Left Termination proof of /tmp/tmpgoPfJ3/cconfdel.pl

Left Termination of the query pattern conf_in_1(g) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 AND

            ↳9 PiDP

                ↳10 UsableRulesProof (⇔)

                ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                ↳13 QDP

                ↳14 NonTerminationProof (⇔)

                ↳15 NO

            ↳16 PiDP

                ↳17 UsableRulesProof (⇔)

                ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                ↳20 QDP

                ↳21 NonTerminationProof (⇔)

                ↳22 NO

            ↳23 PiDP

                ↳24 UsableRulesProof (⇔)

                ↳25 PiDP

                ↳26 PiDPToQDPProof (⇐)

                ↳27 QDP

                ↳28 Narrowing (⇐)

                ↳29 QDP

                ↳30 Narrowing (⇐)

                ↳31 QDP

                ↳32 UsableRulesProof (⇔)

                ↳33 QDP

                ↳34 QReductionProof (⇔)

                ↳35 QDP

                ↳36 NonTerminationProof (⇔)

                ↳37 NO

            ↳38 PiDP

                ↳39 UsableRulesProof (⇔)

                ↳40 PiDP

                ↳41 PiDPToQDPProof (⇐)

                ↳42 QDP

                ↳43 QDPSizeChangeProof (⇔)

                ↳44 YES

            ↳45 PiDP

                ↳46 UsableRulesProof (⇔)

                ↳47 PiDP

                ↳48 PiDPToQDPProof (⇐)

                ↳49 QDP

                ↳50 QDPSizeChangeProof (⇔)

                ↳51 YES

            ↳52 PiDP

                ↳53 UsableRulesProof (⇔)

                ↳54 PiDP

                ↳55 PiDPToQDPProof (⇐)

                ↳56 QDP

                ↳57 QDPSizeChangeProof (⇔)

                ↳58 YES

    ↳59 PrologToPiTRSProof (⇐)

        ↳60 PiTRS

        ↳61 DependencyPairsProof (⇔)

        ↳62 PiDP

        ↳63 DependencyGraphProof (⇔)

        ↳64 AND

            ↳65 PiDP

                ↳66 UsableRulesProof (⇔)

                ↳67 PiDP

                ↳68 PiDPToQDPProof (⇐)

                ↳69 QDP

                ↳70 NonTerminationProof (⇔)

                ↳71 NO

            ↳72 PiDP

                ↳73 UsableRulesProof (⇔)

                ↳74 PiDP

                ↳75 PiDPToQDPProof (⇐)

                ↳76 QDP

                ↳77 NonTerminationProof (⇔)

                ↳78 NO

            ↳79 PiDP

                ↳80 UsableRulesProof (⇔)

                ↳81 PiDP

                ↳82 PiDPToQDPProof (⇐)

                ↳83 QDP

                ↳84 Narrowing (⇐)

                ↳85 QDP

                ↳86 Narrowing (⇐)

                ↳87 QDP

                ↳88 UsableRulesProof (⇔)

                ↳89 QDP

                ↳90 QReductionProof (⇔)

                ↳91 QDP

                ↳92 NonTerminationProof (⇔)

                ↳93 NO

            ↳94 PiDP

                ↳95 UsableRulesProof (⇔)

                ↳96 PiDP

                ↳97 PiDPToQDPProof (⇐)

                ↳98 QDP

                ↳99 QDPSizeChangeProof (⇔)

                ↳100 YES

            ↳101 PiDP

                ↳102 UsableRulesProof (⇔)

                ↳103 PiDP

                ↳104 PiDPToQDPProof (⇐)

                ↳105 QDP

                ↳106 QDPSizeChangeProof (⇔)

                ↳107 YES

            ↳108 PiDP

                ↳109 UsableRulesProof (⇔)

                ↳110 PiDP

                ↳111 PiDPToQDPProof (⇐)

                ↳112 QDP

                ↳113 QDPSizeChangeProof (⇔)

                ↳114 YES

(0) Obligation:

Clauses:

conf(X) :- ','(del2(X, Z), ','(del(U, Y, Z), conf(Y))).
del2(X, Y) :- ','(del(U, X, Z), del(V, Z, Y)).
del(X, cons(X, T), T).
del(X, cons(H, T), cons(H, T1)) :- del(X, T, T1).
s2l(s(X), cons(Y, Xs)) :- s2l(X, Xs).
s2l(0, nil).
goal(X) :- ','(s2l(X, XS), conf(XS)).

Queries:

conf(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: conf(T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: conf(T2), SCOPE: 1, CLAUSE: 0\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(del2(T4, X3), ','(del(X4, X5, X3), conf(X5)))\n\nT4 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(del2(T4, X3), ','(del(X4, X5, X3), conf(X5))), SCOPE: 2, CLAUSE: 1\n\nT4 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(del(X23, T9, X24), ','(del(X25, X24, X26), ','(del(X4, X5, X26), conf(X5))))\n\nT9 is ground\nX4 is free; \nX5 is free; \nX26 is free; \nX23 is free; \nX24 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: del(X23, T9, X24)\n\nT9 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(del(X25, T13, X26), ','(del(X4, X5, X26), conf(X5)))\n\nT13 is ground\nX4 is free; \nX5 is free; \nX26 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: del(X23, T9, X24), SCOPE: 3, CLAUSE: 2 | del(X23, T9, X24), SCOPE: 3, CLAUSE: 3\n\nT9 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: del(X23, T9, X24), SCOPE: 3, CLAUSE: 2\n\nT9 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: del(X23, T9, X24), SCOPE: 3, CLAUSE: 3\n\nT9 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: del(X72, T26, X73)\n\nT26 is ground\nX72 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: del(X25, T13, X26)\n\nT13 is ground\nX26 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(del(X4, X5, T32), conf(X5))\n\nT32 is ground\nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: del(X4, X5, T32)\n\nT32 is ground\nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: conf(T37)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: del(X4, X5, T32), SCOPE: 4, CLAUSE: 2 | del(X4, X5, T32), SCOPE: 4, CLAUSE: 3\n\nT32 is ground\nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: del(X4, X5, T32), SCOPE: 4, CLAUSE: 2\n\nT32 is ground\nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: del(X4, X5, T32), SCOPE: 4, CLAUSE: 3\n\nT32 is ground\nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: del(X147, X149, T46)\n\nT46 is ground\nX147 is free; \nX149 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: conf(T37), SCOPE: 5, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(del2(T54, X174), ','(del(X175, X176, X174), conf(X176)))\n\nX174 is free\nX175 is free; \nX176 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: del2(T54, X174)\n\nX174 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(del(X175, X176, T56), conf(X176))\n\nX175 is free\nX176 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: del2(T54, X174), SCOPE: 6, CLAUSE: 1\n\nX174 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(del(X206, T64, X207), del(X208, X207, X209))\n\nX209 is free\nX206 is free; \nX207 is free; \nX208 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: del(X206, T64, X207)\n\nX206 is free\nX207 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: del(X208, T68, X209)\n\nX209 is free\nX208 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: del(X208, T68, X209), SCOPE: 7, CLAUSE: 2 | del(X208, T68, X209), SCOPE: 7, CLAUSE: 3\n\nX209 is free\nX208 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: del(X208, T68, X209), SCOPE: 7, CLAUSE: 2\n\nX209 is free\nX208 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: del(X208, T68, X209), SCOPE: 7, CLAUSE: 3\n\nX209 is free\nX208 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: del(X261, T84, X262)\n\nX261 is free\nX262 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: del(X175, X176, T56)\n\nX175 is free\nX176 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: conf(T89)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: del(X175, X176, T56), SCOPE: 8, CLAUSE: 2 | del(X175, X176, T56), SCOPE: 8, CLAUSE: 3\n\nX175 is free\nX176 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: del(X175, X176, T56), SCOPE: 8, CLAUSE: 2\n\nX175 is free\nX176 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: del(X175, X176, T56), SCOPE: 8, CLAUSE: 3\n\nX175 is free\nX176 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: del(X324, X326, T99)\n\nX324 is free\nX326 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 52 [arrowhead = none ];
52 -> 2;

53 [label="ONLY EVAL with clause\nconf(X2) :- ','(del2(X2, X3), ','(del(X4, X5, X3), conf(X5))).\nand substitution\nT2 -> T4\nX2 -> T4", fontsize=14, style = filled, fillcolor = white];
2 -> 53 [arrowhead = none ];
53 -> 3;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 54 [arrowhead = none ];
54 -> 4;

55 [label="ONLY EVAL with clause\ndel2(X21, X22) :- ','(del(X23, X21, X24), del(X25, X24, X22)).\nand substitution\nT4 -> T9\nX21 -> T9\nX3 -> X26\nX22 -> X26", fontsize=14, style = filled, fillcolor = white];
4 -> 55 [arrowhead = none ];
55 -> 5;

56 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
5 -> 56 [arrowhead = none ];
56 -> 6;

57 [label="SPLIT 2\nnew knowledge:\nT12 is ground\nT13 is ground\nreplacements:\nX23 -> T12\nX24 -> T13", fontsize=14, style = filled, fillcolor = violet];
5 -> 57 [arrowhead = none ];
57 -> 7;

58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 58 [arrowhead = none ];
58 -> 8;

59 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 59 [arrowhead = none ];
59 -> 16;

60 [label="SPLIT 2\nnew knowledge:\nT31 is ground\nT32 is ground\nreplacements:\nX25 -> T31\nX26 -> T32", fontsize=14, style = filled, fillcolor = violet];
7 -> 60 [arrowhead = none ];
60 -> 17;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 61 [arrowhead = none ];
61 -> 9;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 62 [arrowhead = none ];
62 -> 10;

63 [label="EVAL with clause\ndel(X51, cons(X51, X52), X52).\nand substitution\nX23 -> X53\nX51 -> X53\nX52 -> T20\nT9 -> cons(X53, T20)\nX24 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 63 [arrowhead = none ];
63 -> 11;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 64 [arrowhead = none ];
64 -> 12;

65 [label="EVAL with clause\ndel(X68, cons(X69, X70), cons(X69, X71)) :- del(X68, X70, X71).\nand substitution\nX23 -> X72\nX68 -> X72\nX69 -> T25\nX70 -> T26\nT9 -> cons(T25, T26)\nX71 -> X73\nX24 -> cons(T25, X73)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 65 [arrowhead = none ];
65 -> 14;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 66 [arrowhead = none ];
66 -> 15;

67 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 67 [arrowhead = none ];
67 -> 13;

68 [label="INSTANCE with matching:\nX23 -> X72\nT9 -> T26\nX24 -> X73", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 68 [arrowhead = none , style=dashed];
68 -> 6[style=dashed];

69 [label="INSTANCE with matching:\nX23 -> X25\nT9 -> T13\nX24 -> X26", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 69 [arrowhead = none , style=dashed];
69 -> 6[style=dashed];

70 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 70 [arrowhead = none ];
70 -> 18;

71 [label="SPLIT 2\nreplacements:\nX4 -> T36\nX5 -> T37", fontsize=14, style = filled, fillcolor = violet];
17 -> 71 [arrowhead = none ];
71 -> 19;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 72 [arrowhead = none ];
72 -> 20;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 73 [arrowhead = none ];
73 -> 27;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 74 [arrowhead = none ];
74 -> 21;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 75 [arrowhead = none ];
75 -> 22;

76 [label="ONLY EVAL with clause\ndel(X125, cons(X125, X126), X126).\nand substitution\nX4 -> X127\nX125 -> X127\nX126 -> T43\nX5 -> cons(X127, T43)\nT32 -> T43\nX128 -> T43", fontsize=14, style = filled, fillcolor = white];
21 -> 76 [arrowhead = none ];
76 -> 23;

77 [label="EVAL with clause\ndel(X143, cons(X144, X145), cons(X144, X146)) :- del(X143, X145, X146).\nand substitution\nX4 -> X147\nX143 -> X147\nX144 -> X148\nX145 -> X149\nX5 -> cons(X148, X149)\nX146 -> T46\nT32 -> cons(X148, T46)", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 77 [arrowhead = none ];
77 -> 25;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 78 [arrowhead = none ];
78 -> 26;

79 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 79 [arrowhead = none ];
79 -> 24;

80 [label="INSTANCE with matching:\nX4 -> X147\nX5 -> X149\nT32 -> T46", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 80 [arrowhead = none , style=dashed];
80 -> 18[style=dashed];

81 [label="ONLY EVAL with clause\nconf(X173) :- ','(del2(X173, X174), ','(del(X175, X176, X174), conf(X176))).\nand substitution\nT37 -> T54\nX173 -> T54\nT53 -> T54", fontsize=14, style = filled, fillcolor = white];
27 -> 81 [arrowhead = none ];
81 -> 28;

82 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
28 -> 82 [arrowhead = none ];
82 -> 29;

83 [label="SPLIT 2\nreplacements:\nX174 -> T56", fontsize=14, style = filled, fillcolor = violet];
28 -> 83 [arrowhead = none ];
83 -> 30;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 84 [arrowhead = none ];
84 -> 31;

85 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
30 -> 85 [arrowhead = none ];
85 -> 43;

86 [label="SPLIT 2\nreplacements:\nX175 -> T88\nX176 -> T89", fontsize=14, style = filled, fillcolor = violet];
30 -> 86 [arrowhead = none ];
86 -> 44;

87 [label="ONLY EVAL with clause\ndel2(X204, X205) :- ','(del(X206, X204, X207), del(X208, X207, X205)).\nand substitution\nT54 -> T64\nX204 -> T64\nX174 -> X209\nX205 -> X209\nT63 -> T64", fontsize=14, style = filled, fillcolor = white];
31 -> 87 [arrowhead = none ];
87 -> 32;

88 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
32 -> 88 [arrowhead = none ];
88 -> 33;

89 [label="SPLIT 2\nreplacements:\nX206 -> T67\nX207 -> T68", fontsize=14, style = filled, fillcolor = violet];
32 -> 89 [arrowhead = none ];
89 -> 34;

90 [label="INSTANCE with matching:\nX208 -> X206\nT68 -> T64\nX209 -> X207", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 90 [arrowhead = none , style=dashed];
90 -> 34[style=dashed];

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 91 [arrowhead = none ];
91 -> 35;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 92 [arrowhead = none ];
92 -> 36;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 93 [arrowhead = none ];
93 -> 37;

94 [label="EVAL with clause\ndel(X240, cons(X240, X241), X241).\nand substitution\nX208 -> X242\nX240 -> X242\nX241 -> T77\nT68 -> cons(X242, T77)\nX209 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 94 [arrowhead = none ];
94 -> 38;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 95 [arrowhead = none ];
95 -> 39;

96 [label="EVAL with clause\ndel(X257, cons(X258, X259), cons(X258, X260)) :- del(X257, X259, X260).\nand substitution\nX208 -> X261\nX257 -> X261\nX258 -> T82\nX259 -> T84\nT68 -> cons(T82, T84)\nX260 -> X262\nX209 -> cons(T82, X262)\nT83 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 96 [arrowhead = none ];
96 -> 41;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 97 [arrowhead = none ];
97 -> 42;

98 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 98 [arrowhead = none ];
98 -> 40;

99 [label="INSTANCE with matching:\nX206 -> X261\nT64 -> T84\nX207 -> X262", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 99 [arrowhead = none , style=dashed];
99 -> 33[style=dashed];

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 100 [arrowhead = none ];
100 -> 45;

101 [label="INSTANCE with matching:\nT37 -> T89", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 101 [arrowhead = none , style=dashed];
101 -> 19[style=dashed];

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 102 [arrowhead = none ];
102 -> 46;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 103 [arrowhead = none ];
103 -> 47;

104 [label="ONLY EVAL with clause\ndel(X302, cons(X302, X303), X303).\nand substitution\nX175 -> X304\nX302 -> X304\nX303 -> T95\nX176 -> cons(X304, T95)\nT56 -> T95\nX305 -> T95", fontsize=14, style = filled, fillcolor = white];
46 -> 104 [arrowhead = none ];
104 -> 48;

105 [label="EVAL with clause\ndel(X320, cons(X321, X322), cons(X321, X323)) :- del(X320, X322, X323).\nand substitution\nX175 -> X324\nX320 -> X324\nX321 -> X325\nX322 -> X326\nX176 -> cons(X325, X326)\nX323 -> T99\nT56 -> cons(X325, T99)\nT98 -> T99", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 105 [arrowhead = none ];
105 -> 50;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 106 [arrowhead = none ];
106 -> 51;

107 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 107 [arrowhead = none ];
107 -> 49;

108 [label="INSTANCE with matching:\nX175 -> X324\nX176 -> X326\nT56 -> T99", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 108 [arrowhead = none , style=dashed];
108 -> 43[style=dashed];

}

(2) Obligation:

Clauses:

del6(X53, cons(X53, T20), T20).
del6(X72, cons(T25, T26), cons(T25, X73)) :- del6(X72, T26, X73).
del18(X127, cons(X127, T43), T43).
del18(X147, cons(X148, X149), cons(X148, T46)) :- del18(X147, X149, T46).
del34(X242, cons(X242, T77), T77).
del34(X261, cons(T82, T84), cons(T82, X262)) :- del34(X261, T84, X262).
conf19(T54) :- del229(T54, X174).
conf19(T54) :- ','(del229(T54, T56), del43(X175, X176, T56)).
conf19(T54) :- ','(del229(T54, T56), ','(del43(T88, T89, T56), conf19(T89))).
del43(X304, cons(X304, T95), T95).
del43(X324, cons(X325, X326), cons(X325, T99)) :- del43(X324, X326, T99).
del229(T64, X209) :- del34(X206, T64, X207).
del229(T64, X209) :- ','(del34(T67, T64, T68), del34(X208, T68, X209)).
conf1(T9) :- del6(X23, T9, X24).
conf1(T9) :- ','(del6(T12, T9, T13), del6(X25, T13, X26)).
conf1(T9) :- ','(del6(T12, T9, T13), ','(del6(T31, T13, T32), del18(X4, X5, T32))).
conf1(T9) :- ','(del6(T12, T9, T13), ','(del6(T31, T13, T32), ','(del18(T36, T37, T32), conf19(T37)))).

Queries:

conf1(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
conf1_in: (b)
del6_in: (f,b,f)
del18_in: (f,f,b)
conf19_in: (b) (f)
del229_in: (b,f) (f,f)
del34_in: (f,b,f) (f,f,f)
del43_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(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:

CONF1_IN_G(T9) → U13_G(T9, del6_in_aga(X23, T9, X24))
CONF1_IN_G(T9) → DEL6_IN_AGA(X23, T9, X24)
DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → U1_AGA(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)
CONF1_IN_G(T9) → U14_G(T9, del6_in_aga(T12, T9, T13))
U14_G(T9, del6_out_aga(T12, T9, T13)) → U15_G(T9, del6_in_aga(X25, T13, X26))
U14_G(T9, del6_out_aga(T12, T9, T13)) → DEL6_IN_AGA(X25, T13, X26)
U14_G(T9, del6_out_aga(T12, T9, T13)) → U16_G(T9, del6_in_aga(T31, T13, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → U17_G(T9, del18_in_aag(X4, X5, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → DEL18_IN_AAG(X4, X5, T32)
DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → U2_AAG(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)
U16_G(T9, del6_out_aga(T31, T13, T32)) → U18_G(T9, del18_in_aag(T36, T37, T32))
U18_G(T9, del18_out_aag(T36, T37, T32)) → U19_G(T9, conf19_in_g(T37))
U18_G(T9, del18_out_aag(T36, T37, T32)) → CONF19_IN_G(T37)
CONF19_IN_G(T54) → U4_G(T54, del229_in_ga(T54, X174))
CONF19_IN_G(T54) → DEL229_IN_GA(T54, X174)
DEL229_IN_GA(T64, X209) → U10_GA(T64, X209, del34_in_aga(X206, T64, X207))
DEL229_IN_GA(T64, X209) → DEL34_IN_AGA(X206, T64, X207)
DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → U3_AGA(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)
DEL229_IN_GA(T64, X209) → U11_GA(T64, X209, del34_in_aga(T67, T64, T68))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → U12_GA(T64, X209, del34_in_aga(X208, T68, X209))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → DEL34_IN_AGA(X208, T68, X209)
CONF19_IN_G(T54) → U5_G(T54, del229_in_ga(T54, T56))
U5_G(T54, del229_out_ga(T54, T56)) → U6_G(T54, del43_in_aaa(X175, X176, T56))
U5_G(T54, del229_out_ga(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → U9_AAA(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)
U5_G(T54, del229_out_ga(T54, T56)) → U7_G(T54, del43_in_aaa(T88, T89, T56))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → U8_G(T54, conf19_in_a(T89))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U4_A(T54, del229_in_aa(T54, X174))
CONF19_IN_A(T54) → DEL229_IN_AA(T54, X174)
DEL229_IN_AA(T64, X209) → U10_AA(T64, X209, del34_in_aaa(X206, T64, X207))
DEL229_IN_AA(T64, X209) → DEL34_IN_AAA(X206, T64, X207)
DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → U3_AAA(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)
DEL229_IN_AA(T64, X209) → U11_AA(T64, X209, del34_in_aaa(T67, T64, T68))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_AA(T64, X209, del34_in_aaa(X208, T68, X209))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → DEL34_IN_AAA(X208, T68, X209)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))
U5_A(T54, del229_out_aa(T54, T56)) → U6_A(T54, del43_in_aaa(X175, X176, T56))
U5_A(T54, del229_out_aa(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → U8_A(T54, conf19_in_a(T89))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1) = conf1_in_g(x1)
U13_g(x1, x2) = U13_g(x2)
del6_in_aga(x1, x2, x3) = del6_in_aga(x2)
cons(x1, x2) = cons(x2)
del6_out_aga(x1, x2, x3) = del6_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x5)
conf1_out_g(x1) = conf1_out_g
U14_g(x1, x2) = U14_g(x2)
U15_g(x1, x2) = U15_g(x2)
U16_g(x1, x2) = U16_g(x2)
U17_g(x1, x2) = U17_g(x2)
del18_in_aag(x1, x2, x3) = del18_in_aag(x3)
del18_out_aag(x1, x2, x3) = del18_out_aag(x2)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
U18_g(x1, x2) = U18_g(x2)
U19_g(x1, x2) = U19_g(x2)
conf19_in_g(x1) = conf19_in_g(x1)
U4_g(x1, x2) = U4_g(x2)
del229_in_ga(x1, x2) = del229_in_ga(x1)
U10_ga(x1, x2, x3) = U10_ga(x3)
del34_in_aga(x1, x2, x3) = del34_in_aga(x2)
del34_out_aga(x1, x2, x3) = del34_out_aga(x3)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x5)
del229_out_ga(x1, x2) = del229_out_ga
U11_ga(x1, x2, x3) = U11_ga(x3)
U12_ga(x1, x2, x3) = U12_ga(x3)
conf19_out_g(x1) = conf19_out_g
U5_g(x1, x2) = U5_g(x2)
U6_g(x1, x2) = U6_g(x2)
del43_in_aaa(x1, x2, x3) = del43_in_aaa
del43_out_aaa(x1, x2, x3) = del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5)
U7_g(x1, x2) = U7_g(x2)
U8_g(x1, x2) = U8_g(x2)
conf19_in_a(x1) = conf19_in_a
U4_a(x1, x2) = U4_a(x2)
del229_in_aa(x1, x2) = del229_in_aa
U10_aa(x1, x2, x3) = U10_aa(x3)
del34_in_aaa(x1, x2, x3) = del34_in_aaa
del34_out_aaa(x1, x2, x3) = del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
del229_out_aa(x1, x2) = del229_out_aa
U11_aa(x1, x2, x3) = U11_aa(x3)
U12_aa(x1, x2, x3) = U12_aa(x3)
conf19_out_a(x1) = conf19_out_a
U5_a(x1, x2) = U5_a(x2)
U6_a(x1, x2) = U6_a(x2)
U7_a(x1, x2) = U7_a(x2)
U8_a(x1, x2) = U8_a(x2)
CONF1_IN_G(x1) = CONF1_IN_G(x1)
U13_G(x1, x2) = U13_G(x2)
DEL6_IN_AGA(x1, x2, x3) = DEL6_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x5)
U14_G(x1, x2) = U14_G(x2)
U15_G(x1, x2) = U15_G(x2)
U16_G(x1, x2) = U16_G(x2)
U17_G(x1, x2) = U17_G(x2)
DEL18_IN_AAG(x1, x2, x3) = DEL18_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5)
U18_G(x1, x2) = U18_G(x2)
U19_G(x1, x2) = U19_G(x2)
CONF19_IN_G(x1) = CONF19_IN_G(x1)
U4_G(x1, x2) = U4_G(x2)
DEL229_IN_GA(x1, x2) = DEL229_IN_GA(x1)
U10_GA(x1, x2, x3) = U10_GA(x3)
DEL34_IN_AGA(x1, x2, x3) = DEL34_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x5)
U11_GA(x1, x2, x3) = U11_GA(x3)
U12_GA(x1, x2, x3) = U12_GA(x3)
U5_G(x1, x2) = U5_G(x2)
U6_G(x1, x2) = U6_G(x2)
DEL43_IN_AAA(x1, x2, x3) = DEL43_IN_AAA
U9_AAA(x1, x2, x3, x4, x5) = U9_AAA(x5)
U7_G(x1, x2) = U7_G(x2)
U8_G(x1, x2) = U8_G(x2)
CONF19_IN_A(x1) = CONF19_IN_A
U4_A(x1, x2) = U4_A(x2)
DEL229_IN_AA(x1, x2) = DEL229_IN_AA
U10_AA(x1, x2, x3) = U10_AA(x3)
DEL34_IN_AAA(x1, x2, x3) = DEL34_IN_AAA
U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5)
U11_AA(x1, x2, x3) = U11_AA(x3)
U12_AA(x1, x2, x3) = U12_AA(x3)
U5_A(x1, x2) = U5_A(x2)
U6_A(x1, x2) = U6_A(x2)
U7_A(x1, x2) = U7_A(x2)
U8_A(x1, x2) = U8_A(x2)

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

(6) Obligation:

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

CONF1_IN_G(T9) → U13_G(T9, del6_in_aga(X23, T9, X24))
CONF1_IN_G(T9) → DEL6_IN_AGA(X23, T9, X24)
DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → U1_AGA(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)
CONF1_IN_G(T9) → U14_G(T9, del6_in_aga(T12, T9, T13))
U14_G(T9, del6_out_aga(T12, T9, T13)) → U15_G(T9, del6_in_aga(X25, T13, X26))
U14_G(T9, del6_out_aga(T12, T9, T13)) → DEL6_IN_AGA(X25, T13, X26)
U14_G(T9, del6_out_aga(T12, T9, T13)) → U16_G(T9, del6_in_aga(T31, T13, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → U17_G(T9, del18_in_aag(X4, X5, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → DEL18_IN_AAG(X4, X5, T32)
DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → U2_AAG(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)
U16_G(T9, del6_out_aga(T31, T13, T32)) → U18_G(T9, del18_in_aag(T36, T37, T32))
U18_G(T9, del18_out_aag(T36, T37, T32)) → U19_G(T9, conf19_in_g(T37))
U18_G(T9, del18_out_aag(T36, T37, T32)) → CONF19_IN_G(T37)
CONF19_IN_G(T54) → U4_G(T54, del229_in_ga(T54, X174))
CONF19_IN_G(T54) → DEL229_IN_GA(T54, X174)
DEL229_IN_GA(T64, X209) → U10_GA(T64, X209, del34_in_aga(X206, T64, X207))
DEL229_IN_GA(T64, X209) → DEL34_IN_AGA(X206, T64, X207)
DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → U3_AGA(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)
DEL229_IN_GA(T64, X209) → U11_GA(T64, X209, del34_in_aga(T67, T64, T68))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → U12_GA(T64, X209, del34_in_aga(X208, T68, X209))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → DEL34_IN_AGA(X208, T68, X209)
CONF19_IN_G(T54) → U5_G(T54, del229_in_ga(T54, T56))
U5_G(T54, del229_out_ga(T54, T56)) → U6_G(T54, del43_in_aaa(X175, X176, T56))
U5_G(T54, del229_out_ga(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → U9_AAA(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)
U5_G(T54, del229_out_ga(T54, T56)) → U7_G(T54, del43_in_aaa(T88, T89, T56))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → U8_G(T54, conf19_in_a(T89))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U4_A(T54, del229_in_aa(T54, X174))
CONF19_IN_A(T54) → DEL229_IN_AA(T54, X174)
DEL229_IN_AA(T64, X209) → U10_AA(T64, X209, del34_in_aaa(X206, T64, X207))
DEL229_IN_AA(T64, X209) → DEL34_IN_AAA(X206, T64, X207)
DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → U3_AAA(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)
DEL229_IN_AA(T64, X209) → U11_AA(T64, X209, del34_in_aaa(T67, T64, T68))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_AA(T64, X209, del34_in_aaa(X208, T68, X209))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → DEL34_IN_AAA(X208, T68, X209)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))
U5_A(T54, del229_out_aa(T54, T56)) → U6_A(T54, del43_in_aaa(X175, X176, T56))
U5_A(T54, del229_out_aa(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → U8_A(T54, conf19_in_a(T89))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 39 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(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:

DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x2)
DEL34_IN_AAA(x1, x2, x3) = DEL34_IN_AAA

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:

DEL34_IN_AAA → DEL34_IN_AAA

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

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = DEL34_IN_AAA evaluates to t =DEL34_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DEL34_IN_AAA to DEL34_IN_AAA.

(15) NO

(16) Obligation:

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

DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(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:

DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x2)
DEL43_IN_AAA(x1, x2, x3) = DEL43_IN_AAA

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

(19) PiDPToQDPProof (SOUND 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:

DEL43_IN_AAA → DEL43_IN_AAA

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

(21) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = DEL43_IN_AAA evaluates to t =DEL43_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DEL43_IN_AAA to DEL43_IN_AAA.

(22) NO

(23) Obligation:

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

U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(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:

U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))

The TRS R consists of the following rules:

del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))

The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x2)
del43_in_aaa(x1, x2, x3) = del43_in_aaa
del43_out_aaa(x1, x2, x3) = del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5)
del229_in_aa(x1, x2) = del229_in_aa
U10_aa(x1, x2, x3) = U10_aa(x3)
del34_in_aaa(x1, x2, x3) = del34_in_aaa
del34_out_aaa(x1, x2, x3) = del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
del229_out_aa(x1, x2) = del229_out_aa
U11_aa(x1, x2, x3) = U11_aa(x3)
U12_aa(x1, x2, x3) = U12_aa(x3)
CONF19_IN_A(x1) = CONF19_IN_A
U5_A(x1, x2) = U5_A(x2)
U7_A(x1, x2) = U7_A(x2)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

U5_A(del229_out_aa) → U7_A(del43_in_aaa)
U7_A(del43_out_aaa) → CONF19_IN_A
CONF19_IN_A → U5_A(del229_in_aa)

The TRS R consists of the following rules:

del43_in_aaa → del43_out_aaa
del43_in_aaa → U9_aaa(del43_in_aaa)
del229_in_aa → U10_aa(del34_in_aaa)
del229_in_aa → U11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaa → del34_out_aaa
del34_in_aaa → U3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(28) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U5_A(del229_out_aa) → U7_A(del43_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))

(29) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
CONF19_IN_A → U5_A(del229_in_aa)
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))

The TRS R consists of the following rules:

del43_in_aaa → del43_out_aaa
del43_in_aaa → U9_aaa(del43_in_aaa)
del229_in_aa → U10_aa(del34_in_aaa)
del229_in_aa → U11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaa → del34_out_aaa
del34_in_aaa → U3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(30) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule CONF19_IN_A → U5_A(del229_in_aa) at position [0] we obtained the following new rules [LPAR04]:

CONF19_IN_A → U5_A(U10_aa(del34_in_aaa))
CONF19_IN_A → U5_A(U11_aa(del34_in_aaa))

(31) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_A → U5_A(U10_aa(del34_in_aaa))
CONF19_IN_A → U5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del43_in_aaa → del43_out_aaa
del43_in_aaa → U9_aaa(del43_in_aaa)
del229_in_aa → U10_aa(del34_in_aaa)
del229_in_aa → U11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaa → del34_out_aaa
del34_in_aaa → U3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(32) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(33) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_A → U5_A(U10_aa(del34_in_aaa))
CONF19_IN_A → U5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del34_in_aaa → del34_out_aaa
del34_in_aaa → U3_aaa(del34_in_aaa)
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
del43_in_aaa → del43_out_aaa
del43_in_aaa → U9_aaa(del43_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(34) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

del229_in_aa

(35) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_A → U5_A(U10_aa(del34_in_aaa))
CONF19_IN_A → U5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del34_in_aaa → del34_out_aaa
del34_in_aaa → U3_aaa(del34_in_aaa)
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
del43_in_aaa → del43_out_aaa
del43_in_aaa → U9_aaa(del43_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa

The set Q consists of the following terms:

del43_in_aaa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(36) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U5_A(U10_aa(del34_in_aaa)) evaluates to t =U5_A(U10_aa(del34_in_aaa))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

U5_A(U10_aa(del34_in_aaa)) → U5_A(U10_aa(del34_out_aaa))
with rule del34_in_aaa → del34_out_aaa at position [0,0] and matcher [ ]

U5_A(U10_aa(del34_out_aaa)) → U5_A(del229_out_aa)
with rule U10_aa(del34_out_aaa) → del229_out_aa at position [0] and matcher [ ]

U5_A(del229_out_aa) → U7_A(del43_out_aaa)
with rule U5_A(del229_out_aa) → U7_A(del43_out_aaa) at position [] and matcher [ ]

U7_A(del43_out_aaa) → CONF19_IN_A
with rule U7_A(del43_out_aaa) → CONF19_IN_A at position [] and matcher [ ]

CONF19_IN_A → U5_A(U10_aa(del34_in_aaa))
with rule CONF19_IN_A → U5_A(U10_aa(del34_in_aaa))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(37) NO

(38) Obligation:

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

DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(39) UsableRulesProof (EQUIVALENT transformation)

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

(40) Obligation:

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

DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x2)
DEL34_IN_AGA(x1, x2, x3) = DEL34_IN_AGA(x2)

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

(41) PiDPToQDPProof (SOUND transformation)

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

(42) Obligation:

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

DEL34_IN_AGA(cons(T84)) → DEL34_IN_AGA(T84)

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

(43) 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:

DEL34_IN_AGA(cons(T84)) → DEL34_IN_AGA(T84)
The graph contains the following edges 1 > 1

(44) YES

(45) Obligation:

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

DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(46) UsableRulesProof (EQUIVALENT transformation)

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

(47) Obligation:

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

DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x2)
DEL18_IN_AAG(x1, x2, x3) = DEL18_IN_AAG(x3)

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

(48) PiDPToQDPProof (SOUND transformation)

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

(49) Obligation:

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

DEL18_IN_AAG(cons(T46)) → DEL18_IN_AAG(T46)

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

(50) 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:

DEL18_IN_AAG(cons(T46)) → DEL18_IN_AAG(T46)
The graph contains the following edges 1 > 1

(51) YES

(52) Obligation:

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

DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(53) UsableRulesProof (EQUIVALENT transformation)

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

(54) Obligation:

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

DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x2)
DEL6_IN_AGA(x1, x2, x3) = DEL6_IN_AGA(x2)

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

(55) PiDPToQDPProof (SOUND transformation)

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

(56) Obligation:

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

DEL6_IN_AGA(cons(T26)) → DEL6_IN_AGA(T26)

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

(57) 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:

DEL6_IN_AGA(cons(T26)) → DEL6_IN_AGA(T26)
The graph contains the following edges 1 > 1

(58) YES

(59) PrologToPiTRSProof (SOUND transformation)

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(60) Obligation:

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

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(61) 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:

CONF1_IN_G(T9) → U13_G(T9, del6_in_aga(X23, T9, X24))
CONF1_IN_G(T9) → DEL6_IN_AGA(X23, T9, X24)
DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → U1_AGA(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)
CONF1_IN_G(T9) → U14_G(T9, del6_in_aga(T12, T9, T13))
U14_G(T9, del6_out_aga(T12, T9, T13)) → U15_G(T9, del6_in_aga(X25, T13, X26))
U14_G(T9, del6_out_aga(T12, T9, T13)) → DEL6_IN_AGA(X25, T13, X26)
U14_G(T9, del6_out_aga(T12, T9, T13)) → U16_G(T9, del6_in_aga(T31, T13, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → U17_G(T9, del18_in_aag(X4, X5, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → DEL18_IN_AAG(X4, X5, T32)
DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → U2_AAG(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)
U16_G(T9, del6_out_aga(T31, T13, T32)) → U18_G(T9, del18_in_aag(T36, T37, T32))
U18_G(T9, del18_out_aag(T36, T37, T32)) → U19_G(T9, conf19_in_g(T37))
U18_G(T9, del18_out_aag(T36, T37, T32)) → CONF19_IN_G(T37)
CONF19_IN_G(T54) → U4_G(T54, del229_in_ga(T54, X174))
CONF19_IN_G(T54) → DEL229_IN_GA(T54, X174)
DEL229_IN_GA(T64, X209) → U10_GA(T64, X209, del34_in_aga(X206, T64, X207))
DEL229_IN_GA(T64, X209) → DEL34_IN_AGA(X206, T64, X207)
DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → U3_AGA(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)
DEL229_IN_GA(T64, X209) → U11_GA(T64, X209, del34_in_aga(T67, T64, T68))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → U12_GA(T64, X209, del34_in_aga(X208, T68, X209))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → DEL34_IN_AGA(X208, T68, X209)
CONF19_IN_G(T54) → U5_G(T54, del229_in_ga(T54, T56))
U5_G(T54, del229_out_ga(T54, T56)) → U6_G(T54, del43_in_aaa(X175, X176, T56))
U5_G(T54, del229_out_ga(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → U9_AAA(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)
U5_G(T54, del229_out_ga(T54, T56)) → U7_G(T54, del43_in_aaa(T88, T89, T56))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → U8_G(T54, conf19_in_a(T89))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U4_A(T54, del229_in_aa(T54, X174))
CONF19_IN_A(T54) → DEL229_IN_AA(T54, X174)
DEL229_IN_AA(T64, X209) → U10_AA(T64, X209, del34_in_aaa(X206, T64, X207))
DEL229_IN_AA(T64, X209) → DEL34_IN_AAA(X206, T64, X207)
DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → U3_AAA(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)
DEL229_IN_AA(T64, X209) → U11_AA(T64, X209, del34_in_aaa(T67, T64, T68))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_AA(T64, X209, del34_in_aaa(X208, T68, X209))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → DEL34_IN_AAA(X208, T68, X209)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))
U5_A(T54, del229_out_aa(T54, T56)) → U6_A(T54, del43_in_aaa(X175, X176, T56))
U5_A(T54, del229_out_aa(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → U8_A(T54, conf19_in_a(T89))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1) = conf1_in_g(x1)
U13_g(x1, x2) = U13_g(x1, x2)
del6_in_aga(x1, x2, x3) = del6_in_aga(x2)
cons(x1, x2) = cons(x2)
del6_out_aga(x1, x2, x3) = del6_out_aga(x2, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x3, x5)
conf1_out_g(x1) = conf1_out_g(x1)
U14_g(x1, x2) = U14_g(x1, x2)
U15_g(x1, x2) = U15_g(x1, x2)
U16_g(x1, x2) = U16_g(x1, x2)
U17_g(x1, x2) = U17_g(x1, x2)
del18_in_aag(x1, x2, x3) = del18_in_aag(x3)
del18_out_aag(x1, x2, x3) = del18_out_aag(x2, x3)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5)
U18_g(x1, x2) = U18_g(x1, x2)
U19_g(x1, x2) = U19_g(x1, x2)
conf19_in_g(x1) = conf19_in_g(x1)
U4_g(x1, x2) = U4_g(x1, x2)
del229_in_ga(x1, x2) = del229_in_ga(x1)
U10_ga(x1, x2, x3) = U10_ga(x1, x3)
del34_in_aga(x1, x2, x3) = del34_in_aga(x2)
del34_out_aga(x1, x2, x3) = del34_out_aga(x2, x3)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x3, x5)
del229_out_ga(x1, x2) = del229_out_ga(x1)
U11_ga(x1, x2, x3) = U11_ga(x1, x3)
U12_ga(x1, x2, x3) = U12_ga(x1, x3)
conf19_out_g(x1) = conf19_out_g(x1)
U5_g(x1, x2) = U5_g(x1, x2)
U6_g(x1, x2) = U6_g(x1, x2)
del43_in_aaa(x1, x2, x3) = del43_in_aaa
del43_out_aaa(x1, x2, x3) = del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5)
U7_g(x1, x2) = U7_g(x1, x2)
U8_g(x1, x2) = U8_g(x1, x2)
conf19_in_a(x1) = conf19_in_a
U4_a(x1, x2) = U4_a(x2)
del229_in_aa(x1, x2) = del229_in_aa
U10_aa(x1, x2, x3) = U10_aa(x3)
del34_in_aaa(x1, x2, x3) = del34_in_aaa
del34_out_aaa(x1, x2, x3) = del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
del229_out_aa(x1, x2) = del229_out_aa
U11_aa(x1, x2, x3) = U11_aa(x3)
U12_aa(x1, x2, x3) = U12_aa(x3)
conf19_out_a(x1) = conf19_out_a
U5_a(x1, x2) = U5_a(x2)
U6_a(x1, x2) = U6_a(x2)
U7_a(x1, x2) = U7_a(x2)
U8_a(x1, x2) = U8_a(x2)
CONF1_IN_G(x1) = CONF1_IN_G(x1)
U13_G(x1, x2) = U13_G(x1, x2)
DEL6_IN_AGA(x1, x2, x3) = DEL6_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x3, x5)
U14_G(x1, x2) = U14_G(x1, x2)
U15_G(x1, x2) = U15_G(x1, x2)
U16_G(x1, x2) = U16_G(x1, x2)
U17_G(x1, x2) = U17_G(x1, x2)
DEL18_IN_AAG(x1, x2, x3) = DEL18_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x4, x5)
U18_G(x1, x2) = U18_G(x1, x2)
U19_G(x1, x2) = U19_G(x1, x2)
CONF19_IN_G(x1) = CONF19_IN_G(x1)
U4_G(x1, x2) = U4_G(x1, x2)
DEL229_IN_GA(x1, x2) = DEL229_IN_GA(x1)
U10_GA(x1, x2, x3) = U10_GA(x1, x3)
DEL34_IN_AGA(x1, x2, x3) = DEL34_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x3, x5)
U11_GA(x1, x2, x3) = U11_GA(x1, x3)
U12_GA(x1, x2, x3) = U12_GA(x1, x3)
U5_G(x1, x2) = U5_G(x1, x2)
U6_G(x1, x2) = U6_G(x1, x2)
DEL43_IN_AAA(x1, x2, x3) = DEL43_IN_AAA
U9_AAA(x1, x2, x3, x4, x5) = U9_AAA(x5)
U7_G(x1, x2) = U7_G(x1, x2)
U8_G(x1, x2) = U8_G(x1, x2)
CONF19_IN_A(x1) = CONF19_IN_A
U4_A(x1, x2) = U4_A(x2)
DEL229_IN_AA(x1, x2) = DEL229_IN_AA
U10_AA(x1, x2, x3) = U10_AA(x3)
DEL34_IN_AAA(x1, x2, x3) = DEL34_IN_AAA
U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5)
U11_AA(x1, x2, x3) = U11_AA(x3)
U12_AA(x1, x2, x3) = U12_AA(x3)
U5_A(x1, x2) = U5_A(x2)
U6_A(x1, x2) = U6_A(x2)
U7_A(x1, x2) = U7_A(x2)
U8_A(x1, x2) = U8_A(x2)

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

(62) Obligation:

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

CONF1_IN_G(T9) → U13_G(T9, del6_in_aga(X23, T9, X24))
CONF1_IN_G(T9) → DEL6_IN_AGA(X23, T9, X24)
DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → U1_AGA(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)
CONF1_IN_G(T9) → U14_G(T9, del6_in_aga(T12, T9, T13))
U14_G(T9, del6_out_aga(T12, T9, T13)) → U15_G(T9, del6_in_aga(X25, T13, X26))
U14_G(T9, del6_out_aga(T12, T9, T13)) → DEL6_IN_AGA(X25, T13, X26)
U14_G(T9, del6_out_aga(T12, T9, T13)) → U16_G(T9, del6_in_aga(T31, T13, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → U17_G(T9, del18_in_aag(X4, X5, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → DEL18_IN_AAG(X4, X5, T32)
DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → U2_AAG(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)
U16_G(T9, del6_out_aga(T31, T13, T32)) → U18_G(T9, del18_in_aag(T36, T37, T32))
U18_G(T9, del18_out_aag(T36, T37, T32)) → U19_G(T9, conf19_in_g(T37))
U18_G(T9, del18_out_aag(T36, T37, T32)) → CONF19_IN_G(T37)
CONF19_IN_G(T54) → U4_G(T54, del229_in_ga(T54, X174))
CONF19_IN_G(T54) → DEL229_IN_GA(T54, X174)
DEL229_IN_GA(T64, X209) → U10_GA(T64, X209, del34_in_aga(X206, T64, X207))
DEL229_IN_GA(T64, X209) → DEL34_IN_AGA(X206, T64, X207)
DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → U3_AGA(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)
DEL229_IN_GA(T64, X209) → U11_GA(T64, X209, del34_in_aga(T67, T64, T68))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → U12_GA(T64, X209, del34_in_aga(X208, T68, X209))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → DEL34_IN_AGA(X208, T68, X209)
CONF19_IN_G(T54) → U5_G(T54, del229_in_ga(T54, T56))
U5_G(T54, del229_out_ga(T54, T56)) → U6_G(T54, del43_in_aaa(X175, X176, T56))
U5_G(T54, del229_out_ga(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → U9_AAA(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)
U5_G(T54, del229_out_ga(T54, T56)) → U7_G(T54, del43_in_aaa(T88, T89, T56))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → U8_G(T54, conf19_in_a(T89))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U4_A(T54, del229_in_aa(T54, X174))
CONF19_IN_A(T54) → DEL229_IN_AA(T54, X174)
DEL229_IN_AA(T64, X209) → U10_AA(T64, X209, del34_in_aaa(X206, T64, X207))
DEL229_IN_AA(T64, X209) → DEL34_IN_AAA(X206, T64, X207)
DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → U3_AAA(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)
DEL229_IN_AA(T64, X209) → U11_AA(T64, X209, del34_in_aaa(T67, T64, T68))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_AA(T64, X209, del34_in_aaa(X208, T68, X209))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → DEL34_IN_AAA(X208, T68, X209)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))
U5_A(T54, del229_out_aa(T54, T56)) → U6_A(T54, del43_in_aaa(X175, X176, T56))
U5_A(T54, del229_out_aa(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → U8_A(T54, conf19_in_a(T89))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(63) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 39 less nodes.

(64) Complex Obligation (AND)

(65) Obligation:

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

DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(66) UsableRulesProof (EQUIVALENT transformation)

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

(67) Obligation:

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

DEL34_IN_AAA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x2)
DEL34_IN_AAA(x1, x2, x3) = DEL34_IN_AAA

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

(68) PiDPToQDPProof (SOUND transformation)

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

(69) Obligation:

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

DEL34_IN_AAA → DEL34_IN_AAA

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

(70) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DEL34_IN_AAA to DEL34_IN_AAA.

(71) NO

(72) Obligation:

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

DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(73) UsableRulesProof (EQUIVALENT transformation)

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

(74) Obligation:

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

DEL43_IN_AAA(X324, cons(X325, X326), cons(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x2)
DEL43_IN_AAA(x1, x2, x3) = DEL43_IN_AAA

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

(75) PiDPToQDPProof (SOUND transformation)

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

(76) Obligation:

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

DEL43_IN_AAA → DEL43_IN_AAA

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

(77) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DEL43_IN_AAA to DEL43_IN_AAA.

(78) NO

(79) Obligation:

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

U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(80) UsableRulesProof (EQUIVALENT transformation)

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

(81) Obligation:

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

U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))

The TRS R consists of the following rules:

del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))

(82) PiDPToQDPProof (SOUND transformation)

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

(83) Obligation:

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

U5_A(del229_out_aa) → U7_A(del43_in_aaa)
U7_A(del43_out_aaa) → CONF19_IN_A
CONF19_IN_A → U5_A(del229_in_aa)

The TRS R consists of the following rules:

del43_in_aaa → del43_out_aaa
del43_in_aaa → U9_aaa(del43_in_aaa)
del229_in_aa → U10_aa(del34_in_aaa)
del229_in_aa → U11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaa → del34_out_aaa
del34_in_aaa → U3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(84) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U5_A(del229_out_aa) → U7_A(del43_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))

(85) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
CONF19_IN_A → U5_A(del229_in_aa)
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))

The TRS R consists of the following rules:

del43_in_aaa → del43_out_aaa
del43_in_aaa → U9_aaa(del43_in_aaa)
del229_in_aa → U10_aa(del34_in_aaa)
del229_in_aa → U11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaa → del34_out_aaa
del34_in_aaa → U3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(86) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule CONF19_IN_A → U5_A(del229_in_aa) at position [0] we obtained the following new rules [LPAR04]:

CONF19_IN_A → U5_A(U10_aa(del34_in_aaa))
CONF19_IN_A → U5_A(U11_aa(del34_in_aaa))

(87) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_A → U5_A(U10_aa(del34_in_aaa))
CONF19_IN_A → U5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del43_in_aaa → del43_out_aaa
del43_in_aaa → U9_aaa(del43_in_aaa)
del229_in_aa → U10_aa(del34_in_aaa)
del229_in_aa → U11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaa → del34_out_aaa
del34_in_aaa → U3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(88) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(89) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_A → U5_A(U10_aa(del34_in_aaa))
CONF19_IN_A → U5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del34_in_aaa → del34_out_aaa
del34_in_aaa → U3_aaa(del34_in_aaa)
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
del43_in_aaa → del43_out_aaa
del43_in_aaa → U9_aaa(del43_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(90) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

del229_in_aa

(91) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_A → U5_A(U10_aa(del34_in_aaa))
CONF19_IN_A → U5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del34_in_aaa → del34_out_aaa
del34_in_aaa → U3_aaa(del34_in_aaa)
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
del43_in_aaa → del43_out_aaa
del43_in_aaa → U9_aaa(del43_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa

The set Q consists of the following terms:

del43_in_aaa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(92) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

(93) NO

(94) Obligation:

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

DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(95) UsableRulesProof (EQUIVALENT transformation)

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

(96) Obligation:

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

DEL34_IN_AGA(X261, cons(T82, T84), cons(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x2)
DEL34_IN_AGA(x1, x2, x3) = DEL34_IN_AGA(x2)

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

(97) PiDPToQDPProof (SOUND transformation)

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

(98) Obligation:

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

DEL34_IN_AGA(cons(T84)) → DEL34_IN_AGA(T84)

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

(99) 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:

DEL34_IN_AGA(cons(T84)) → DEL34_IN_AGA(T84)
The graph contains the following edges 1 > 1

(100) YES

(101) Obligation:

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

DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(102) UsableRulesProof (EQUIVALENT transformation)

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

(103) Obligation:

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

DEL18_IN_AAG(X147, cons(X148, X149), cons(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x2)
DEL18_IN_AAG(x1, x2, x3) = DEL18_IN_AAG(x3)

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

(104) PiDPToQDPProof (SOUND transformation)

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

(105) Obligation:

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

DEL18_IN_AAG(cons(T46)) → DEL18_IN_AAG(T46)

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

(106) 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:

DEL18_IN_AAG(cons(T46)) → DEL18_IN_AAG(T46)
The graph contains the following edges 1 > 1

(107) YES

(108) Obligation:

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

DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, cons(X53, T20), T20) → del6_out_aga(X53, cons(X53, T20), T20)
del6_in_aga(X72, cons(T25, T26), cons(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, cons(T25, T26), cons(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, cons(X127, T43), T43) → del18_out_aag(X127, cons(X127, T43), T43)
del18_in_aag(X147, cons(X148, X149), cons(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, cons(X148, X149), cons(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, cons(X242, T77), T77) → del34_out_aga(X242, cons(X242, T77), T77)
del34_in_aga(X261, cons(T82, T84), cons(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, cons(T82, T84), cons(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, cons(X304, T95), T95) → del43_out_aaa(X304, cons(X304, T95), T95)
del43_in_aaa(X324, cons(X325, X326), cons(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, cons(X325, X326), cons(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, cons(X242, T77), T77) → del34_out_aaa(X242, cons(X242, T77), T77)
del34_in_aaa(X261, cons(T82, T84), cons(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, cons(T82, T84), cons(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

(109) UsableRulesProof (EQUIVALENT transformation)

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

(110) Obligation:

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

DEL6_IN_AGA(X72, cons(T25, T26), cons(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x2)
DEL6_IN_AGA(x1, x2, x3) = DEL6_IN_AGA(x2)

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

(111) PiDPToQDPProof (SOUND transformation)

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

(112) Obligation:

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

DEL6_IN_AGA(cons(T26)) → DEL6_IN_AGA(T26)

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

(113) 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:

DEL6_IN_AGA(cons(T26)) → DEL6_IN_AGA(T26)
The graph contains the following edges 1 > 1