2, x3, x4, x5)  =  U9_gggg(x5)
deleteD_in_gga(x1, x2, x3)  =  deleteD_in_gga(x1, x2)
deleteD_out_gga(x1, x2, x3)  =  deleteD_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
U10_gggg(x1, x2, x3, x4, x5, x6)  =  U10_gggg(x2, x3, x5, x6)
U11_gggg(x1, x2, x3, x4, x5)  =  U11_gggg(x5)

(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:

REACHE_IN_GGGG(T46, T47, .(T48, T49), T14) → U5_GGGG(T46, T47, T48, T49, T14, memberA_in_ggg(T46, T47, T49))
REACHE_IN_GGGG(T46, T47, .(T48, T49), T14) → MEMBERA_IN_GGG(T46, T47, T49)
MEMBERA_IN_GGG(T79, T80, .(T81, T82)) → U1_GGG(T79, T80, T81, T82, memberA_in_ggg(T79, T80, T82))
MEMBERA_IN_GGG(T79, T80, .(T81, T82)) → MEMBERA_IN_GGG(T79, T80, T82)
REACHE_IN_GGGG(T103, T104, T105, T106) → U6_GGGG(T103, T104, T105, T106, memberB_in_gag(T103, X81, T105))
REACHE_IN_GGGG(T103, T104, T105, T106) → MEMBERB_IN_GAG(T103, X81, T105)
MEMBERB_IN_GAG(T141, X121, .(T142, T143)) → U2_GAG(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
MEMBERB_IN_GAG(T141, X121, .(T142, T143)) → MEMBERB_IN_GAG(T141, X121, T143)
REACHE_IN_GGGG(T103, T104, T105, T106) → U7_GGGG(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_GGGG(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_GGGG(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
U7_GGGG(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → MEMBERC_IN_GG(T113, T106)
MEMBERC_IN_GG(T174, .(T175, T176)) → U3_GG(T174, T175, T176, memberC_in_gg(T174, T176))
MEMBERC_IN_GG(T174, .(T175, T176)) → MEMBERC_IN_GG(T174, T176)
U8_GGGG(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U9_GGGG(T103, T104, T105, T106, deleteD_in_gga(T113, T106, X82))
U8_GGGG(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → DELETED_IN_GGA(T113, T106, X82)
DELETED_IN_GGA(T206, .(T207, T208), .(T207, X191)) → U4_GGA(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
DELETED_IN_GGA(T206, .(T207, T208), .(T207, X191)) → DELETED_IN_GGA(T206, T208, X191)
U8_GGGG(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_GGGG(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_GGGG(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → U11_GGGG(T103, T104, T105, T106, reachE_in_gggg(T113, T104, T105, T185))
U10_GGGG(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → REACHE_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

reachE_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachE_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachE_in_gggg(T46, T47, .(T48, T49), T14) → U5_gggg(T46, T47, T48, T49, T14, memberA_in_ggg(T46, T47, T49))
memberA_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberA_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberA_in_ggg(T79, T80, .(T81, T82)) → U1_ggg(T79, T80, T81, T82, memberA_in_ggg(T79, T80, T82))
U1_ggg(T79, T80, T81, T82, memberA_out_ggg(T79, T80, T82)) → memberA_out_ggg(T79, T80, .(T81, T82))
U5_gggg(T46, T47, T48, T49, T14, memberA_out_ggg(T46, T47, T49)) → reachE_out_gggg(T46, T47, .(T48, T49), T14)
reachE_in_gggg(T103, T104, T105, T106) → U6_gggg(T103, T104, T105, T106, memberB_in_gag(T103, X81, T105))
memberB_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberB_in_gag(T141, X121, .(T142, T143)) → U2_gag(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
U2_gag(T141, X121, T142, T143, memberB_out_gag(T141, X121, T143)) → memberB_out_gag(T141, X121, .(T142, T143))
U6_gggg(T103, T104, T105, T106, memberB_out_gag(T103, X81, T105)) → reachE_out_gggg(T103, T104, T105, T106)
reachE_in_gggg(T103, T104, T105, T106) → U7_gggg(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_gggg(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_gggg(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg(T166, .(T166, T167))
memberC_in_gg(T174, .(T175, T176)) → U3_gg(T174, T175, T176, memberC_in_gg(T174, T176))
U3_gg(T174, T175, T176, memberC_out_gg(T174, T176)) → memberC_out_gg(T174, .(T175, T176))
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U9_gggg(T103, T104, T105, T106, deleteD_in_gga(T113, T106, X82))
deleteD_in_gga(T198, .(T198, T199), T199) → deleteD_out_gga(T198, .(T198, T199), T199)
deleteD_in_gga(T206, .(T207, T208), .(T207, X191)) → U4_gga(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
U4_gga(T206, T207, T208, X191, deleteD_out_gga(T206, T208, X191)) → deleteD_out_gga(T206, .(T207, T208), .(T207, X191))
U9_gggg(T103, T104, T105, T106, deleteD_out_gga(T113, T106, X82)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_gggg(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_gggg(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → U11_gggg(T103, T104, T105, T106, reachE_in_gggg(T113, T104, T105, T185))
U11_gggg(T103, T104, T105, T106, reachE_out_gggg(T113, T104, T105, T185)) → reachE_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachE_in_gggg(x1, x2, x3, x4)  =  reachE_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachE_out_gggg(x1, x2, x3, x4)  =  reachE_out_gggg
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x6)
memberA_in_ggg(x1, x2, x3)  =  memberA_in_ggg(x1, x2, x3)
memberA_out_ggg(x1, x2, x3)  =  memberA_out_ggg
U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
U6_gggg(x1, x2, x3, x4, x5)  =  U6_gggg(x5)
memberB_in_gag(x1, x2, x3)  =  memberB_in_gag(x1, x3)
memberB_out_gag(x1, x2, x3)  =  memberB_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U7_gggg(x1, x2, x3, x4, x5)  =  U7_gggg(x2, x3, x4, x5)
U8_gggg(x1, x2, x3, x4, x5, x6)  =  U8_gggg(x2, x3, x4, x5, x6)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U9_gggg(x1, x2, x3, x4, x5)  =  U9_gggg(x5)
deleteD_in_gga(x1, x2, x3)  =  deleteD_in_gga(x1, x2)
deleteD_out_gga(x1, x2, x3)  =  deleteD_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
U10_gggg(x1, x2, x3, x4, x5, x6)  =  U10_gggg(x2, x3, x5, x6)
U11_gggg(x1, x2, x3, x4, x5)  =  U11_gggg(x5)
REACHE_IN_GGGG(x1, x2, x3, x4)  =  REACHE_IN_GGGG(x1, x2, x3, x4)
U5_GGGG(x1, x2, x3, x4, x5, x6)  =  U5_GGGG(x6)
MEMBERA_IN_GGG(x1, x2, x3)  =  MEMBERA_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3, x4, x5)  =  U1_GGG(x5)
U6_GGGG(x1, x2, x3, x4, x5)  =  U6_GGGG(x5)
MEMBERB_IN_GAG(x1, x2, x3)  =  MEMBERB_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x5)
U7_GGGG(x1, x2, x3, x4, x5)  =  U7_GGGG(x2, x3, x4, x5)
U8_GGGG(x1, x2, x3, x4, x5, x6)  =  U8_GGGG(x2, x3, x4, x5, x6)
MEMBERC_IN_GG(x1, x2)  =  MEMBERC_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x4)
U9_GGGG(x1, x2, x3, x4, x5)  =  U9_GGGG(x5)
DELETED_IN_GGA(x1, x2, x3)  =  DELETED_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x2, x5)
U10_GGGG(x1, x2, x3, x4, x5, x6)  =  U10_GGGG(x2, x3, x5, x6)
U11_GGGG(x1, x2, x3, x4, x5)  =  U11_GGGG(x5)

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

(6) Obligation:

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

REACHE_IN_GGGG(T46, T47, .(T48, T49), T14) → U5_GGGG(T46, T47, T48, T49, T14, memberA_in_ggg(T46, T47, T49))
REACHE_IN_GGGG(T46, T47, .(T48, T49), T14) → MEMBERA_IN_GGG(T46, T47, T49)
MEMBERA_IN_GGG(T79, T80, .(T81, T82)) → U1_GGG(T79, T80, T81, T82, memberA_in_ggg(T79, T80, T82))
MEMBERA_IN_GGG(T79, T80, .(T81, T82)) → MEMBERA_IN_GGG(T79, T80, T82)
REACHE_IN_GGGG(T103, T104, T105, T106) → U6_GGGG(T103, T104, T105, T106, memberB_in_gag(T103, X81, T105))
REACHE_IN_GGGG(T103, T104, T105, T106) → MEMBERB_IN_GAG(T103, X81, T105)
MEMBERB_IN_GAG(T141, X121, .(T142, T143)) → U2_GAG(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
MEMBERB_IN_GAG(T141, X121, .(T142, T143)) → MEMBERB_IN_GAG(T141, X121, T143)
REACHE_IN_GGGG(T103, T104, T105, T106) → U7_GGGG(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_GGGG(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_GGGG(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
U7_GGGG(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → MEMBERC_IN_GG(T113, T106)
MEMBERC_IN_GG(T174, .(T175, T176)) → U3_GG(T174, T175, T176, memberC_in_gg(T174, T176))
MEMBERC_IN_GG(T174, .(T175, T176)) → MEMBERC_IN_GG(T174, T176)
U8_GGGG(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U9_GGGG(T103, T104, T105, T106, deleteD_in_gga(T113, T106, X82))
U8_GGGG(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → DELETED_IN_GGA(T113, T106, X82)
DELETED_IN_GGA(T206, .(T207, T208), .(T207, X191)) → U4_GGA(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
DELETED_IN_GGA(T206, .(T207, T208), .(T207, X191)) → DELETED_IN_GGA(T206, T208, X191)
U8_GGGG(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_GGGG(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_GGGG(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → U11_GGGG(T103, T104, T105, T106, reachE_in_gggg(T113, T104, T105, T185))
U10_GGGG(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → REACHE_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

reachE_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachE_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachE_in_gggg(T46, T47, .(T48, T49), T14) → U5_gggg(T46, T47, T48, T49, T14, memberA_in_ggg(T46, T47, T49))
memberA_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberA_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberA_in_ggg(T79, T80, .(T81, T82)) → U1_ggg(T79, T80, T81, T82, memberA_in_ggg(T79, T80, T82))
U1_ggg(T79, T80, T81, T82, memberA_out_ggg(T79, T80, T82)) → memberA_out_ggg(T79, T80, .(T81, T82))
U5_gggg(T46, T47, T48, T49, T14, memberA_out_ggg(T46, T47, T49)) → reachE_out_gggg(T46, T47, .(T48, T49), T14)
reachE_in_gggg(T103, T104, T105, T106) → U6_gggg(T103, T104, T105, T106, memberB_in_gag(T103, X81, T105))
memberB_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberB_in_gag(T141, X121, .(T142, T143)) → U2_gag(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
U2_gag(T141, X121, T142, T143, memberB_out_gag(T141, X121, T143)) → memberB_out_gag(T141, X121, .(T142, T143))
U6_gggg(T103, T104, T105, T106, memberB_out_gag(T103, X81, T105)) → reachE_out_gggg(T103, T104, T105, T106)
reachE_in_gggg(T103, T104, T105, T106) → U7_gggg(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_gggg(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_gggg(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg(T166, .(T166, T167))
memberC_in_gg(T174, .(T175, T176)) → U3_gg(T174, T175, T176, memberC_in_gg(T174, T176))
U3_gg(T174, T175, T176, memberC_out_gg(T174, T176)) → memberC_out_gg(T174, .(T175, T176))
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U9_gggg(T103, T104, T105, T106, deleteD_in_gga(T113, T106, X82))
deleteD_in_gga(T198, .(T198, T199), T199) → deleteD_out_gga(T198, .(T198, T199), T199)
deleteD_in_gga(T206, .(T207, T208), .(T207, X191)) → U4_gga(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
U4_gga(T206, T207, T208, X191, deleteD_out_gga(T206, T208, X191)) → deleteD_out_gga(T206, .(T207, T208), .(T207, X191))
U9_gggg(T103, T104, T105, T106, deleteD_out_gga(T113, T106, X82)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_gggg(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_gggg(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → U11_gggg(T103, T104, T105, T106, reachE_in_gggg(T113, T104, T105, T185))
U11_gggg(T103, T104, T105, T106, reachE_out_gggg(T113, T104, T105, T185)) → reachE_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachE_in_gggg(x1, x2, x3, x4)  =  reachE_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachE_out_gggg(x1, x2, x3, x4)  =  reachE_out_gggg
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x6)
memberA_in_ggg(x1, x2, x3)  =  memberA_in_ggg(x1, x2, x3)
memberA_out_ggg(x1, x2, x3)  =  memberA_out_ggg
U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
U6_gggg(x1, x2, x3, x4, x5)  =  U6_gggg(x5)
memberB_in_gag(x1, x2, x3)  =  memberB_in_gag(x1, x3)
memberB_out_gag(x1, x2, x3)  =  memberB_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U7_gggg(x1, x2, x3, x4, x5)  =  U7_gggg(x2, x3, x4, x5)
U8_gggg(x1, x2, x3, x4, x5, x6)  =  U8_gggg(x2, x3, x4, x5, x6)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U9_gggg(x1, x2, x3, x4, x5)  =  U9_gggg(x5)
deleteD_in_gga(x1, x2, x3)  =  deleteD_in_gga(x1, x2)
deleteD_out_gga(x1, x2, x3)  =  deleteD_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
U10_gggg(x1, x2, x3, x4, x5, x6)  =  U10_gggg(x2, x3, x5, x6)
U11_gggg(x1, x2, x3, x4, x5)  =  U11_gggg(x5)
REACHE_IN_GGGG(x1, x2, x3, x4)  =  REACHE_IN_GGGG(x1, x2, x3, x4)
U5_GGGG(x1, x2, x3, x4, x5, x6)  =  U5_GGGG(x6)
MEMBERA_IN_GGG(x1, x2, x3)  =  MEMBERA_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3, x4, x5)  =  U1_GGG(x5)
U6_GGGG(x1, x2, x3, x4, x5)  =  U6_GGGG(x5)
MEMBERB_IN_GAG(x1, x2, x3)  =  MEMBERB_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x5)
U7_GGGG(x1, x2, x3, x4, x5)  =  U7_GGGG(x2, x3, x4, x5)
U8_GGGG(x1, x2, x3, x4, x5, x6)  =  U8_GGGG(x2, x3, x4, x5, x6)
MEMBERC_IN_GG(x1, x2)  =  MEMBERC_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x4)
U9_GGGG(x1, x2, x3, x4, x5)  =  U9_GGGG(x5)
DELETED_IN_GGA(x1, x2, x3)  =  DELETED_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x2, x5)
U10_GGGG(x1, x2, x3, x4, x5, x6)  =  U10_GGGG(x2, x3, x5, x6)
U11_GGGG(x1, x2, x3, x4, x5)  =  U11_GGGG(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 12 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

DELETED_IN_GGA(T206, .(T207, T208), .(T207, X191)) → DELETED_IN_GGA(T206, T208, X191)

The TRS R consists of the following rules:

reachE_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachE_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachE_in_gggg(T46, T47, .(T48, T49), T14) → U5_gggg(T46, T47, T48, T49, T14, memberA_in_ggg(T46, T47, T49))
memberA_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberA_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberA_in_ggg(T79, T80, .(T81, T82)) → U1_ggg(T79, T80, T81, T82, memberA_in_ggg(T79, T80, T82))
U1_ggg(T79, T80, T81, T82, memberA_out_ggg(T79, T80, T82)) → memberA_out_ggg(T79, T80, .(T81, T82))
U5_gggg(T46, T47, T48, T49, T14, memberA_out_ggg(T46, T47, T49)) → reachE_out_gggg(T46, T47, .(T48, T49), T14)
reachE_in_gggg(T103, T104, T105, T106) → U6_gggg(T103, T104, T105, T106, memberB_in_gag(T103, X81, T105))
memberB_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberB_in_gag(T141, X121, .(T142, T143)) → U2_gag(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
U2_gag(T141, X121, T142, T143, memberB_out_gag(T141, X121, T143)) → memberB_out_gag(T141, X121, .(T142, T143))
U6_gggg(T103, T104, T105, T106, memberB_out_gag(T103, X81, T105)) → reachE_out_gggg(T103, T104, T105, T106)
reachE_in_gggg(T103, T104, T105, T106) → U7_gggg(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_gggg(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_gggg(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg(T166, .(T166, T167))
memberC_in_gg(T174, .(T175, T176)) → U3_gg(T174, T175, T176, memberC_in_gg(T174, T176))
U3_gg(T174, T175, T176, memberC_out_gg(T174, T176)) → memberC_out_gg(T174, .(T175, T176))
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U9_gggg(T103, T104, T105, T106, deleteD_in_gga(T113, T106, X82))
deleteD_in_gga(T198, .(T198, T199), T199) → deleteD_out_gga(T198, .(T198, T199), T199)
deleteD_in_gga(T206, .(T207, T208), .(T207, X191)) → U4_gga(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
U4_gga(T206, T207, T208, X191, deleteD_out_gga(T206, T208, X191)) → deleteD_out_gga(T206, .(T207, T208), .(T207, X191))
U9_gggg(T103, T104, T105, T106, deleteD_out_gga(T113, T106, X82)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_gggg(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_gggg(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → U11_gggg(T103, T104, T105, T106, reachE_in_gggg(T113, T104, T105, T185))
U11_gggg(T103, T104, T105, T106, reachE_out_gggg(T113, T104, T105, T185)) → reachE_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachE_in_gggg(x1, x2, x3, x4)  =  reachE_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachE_out_gggg(x1, x2, x3, x4)  =  reachE_out_gggg
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x6)
memberA_in_ggg(x1, x2, x3)  =  memberA_in_ggg(x1, x2, x3)
memberA_out_ggg(x1, x2, x3)  =  memberA_out_ggg
U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
U6_gggg(x1, x2, x3, x4, x5)  =  U6_gggg(x5)
memberB_in_gag(x1, x2, x3)  =  memberB_in_gag(x1, x3)
memberB_out_gag(x1, x2, x3)  =  memberB_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U7_gggg(x1, x2, x3, x4, x5)  =  U7_gggg(x2, x3, x4, x5)
U8_gggg(x1, x2, x3, x4, x5, x6)  =  U8_gggg(x2, x3, x4, x5, x6)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U9_gggg(x1, x2, x3, x4, x5)  =  U9_gggg(x5)
deleteD_in_gga(x1, x2, x3)  =  deleteD_in_gga(x1, x2)
deleteD_out_gga(x1, x2, x3)  =  deleteD_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
U10_gggg(x1, x2, x3, x4, x5, x6)  =  U10_gggg(x2, x3, x5, x6)
U11_gggg(x1, x2, x3, x4, x5)  =  U11_gggg(x5)
DELETED_IN_GGA(x1, x2, x3)  =  DELETED_IN_GGA(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:

DELETED_IN_GGA(T206, .(T207, T208), .(T207, X191)) → DELETED_IN_GGA(T206, T208, X191)

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

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

(12) PiDPToQDPProof (SOUND 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:

DELETED_IN_GGA(T206, .(T207, T208)) → DELETED_IN_GGA(T206, T208)

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:

• DELETED_IN_GGA(T206, .(T207, T208)) → DELETED_IN_GGA(T206, T208)
  The graph contains the following edges 1 >= 1, 2 > 2

(15) YES

(16) Obligation:

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

MEMBERC_IN_GG(T174, .(T175, T176)) → MEMBERC_IN_GG(T174, T176)

The TRS R consists of the following rules:

reachE_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachE_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachE_in_gggg(T46, T47, .(T48, T49), T14) → U5_gggg(T46, T47, T48, T49, T14, memberA_in_ggg(T46, T47, T49))
memberA_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberA_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberA_in_ggg(T79, T80, .(T81, T82)) → U1_ggg(T79, T80, T81, T82, memberA_in_ggg(T79, T80, T82))
U1_ggg(T79, T80, T81, T82, memberA_out_ggg(T79, T80, T82)) → memberA_out_ggg(T79, T80, .(T81, T82))
U5_gggg(T46, T47, T48, T49, T14, memberA_out_ggg(T46, T47, T49)) → reachE_out_gggg(T46, T47, .(T48, T49), T14)
reachE_in_gggg(T103, T104, T105, T106) → U6_gggg(T103, T104, T105, T106, memberB_in_gag(T103, X81, T105))
memberB_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberB_in_gag(T141, X121, .(T142, T143)) → U2_gag(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
U2_gag(T141, X121, T142, T143, memberB_out_gag(T141, X121, T143)) → memberB_out_gag(T141, X121, .(T142, T143))
U6_gggg(T103, T104, T105, T106, memberB_out_gag(T103, X81, T105)) → reachE_out_gggg(T103, T104, T105, T106)
reachE_in_gggg(T103, T104, T105, T106) → U7_gggg(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_gggg(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_gggg(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg(T166, .(T166, T167))
memberC_in_gg(T174, .(T175, T176)) → U3_gg(T174, T175, T176, memberC_in_gg(T174, T176))
U3_gg(T174, T175, T176, memberC_out_gg(T174, T176)) → memberC_out_gg(T174, .(T175, T176))
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U9_gggg(T103, T104, T105, T106, deleteD_in_gga(T113, T106, X82))
deleteD_in_gga(T198, .(T198, T199), T199) → deleteD_out_gga(T198, .(T198, T199), T199)
deleteD_in_gga(T206, .(T207, T208), .(T207, X191)) → U4_gga(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
U4_gga(T206, T207, T208, X191, deleteD_out_gga(T206, T208, X191)) → deleteD_out_gga(T206, .(T207, T208), .(T207, X191))
U9_gggg(T103, T104, T105, T106, deleteD_out_gga(T113, T106, X82)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_gggg(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_gggg(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → U11_gggg(T103, T104, T105, T106, reachE_in_gggg(T113, T104, T105, T185))
U11_gggg(T103, T104, T105, T106, reachE_out_gggg(T113, T104, T105, T185)) → reachE_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachE_in_gggg(x1, x2, x3, x4)  =  reachE_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachE_out_gggg(x1, x2, x3, x4)  =  reachE_out_gggg
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x6)
memberA_in_ggg(x1, x2, x3)  =  memberA_in_ggg(x1, x2, x3)
memberA_out_ggg(x1, x2, x3)  =  memberA_out_ggg
U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
U6_gggg(x1, x2, x3, x4, x5)  =  U6_gggg(x5)
memberB_in_gag(x1, x2, x3)  =  memberB_in_gag(x1, x3)
memberB_out_gag(x1, x2, x3)  =  memberB_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U7_gggg(x1, x2, x3, x4, x5)  =  U7_gggg(x2, x3, x4, x5)
U8_gggg(x1, x2, x3, x4, x5, x6)  =  U8_gggg(x2, x3, x4, x5, x6)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U9_gggg(x1, x2, x3, x4, x5)  =  U9_gggg(x5)
deleteD_in_gga(x1, x2, x3)  =  deleteD_in_gga(x1, x2)
deleteD_out_gga(x1, x2, x3)  =  deleteD_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
U10_gggg(x1, x2, x3, x4, x5, x6)  =  U10_gggg(x2, x3, x5, x6)
U11_gggg(x1, x2, x3, x4, x5)  =  U11_gggg(x5)
MEMBERC_IN_GG(x1, x2)  =  MEMBERC_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:

MEMBERC_IN_GG(T174, .(T175, T176)) → MEMBERC_IN_GG(T174, T176)

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:

MEMBERC_IN_GG(T174, .(T175, T176)) → MEMBERC_IN_GG(T174, T176)

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:

• MEMBERC_IN_GG(T174, .(T175, T176)) → MEMBERC_IN_GG(T174, T176)
  The graph contains the following edges 1 >= 1, 2 > 2

(22) YES

(23) Obligation:

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

MEMBERB_IN_GAG(T141, X121, .(T142, T143)) → MEMBERB_IN_GAG(T141, X121, T143)

The TRS R consists of the following rules:

reachE_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachE_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachE_in_gggg(T46, T47, .(T48, T49), T14) → U5_gggg(T46, T47, T48, T49, T14, memberA_in_ggg(T46, T47, T49))
memberA_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberA_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberA_in_ggg(T79, T80, .(T81, T82)) → U1_ggg(T79, T80, T81, T82, memberA_in_ggg(T79, T80, T82))
U1_ggg(T79, T80, T81, T82, memberA_out_ggg(T79, T80, T82)) → memberA_out_ggg(T79, T80, .(T81, T82))
U5_gggg(T46, T47, T48, T49, T14, memberA_out_ggg(T46, T47, T49)) → reachE_out_gggg(T46, T47, .(T48, T49), T14)
reachE_in_gggg(T103, T104, T105, T106) → U6_gggg(T103, T104, T105, T106, memberB_in_gag(T103, X81, T105))
memberB_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberB_in_gag(T141, X121, .(T142, T143)) → U2_gag(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
U2_gag(T141, X121, T142, T143, memberB_out_gag(T141, X121, T143)) → memberB_out_gag(T141, X121, .(T142, T143))
U6_gggg(T103, T104, T105, T106, memberB_out_gag(T103, X81, T105)) → reachE_out_gggg(T103, T104, T105, T106)
reachE_in_gggg(T103, T104, T105, T106) → U7_gggg(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_gggg(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_gggg(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg(T166, .(T166, T167))
memberC_in_gg(T174, .(T175, T176)) → U3_gg(T174, T175, T176, memberC_in_gg(T174, T176))
U3_gg(T174, T175, T176, memberC_out_gg(T174, T176)) → memberC_out_gg(T174, .(T175, T176))
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U9_gggg(T103, T104, T105, T106, deleteD_in_gga(T113, T106, X82))
deleteD_in_gga(T198, .(T198, T199), T199) → deleteD_out_gga(T198, .(T198, T199), T199)
deleteD_in_gga(T206, .(T207, T208), .(T207, X191)) → U4_gga(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
U4_gga(T206, T207, T208, X191, deleteD_out_gga(T206, T208, X191)) → deleteD_out_gga(T206, .(T207, T208), .(T207, X191))
U9_gggg(T103, T104, T105, T106, deleteD_out_gga(T113, T106, X82)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_gggg(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_gggg(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → U11_gggg(T103, T104, T105, T106, reachE_in_gggg(T113, T104, T105, T185))
U11_gggg(T103, T104, T105, T106, reachE_out_gggg(T113, T104, T105, T185)) → reachE_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachE_in_gggg(x1, x2, x3, x4)  =  reachE_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachE_out_gggg(x1, x2, x3, x4)  =  reachE_out_gggg
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x6)
memberA_in_ggg(x1, x2, x3)  =  memberA_in_ggg(x1, x2, x3)
memberA_out_ggg(x1, x2, x3)  =  memberA_out_ggg
U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
U6_gggg(x1, x2, x3, x4, x5)  =  U6_gggg(x5)
memberB_in_gag(x1, x2, x3)  =  memberB_in_gag(x1, x3)
memberB_out_gag(x1, x2, x3)  =  memberB_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U7_gggg(x1, x2, x3, x4, x5)  =  U7_gggg(x2, x3, x4, x5)
U8_gggg(x1, x2, x3, x4, x5, x6)  =  U8_gggg(x2, x3, x4, x5, x6)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U9_gggg(x1, x2, x3, x4, x5)  =  U9_gggg(x5)
deleteD_in_gga(x1, x2, x3)  =  deleteD_in_gga(x1, x2)
deleteD_out_gga(x1, x2, x3)  =  deleteD_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
U10_gggg(x1, x2, x3, x4, x5, x6)  =  U10_gggg(x2, x3, x5, x6)
U11_gggg(x1, x2, x3, x4, x5)  =  U11_gggg(x5)
MEMBERB_IN_GAG(x1, x2, x3)  =  MEMBERB_IN_GAG(x1, x3)

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:

MEMBERB_IN_GAG(T141, X121, .(T142, T143)) → MEMBERB_IN_GAG(T141, X121, T143)

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

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:

MEMBERB_IN_GAG(T141, .(T142, T143)) → MEMBERB_IN_GAG(T141, T143)

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:

• MEMBERB_IN_GAG(T141, .(T142, T143)) → MEMBERB_IN_GAG(T141, T143)
  The graph contains the following edges 1 >= 1, 2 > 2

(29) YES

(30) Obligation:

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

MEMBERA_IN_GGG(T79, T80, .(T81, T82)) → MEMBERA_IN_GGG(T79, T80, T82)

The TRS R consists of the following rules:

reachE_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachE_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachE_in_gggg(T46, T47, .(T48, T49), T14) → U5_gggg(T46, T47, T48, T49, T14, memberA_in_ggg(T46, T47, T49))
memberA_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberA_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberA_in_ggg(T79, T80, .(T81, T82)) → U1_ggg(T79, T80, T81, T82, memberA_in_ggg(T79, T80, T82))
U1_ggg(T79, T80, T81, T82, memberA_out_ggg(T79, T80, T82)) → memberA_out_ggg(T79, T80, .(T81, T82))
U5_gggg(T46, T47, T48, T49, T14, memberA_out_ggg(T46, T47, T49)) → reachE_out_gggg(T46, T47, .(T48, T49), T14)
reachE_in_gggg(T103, T104, T105, T106) → U6_gggg(T103, T104, T105, T106, memberB_in_gag(T103, X81, T105))
memberB_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberB_in_gag(T141, X121, .(T142, T143)) → U2_gag(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
U2_gag(T141, X121, T142, T143, memberB_out_gag(T141, X121, T143)) → memberB_out_gag(T141, X121, .(T142, T143))
U6_gggg(T103, T104, T105, T106, memberB_out_gag(T103, X81, T105)) → reachE_out_gggg(T103, T104, T105, T106)
reachE_in_gggg(T103, T104, T105, T106) → U7_gggg(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_gggg(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_gggg(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg(T166, .(T166, T167))
memberC_in_gg(T174, .(T175, T176)) → U3_gg(T174, T175, T176, memberC_in_gg(T174, T176))
U3_gg(T174, T175, T176, memberC_out_gg(T174, T176)) → memberC_out_gg(T174, .(T175, T176))
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U9_gggg(T103, T104, T105, T106, deleteD_in_gga(T113, T106, X82))
deleteD_in_gga(T198, .(T198, T199), T199) → deleteD_out_gga(T198, .(T198, T199), T199)
deleteD_in_gga(T206, .(T207, T208), .(T207, X191)) → U4_gga(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
U4_gga(T206, T207, T208, X191, deleteD_out_gga(T206, T208, X191)) → deleteD_out_gga(T206, .(T207, T208), .(T207, X191))
U9_gggg(T103, T104, T105, T106, deleteD_out_gga(T113, T106, X82)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_gggg(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_gggg(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → U11_gggg(T103, T104, T105, T106, reachE_in_gggg(T113, T104, T105, T185))
U11_gggg(T103, T104, T105, T106, reachE_out_gggg(T113, T104, T105, T185)) → reachE_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachE_in_gggg(x1, x2, x3, x4)  =  reachE_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachE_out_gggg(x1, x2, x3, x4)  =  reachE_out_gggg
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x6)
memberA_in_ggg(x1, x2, x3)  =  memberA_in_ggg(x1, x2, x3)
memberA_out_ggg(x1, x2, x3)  =  memberA_out_ggg
U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
U6_gggg(x1, x2, x3, x4, x5)  =  U6_gggg(x5)
memberB_in_gag(x1, x2, x3)  =  memberB_in_gag(x1, x3)
memberB_out_gag(x1, x2, x3)  =  memberB_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U7_gggg(x1, x2, x3, x4, x5)  =  U7_gggg(x2, x3, x4, x5)
U8_gggg(x1, x2, x3, x4, x5, x6)  =  U8_gggg(x2, x3, x4, x5, x6)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U9_gggg(x1, x2, x3, x4, x5)  =  U9_gggg(x5)
deleteD_in_gga(x1, x2, x3)  =  deleteD_in_gga(x1, x2)
deleteD_out_gga(x1, x2, x3)  =  deleteD_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
U10_gggg(x1, x2, x3, x4, x5, x6)  =  U10_gggg(x2, x3, x5, x6)
U11_gggg(x1, x2, x3, x4, x5)  =  U11_gggg(x5)
MEMBERA_IN_GGG(x1, x2, x3)  =  MEMBERA_IN_GGG(x1, x2, x3)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

MEMBERA_IN_GGG(T79, T80, .(T81, T82)) → MEMBERA_IN_GGG(T79, T80, T82)

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

(33) PiDPToQDPProof (EQUIVALENT transformation)

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

(34) Obligation:

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

MEMBERA_IN_GGG(T79, T80, .(T81, T82)) → MEMBERA_IN_GGG(T79, T80, T82)

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

(35) 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:

• MEMBERA_IN_GGG(T79, T80, .(T81, T82)) → MEMBERA_IN_GGG(T79, T80, T82)
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3

(36) YES

(37) Obligation:

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

REACHE_IN_GGGG(T103, T104, T105, T106) → U7_GGGG(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_GGGG(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_GGGG(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
U8_GGGG(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_GGGG(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_GGGG(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → REACHE_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

reachE_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachE_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachE_in_gggg(T46, T47, .(T48, T49), T14) → U5_gggg(T46, T47, T48, T49, T14, memberA_in_ggg(T46, T47, T49))
memberA_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberA_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberA_in_ggg(T79, T80, .(T81, T82)) → U1_ggg(T79, T80, T81, T82, memberA_in_ggg(T79, T80, T82))
U1_ggg(T79, T80, T81, T82, memberA_out_ggg(T79, T80, T82)) → memberA_out_ggg(T79, T80, .(T81, T82))
U5_gggg(T46, T47, T48, T49, T14, memberA_out_ggg(T46, T47, T49)) → reachE_out_gggg(T46, T47, .(T48, T49), T14)
reachE_in_gggg(T103, T104, T105, T106) → U6_gggg(T103, T104, T105, T106, memberB_in_gag(T103, X81, T105))
memberB_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberB_in_gag(T141, X121, .(T142, T143)) → U2_gag(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
U2_gag(T141, X121, T142, T143, memberB_out_gag(T141, X121, T143)) → memberB_out_gag(T141, X121, .(T142, T143))
U6_gggg(T103, T104, T105, T106, memberB_out_gag(T103, X81, T105)) → reachE_out_gggg(T103, T104, T105, T106)
reachE_in_gggg(T103, T104, T105, T106) → U7_gggg(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_gggg(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_gggg(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg(T166, .(T166, T167))
memberC_in_gg(T174, .(T175, T176)) → U3_gg(T174, T175, T176, memberC_in_gg(T174, T176))
U3_gg(T174, T175, T176, memberC_out_gg(T174, T176)) → memberC_out_gg(T174, .(T175, T176))
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U9_gggg(T103, T104, T105, T106, deleteD_in_gga(T113, T106, X82))
deleteD_in_gga(T198, .(T198, T199), T199) → deleteD_out_gga(T198, .(T198, T199), T199)
deleteD_in_gga(T206, .(T207, T208), .(T207, X191)) → U4_gga(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
U4_gga(T206, T207, T208, X191, deleteD_out_gga(T206, T208, X191)) → deleteD_out_gga(T206, .(T207, T208), .(T207, X191))
U9_gggg(T103, T104, T105, T106, deleteD_out_gga(T113, T106, X82)) → reachE_out_gggg(T103, T104, T105, T106)
U8_gggg(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_gggg(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_gggg(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → U11_gggg(T103, T104, T105, T106, reachE_in_gggg(T113, T104, T105, T185))
U11_gggg(T103, T104, T105, T106, reachE_out_gggg(T113, T104, T105, T185)) → reachE_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachE_in_gggg(x1, x2, x3, x4)  =  reachE_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachE_out_gggg(x1, x2, x3, x4)  =  reachE_out_gggg
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x6)
memberA_in_ggg(x1, x2, x3)  =  memberA_in_ggg(x1, x2, x3)
memberA_out_ggg(x1, x2, x3)  =  memberA_out_ggg
U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
U6_gggg(x1, x2, x3, x4, x5)  =  U6_gggg(x5)
memberB_in_gag(x1, x2, x3)  =  memberB_in_gag(x1, x3)
memberB_out_gag(x1, x2, x3)  =  memberB_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U7_gggg(x1, x2, x3, x4, x5)  =  U7_gggg(x2, x3, x4, x5)
U8_gggg(x1, x2, x3, x4, x5, x6)  =  U8_gggg(x2, x3, x4, x5, x6)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U9_gggg(x1, x2, x3, x4, x5)  =  U9_gggg(x5)
deleteD_in_gga(x1, x2, x3)  =  deleteD_in_gga(x1, x2)
deleteD_out_gga(x1, x2, x3)  =  deleteD_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
U10_gggg(x1, x2, x3, x4, x5, x6)  =  U10_gggg(x2, x3, x5, x6)
U11_gggg(x1, x2, x3, x4, x5)  =  U11_gggg(x5)
REACHE_IN_GGGG(x1, x2, x3, x4)  =  REACHE_IN_GGGG(x1, x2, x3, x4)
U7_GGGG(x1, x2, x3, x4, x5)  =  U7_GGGG(x2, x3, x4, x5)
U8_GGGG(x1, x2, x3, x4, x5, x6)  =  U8_GGGG(x2, x3, x4, x5, x6)
U10_GGGG(x1, x2, x3, x4, x5, x6)  =  U10_GGGG(x2, x3, x5, x6)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

REACHE_IN_GGGG(T103, T104, T105, T106) → U7_GGGG(T103, T104, T105, T106, memberB_in_gag(T103, T113, T105))
U7_GGGG(T103, T104, T105, T106, memberB_out_gag(T103, T113, T105)) → U8_GGGG(T103, T104, T105, T106, T113, memberC_in_gg(T113, T106))
U8_GGGG(T103, T104, T105, T106, T113, memberC_out_gg(T113, T106)) → U10_GGGG(T103, T104, T105, T106, T113, deleteD_in_gga(T113, T106, T185))
U10_GGGG(T103, T104, T105, T106, T113, deleteD_out_gga(T113, T106, T185)) → REACHE_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

memberB_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberB_in_gag(T141, X121, .(T142, T143)) → U2_gag(T141, X121, T142, T143, memberB_in_gag(T141, X121, T143))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg(T166, .(T166, T167))
memberC_in_gg(T174, .(T175, T176)) → U3_gg(T174, T175, T176, memberC_in_gg(T174, T176))
deleteD_in_gga(T198, .(T198, T199), T199) → deleteD_out_gga(T198, .(T198, T199), T199)
deleteD_in_gga(T206, .(T207, T208), .(T207, X191)) → U4_gga(T206, T207, T208, X191, deleteD_in_gga(T206, T208, X191))
U2_gag(T141, X121, T142, T143, memberB_out_gag(T141, X121, T143)) → memberB_out_gag(T141, X121, .(T142, T143))
U3_gg(T174, T175, T176, memberC_out_gg(T174, T176)) → memberC_out_gg(T174, .(T175, T176))
U4_gga(T206, T207, T208, X191, deleteD_out_gga(T206, T208, X191)) → deleteD_out_gga(T206, .(T207, T208), .(T207, X191))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
memberB_in_gag(x1, x2, x3)  =  memberB_in_gag(x1, x3)
memberB_out_gag(x1, x2, x3)  =  memberB_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
memberC_in_gg(x1, x2)  =  memberC_in_gg(x1, x2)
memberC_out_gg(x1, x2)  =  memberC_out_gg
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
deleteD_in_gga(x1, x2, x3)  =  deleteD_in_gga(x1, x2)
deleteD_out_gga(x1, x2, x3)  =  deleteD_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x5)
REACHE_IN_GGGG(x1, x2, x3, x4)  =  REACHE_IN_GGGG(x1, x2, x3, x4)
U7_GGGG(x1, x2, x3, x4, x5)  =  U7_GGGG(x2, x3, x4, x5)
U8_GGGG(x1, x2, x3, x4, x5, x6)  =  U8_GGGG(x2, x3, x4, x5, x6)
U10_GGGG(x1, x2, x3, x4, x5, x6)  =  U10_GGGG(x2, x3, x5, x6)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

REACHE_IN_GGGG(T103, T104, T105, T106) → U7_GGGG(T104, T105, T106, memberB_in_gag(T103, T105))
U7_GGGG(T104, T105, T106, memberB_out_gag(T113)) → U8_GGGG(T104, T105, T106, T113, memberC_in_gg(T113, T106))
U8_GGGG(T104, T105, T106, T113, memberC_out_gg) → U10_GGGG(T104, T105, T113, deleteD_in_gga(T113, T106))
U10_GGGG(T104, T105, T113, deleteD_out_gga(T185)) → REACHE_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

memberB_in_gag(T132, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T133)
memberB_in_gag(T141, .(T142, T143)) → U2_gag(memberB_in_gag(T141, T143))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg
memberC_in_gg(T174, .(T175, T176)) → U3_gg(memberC_in_gg(T174, T176))
deleteD_in_gga(T198, .(T198, T199)) → deleteD_out_gga(T199)
deleteD_in_gga(T206, .(T207, T208)) → U4_gga(T207, deleteD_in_gga(T206, T208))
U2_gag(memberB_out_gag(X121)) → memberB_out_gag(X121)
U3_gg(memberC_out_gg) → memberC_out_gg
U4_gga(T207, deleteD_out_gga(X191)) → deleteD_out_gga(.(T207, X191))

The set Q consists of the following terms:

memberB_in_gag(x0, x1)
memberC_in_gg(x0, x1)
deleteD_in_gga(x0, x1)
U2_gag(x0)
U3_gg(x0)
U4_gga(x0, x1)

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

REACHE_IN_GGGG(T103, T104, T105, T106) → U7_GGGG(T104, T105, T106, memberB_in_gag(T103, T105))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂   
POL(REACHE_IN_GGGG(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U10_GGGG(x₁, x₂, x₃, x₄)) = x₄   
POL(U2_gag(x₁)) = 0   
POL(U3_gg(x₁)) = 0   
POL(U4_gga(x₁, x₂)) = 1 + x₂   
POL(U7_GGGG(x₁, x₂, x₃, x₄)) = x₃   
POL(U8_GGGG(x₁, x₂, x₃, x₄, x₅)) = x₃   
POL([]) = 0   
POL(deleteD_in_gga(x₁, x₂)) = x₂   
POL(deleteD_out_gga(x₁)) = 1 + x₁   
POL(memberB_in_gag(x₁, x₂)) = 0   
POL(memberB_out_gag(x₁)) = 0   
POL(memberC_in_gg(x₁, x₂)) = 0   
POL(memberC_out_gg) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

deleteD_in_gga(T198, .(T198, T199)) → deleteD_out_gga(T199)
deleteD_in_gga(T206, .(T207, T208)) → U4_gga(T207, deleteD_in_gga(T206, T208))
U4_gga(T207, deleteD_out_gga(X191)) → deleteD_out_gga(.(T207, X191))

(43) Obligation:

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

U7_GGGG(T104, T105, T106, memberB_out_gag(T113)) → U8_GGGG(T104, T105, T106, T113, memberC_in_gg(T113, T106))
U8_GGGG(T104, T105, T106, T113, memberC_out_gg) → U10_GGGG(T104, T105, T113, deleteD_in_gga(T113, T106))
U10_GGGG(T104, T105, T113, deleteD_out_gga(T185)) → REACHE_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

memberB_in_gag(T132, .(.(T132, .(T133, [])), T134)) → memberB_out_gag(T133)
memberB_in_gag(T141, .(T142, T143)) → U2_gag(memberB_in_gag(T141, T143))
memberC_in_gg(T166, .(T166, T167)) → memberC_out_gg
memberC_in_gg(T174, .(T175, T176)) → U3_gg(memberC_in_gg(T174, T176))
deleteD_in_gga(T198, .(T198, T199)) → deleteD_out_gga(T199)
deleteD_in_gga(T206, .(T207, T208)) → U4_gga(T207, deleteD_in_gga(T206, T208))
U2_gag(memberB_out_gag(X121)) → memberB_out_gag(X121)
U3_gg(memberC_out_gg) → memberC_out_gg
U4_gga(T207, deleteD_out_gga(X191)) → deleteD_out_gga(.(T207, X191))

The set Q consists of the following terms:

memberB_in_gag(x0, x1)
memberC_in_gg(x0, x1)
deleteD_in_gga(x0, x1)
U2_gag(x0)
U3_gg(x0)
U4_gga(x0, x1)

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

(44) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(45) TRUE