Left Termination proof of /tmp/tmpbjyaT0/pl7.6.2c.pl

Left Termination of the query pattern reach_in_4(g, g, g, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇐)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇔)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇐)

        ↳25 QDP

        ↳26 QDPSizeChangeProof (⇔)

        ↳27 YES

    ↳28 PiDP

        ↳29 UsableRulesProof (⇔)

        ↳30 PiDP

        ↳31 PiDPToQDPProof (⇔)

        ↳32 QDP

        ↳33 QDPSizeChangeProof (⇔)

        ↳34 YES

    ↳35 PiDP

        ↳36 PiDPToQDPProof (⇐)

        ↳37 QDP

        ↳38 QDPOrderProof (⇔)

        ↳39 QDP

        ↳40 DependencyGraphProof (⇔)

        ↳41 TRUE

(0) Obligation:

Clauses:

reach(X, Y, E, L) :- member(.(X, .(Y, [])), E).
reach(X, Z, E, L) :- ','(member1(.(X, .(Y, [])), E), ','(member(Y, L), ','(delete(Y, L, V1), reach(Y, Z, E, V1)))).
member(H, .(H, L)).
member(X, .(H, L)) :- member(X, L).
member1(H, .(H, L)).
member1(X, .(H, L)) :- member1(X, L).
delete(X, .(X, Y), Y).
delete(X, .(H, T1), .(H, T2)) :- delete(X, T1, T2).

Queries:

reach(g,g,g,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

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: ','(member1(.(T101, .(X91, [])), T103), ','(member(X91, T104), ','(delete(X91, T104, X92), reach(X91, T102, T103, X92))))\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nX91 is free; \nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: member1(.(T101, .(X91, [])), T103)\n\nT101 is ground\nT103 is ground\nX91 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(member(T109, T104), ','(delete(T109, T104, X92), reach(T109, T102, T103, X92)))\n\nT102 is ground\nT103 is ground\nT104 is ground\nT109 is ground\nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: member1(.(T101, .(X91, [])), T103), SCOPE: 4, CLAUSE: 4 | member1(.(T101, .(X91, [])), T103), SCOPE: 4, CLAUSE: 5\n\nT101 is ground\nT103 is ground\nX91 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: member1(.(T101, .(X91, [])), T103), SCOPE: 4, CLAUSE: 4\n\nT101 is ground\nT103 is ground\nX91 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: member1(.(T101, .(X91, [])), T103), SCOPE: 4, CLAUSE: 5\n\nT101 is ground\nT103 is ground\nX91 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: member1(.(T130, .(X133, [])), T132)\n\nT130 is ground\nT132 is ground\nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: member(T109, T104)\n\nT104 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(delete(T109, T104, X92), reach(T109, T102, T103, X92))\n\nT102 is ground\nT103 is ground\nT104 is ground\nT109 is ground\nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: member(T109, T104), SCOPE: 5, CLAUSE: 2 | member(T109, T104), SCOPE: 5, CLAUSE: 3\n\nT104 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: member(T109, T104), SCOPE: 5, CLAUSE: 2\n\nT104 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: member(T109, T104), SCOPE: 5, CLAUSE: 3\n\nT104 is ground\nT109 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(T161, T163)\n\nT161 is ground\nT163 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: delete(T109, T104, X92)\n\nT104 is ground\nT109 is ground\nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: reach(T109, T102, T103, T172)\n\nT102 is ground\nT103 is ground\nT109 is ground\nT172 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: delete(T109, T104, X92), SCOPE: 6, CLAUSE: 6 | delete(T109, T104, X92), SCOPE: 6, CLAUSE: 7\n\nT104 is ground\nT109 is ground\nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: delete(T109, T104, X92), SCOPE: 6, CLAUSE: 6\n\nT104 is ground\nT109 is ground\nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: delete(T109, T104, X92), SCOPE: 6, CLAUSE: 7\n\nT104 is ground\nT109 is ground\nX92 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(T193, T195, X213)\n\nT193 is ground\nT195 is ground\nX213 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 substitution\nT3 -> T11\nX5 -> T11\nT4 -> T12\nX6 -> T12\nT5 -> T13\nX7 -> T13\nT6 -> T14\nX8 -> T14", fontsize=14, style = filled, fillcolor = white];
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 substitution\nT11 -> 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 = lightblue];
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(X33, .(X34, X35)) :- member(X33, X35).\nand substitution\nT11 -> T46\nT12 -> T47\nX33 -> .(T46, .(T47, []))\nX34 -> T48\nX35 -> 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 = lightblue];
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(X51, .(X51, X52)).\nand substitution\nT46 -> T68\nT47 -> T69\nX51 -> .(T68, .(T69, []))\nX52 -> 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 = lightblue];
15 -> 71 [arrowhead = none ];
71 -> 18;

72 [label="EVAL with clause\nmember(X61, .(X62, X63)) :- member(X61, X63).\nand substitution\nT46 -> T79\nT47 -> T80\nX61 -> .(T79, .(T80, []))\nX62 -> T81\nX63 -> 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 = lightblue];
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(X87, X88, X89, X90) :- ','(member1(.(X87, .(X91, [])), X89), ','(member(X91, X90), ','(delete(X91, X90, X92), reach(X91, X88, X89, X92)))).\nand substitution\nT11 -> T101\nX87 -> T101\nT12 -> T102\nX88 -> T102\nT13 -> T103\nX89 -> T103\nT14 -> T104\nX90 -> T104", fontsize=14, style = filled, fillcolor = white];
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:\nT109 is ground\nreplacements:\nX91 -> T109", 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", 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\nmember1(X117, .(X117, X118)).\nand substitution\nT101 -> T122\nX91 -> X119\nX117 -> .(T122, .(X119, []))\nX118 -> T123\nT103 -> .(.(T122, .(X119, [])), T123)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 84 [arrowhead = none ];
84 -> 29;

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

86 [label="EVAL with clause\nmember1(X130, .(X131, X132)) :- member1(X130, X132).\nand substitution\nT101 -> T130\nX91 -> X133\nX130 -> .(T130, .(X133, []))\nX131 -> T131\nX132 -> T132\nT103 -> .(T131, T132)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 86 [arrowhead = none ];
86 -> 32;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT101 -> T130\nX91 -> X133\nT103 -> T132", 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:\nT172 is ground\nreplacements:\nX92 -> T172", 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(X159, .(X159, X160)).\nand substitution\nT109 -> T153\nX159 -> T153\nX160 -> T154\nT104 -> .(T153, T154)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 95 [arrowhead = none ];
95 -> 39;

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

97 [label="EVAL with clause\nmember(X169, .(X170, X171)) :- member(X169, X171).\nand substitution\nT109 -> T161\nX169 -> T161\nX170 -> T162\nX171 -> T163\nT104 -> .(T162, T163)", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 97 [arrowhead = none ];
97 -> 42;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT109 -> T161\nT104 -> T163", 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 -> T109\nT4 -> T102\nT5 -> T103\nT6 -> T172", 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(X195, .(X195, X196), X196).\nand substitution\nT109 -> T185\nX195 -> T185\nX196 -> T186\nT104 -> .(T185, T186)\nX92 -> T186", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 105 [arrowhead = none ];
105 -> 49;

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

107 [label="EVAL with clause\ndelete(X209, .(X210, X211), .(X210, X212)) :- delete(X209, X211, X212).\nand substitution\nT109 -> T193\nX209 -> T193\nX210 -> T194\nX211 -> T195\nT104 -> .(T194, T195)\nX212 -> X213\nX92 -> .(T194, X213)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 107 [arrowhead = none ];
107 -> 52;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT109 -> T193\nT104 -> T195\nX92 -> X213", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 110 [arrowhead = none , style=dashed];
110 -> 44[style=dashed];

}

(2) Obligation:

Triples:

member12(T79, T80, .(T81, T82)) :- member12(T79, T80, T82).
member124(T130, X133, .(T131, T132)) :- member124(T130, X133, T132).
member34(T161, .(T162, T163)) :- member34(T161, T163).
delete44(T193, .(T194, T195), .(T194, X213)) :- delete44(T193, T195, X213).
reach1(T46, T47, .(T48, T49), T14) :- member12(T46, T47, T49).
reach1(T101, T102, T103, T104) :- member124(T101, X91, T103).
reach1(T101, T102, T103, T104) :- ','(member1c24(T101, T109, T103), member34(T109, T104)).
reach1(T101, T102, T103, T104) :- ','(member1c24(T101, T109, T103), ','(memberc34(T109, T104), delete44(T109, T104, X92))).
reach1(T101, T102, T103, T104) :- ','(member1c24(T101, T109, T103), ','(memberc34(T109, T104), ','(deletec44(T109, T104, T172), reach1(T109, T102, T103, T172)))).

Clauses:

memberc12(T68, T69, .(.(T68, .(T69, [])), T70)).
memberc12(T79, T80, .(T81, T82)) :- memberc12(T79, T80, T82).
member1c24(T122, X119, .(.(T122, .(X119, [])), T123)).
member1c24(T130, X133, .(T131, T132)) :- member1c24(T130, X133, T132).
memberc34(T153, .(T153, T154)).
memberc34(T161, .(T162, T163)) :- memberc34(T161, T163).
reachc1(T27, T28, .(.(T27, .(T28, [])), T29), T14).
reachc1(T46, T47, .(T48, T49), T14) :- memberc12(T46, T47, T49).
reachc1(T101, T102, T103, T104) :- ','(member1c24(T101, T109, T103), ','(memberc34(T109, T104), ','(deletec44(T109, T104, T172), reachc1(T109, T102, T103, T172)))).
deletec44(T185, .(T185, T186), T186).
deletec44(T193, .(T194, T195), .(T194, X213)) :- deletec44(T193, T195, X213).

Afs:

reach1(x1, x2, x3, x4) = reach1(x1, x2, x3, x4)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
reach1_in: (b,b,b,b)
member12_in: (b,b,b)
member124_in: (b,f,b)
member1c24_in: (b,f,b)
member34_in: (b,b)
memberc34_in: (b,b)
delete44_in: (b,b,f)
deletec44_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REACH1_IN_GGGG(T46, T47, .(T48, T49), T14) → U5_GGGG(T46, T47, T48, T49, T14, member12_in_ggg(T46, T47, T49))
REACH1_IN_GGGG(T46, T47, .(T48, T49), T14) → MEMBER12_IN_GGG(T46, T47, T49)
MEMBER12_IN_GGG(T79, T80, .(T81, T82)) → U1_GGG(T79, T80, T81, T82, member12_in_ggg(T79, T80, T82))
MEMBER12_IN_GGG(T79, T80, .(T81, T82)) → MEMBER12_IN_GGG(T79, T80, T82)
REACH1_IN_GGGG(T101, T102, T103, T104) → U6_GGGG(T101, T102, T103, T104, member124_in_gag(T101, X91, T103))
REACH1_IN_GGGG(T101, T102, T103, T104) → MEMBER124_IN_GAG(T101, X91, T103)
MEMBER124_IN_GAG(T130, X133, .(T131, T132)) → U2_GAG(T130, X133, T131, T132, member124_in_gag(T130, X133, T132))
MEMBER124_IN_GAG(T130, X133, .(T131, T132)) → MEMBER124_IN_GAG(T130, X133, T132)
REACH1_IN_GGGG(T101, T102, T103, T104) → U7_GGGG(T101, T102, T103, T104, member1c24_in_gag(T101, T109, T103))
U7_GGGG(T101, T102, T103, T104, member1c24_out_gag(T101, T109, T103)) → U8_GGGG(T101, T102, T103, T104, member34_in_gg(T109, T104))
U7_GGGG(T101, T102, T103, T104, member1c24_out_gag(T101, T109, T103)) → MEMBER34_IN_GG(T109, T104)
MEMBER34_IN_GG(T161, .(T162, T163)) → U3_GG(T161, T162, T163, member34_in_gg(T161, T163))
MEMBER34_IN_GG(T161, .(T162, T163)) → MEMBER34_IN_GG(T161, T163)
U7_GGGG(T101, T102, T103, T104, member1c24_out_gag(T101, T109, T103)) → U9_GGGG(T101, T102, T103, T104, T109, memberc34_in_gg(T109, T104))
U9_GGGG(T101, T102, T103, T104, T109, memberc34_out_gg(T109, T104)) → U10_GGGG(T101, T102, T103, T104, delete44_in_gga(T109, T104, X92))
U9_GGGG(T101, T102, T103, T104, T109, memberc34_out_gg(T109, T104)) → DELETE44_IN_GGA(T109, T104, X92)
DELETE44_IN_GGA(T193, .(T194, T195), .(T194, X213)) → U4_GGA(T193, T194, T195, X213, delete44_in_gga(T193, T195, X213))
DELETE44_IN_GGA(T193, .(T194, T195), .(T194, X213)) → DELETE44_IN_GGA(T193, T195, X213)
U9_GGGG(T101, T102, T103, T104, T109, memberc34_out_gg(T109, T104)) → U11_GGGG(T101, T102, T103, T104, T109, deletec44_in_gga(T109, T104, T172))
U11_GGGG(T101, T102, T103, T104, T109, deletec44_out_gga(T109, T104, T172)) → U12_GGGG(T101, T102, T103, T104, reach1_in_gggg(T109, T102, T103, T172))
U11_GGGG(T101, T102, T103, T104, T109, deletec44_out_gga(T109, T104, T172)) → REACH1_IN_GGGG(T109, T102, T103, T172)

The TRS R consists of the following rules:

member1c24_in_gag(T122, X119, .(.(T122, .(X119, [])), T123)) → member1c24_out_gag(T122, X119, .(.(T122, .(X119, [])), T123))
member1c24_in_gag(T130, X133, .(T131, T132)) → U15_gag(T130, X133, T131, T132, member1c24_in_gag(T130, X133, T132))
U15_gag(T130, X133, T131, T132, member1c24_out_gag(T130, X133, T132)) → member1c24_out_gag(T130, X133, .(T131, T132))
memberc34_in_gg(T153, .(T153, T154)) → memberc34_out_gg(T153, .(T153, T154))
memberc34_in_gg(T161, .(T162, T163)) → U16_gg(T161, T162, T163, memberc34_in_gg(T161, T163))
U16_gg(T161, T162, T163, memberc34_out_gg(T161, T163)) → memberc34_out_gg(T161, .(T162, T163))
deletec44_in_gga(T185, .(T185, T186), T186) → deletec44_out_gga(T185, .(T185, T186), T186)
deletec44_in_gga(T193, .(T194, T195), .(T194, X213)) → U22_gga(T193, T194, T195, X213, deletec44_in_gga(T193, T195, X213))
U22_gga(T193, T194, T195, X213, deletec44_out_gga(T193, T195, X213)) → deletec44_out_gga(T193, .(T194, T195), .(T194, X213))

The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U15_gag(x1, x2, x3, x4, x5) = U15_gag(x1, x3, x4, x5)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U16_gg(x1, x2, x3, x4) = U16_gg(x1, x2, x3, x4)
delete44_in_gga(x1, x2, x3) = delete44_in_gga(x1, x2)
deletec44_in_gga(x1, x2, x3) = deletec44_in_gga(x1, x2)
deletec44_out_gga(x1, x2, x3) = deletec44_out_gga(x1, x2, x3)
U22_gga(x1, x2, x3, x4, x5) = U22_gga(x1, x2, x3, x5)
REACH1_IN_GGGG(x1, x2, x3, x4) = REACH1_IN_GGGG(x1, x2, x3, x4)
U5_GGGG(x1, x2, x3, x4, x5, x6) = U5_GGGG(x1, x2, x3, x4, x5, x6)
MEMBER12_IN_GGG(x1, x2, x3) = MEMBER12_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3, x4, x5) = U1_GGG(x1, x2, x3, x4, x5)
U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x1, x2, x3, x4, x5)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x1, x3, x4, x5)
U7_GGGG(x1, x2, x3, x4, x5) = U7_GGGG(x1, x2, x3, x4, x5)
U8_GGGG(x1, x2, x3, x4, x5) = U8_GGGG(x1, x2, x3, x4, x5)
MEMBER34_IN_GG(x1, x2) = MEMBER34_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4) = U3_GG(x1, x2, x3, x4)
U9_GGGG(x1, x2, x3, x4, x5, x6) = U9_GGGG(x1, x2, x3, x4, x5, x6)
U10_GGGG(x1, x2, x3, x4, x5) = U10_GGGG(x1, x2, x3, x4, x5)
DELETE44_IN_GGA(x1, x2, x3) = DELETE44_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5)
U11_GGGG(x1, x2, x3, x4, x5, x6) = U11_GGGG(x1, x2, x3, x4, x5, x6)
U12_GGGG(x1, x2, x3, x4, x5) = U12_GGGG(x1, x2, x3, x4, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

REACH1_IN_GGGG(T46, T47, .(T48, T49), T14) → U5_GGGG(T46, T47, T48, T49, T14, member12_in_ggg(T46, T47, T49))
REACH1_IN_GGGG(T46, T47, .(T48, T49), T14) → MEMBER12_IN_GGG(T46, T47, T49)
MEMBER12_IN_GGG(T79, T80, .(T81, T82)) → U1_GGG(T79, T80, T81, T82, member12_in_ggg(T79, T80, T82))
MEMBER12_IN_GGG(T79, T80, .(T81, T82)) → MEMBER12_IN_GGG(T79, T80, T82)
REACH1_IN_GGGG(T101, T102, T103, T104) → U6_GGGG(T101, T102, T103, T104, member124_in_gag(T101, X91, T103))
REACH1_IN_GGGG(T101, T102, T103, T104) → MEMBER124_IN_GAG(T101, X91, T103)
MEMBER124_IN_GAG(T130, X133, .(T131, T132)) → U2_GAG(T130, X133, T131, T132, member124_in_gag(T130, X133, T132))
MEMBER124_IN_GAG(T130, X133, .(T131, T132)) → MEMBER124_IN_GAG(T130, X133, T132)
REACH1_IN_GGGG(T101, T102, T103, T104) → U7_GGGG(T101, T102, T103, T104, member1c24_in_gag(T101, T109, T103))
U7_GGGG(T101, T102, T103, T104, member1c24_out_gag(T101, T109, T103)) → U8_GGGG(T101, T102, T103, T104, member34_in_gg(T109, T104))
U7_GGGG(T101, T102, T103, T104, member1c24_out_gag(T101, T109, T103)) → MEMBER34_IN_GG(T109, T104)
MEMBER34_IN_GG(T161, .(T162, T163)) → U3_GG(T161, T162, T163, member34_in_gg(T161, T163))
MEMBER34_IN_GG(T161, .(T162, T163)) → MEMBER34_IN_GG(T161, T163)
U7_GGGG(T101, T102, T103, T104, member1c24_out_gag(T101, T109, T103)) → U9_GGGG(T101, T102, T103, T104, T109, memberc34_in_gg(T109, T104))
U9_GGGG(T101, T102, T103, T104, T109, memberc34_out_gg(T109, T104)) → U10_GGGG(T101, T102, T103, T104, delete44_in_gga(T109, T104, X92))
U9_GGGG(T101, T102, T103, T104, T109, memberc34_out_gg(T109, T104)) → DELETE44_IN_GGA(T109, T104, X92)
DELETE44_IN_GGA(T193, .(T194, T195), .(T194, X213)) → U4_GGA(T193, T194, T195, X213, delete44_in_gga(T193, T195, X213))
DELETE44_IN_GGA(T193, .(T194, T195), .(T194, X213)) → DELETE44_IN_GGA(T193, T195, X213)
U9_GGGG(T101, T102, T103, T104, T109, memberc34_out_gg(T109, T104)) → U11_GGGG(T101, T102, T103, T104, T109, deletec44_in_gga(T109, T104, T172))
U11_GGGG(T101, T102, T103, T104, T109, deletec44_out_gga(T109, T104, T172)) → U12_GGGG(T101, T102, T103, T104, reach1_in_gggg(T109, T102, T103, T172))
U11_GGGG(T101, T102, T103, T104, T109, deletec44_out_gga(T109, T104, T172)) → REACH1_IN_GGGG(T109, T102, T103, T172)

The TRS R consists of the following rules:

member1c24_in_gag(T122, X119, .(.(T122, .(X119, [])), T123)) → member1c24_out_gag(T122, X119, .(.(T122, .(X119, [])), T123))
member1c24_in_gag(T130, X133, .(T131, T132)) → U15_gag(T130, X133, T131, T132, member1c24_in_gag(T130, X133, T132))
U15_gag(T130, X133, T131, T132, member1c24_out_gag(T130, X133, T132)) → member1c24_out_gag(T130, X133, .(T131, T132))
memberc34_in_gg(T153, .(T153, T154)) → memberc34_out_gg(T153, .(T153, T154))
memberc34_in_gg(T161, .(T162, T163)) → U16_gg(T161, T162, T163, memberc34_in_gg(T161, T163))
U16_gg(T161, T162, T163, memberc34_out_gg(T161, T163)) → memberc34_out_gg(T161, .(T162, T163))
deletec44_in_gga(T185, .(T185, T186), T186) → deletec44_out_gga(T185, .(T185, T186), T186)
deletec44_in_gga(T193, .(T194, T195), .(T194, X213)) → U22_gga(T193, T194, T195, X213, deletec44_in_gga(T193, T195, X213))
U22_gga(T193, T194, T195, X213, deletec44_out_gga(T193, T195, X213)) → deletec44_out_gga(T193, .(T194, T195), .(T194, X213))

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 13 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

DELETE44_IN_GGA(T193, .(T194, T195), .(T194, X213)) → DELETE44_IN_GGA(T193, T195, X213)

The TRS R consists of the following rules:

member1c24_in_gag(T122, X119, .(.(T122, .(X119, [])), T123)) → member1c24_out_gag(T122, X119, .(.(T122, .(X119, [])), T123))
member1c24_in_gag(T130, X133, .(T131, T132)) → U15_gag(T130, X133, T131, T132, member1c24_in_gag(T130, X133, T132))
U15_gag(T130, X133, T131, T132, member1c24_out_gag(T130, X133, T132)) → member1c24_out_gag(T130, X133, .(T131, T132))
memberc34_in_gg(T153, .(T153, T154)) → memberc34_out_gg(T153, .(T153, T154))
memberc34_in_gg(T161, .(T162, T163)) → U16_gg(T161, T162, T163, memberc34_in_gg(T161, T163))
U16_gg(T161, T162, T163, memberc34_out_gg(T161, T163)) → memberc34_out_gg(T161, .(T162, T163))
deletec44_in_gga(T185, .(T185, T186), T186) → deletec44_out_gga(T185, .(T185, T186), T186)
deletec44_in_gga(T193, .(T194, T195), .(T194, X213)) → U22_gga(T193, T194, T195, X213, deletec44_in_gga(T193, T195, X213))
U22_gga(T193, T194, T195, X213, deletec44_out_gga(T193, T195, X213)) → deletec44_out_gga(T193, .(T194, T195), .(T194, X213))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U15_gag(x1, x2, x3, x4, x5) = U15_gag(x1, x3, x4, x5)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U16_gg(x1, x2, x3, x4) = U16_gg(x1, x2, x3, x4)
deletec44_in_gga(x1, x2, x3) = deletec44_in_gga(x1, x2)
deletec44_out_gga(x1, x2, x3) = deletec44_out_gga(x1, x2, x3)
U22_gga(x1, x2, x3, x4, x5) = U22_gga(x1, x2, x3, x5)
DELETE44_IN_GGA(x1, x2, x3) = DELETE44_IN_GGA(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:

DELETE44_IN_GGA(T193, .(T194, T195), .(T194, X213)) → DELETE44_IN_GGA(T193, T195, X213)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
DELETE44_IN_GGA(x1, x2, x3) = DELETE44_IN_GGA(x1, x2)

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

(10) PiDPToQDPProof (SOUND 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:

DELETE44_IN_GGA(T193, .(T194, T195)) → DELETE44_IN_GGA(T193, T195)

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:

DELETE44_IN_GGA(T193, .(T194, T195)) → DELETE44_IN_GGA(T193, T195)
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:

MEMBER34_IN_GG(T161, .(T162, T163)) → MEMBER34_IN_GG(T161, T163)

The TRS R consists of the following rules:

member1c24_in_gag(T122, X119, .(.(T122, .(X119, [])), T123)) → member1c24_out_gag(T122, X119, .(.(T122, .(X119, [])), T123))
member1c24_in_gag(T130, X133, .(T131, T132)) → U15_gag(T130, X133, T131, T132, member1c24_in_gag(T130, X133, T132))
U15_gag(T130, X133, T131, T132, member1c24_out_gag(T130, X133, T132)) → member1c24_out_gag(T130, X133, .(T131, T132))
memberc34_in_gg(T153, .(T153, T154)) → memberc34_out_gg(T153, .(T153, T154))
memberc34_in_gg(T161, .(T162, T163)) → U16_gg(T161, T162, T163, memberc34_in_gg(T161, T163))
U16_gg(T161, T162, T163, memberc34_out_gg(T161, T163)) → memberc34_out_gg(T161, .(T162, T163))
deletec44_in_gga(T185, .(T185, T186), T186) → deletec44_out_gga(T185, .(T185, T186), T186)
deletec44_in_gga(T193, .(T194, T195), .(T194, X213)) → U22_gga(T193, T194, T195, X213, deletec44_in_gga(T193, T195, X213))
U22_gga(T193, T194, T195, X213, deletec44_out_gga(T193, T195, X213)) → deletec44_out_gga(T193, .(T194, T195), .(T194, X213))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U15_gag(x1, x2, x3, x4, x5) = U15_gag(x1, x3, x4, x5)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U16_gg(x1, x2, x3, x4) = U16_gg(x1, x2, x3, x4)
deletec44_in_gga(x1, x2, x3) = deletec44_in_gga(x1, x2)
deletec44_out_gga(x1, x2, x3) = deletec44_out_gga(x1, x2, x3)
U22_gga(x1, x2, x3, x4, x5) = U22_gga(x1, x2, x3, x5)
MEMBER34_IN_GG(x1, x2) = MEMBER34_IN_GG(x1, x2)

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:

MEMBER34_IN_GG(T161, .(T162, T163)) → MEMBER34_IN_GG(T161, T163)

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

(17) PiDPToQDPProof (EQUIVALENT 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:

MEMBER34_IN_GG(T161, .(T162, T163)) → MEMBER34_IN_GG(T161, T163)

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:

MEMBER34_IN_GG(T161, .(T162, T163)) → MEMBER34_IN_GG(T161, T163)
The graph contains the following edges 1 >= 1, 2 > 2

(20) YES

(21) Obligation:

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

MEMBER124_IN_GAG(T130, X133, .(T131, T132)) → MEMBER124_IN_GAG(T130, X133, T132)

The TRS R consists of the following rules:

member1c24_in_gag(T122, X119, .(.(T122, .(X119, [])), T123)) → member1c24_out_gag(T122, X119, .(.(T122, .(X119, [])), T123))
member1c24_in_gag(T130, X133, .(T131, T132)) → U15_gag(T130, X133, T131, T132, member1c24_in_gag(T130, X133, T132))
U15_gag(T130, X133, T131, T132, member1c24_out_gag(T130, X133, T132)) → member1c24_out_gag(T130, X133, .(T131, T132))
memberc34_in_gg(T153, .(T153, T154)) → memberc34_out_gg(T153, .(T153, T154))
memberc34_in_gg(T161, .(T162, T163)) → U16_gg(T161, T162, T163, memberc34_in_gg(T161, T163))
U16_gg(T161, T162, T163, memberc34_out_gg(T161, T163)) → memberc34_out_gg(T161, .(T162, T163))
deletec44_in_gga(T185, .(T185, T186), T186) → deletec44_out_gga(T185, .(T185, T186), T186)
deletec44_in_gga(T193, .(T194, T195), .(T194, X213)) → U22_gga(T193, T194, T195, X213, deletec44_in_gga(T193, T195, X213))
U22_gga(T193, T194, T195, X213, deletec44_out_gga(T193, T195, X213)) → deletec44_out_gga(T193, .(T194, T195), .(T194, X213))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U15_gag(x1, x2, x3, x4, x5) = U15_gag(x1, x3, x4, x5)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U16_gg(x1, x2, x3, x4) = U16_gg(x1, x2, x3, x4)
deletec44_in_gga(x1, x2, x3) = deletec44_in_gga(x1, x2)
deletec44_out_gga(x1, x2, x3) = deletec44_out_gga(x1, x2, x3)
U22_gga(x1, x2, x3, x4, x5) = U22_gga(x1, x2, x3, x5)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

MEMBER124_IN_GAG(T130, X133, .(T131, T132)) → MEMBER124_IN_GAG(T130, X133, T132)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

MEMBER124_IN_GAG(T130, .(T131, T132)) → MEMBER124_IN_GAG(T130, T132)

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

(26) 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:

MEMBER124_IN_GAG(T130, .(T131, T132)) → MEMBER124_IN_GAG(T130, T132)
The graph contains the following edges 1 >= 1, 2 > 2

(27) YES

(28) Obligation:

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

MEMBER12_IN_GGG(T79, T80, .(T81, T82)) → MEMBER12_IN_GGG(T79, T80, T82)

The TRS R consists of the following rules:

member1c24_in_gag(T122, X119, .(.(T122, .(X119, [])), T123)) → member1c24_out_gag(T122, X119, .(.(T122, .(X119, [])), T123))
member1c24_in_gag(T130, X133, .(T131, T132)) → U15_gag(T130, X133, T131, T132, member1c24_in_gag(T130, X133, T132))
U15_gag(T130, X133, T131, T132, member1c24_out_gag(T130, X133, T132)) → member1c24_out_gag(T130, X133, .(T131, T132))
memberc34_in_gg(T153, .(T153, T154)) → memberc34_out_gg(T153, .(T153, T154))
memberc34_in_gg(T161, .(T162, T163)) → U16_gg(T161, T162, T163, memberc34_in_gg(T161, T163))
U16_gg(T161, T162, T163, memberc34_out_gg(T161, T163)) → memberc34_out_gg(T161, .(T162, T163))
deletec44_in_gga(T185, .(T185, T186), T186) → deletec44_out_gga(T185, .(T185, T186), T186)
deletec44_in_gga(T193, .(T194, T195), .(T194, X213)) → U22_gga(T193, T194, T195, X213, deletec44_in_gga(T193, T195, X213))
U22_gga(T193, T194, T195, X213, deletec44_out_gga(T193, T195, X213)) → deletec44_out_gga(T193, .(T194, T195), .(T194, X213))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U15_gag(x1, x2, x3, x4, x5) = U15_gag(x1, x3, x4, x5)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U16_gg(x1, x2, x3, x4) = U16_gg(x1, x2, x3, x4)
deletec44_in_gga(x1, x2, x3) = deletec44_in_gga(x1, x2)
deletec44_out_gga(x1, x2, x3) = deletec44_out_gga(x1, x2, x3)
U22_gga(x1, x2, x3, x4, x5) = U22_gga(x1, x2, x3, x5)
MEMBER12_IN_GGG(x1, x2, x3) = MEMBER12_IN_GGG(x1, x2, x3)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

MEMBER12_IN_GGG(T79, T80, .(T81, T82)) → MEMBER12_IN_GGG(T79, T80, T82)

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

(31) PiDPToQDPProof (EQUIVALENT transformation)

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

(32) Obligation:

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

MEMBER12_IN_GGG(T79, T80, .(T81, T82)) → MEMBER12_IN_GGG(T79, T80, T82)

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

(33) 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:

MEMBER12_IN_GGG(T79, T80, .(T81, T82)) → MEMBER12_IN_GGG(T79, T80, T82)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3

(34) YES

(35) Obligation:

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

REACH1_IN_GGGG(T101, T102, T103, T104) → U7_GGGG(T101, T102, T103, T104, member1c24_in_gag(T101, T109, T103))
U7_GGGG(T101, T102, T103, T104, member1c24_out_gag(T101, T109, T103)) → U9_GGGG(T101, T102, T103, T104, T109, memberc34_in_gg(T109, T104))
U9_GGGG(T101, T102, T103, T104, T109, memberc34_out_gg(T109, T104)) → U11_GGGG(T101, T102, T103, T104, T109, deletec44_in_gga(T109, T104, T172))
U11_GGGG(T101, T102, T103, T104, T109, deletec44_out_gga(T109, T104, T172)) → REACH1_IN_GGGG(T109, T102, T103, T172)

The TRS R consists of the following rules:

member1c24_in_gag(T122, X119, .(.(T122, .(X119, [])), T123)) → member1c24_out_gag(T122, X119, .(.(T122, .(X119, [])), T123))
member1c24_in_gag(T130, X133, .(T131, T132)) → U15_gag(T130, X133, T131, T132, member1c24_in_gag(T130, X133, T132))
U15_gag(T130, X133, T131, T132, member1c24_out_gag(T130, X133, T132)) → member1c24_out_gag(T130, X133, .(T131, T132))
memberc34_in_gg(T153, .(T153, T154)) → memberc34_out_gg(T153, .(T153, T154))
memberc34_in_gg(T161, .(T162, T163)) → U16_gg(T161, T162, T163, memberc34_in_gg(T161, T163))
U16_gg(T161, T162, T163, memberc34_out_gg(T161, T163)) → memberc34_out_gg(T161, .(T162, T163))
deletec44_in_gga(T185, .(T185, T186), T186) → deletec44_out_gga(T185, .(T185, T186), T186)
deletec44_in_gga(T193, .(T194, T195), .(T194, X213)) → U22_gga(T193, T194, T195, X213, deletec44_in_gga(T193, T195, X213))
U22_gga(T193, T194, T195, X213, deletec44_out_gga(T193, T195, X213)) → deletec44_out_gga(T193, .(T194, T195), .(T194, X213))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U15_gag(x1, x2, x3, x4, x5) = U15_gag(x1, x3, x4, x5)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U16_gg(x1, x2, x3, x4) = U16_gg(x1, x2, x3, x4)
deletec44_in_gga(x1, x2, x3) = deletec44_in_gga(x1, x2)
deletec44_out_gga(x1, x2, x3) = deletec44_out_gga(x1, x2, x3)
U22_gga(x1, x2, x3, x4, x5) = U22_gga(x1, x2, x3, x5)
REACH1_IN_GGGG(x1, x2, x3, x4) = REACH1_IN_GGGG(x1, x2, x3, x4)
U7_GGGG(x1, x2, x3, x4, x5) = U7_GGGG(x1, x2, x3, x4, x5)
U9_GGGG(x1, x2, x3, x4, x5, x6) = U9_GGGG(x1, x2, x3, x4, x5, x6)
U11_GGGG(x1, x2, x3, x4, x5, x6) = U11_GGGG(x1, x2, x3, x4, x5, x6)

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

(36) PiDPToQDPProof (SOUND transformation)

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

(37) Obligation:

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

REACH1_IN_GGGG(T101, T102, T103, T104) → U7_GGGG(T101, T102, T103, T104, member1c24_in_gag(T101, T103))
U7_GGGG(T101, T102, T103, T104, member1c24_out_gag(T101, T109, T103)) → U9_GGGG(T101, T102, T103, T104, T109, memberc34_in_gg(T109, T104))
U9_GGGG(T101, T102, T103, T104, T109, memberc34_out_gg(T109, T104)) → U11_GGGG(T101, T102, T103, T104, T109, deletec44_in_gga(T109, T104))
U11_GGGG(T101, T102, T103, T104, T109, deletec44_out_gga(T109, T104, T172)) → REACH1_IN_GGGG(T109, T102, T103, T172)

The TRS R consists of the following rules:

member1c24_in_gag(T122, .(.(T122, .(X119, [])), T123)) → member1c24_out_gag(T122, X119, .(.(T122, .(X119, [])), T123))
member1c24_in_gag(T130, .(T131, T132)) → U15_gag(T130, T131, T132, member1c24_in_gag(T130, T132))
U15_gag(T130, T131, T132, member1c24_out_gag(T130, X133, T132)) → member1c24_out_gag(T130, X133, .(T131, T132))
memberc34_in_gg(T153, .(T153, T154)) → memberc34_out_gg(T153, .(T153, T154))
memberc34_in_gg(T161, .(T162, T163)) → U16_gg(T161, T162, T163, memberc34_in_gg(T161, T163))
U16_gg(T161, T162, T163, memberc34_out_gg(T161, T163)) → memberc34_out_gg(T161, .(T162, T163))
deletec44_in_gga(T185, .(T185, T186)) → deletec44_out_gga(T185, .(T185, T186), T186)
deletec44_in_gga(T193, .(T194, T195)) → U22_gga(T193, T194, T195, deletec44_in_gga(T193, T195))
U22_gga(T193, T194, T195, deletec44_out_gga(T193, T195, X213)) → deletec44_out_gga(T193, .(T194, T195), .(T194, X213))

The set Q consists of the following terms:

member1c24_in_gag(x0, x1)
U15_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U16_gg(x0, x1, x2, x3)
deletec44_in_gga(x0, x1)
U22_gga(x0, x1, x2, x3)

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U7_GGGG(T101, T102, T103, T104, member1c24_out_gag(T101, T109, T103)) → U9_GGGG(T101, T102, T103, T104, T109, memberc34_in_gg(T109, T104))

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

POL(.(x₁, x₂)) = 1 + x₂
POL(REACH1_IN_GGGG(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U11_GGGG(x₁, x₂, x₃, x₄, x₅, x₆)) = x₆
POL(U15_gag(x₁, x₂, x₃, x₄)) = 0
POL(U16_gg(x₁, x₂, x₃, x₄)) = 0
POL(U22_gga(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U7_GGGG(x₁, x₂, x₃, x₄, x₅)) = 1 + x₄
POL(U9_GGGG(x₁, x₂, x₃, x₄, x₅, x₆)) = x₄
POL([]) = 0
POL(deletec44_in_gga(x₁, x₂)) = x₂
POL(deletec44_out_gga(x₁, x₂, x₃)) = 1 + x₃
POL(member1c24_in_gag(x₁, x₂)) = 0
POL(member1c24_out_gag(x₁, x₂, x₃)) = 0
POL(memberc34_in_gg(x₁, x₂)) = 0
POL(memberc34_out_gg(x₁, x₂)) = 0

The following usable rules [FROCOS05] were oriented:

deletec44_in_gga(T185, .(T185, T186)) → deletec44_out_gga(T185, .(T185, T186), T186)
deletec44_in_gga(T193, .(T194, T195)) → U22_gga(T193, T194, T195, deletec44_in_gga(T193, T195))
U22_gga(T193, T194, T195, deletec44_out_gga(T193, T195, X213)) → deletec44_out_gga(T193, .(T194, T195), .(T194, X213))

(39) Obligation:

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

REACH1_IN_GGGG(T101, T102, T103, T104) → U7_GGGG(T101, T102, T103, T104, member1c24_in_gag(T101, T103))
U9_GGGG(T101, T102, T103, T104, T109, memberc34_out_gg(T109, T104)) → U11_GGGG(T101, T102, T103, T104, T109, deletec44_in_gga(T109, T104))
U11_GGGG(T101, T102, T103, T104, T109, deletec44_out_gga(T109, T104, T172)) → REACH1_IN_GGGG(T109, T102, T103, T172)

The TRS R consists of the following rules:

member1c24_in_gag(T122, .(.(T122, .(X119, [])), T123)) → member1c24_out_gag(T122, X119, .(.(T122, .(X119, [])), T123))
member1c24_in_gag(T130, .(T131, T132)) → U15_gag(T130, T131, T132, member1c24_in_gag(T130, T132))
U15_gag(T130, T131, T132, member1c24_out_gag(T130, X133, T132)) → member1c24_out_gag(T130, X133, .(T131, T132))
memberc34_in_gg(T153, .(T153, T154)) → memberc34_out_gg(T153, .(T153, T154))
memberc34_in_gg(T161, .(T162, T163)) → U16_gg(T161, T162, T163, memberc34_in_gg(T161, T163))
U16_gg(T161, T162, T163, memberc34_out_gg(T161, T163)) → memberc34_out_gg(T161, .(T162, T163))
deletec44_in_gga(T185, .(T185, T186)) → deletec44_out_gga(T185, .(T185, T186), T186)
deletec44_in_gga(T193, .(T194, T195)) → U22_gga(T193, T194, T195, deletec44_in_gga(T193, T195))
U22_gga(T193, T194, T195, deletec44_out_gga(T193, T195, X213)) → deletec44_out_gga(T193, .(T194, T195), .(T194, X213))

The set Q consists of the following terms:

member1c24_in_gag(x0, x1)
U15_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U16_gg(x0, x1, x2, x3)
deletec44_in_gga(x0, x1)
U22_gga(x0, x1, x2, x3)

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

(40) DependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) PiDPToQDPProof (SOUND transformation)

(25) Obligation:

(26) QDPSizeChangeProof (EQUIVALENT transformation)

(27) YES

(28) Obligation:

(29) UsableRulesProof (EQUIVALENT transformation)

(30) Obligation:

(31) PiDPToQDPProof (EQUIVALENT transformation)

(32) Obligation:

(33) QDPSizeChangeProof (EQUIVALENT transformation)

(34) YES

(35) Obligation:

(36) PiDPToQDPProof (SOUND transformation)

(37) Obligation:

(38) QDPOrderProof (EQUIVALENT transformation)

(39) Obligation:

(40) DependencyGraphProof (EQUIVALENT transformation)

(41) TRUE