Left Termination of the query pattern conf_in_1(g) w.r.t. the given Prolog program could successfully be proven:

(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, .(X, T), T).
del(X, .(H, T), .(H, T1)) :- del(X, T, T1).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).
goal(X) :- ','(s2l(X, XS), conf(XS)).

Queries:

conf(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT 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, .(X51, X52), X52).\nand substitution\nX23 -> X53\nX51 -> X53\nX52 -> T20\nT9 -> .(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, .(X69, X70), .(X69, X71)) :- del(X68, X70, X71).\nand substitution\nX23 -> X72\nX68 -> X72\nX69 -> T25\nX70 -> T26\nT9 -> .(T25, T26)\nX71 -> X73\nX24 -> .(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, .(X125, X126), X126).\nand substitution\nX4 -> X127\nX125 -> X127\nX126 -> T43\nX5 -> .(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, .(X144, X145), .(X144, X146)) :- del(X143, X145, X146).\nand substitution\nX4 -> X147\nX143 -> X147\nX144 -> X148\nX145 -> X149\nX5 -> .(X148, X149)\nX146 -> T46\nT32 -> .(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, .(X240, X241), X241).\nand substitution\nX208 -> X242\nX240 -> X242\nX241 -> T77\nT68 -> .(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, .(X258, X259), .(X258, X260)) :- del(X257, X259, X260).\nand substitution\nX208 -> X261\nX257 -> X261\nX258 -> T82\nX259 -> T84\nT68 -> .(T82, T84)\nX260 -> X262\nX209 -> .(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, .(X302, X303), X303).\nand substitution\nX175 -> X304\nX302 -> X304\nX303 -> T95\nX176 -> .(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, .(X321, X322), .(X321, X323)) :- del(X320, X322, X323).\nand substitution\nX175 -> X324\nX320 -> X324\nX321 -> X325\nX322 -> X326\nX176 -> .(X325, X326)\nX323 -> T99\nT56 -> .(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:

Triples:

del6(X72, .(T25, T26), .(T25, X73)) :- del6(X72, T26, X73).
del18(X147, .(X148, X149), .(X148, T46)) :- del18(X147, X149, T46).
del34(X261, .(T82, T84), .(T82, X262)) :- del34(X261, T84, X262).
conf19(T64) :- del34(X206, T64, X207).
conf19(T64) :- ','(delc34(T67, T64, T68), del34(X208, T68, X209)).
conf19(T54) :- ','(del2c29(T54, T56), del43(X175, X176, T56)).
conf19(T54) :- ','(del2c29(T54, T56), ','(delc43(T88, T89, T56), conf19(T89))).
del43(X324, .(X325, X326), .(X325, T99)) :- del43(X324, X326, T99).
conf1(T9) :- del6(X23, T9, X24).
conf1(T9) :- ','(delc6(T12, T9, T13), del6(X25, T13, X26)).
conf1(T9) :- ','(delc6(T12, T9, T13), ','(delc6(T31, T13, T32), del18(X4, X5, T32))).
conf1(T9) :- ','(delc6(T12, T9, T13), ','(delc6(T31, T13, T32), ','(delc18(T36, T37, T32), conf19(T37)))).

Clauses:

delc6(X53, .(X53, T20), T20).
delc6(X72, .(T25, T26), .(T25, X73)) :- delc6(X72, T26, X73).
delc18(X127, .(X127, T43), T43).
delc18(X147, .(X148, X149), .(X148, T46)) :- delc18(X147, X149, T46).
delc34(X242, .(X242, T77), T77).
delc34(X261, .(T82, T84), .(T82, X262)) :- delc34(X261, T84, X262).
confc19(T54) :- ','(del2c29(T54, T56), ','(delc43(T88, T89, T56), confc19(T89))).
delc43(X304, .(X304, T95), T95).
delc43(X324, .(X325, X326), .(X325, T99)) :- delc43(X324, X326, T99).
del2c29(T64, X209) :- ','(delc34(T67, T64, T68), delc34(X208, T68, X209)).

Afs:

conf1(x1)  =  conf1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
conf1_in: (b)
del6_in: (f,b,f)
delc6_in: (f,b,f)
del18_in: (f,f,b)
delc18_in: (f,f,b)
conf19_in: (b)
del34_in: (f,b,f)
delc34_in: (f,b,f)
del2c29_in: (b,f)
del43_in: (f,f,b)
delc43_in: (f,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

CONF1_IN_G(T9) → U12_G(T9, del6_in_aga(X23, T9, X24))
CONF1_IN_G(T9) → DEL6_IN_AGA(X23, T9, X24)
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → U1_AGA(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)
CONF1_IN_G(T9) → U13_G(T9, delc6_in_aga(T12, T9, T13))
U13_G(T9, delc6_out_aga(T12, T9, T13)) → U14_G(T9, del6_in_aga(X25, T13, X26))
U13_G(T9, delc6_out_aga(T12, T9, T13)) → DEL6_IN_AGA(X25, T13, X26)
U13_G(T9, delc6_out_aga(T12, T9, T13)) → U15_G(T9, delc6_in_aga(T31, T13, T32))
U15_G(T9, delc6_out_aga(T31, T13, T32)) → U16_G(T9, del18_in_aag(X4, X5, T32))
U15_G(T9, delc6_out_aga(T31, T13, T32)) → DEL18_IN_AAG(X4, X5, T32)
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → U2_AAG(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)
U15_G(T9, delc6_out_aga(T31, T13, T32)) → U17_G(T9, delc18_in_aag(T36, T37, T32))
U17_G(T9, delc18_out_aag(T36, T37, T32)) → U18_G(T9, conf19_in_g(T37))
U17_G(T9, delc18_out_aag(T36, T37, T32)) → CONF19_IN_G(T37)
CONF19_IN_G(T64) → U4_G(T64, del34_in_aga(X206, T64, X207))
CONF19_IN_G(T64) → DEL34_IN_AGA(X206, T64, X207)
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → U3_AGA(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)
CONF19_IN_G(T64) → U5_G(T64, delc34_in_aga(T67, T64, T68))
U5_G(T64, delc34_out_aga(T67, T64, T68)) → U6_G(T64, del34_in_aga(X208, T68, X209))
U5_G(T64, delc34_out_aga(T67, T64, T68)) → DEL34_IN_AGA(X208, T68, X209)
CONF19_IN_G(T54) → U7_G(T54, del2c29_in_ga(T54, T56))
U7_G(T54, del2c29_out_ga(T54, T56)) → U8_G(T54, del43_in_aag(X175, X176, T56))
U7_G(T54, del2c29_out_ga(T54, T56)) → DEL43_IN_AAG(X175, X176, T56)
DEL43_IN_AAG(X324, .(X325, X326), .(X325, T99)) → U11_AAG(X324, X325, X326, T99, del43_in_aag(X324, X326, T99))
DEL43_IN_AAG(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAG(X324, X326, T99)
U7_G(T54, del2c29_out_ga(T54, T56)) → U9_G(T54, delc43_in_aag(T88, T89, T56))
U9_G(T54, delc43_out_aag(T88, T89, T56)) → U10_G(T54, conf19_in_g(T89))
U9_G(T54, delc43_out_aag(T88, T89, T56)) → CONF19_IN_G(T89)

The TRS R consists of the following rules:

delc6_in_aga(X53, .(X53, T20), T20) → delc6_out_aga(X53, .(X53, T20), T20)
delc6_in_aga(X72, .(T25, T26), .(T25, X73)) → U20_aga(X72, T25, T26, X73, delc6_in_aga(X72, T26, X73))
U20_aga(X72, T25, T26, X73, delc6_out_aga(X72, T26, X73)) → delc6_out_aga(X72, .(T25, T26), .(T25, X73))
delc18_in_aag(X127, .(X127, T43), T43) → delc18_out_aag(X127, .(X127, T43), T43)
delc18_in_aag(X147, .(X148, X149), .(X148, T46)) → U21_aag(X147, X148, X149, T46, delc18_in_aag(X147, X149, T46))
U21_aag(X147, X148, X149, T46, delc18_out_aag(X147, X149, T46)) → delc18_out_aag(X147, .(X148, X149), .(X148, T46))
delc34_in_aga(X242, .(X242, T77), T77) → delc34_out_aga(X242, .(X242, T77), T77)
delc34_in_aga(X261, .(T82, T84), .(T82, X262)) → U22_aga(X261, T82, T84, X262, delc34_in_aga(X261, T84, X262))
U22_aga(X261, T82, T84, X262, delc34_out_aga(X261, T84, X262)) → delc34_out_aga(X261, .(T82, T84), .(T82, X262))
del2c29_in_ga(T64, X209) → U27_ga(T64, X209, delc34_in_aga(T67, T64, T68))
U27_ga(T64, X209, delc34_out_aga(T67, T64, T68)) → U28_ga(T64, X209, delc34_in_aga(X208, T68, X209))
U28_ga(T64, X209, delc34_out_aga(X208, T68, X209)) → del2c29_out_ga(T64, X209)
delc43_in_aag(X304, .(X304, T95), T95) → delc43_out_aag(X304, .(X304, T95), T95)
delc43_in_aag(X324, .(X325, X326), .(X325, T99)) → U26_aag(X324, X325, X326, T99, delc43_in_aag(X324, X326, T99))
U26_aag(X324, X325, X326, T99, delc43_out_aag(X324, X326, T99)) → delc43_out_aag(X324, .(X325, X326), .(X325, T99))

The argument filtering Pi contains the following mapping:
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
delc6_in_aga(x1, x2, x3)  =  delc6_in_aga(x2)
delc6_out_aga(x1, x2, x3)  =  delc6_out_aga(x2, x3)
U20_aga(x1, x2, x3, x4, x5)  =  U20_aga(x3, x5)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
delc18_in_aag(x1, x2, x3)  =  delc18_in_aag(x3)
delc18_out_aag(x1, x2, x3)  =  delc18_out_aag(x2, x3)
U21_aag(x1, x2, x3, x4, x5)  =  U21_aag(x4, x5)
conf19_in_g(x1)  =  conf19_in_g(x1)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
delc34_in_aga(x1, x2, x3)  =  delc34_in_aga(x2)
delc34_out_aga(x1, x2, x3)  =  delc34_out_aga(x2, x3)
U22_aga(x1, x2, x3, x4, x5)  =  U22_aga(x3, x5)
del2c29_in_ga(x1, x2)  =  del2c29_in_ga(x1)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
del2c29_out_ga(x1, x2)  =  del2c29_out_ga(x1, x2)
del43_in_aag(x1, x2, x3)  =  del43_in_aag(x3)
delc43_in_aag(x1, x2, x3)  =  delc43_in_aag(x3)
delc43_out_aag(x1, x2, x3)  =  delc43_out_aag(x2, x3)
U26_aag(x1, x2, x3, x4, x5)  =  U26_aag(x4, x5)
CONF1_IN_G(x1)  =  CONF1_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
DEL6_IN_AGA(x1, x2, x3)  =  DEL6_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
DEL18_IN_AAG(x1, x2, x3)  =  DEL18_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x4, x5)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1, x2)  =  U18_G(x1, x2)
CONF19_IN_G(x1)  =  CONF19_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x1, x2)
DEL34_IN_AGA(x1, x2, x3)  =  DEL34_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x3, x5)
U5_G(x1, x2)  =  U5_G(x1, x2)
U6_G(x1, x2)  =  U6_G(x1, x2)
U7_G(x1, x2)  =  U7_G(x1, x2)
U8_G(x1, x2)  =  U8_G(x1, x2)
DEL43_IN_AAG(x1, x2, x3)  =  DEL43_IN_AAG(x3)
U11_AAG(x1, x2, x3, x4, x5)  =  U11_AAG(x4, x5)
U9_G(x1, x2)  =  U9_G(x1, x2)
U10_G(x1, x2)  =  U10_G(x1, x2)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

CONF1_IN_G(T9) → U12_G(T9, del6_in_aga(X23, T9, X24))
CONF1_IN_G(T9) → DEL6_IN_AGA(X23, T9, X24)
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → U1_AGA(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)
CONF1_IN_G(T9) → U13_G(T9, delc6_in_aga(T12, T9, T13))
U13_G(T9, delc6_out_aga(T12, T9, T13)) → U14_G(T9, del6_in_aga(X25, T13, X26))
U13_G(T9, delc6_out_aga(T12, T9, T13)) → DEL6_IN_AGA(X25, T13, X26)
U13_G(T9, delc6_out_aga(T12, T9, T13)) → U15_G(T9, delc6_in_aga(T31, T13, T32))
U15_G(T9, delc6_out_aga(T31, T13, T32)) → U16_G(T9, del18_in_aag(X4, X5, T32))
U15_G(T9, delc6_out_aga(T31, T13, T32)) → DEL18_IN_AAG(X4, X5, T32)
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → U2_AAG(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)
U15_G(T9, delc6_out_aga(T31, T13, T32)) → U17_G(T9, delc18_in_aag(T36, T37, T32))
U17_G(T9, delc18_out_aag(T36, T37, T32)) → U18_G(T9, conf19_in_g(T37))
U17_G(T9, delc18_out_aag(T36, T37, T32)) → CONF19_IN_G(T37)
CONF19_IN_G(T64) → U4_G(T64, del34_in_aga(X206, T64, X207))
CONF19_IN_G(T64) → DEL34_IN_AGA(X206, T64, X207)
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → U3_AGA(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)
CONF19_IN_G(T64) → U5_G(T64, delc34_in_aga(T67, T64, T68))
U5_G(T64, delc34_out_aga(T67, T64, T68)) → U6_G(T64, del34_in_aga(X208, T68, X209))
U5_G(T64, delc34_out_aga(T67, T64, T68)) → DEL34_IN_AGA(X208, T68, X209)
CONF19_IN_G(T54) → U7_G(T54, del2c29_in_ga(T54, T56))
U7_G(T54, del2c29_out_ga(T54, T56)) → U8_G(T54, del43_in_aag(X175, X176, T56))
U7_G(T54, del2c29_out_ga(T54, T56)) → DEL43_IN_AAG(X175, X176, T56)
DEL43_IN_AAG(X324, .(X325, X326), .(X325, T99)) → U11_AAG(X324, X325, X326, T99, del43_in_aag(X324, X326, T99))
DEL43_IN_AAG(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAG(X324, X326, T99)
U7_G(T54, del2c29_out_ga(T54, T56)) → U9_G(T54, delc43_in_aag(T88, T89, T56))
U9_G(T54, delc43_out_aag(T88, T89, T56)) → U10_G(T54, conf19_in_g(T89))
U9_G(T54, delc43_out_aag(T88, T89, T56)) → CONF19_IN_G(T89)

The TRS R consists of the following rules:

delc6_in_aga(X53, .(X53, T20), T20) → delc6_out_aga(X53, .(X53, T20), T20)
delc6_in_aga(X72, .(T25, T26), .(T25, X73)) → U20_aga(X72, T25, T26, X73, delc6_in_aga(X72, T26, X73))
U20_aga(X72, T25, T26, X73, delc6_out_aga(X72, T26, X73)) → delc6_out_aga(X72, .(T25, T26), .(T25, X73))
delc18_in_aag(X127, .(X127, T43), T43) → delc18_out_aag(X127, .(X127, T43), T43)
delc18_in_aag(X147, .(X148, X149), .(X148, T46)) → U21_aag(X147, X148, X149, T46, delc18_in_aag(X147, X149, T46))
U21_aag(X147, X148, X149, T46, delc18_out_aag(X147, X149, T46)) → delc18_out_aag(X147, .(X148, X149), .(X148, T46))
delc34_in_aga(X242, .(X242, T77), T77) → delc34_out_aga(X242, .(X242, T77), T77)
delc34_in_aga(X261, .(T82, T84), .(T82, X262)) → U22_aga(X261, T82, T84, X262, delc34_in_aga(X261, T84, X262))
U22_aga(X261, T82, T84, X262, delc34_out_aga(X261, T84, X262)) → delc34_out_aga(X261, .(T82, T84), .(T82, X262))
del2c29_in_ga(T64, X209) → U27_ga(T64, X209, delc34_in_aga(T67, T64, T68))
U27_ga(T64, X209, delc34_out_aga(T67, T64, T68)) → U28_ga(T64, X209, delc34_in_aga(X208, T68, X209))
U28_ga(T64, X209, delc34_out_aga(X208, T68, X209)) → del2c29_out_ga(T64, X209)
delc43_in_aag(X304, .(X304, T95), T95) → delc43_out_aag(X304, .(X304, T95), T95)
delc43_in_aag(X324, .(X325, X326), .(X325, T99)) → U26_aag(X324, X325, X326, T99, delc43_in_aag(X324, X326, T99))
U26_aag(X324, X325, X326, T99, delc43_out_aag(X324, X326, T99)) → delc43_out_aag(X324, .(X325, X326), .(X325, T99))

The argument filtering Pi contains the following mapping:
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
delc6_in_aga(x1, x2, x3)  =  delc6_in_aga(x2)
delc6_out_aga(x1, x2, x3)  =  delc6_out_aga(x2, x3)
U20_aga(x1, x2, x3, x4, x5)  =  U20_aga(x3, x5)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
delc18_in_aag(x1, x2, x3)  =  delc18_in_aag(x3)
delc18_out_aag(x1, x2, x3)  =  delc18_out_aag(x2, x3)
U21_aag(x1, x2, x3, x4, x5)  =  U21_aag(x4, x5)
conf19_in_g(x1)  =  conf19_in_g(x1)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
delc34_in_aga(x1, x2, x3)  =  delc34_in_aga(x2)
delc34_out_aga(x1, x2, x3)  =  delc34_out_aga(x2, x3)
U22_aga(x1, x2, x3, x4, x5)  =  U22_aga(x3, x5)
del2c29_in_ga(x1, x2)  =  del2c29_in_ga(x1)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
del2c29_out_ga(x1, x2)  =  del2c29_out_ga(x1, x2)
del43_in_aag(x1, x2, x3)  =  del43_in_aag(x3)
delc43_in_aag(x1, x2, x3)  =  delc43_in_aag(x3)
delc43_out_aag(x1, x2, x3)  =  delc43_out_aag(x2, x3)
U26_aag(x1, x2, x3, x4, x5)  =  U26_aag(x4, x5)
CONF1_IN_G(x1)  =  CONF1_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
DEL6_IN_AGA(x1, x2, x3)  =  DEL6_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
DEL18_IN_AAG(x1, x2, x3)  =  DEL18_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x4, x5)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1, x2)  =  U18_G(x1, x2)
CONF19_IN_G(x1)  =  CONF19_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x1, x2)
DEL34_IN_AGA(x1, x2, x3)  =  DEL34_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x3, x5)
U5_G(x1, x2)  =  U5_G(x1, x2)
U6_G(x1, x2)  =  U6_G(x1, x2)
U7_G(x1, x2)  =  U7_G(x1, x2)
U8_G(x1, x2)  =  U8_G(x1, x2)
DEL43_IN_AAG(x1, x2, x3)  =  DEL43_IN_AAG(x3)
U11_AAG(x1, x2, x3, x4, x5)  =  U11_AAG(x4, x5)
U9_G(x1, x2)  =  U9_G(x1, x2)
U10_G(x1, x2)  =  U10_G(x1, x2)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 23 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

DEL43_IN_AAG(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAG(X324, X326, T99)

The TRS R consists of the following rules:

delc6_in_aga(X53, .(X53, T20), T20) → delc6_out_aga(X53, .(X53, T20), T20)
delc6_in_aga(X72, .(T25, T26), .(T25, X73)) → U20_aga(X72, T25, T26, X73, delc6_in_aga(X72, T26, X73))
U20_aga(X72, T25, T26, X73, delc6_out_aga(X72, T26, X73)) → delc6_out_aga(X72, .(T25, T26), .(T25, X73))
delc18_in_aag(X127, .(X127, T43), T43) → delc18_out_aag(X127, .(X127, T43), T43)
delc18_in_aag(X147, .(X148, X149), .(X148, T46)) → U21_aag(X147, X148, X149, T46, delc18_in_aag(X147, X149, T46))
U21_aag(X147, X148, X149, T46, delc18_out_aag(X147, X149, T46)) → delc18_out_aag(X147, .(X148, X149), .(X148, T46))
delc34_in_aga(X242, .(X242, T77), T77) → delc34_out_aga(X242, .(X242, T77), T77)
delc34_in_aga(X261, .(T82, T84), .(T82, X262)) → U22_aga(X261, T82, T84, X262, delc34_in_aga(X261, T84, X262))
U22_aga(X261, T82, T84, X262, delc34_out_aga(X261, T84, X262)) → delc34_out_aga(X261, .(T82, T84), .(T82, X262))
del2c29_in_ga(T64, X209) → U27_ga(T64, X209, delc34_in_aga(T67, T64, T68))
U27_ga(T64, X209, delc34_out_aga(T67, T64, T68)) → U28_ga(T64, X209, delc34_in_aga(X208, T68, X209))
U28_ga(T64, X209, delc34_out_aga(X208, T68, X209)) → del2c29_out_ga(T64, X209)
delc43_in_aag(X304, .(X304, T95), T95) → delc43_out_aag(X304, .(X304, T95), T95)
delc43_in_aag(X324, .(X325, X326), .(X325, T99)) → U26_aag(X324, X325, X326, T99, delc43_in_aag(X324, X326, T99))
U26_aag(X324, X325, X326, T99, delc43_out_aag(X324, X326, T99)) → delc43_out_aag(X324, .(X325, X326), .(X325, T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
delc6_in_aga(x1, x2, x3)  =  delc6_in_aga(x2)
delc6_out_aga(x1, x2, x3)  =  delc6_out_aga(x2, x3)
U20_aga(x1, x2, x3, x4, x5)  =  U20_aga(x3, x5)
delc18_in_aag(x1, x2, x3)  =  delc18_in_aag(x3)
delc18_out_aag(x1, x2, x3)  =  delc18_out_aag(x2, x3)
U21_aag(x1, x2, x3, x4, x5)  =  U21_aag(x4, x5)
delc34_in_aga(x1, x2, x3)  =  delc34_in_aga(x2)
delc34_out_aga(x1, x2, x3)  =  delc34_out_aga(x2, x3)
U22_aga(x1, x2, x3, x4, x5)  =  U22_aga(x3, x5)
del2c29_in_ga(x1, x2)  =  del2c29_in_ga(x1)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
del2c29_out_ga(x1, x2)  =  del2c29_out_ga(x1, x2)
delc43_in_aag(x1, x2, x3)  =  delc43_in_aag(x3)
delc43_out_aag(x1, x2, x3)  =  delc43_out_aag(x2, x3)
U26_aag(x1, x2, x3, x4, x5)  =  U26_aag(x4, x5)
DEL43_IN_AAG(x1, x2, x3)  =  DEL43_IN_AAG(x3)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

DEL43_IN_AAG(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAG(X324, X326, T99)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL43_IN_AAG(x1, x2, x3)  =  DEL43_IN_AAG(x3)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

DEL43_IN_AAG(.(T99)) → DEL43_IN_AAG(T99)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• DEL43_IN_AAG(.(T99)) → DEL43_IN_AAG(T99)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

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

The TRS R consists of the following rules:

delc6_in_aga(X53, .(X53, T20), T20) → delc6_out_aga(X53, .(X53, T20), T20)
delc6_in_aga(X72, .(T25, T26), .(T25, X73)) → U20_aga(X72, T25, T26, X73, delc6_in_aga(X72, T26, X73))
U20_aga(X72, T25, T26, X73, delc6_out_aga(X72, T26, X73)) → delc6_out_aga(X72, .(T25, T26), .(T25, X73))
delc18_in_aag(X127, .(X127, T43), T43) → delc18_out_aag(X127, .(X127, T43), T43)
delc18_in_aag(X147, .(X148, X149), .(X148, T46)) → U21_aag(X147, X148, X149, T46, delc18_in_aag(X147, X149, T46))
U21_aag(X147, X148, X149, T46, delc18_out_aag(X147, X149, T46)) → delc18_out_aag(X147, .(X148, X149), .(X148, T46))
delc34_in_aga(X242, .(X242, T77), T77) → delc34_out_aga(X242, .(X242, T77), T77)
delc34_in_aga(X261, .(T82, T84), .(T82, X262)) → U22_aga(X261, T82, T84, X262, delc34_in_aga(X261, T84, X262))
U22_aga(X261, T82, T84, X262, delc34_out_aga(X261, T84, X262)) → delc34_out_aga(X261, .(T82, T84), .(T82, X262))
del2c29_in_ga(T64, X209) → U27_ga(T64, X209, delc34_in_aga(T67, T64, T68))
U27_ga(T64, X209, delc34_out_aga(T67, T64, T68)) → U28_ga(T64, X209, delc34_in_aga(X208, T68, X209))
U28_ga(T64, X209, delc34_out_aga(X208, T68, X209)) → del2c29_out_ga(T64, X209)
delc43_in_aag(X304, .(X304, T95), T95) → delc43_out_aag(X304, .(X304, T95), T95)
delc43_in_aag(X324, .(X325, X326), .(X325, T99)) → U26_aag(X324, X325, X326, T99, delc43_in_aag(X324, X326, T99))
U26_aag(X324, X325, X326, T99, delc43_out_aag(X324, X326, T99)) → delc43_out_aag(X324, .(X325, X326), .(X325, T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
delc6_in_aga(x1, x2, x3)  =  delc6_in_aga(x2)
delc6_out_aga(x1, x2, x3)  =  delc6_out_aga(x2, x3)
U20_aga(x1, x2, x3, x4, x5)  =  U20_aga(x3, x5)
delc18_in_aag(x1, x2, x3)  =  delc18_in_aag(x3)
delc18_out_aag(x1, x2, x3)  =  delc18_out_aag(x2, x3)
U21_aag(x1, x2, x3, x4, x5)  =  U21_aag(x4, x5)
delc34_in_aga(x1, x2, x3)  =  delc34_in_aga(x2)
delc34_out_aga(x1, x2, x3)  =  delc34_out_aga(x2, x3)
U22_aga(x1, x2, x3, x4, x5)  =  U22_aga(x3, x5)
del2c29_in_ga(x1, x2)  =  del2c29_in_ga(x1)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
del2c29_out_ga(x1, x2)  =  del2c29_out_ga(x1, x2)
delc43_in_aag(x1, x2, x3)  =  delc43_in_aag(x3)
delc43_out_aag(x1, x2, x3)  =  delc43_out_aag(x2, x3)
U26_aag(x1, x2, x3, x4, x5)  =  U26_aag(x4, x5)
DEL34_IN_AGA(x1, x2, x3)  =  DEL34_IN_AGA(x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

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

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

DEL34_IN_AGA(.(T84)) → DEL34_IN_AGA(T84)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

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

(20) YES

(21) Obligation:

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

CONF19_IN_G(T54) → U7_G(T54, del2c29_in_ga(T54, T56))
U7_G(T54, del2c29_out_ga(T54, T56)) → U9_G(T54, delc43_in_aag(T88, T89, T56))
U9_G(T54, delc43_out_aag(T88, T89, T56)) → CONF19_IN_G(T89)

The TRS R consists of the following rules:

delc6_in_aga(X53, .(X53, T20), T20) → delc6_out_aga(X53, .(X53, T20), T20)
delc6_in_aga(X72, .(T25, T26), .(T25, X73)) → U20_aga(X72, T25, T26, X73, delc6_in_aga(X72, T26, X73))
U20_aga(X72, T25, T26, X73, delc6_out_aga(X72, T26, X73)) → delc6_out_aga(X72, .(T25, T26), .(T25, X73))
delc18_in_aag(X127, .(X127, T43), T43) → delc18_out_aag(X127, .(X127, T43), T43)
delc18_in_aag(X147, .(X148, X149), .(X148, T46)) → U21_aag(X147, X148, X149, T46, delc18_in_aag(X147, X149, T46))
U21_aag(X147, X148, X149, T46, delc18_out_aag(X147, X149, T46)) → delc18_out_aag(X147, .(X148, X149), .(X148, T46))
delc34_in_aga(X242, .(X242, T77), T77) → delc34_out_aga(X242, .(X242, T77), T77)
delc34_in_aga(X261, .(T82, T84), .(T82, X262)) → U22_aga(X261, T82, T84, X262, delc34_in_aga(X261, T84, X262))
U22_aga(X261, T82, T84, X262, delc34_out_aga(X261, T84, X262)) → delc34_out_aga(X261, .(T82, T84), .(T82, X262))
del2c29_in_ga(T64, X209) → U27_ga(T64, X209, delc34_in_aga(T67, T64, T68))
U27_ga(T64, X209, delc34_out_aga(T67, T64, T68)) → U28_ga(T64, X209, delc34_in_aga(X208, T68, X209))
U28_ga(T64, X209, delc34_out_aga(X208, T68, X209)) → del2c29_out_ga(T64, X209)
delc43_in_aag(X304, .(X304, T95), T95) → delc43_out_aag(X304, .(X304, T95), T95)
delc43_in_aag(X324, .(X325, X326), .(X325, T99)) → U26_aag(X324, X325, X326, T99, delc43_in_aag(X324, X326, T99))
U26_aag(X324, X325, X326, T99, delc43_out_aag(X324, X326, T99)) → delc43_out_aag(X324, .(X325, X326), .(X325, T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
delc6_in_aga(x1, x2, x3)  =  delc6_in_aga(x2)
delc6_out_aga(x1, x2, x3)  =  delc6_out_aga(x2, x3)
U20_aga(x1, x2, x3, x4, x5)  =  U20_aga(x3, x5)
delc18_in_aag(x1, x2, x3)  =  delc18_in_aag(x3)
delc18_out_aag(x1, x2, x3)  =  delc18_out_aag(x2, x3)
U21_aag(x1, x2, x3, x4, x5)  =  U21_aag(x4, x5)
delc34_in_aga(x1, x2, x3)  =  delc34_in_aga(x2)
delc34_out_aga(x1, x2, x3)  =  delc34_out_aga(x2, x3)
U22_aga(x1, x2, x3, x4, x5)  =  U22_aga(x3, x5)
del2c29_in_ga(x1, x2)  =  del2c29_in_ga(x1)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
del2c29_out_ga(x1, x2)  =  del2c29_out_ga(x1, x2)
delc43_in_aag(x1, x2, x3)  =  delc43_in_aag(x3)
delc43_out_aag(x1, x2, x3)  =  delc43_out_aag(x2, x3)
U26_aag(x1, x2, x3, x4, x5)  =  U26_aag(x4, x5)
CONF19_IN_G(x1)  =  CONF19_IN_G(x1)
U7_G(x1, x2)  =  U7_G(x1, x2)
U9_G(x1, x2)  =  U9_G(x1, x2)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

CONF19_IN_G(T54) → U7_G(T54, del2c29_in_ga(T54, T56))
U7_G(T54, del2c29_out_ga(T54, T56)) → U9_G(T54, delc43_in_aag(T88, T89, T56))
U9_G(T54, delc43_out_aag(T88, T89, T56)) → CONF19_IN_G(T89)

The TRS R consists of the following rules:

del2c29_in_ga(T64, X209) → U27_ga(T64, X209, delc34_in_aga(T67, T64, T68))
delc43_in_aag(X304, .(X304, T95), T95) → delc43_out_aag(X304, .(X304, T95), T95)
delc43_in_aag(X324, .(X325, X326), .(X325, T99)) → U26_aag(X324, X325, X326, T99, delc43_in_aag(X324, X326, T99))
U27_ga(T64, X209, delc34_out_aga(T67, T64, T68)) → U28_ga(T64, X209, delc34_in_aga(X208, T68, X209))
U26_aag(X324, X325, X326, T99, delc43_out_aag(X324, X326, T99)) → delc43_out_aag(X324, .(X325, X326), .(X325, T99))
delc34_in_aga(X242, .(X242, T77), T77) → delc34_out_aga(X242, .(X242, T77), T77)
delc34_in_aga(X261, .(T82, T84), .(T82, X262)) → U22_aga(X261, T82, T84, X262, delc34_in_aga(X261, T84, X262))
U28_ga(T64, X209, delc34_out_aga(X208, T68, X209)) → del2c29_out_ga(T64, X209)
U22_aga(X261, T82, T84, X262, delc34_out_aga(X261, T84, X262)) → delc34_out_aga(X261, .(T82, T84), .(T82, X262))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
delc34_in_aga(x1, x2, x3)  =  delc34_in_aga(x2)
delc34_out_aga(x1, x2, x3)  =  delc34_out_aga(x2, x3)
U22_aga(x1, x2, x3, x4, x5)  =  U22_aga(x3, x5)
del2c29_in_ga(x1, x2)  =  del2c29_in_ga(x1)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
del2c29_out_ga(x1, x2)  =  del2c29_out_ga(x1, x2)
delc43_in_aag(x1, x2, x3)  =  delc43_in_aag(x3)
delc43_out_aag(x1, x2, x3)  =  delc43_out_aag(x2, x3)
U26_aag(x1, x2, x3, x4, x5)  =  U26_aag(x4, x5)
CONF19_IN_G(x1)  =  CONF19_IN_G(x1)
U7_G(x1, x2)  =  U7_G(x1, x2)
U9_G(x1, x2)  =  U9_G(x1, x2)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

CONF19_IN_G(T54) → U7_G(T54, del2c29_in_ga(T54))
U7_G(T54, del2c29_out_ga(T54, T56)) → U9_G(T54, delc43_in_aag(T56))
U9_G(T54, delc43_out_aag(T89, T56)) → CONF19_IN_G(T89)

The TRS R consists of the following rules:

del2c29_in_ga(T64) → U27_ga(T64, delc34_in_aga(T64))
delc43_in_aag(T95) → delc43_out_aag(.(T95), T95)
delc43_in_aag(.(T99)) → U26_aag(T99, delc43_in_aag(T99))
U27_ga(T64, delc34_out_aga(T64, T68)) → U28_ga(T64, delc34_in_aga(T68))
U26_aag(T99, delc43_out_aag(X326, T99)) → delc43_out_aag(.(X326), .(T99))
delc34_in_aga(.(T77)) → delc34_out_aga(.(T77), T77)
delc34_in_aga(.(T84)) → U22_aga(T84, delc34_in_aga(T84))
U28_ga(T64, delc34_out_aga(T68, X209)) → del2c29_out_ga(T64, X209)
U22_aga(T84, delc34_out_aga(T84, X262)) → delc34_out_aga(.(T84), .(X262))

The set Q consists of the following terms:

del2c29_in_ga(x0)
delc43_in_aag(x0)
U27_ga(x0, x1)
U26_aag(x0, x1)
delc34_in_aga(x0)
U28_ga(x0, x1)
U22_aga(x0, x1)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U7_G(T54, del2c29_out_ga(T54, T56)) → U9_G(T54, delc43_in_aag(T56))

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

POL(CONF19_IN_G(x1)) =

0

+

[ 1, 1 ]

·

x1

POL(U7_G(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 1, 1 ]

·

x2

POL(del2c29_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(del2c29_out_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U9_G(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 1, 1 ]

·

x2

POL(delc43_in_aag(x1)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(delc43_out_aag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U27_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

POL(delc34_in_aga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(.(x1)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(U26_aag(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(delc34_out_aga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U22_aga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U28_ga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

The following usable rules [FROCOS05] were oriented:

del2c29_in_ga(T64) → U27_ga(T64, delc34_in_aga(T64))
delc43_in_aag(T95) → delc43_out_aag(.(T95), T95)
delc43_in_aag(.(T99)) → U26_aag(T99, delc43_in_aag(T99))
delc34_in_aga(.(T77)) → delc34_out_aga(.(T77), T77)
delc34_in_aga(.(T84)) → U22_aga(T84, delc34_in_aga(T84))
U27_ga(T64, delc34_out_aga(T64, T68)) → U28_ga(T64, delc34_in_aga(T68))
U26_aag(T99, delc43_out_aag(X326, T99)) → delc43_out_aag(.(X326), .(T99))
U28_ga(T64, delc34_out_aga(T68, X209)) → del2c29_out_ga(T64, X209)
U22_aga(T84, delc34_out_aga(T84, X262)) → delc34_out_aga(.(T84), .(X262))

(27) Obligation:

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

CONF19_IN_G(T54) → U7_G(T54, del2c29_in_ga(T54))
U9_G(T54, delc43_out_aag(T89, T56)) → CONF19_IN_G(T89)

The TRS R consists of the following rules:

del2c29_in_ga(T64) → U27_ga(T64, delc34_in_aga(T64))
delc43_in_aag(T95) → delc43_out_aag(.(T95), T95)
delc43_in_aag(.(T99)) → U26_aag(T99, delc43_in_aag(T99))
U27_ga(T64, delc34_out_aga(T64, T68)) → U28_ga(T64, delc34_in_aga(T68))
U26_aag(T99, delc43_out_aag(X326, T99)) → delc43_out_aag(.(X326), .(T99))
delc34_in_aga(.(T77)) → delc34_out_aga(.(T77), T77)
delc34_in_aga(.(T84)) → U22_aga(T84, delc34_in_aga(T84))
U28_ga(T64, delc34_out_aga(T68, X209)) → del2c29_out_ga(T64, X209)
U22_aga(T84, delc34_out_aga(T84, X262)) → delc34_out_aga(.(T84), .(X262))

The set Q consists of the following terms:

del2c29_in_ga(x0)
delc43_in_aag(x0)
U27_ga(x0, x1)
U26_aag(x0, x1)
delc34_in_aga(x0)
U28_ga(x0, x1)
U22_aga(x0, x1)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(29) TRUE

(30) Obligation:

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

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

The TRS R consists of the following rules:

delc6_in_aga(X53, .(X53, T20), T20) → delc6_out_aga(X53, .(X53, T20), T20)
delc6_in_aga(X72, .(T25, T26), .(T25, X73)) → U20_aga(X72, T25, T26, X73, delc6_in_aga(X72, T26, X73))
U20_aga(X72, T25, T26, X73, delc6_out_aga(X72, T26, X73)) → delc6_out_aga(X72, .(T25, T26), .(T25, X73))
delc18_in_aag(X127, .(X127, T43), T43) → delc18_out_aag(X127, .(X127, T43), T43)
delc18_in_aag(X147, .(X148, X149), .(X148, T46)) → U21_aag(X147, X148, X149, T46, delc18_in_aag(X147, X149, T46))
U21_aag(X147, X148, X149, T46, delc18_out_aag(X147, X149, T46)) → delc18_out_aag(X147, .(X148, X149), .(X148, T46))
delc34_in_aga(X242, .(X242, T77), T77) → delc34_out_aga(X242, .(X242, T77), T77)
delc34_in_aga(X261, .(T82, T84), .(T82, X262)) → U22_aga(X261, T82, T84, X262, delc34_in_aga(X261, T84, X262))
U22_aga(X261, T82, T84, X262, delc34_out_aga(X261, T84, X262)) → delc34_out_aga(X261, .(T82, T84), .(T82, X262))
del2c29_in_ga(T64, X209) → U27_ga(T64, X209, delc34_in_aga(T67, T64, T68))
U27_ga(T64, X209, delc34_out_aga(T67, T64, T68)) → U28_ga(T64, X209, delc34_in_aga(X208, T68, X209))
U28_ga(T64, X209, delc34_out_aga(X208, T68, X209)) → del2c29_out_ga(T64, X209)
delc43_in_aag(X304, .(X304, T95), T95) → delc43_out_aag(X304, .(X304, T95), T95)
delc43_in_aag(X324, .(X325, X326), .(X325, T99)) → U26_aag(X324, X325, X326, T99, delc43_in_aag(X324, X326, T99))
U26_aag(X324, X325, X326, T99, delc43_out_aag(X324, X326, T99)) → delc43_out_aag(X324, .(X325, X326), .(X325, T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
delc6_in_aga(x1, x2, x3)  =  delc6_in_aga(x2)
delc6_out_aga(x1, x2, x3)  =  delc6_out_aga(x2, x3)
U20_aga(x1, x2, x3, x4, x5)  =  U20_aga(x3, x5)
delc18_in_aag(x1, x2, x3)  =  delc18_in_aag(x3)
delc18_out_aag(x1, x2, x3)  =  delc18_out_aag(x2, x3)
U21_aag(x1, x2, x3, x4, x5)  =  U21_aag(x4, x5)
delc34_in_aga(x1, x2, x3)  =  delc34_in_aga(x2)
delc34_out_aga(x1, x2, x3)  =  delc34_out_aga(x2, x3)
U22_aga(x1, x2, x3, x4, x5)  =  U22_aga(x3, x5)
del2c29_in_ga(x1, x2)  =  del2c29_in_ga(x1)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
del2c29_out_ga(x1, x2)  =  del2c29_out_ga(x1, x2)
delc43_in_aag(x1, x2, x3)  =  delc43_in_aag(x3)
delc43_out_aag(x1, x2, x3)  =  delc43_out_aag(x2, x3)
U26_aag(x1, x2, x3, x4, x5)  =  U26_aag(x4, x5)
DEL18_IN_AAG(x1, x2, x3)  =  DEL18_IN_AAG(x3)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

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

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

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

DEL18_IN_AAG(.(T46)) → DEL18_IN_AAG(T46)

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

(35) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

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

(36) YES

(37) Obligation:

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

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

The TRS R consists of the following rules:

delc6_in_aga(X53, .(X53, T20), T20) → delc6_out_aga(X53, .(X53, T20), T20)
delc6_in_aga(X72, .(T25, T26), .(T25, X73)) → U20_aga(X72, T25, T26, X73, delc6_in_aga(X72, T26, X73))
U20_aga(X72, T25, T26, X73, delc6_out_aga(X72, T26, X73)) → delc6_out_aga(X72, .(T25, T26), .(T25, X73))
delc18_in_aag(X127, .(X127, T43), T43) → delc18_out_aag(X127, .(X127, T43), T43)
delc18_in_aag(X147, .(X148, X149), .(X148, T46)) → U21_aag(X147, X148, X149, T46, delc18_in_aag(X147, X149, T46))
U21_aag(X147, X148, X149, T46, delc18_out_aag(X147, X149, T46)) → delc18_out_aag(X147, .(X148, X149), .(X148, T46))
delc34_in_aga(X242, .(X242, T77), T77) → delc34_out_aga(X242, .(X242, T77), T77)
delc34_in_aga(X261, .(T82, T84), .(T82, X262)) → U22_aga(X261, T82, T84, X262, delc34_in_aga(X261, T84, X262))
U22_aga(X261, T82, T84, X262, delc34_out_aga(X261, T84, X262)) → delc34_out_aga(X261, .(T82, T84), .(T82, X262))
del2c29_in_ga(T64, X209) → U27_ga(T64, X209, delc34_in_aga(T67, T64, T68))
U27_ga(T64, X209, delc34_out_aga(T67, T64, T68)) → U28_ga(T64, X209, delc34_in_aga(X208, T68, X209))
U28_ga(T64, X209, delc34_out_aga(X208, T68, X209)) → del2c29_out_ga(T64, X209)
delc43_in_aag(X304, .(X304, T95), T95) → delc43_out_aag(X304, .(X304, T95), T95)
delc43_in_aag(X324, .(X325, X326), .(X325, T99)) → U26_aag(X324, X325, X326, T99, delc43_in_aag(X324, X326, T99))
U26_aag(X324, X325, X326, T99, delc43_out_aag(X324, X326, T99)) → delc43_out_aag(X324, .(X325, X326), .(X325, T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
delc6_in_aga(x1, x2, x3)  =  delc6_in_aga(x2)
delc6_out_aga(x1, x2, x3)  =  delc6_out_aga(x2, x3)
U20_aga(x1, x2, x3, x4, x5)  =  U20_aga(x3, x5)
delc18_in_aag(x1, x2, x3)  =  delc18_in_aag(x3)
delc18_out_aag(x1, x2, x3)  =  delc18_out_aag(x2, x3)
U21_aag(x1, x2, x3, x4, x5)  =  U21_aag(x4, x5)
delc34_in_aga(x1, x2, x3)  =  delc34_in_aga(x2)
delc34_out_aga(x1, x2, x3)  =  delc34_out_aga(x2, x3)
U22_aga(x1, x2, x3, x4, x5)  =  U22_aga(x3, x5)
del2c29_in_ga(x1, x2)  =  del2c29_in_ga(x1)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
del2c29_out_ga(x1, x2)  =  del2c29_out_ga(x1, x2)
delc43_in_aag(x1, x2, x3)  =  delc43_in_aag(x3)
delc43_out_aag(x1, x2, x3)  =  delc43_out_aag(x2, x3)
U26_aag(x1, x2, x3, x4, x5)  =  U26_aag(x4, x5)
DEL6_IN_AGA(x1, x2, x3)  =  DEL6_IN_AGA(x2)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

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

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

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

DEL6_IN_AGA(.(T26)) → DEL6_IN_AGA(T26)

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

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

(43) YES