Left Termination of the query pattern palindrome_in_1(g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

palindrome(L) :- ','(halves(L, X1s, X2s, EvenOdd), ','(eq(EvenOdd, even), eq(X1s, X2s))).
palindrome(L) :- ','(halves(L, X1s, X2s, EvenOdd), ','(eq(EvenOdd, odd), last(X1s, X1, X2s))).
halves([], [], [], even).
halves(.(X, []), .(X, []), [], odd).
halves(.(T, .(Y, Xs)), .(T, Ts), .(R, Rs), EvenOdd) :- ','(last(.(Y, Xs), R, Rests), halves(Rests, Ts, Rs, EvenOdd)).
last(.(T, []), T, []).
last(.(H, T), X, .(H, M)) :- last(T, X, M).
eq(X, X).

Queries:

palindrome(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: palindrome(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: palindrome(T1), SCOPE: 1, CLAUSE: 0 | palindrome(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(halves(T3, X4, X5, X6), ','(eq(X6, even), eq(X4, X5))) | palindrome(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX4 is free; \nX5 is free; \nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(halves(T3, X4, X5, X6), ','(eq(X6, even), eq(X4, X5))), SCOPE: 2, CLAUSE: 2 | ','(halves(T3, X4, X5, X6), ','(eq(X6, even), eq(X4, X5))), SCOPE: 2, CLAUSE: 3 | ','(halves(T3, X4, X5, X6), ','(eq(X6, even), eq(X4, X5))), SCOPE: 2, CLAUSE: 4 | ?_2 | palindrome(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX4 is free; \nX5 is free; \nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(halves(T3, X4, X5, X6), ','(eq(X6, even), eq(X4, X5))), SCOPE: 2, CLAUSE: 2\n\nT3 is ground\nX4 is free; \nX5 is free; \nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(halves(T3, X4, X5, X6), ','(eq(X6, even), eq(X4, X5))), SCOPE: 2, CLAUSE: 3 | ','(halves(T3, X4, X5, X6), ','(eq(X6, even), eq(X4, X5))), SCOPE: 2, CLAUSE: 4 | ?_2 | palindrome(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX4 is free; \nX5 is free; \nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(eq(even, even), eq([], []))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(eq(even, even), eq([], [])), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: eq([], [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: eq([], []), SCOPE: 4, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(halves(T3, X4, X5, X6), ','(eq(X6, even), eq(X4, X5))), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nX4 is free; \nX5 is free; \nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(halves(T3, X4, X5, X6), ','(eq(X6, even), eq(X4, X5))), SCOPE: 2, CLAUSE: 4 | ?_2 | palindrome(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX4 is free; \nX5 is free; \nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(eq(odd, even), eq(.(T8, []), []))\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(eq(odd, even), eq(.(T8, []), [])), SCOPE: 5, CLAUSE: 7\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(halves(T3, X4, X5, X6), ','(eq(X6, even), eq(X4, X5))), SCOPE: 2, CLAUSE: 4\n\nT3 is ground\nX4 is free; \nX5 is free; \nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_2 | palindrome(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(last(.(T22, T23), X75, X73), ','(halves(X73, X74, X76, X77), ','(eq(X77, even), eq(.(T21, X74), .(X75, X76)))))\n\nT21 is ground\nT22 is ground\nT23 is ground\nX74 is free; \nX75 is free; \nX76 is free; \nX77 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: last(.(T22, T23), X75, X73)\n\nT22 is ground\nT23 is ground\nX75 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(halves(T27, X74, X76, X77), ','(eq(X77, even), eq(.(T21, X74), .(T26, X76))))\n\nT21 is ground\nT26 is ground\nT27 is ground\nX74 is free; \nX76 is free; \nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: last(.(T22, T23), X75, X73), SCOPE: 6, CLAUSE: 5 | last(.(T22, T23), X75, X73), SCOPE: 6, CLAUSE: 6\n\nT22 is ground\nT23 is ground\nX75 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: last(.(T22, T23), X75, X73), SCOPE: 6, CLAUSE: 5\n\nT22 is ground\nT23 is ground\nX75 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: last(.(T22, T23), X75, X73), SCOPE: 6, CLAUSE: 6\n\nT22 is ground\nT23 is ground\nX75 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: last(T42, X113, X114)\n\nT42 is ground\nX113 is free; \nX114 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: last(T42, X113, X114), SCOPE: 7, CLAUSE: 5 | last(T42, X113, X114), SCOPE: 7, CLAUSE: 6\n\nT42 is ground\nX113 is free; \nX114 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: last(T42, X113, X114), SCOPE: 7, CLAUSE: 5\n\nT42 is ground\nX113 is free; \nX114 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: last(T42, X113, X114), SCOPE: 7, CLAUSE: 6\n\nT42 is ground\nX113 is free; \nX114 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: last(T55, X139, X140)\n\nT55 is ground\nX139 is free; \nX140 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: halves(T27, X74, X76, X77)\n\nT27 is ground\nX74 is free; \nX76 is free; \nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(eq(T60, even), eq(.(T21, T58), .(T26, T59)))\n\nT21 is ground\nT26 is ground\nT58 is ground\nT59 is ground\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: halves(T27, X74, X76, X77), SCOPE: 8, CLAUSE: 2 | halves(T27, X74, X76, X77), SCOPE: 8, CLAUSE: 3 | halves(T27, X74, X76, X77), SCOPE: 8, CLAUSE: 4\n\nT27 is ground\nX74 is free; \nX76 is free; \nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: halves(T27, X74, X76, X77), SCOPE: 8, CLAUSE: 2\n\nT27 is ground\nX74 is free; \nX76 is free; \nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: halves(T27, X74, X76, X77), SCOPE: 8, CLAUSE: 3 | halves(T27, X74, X76, X77), SCOPE: 8, CLAUSE: 4\n\nT27 is ground\nX74 is free; \nX76 is free; \nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: halves(T27, X74, X76, X77), SCOPE: 8, CLAUSE: 3\n\nT27 is ground\nX74 is free; \nX76 is free; \nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: halves(T27, X74, X76, X77), SCOPE: 8, CLAUSE: 4\n\nT27 is ground\nX74 is free; \nX76 is free; \nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(last(.(T73, T74), X185, X183), halves(X183, X184, X186, X187))\n\nT73 is ground\nT74 is ground\nX184 is free; \nX185 is free; \nX186 is free; \nX187 is free; \nX183 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: last(.(T73, T74), X185, X183)\n\nT73 is ground\nT74 is ground\nX185 is free; \nX183 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: halves(T78, X184, X186, X187)\n\nT78 is ground\nX184 is free; \nX186 is free; \nX187 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(eq(T60, even), eq(.(T21, T58), .(T26, T59))), SCOPE: 9, CLAUSE: 7\n\nT21 is ground\nT26 is ground\nT58 is ground\nT59 is ground\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: eq(.(T21, T58), .(T26, T59))\n\nT21 is ground\nT26 is ground\nT58 is ground\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: eq(.(T21, T58), .(T26, T59)), SCOPE: 10, CLAUSE: 7\n\nT21 is ground\nT26 is ground\nT58 is ground\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: palindrome(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(halves(T92, X214, X215, X216), ','(eq(X216, odd), last(X214, X217, X215)))\n\nT92 is ground\nX214 is free; \nX215 is free; \nX216 is free; \nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: halves(T92, X214, X215, X216)\n\nT92 is ground\nX214 is free; \nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(eq(T95, odd), last(T93, X217, T94))\n\nT93 is ground\nT94 is ground\nT95 is ground\nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(eq(T95, odd), last(T93, X217, T94)), SCOPE: 11, CLAUSE: 7\n\nT93 is ground\nT94 is ground\nT95 is ground\nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: last(T93, X217, T94)\n\nT93 is ground\nT94 is ground\nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: last(T93, X217, T94), SCOPE: 12, CLAUSE: 5 | last(T93, X217, T94), SCOPE: 12, CLAUSE: 6\n\nT93 is ground\nT94 is ground\nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: last(T93, X217, T94), SCOPE: 12, CLAUSE: 5\n\nT93 is ground\nT94 is ground\nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: last(T93, X217, T94), SCOPE: 12, CLAUSE: 6\n\nT93 is ground\nT94 is ground\nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: last(T115, X246, T116)\n\nT115 is ground\nT116 is ground\nX246 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 80 [arrowhead = none ];
80 -> 2;

81 [label="ONLY EVAL with clause\npalindrome(X3) :- ','(halves(X3, X4, X5, X6), ','(eq(X6, even), eq(X4, X5))).\nand substitution\nT1 -> T3\nX3 -> T3", fontsize=14, style = filled, fillcolor = white];
2 -> 81 [arrowhead = none ];
81 -> 3;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 82 [arrowhead = none ];
82 -> 4;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 83 [arrowhead = none ];
83 -> 5;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 84 [arrowhead = none ];
84 -> 6;

85 [label="EVAL with clause\nhalves([], [], [], even).\nand substitution\nT3 -> []\nX4 -> []\nX5 -> []\nX6 -> even", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 85 [arrowhead = none ];
85 -> 7;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 86 [arrowhead = none ];
86 -> 8;

87 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 87 [arrowhead = none ];
87 -> 14;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 88 [arrowhead = none ];
88 -> 15;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 89 [arrowhead = none ];
89 -> 9;

90 [label="ONLY EVAL with clause\neq(X9, X9).\nand substitution\nX9 -> even", fontsize=14, style = filled, fillcolor = white];
9 -> 90 [arrowhead = none ];
90 -> 10;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 91 [arrowhead = none ];
91 -> 11;

92 [label="ONLY EVAL with clause\neq(X12, X12).\nand substitution\nX12 -> []", fontsize=14, style = filled, fillcolor = white];
11 -> 92 [arrowhead = none ];
92 -> 12;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 93 [arrowhead = none ];
93 -> 13;

94 [label="EVAL with clause\nhalves(.(X17, []), .(X17, []), [], odd).\nand substitution\nX17 -> T8\nT3 -> .(T8, [])\nX4 -> .(T8, [])\nX5 -> []\nX6 -> odd", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 94 [arrowhead = none ];
94 -> 16;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 95 [arrowhead = none ];
95 -> 17;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 96 [arrowhead = none ];
96 -> 20;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 97 [arrowhead = none ];
97 -> 21;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 98 [arrowhead = none ];
98 -> 18;

99 [label="BACKTRACK\nfor clause: eq(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
18 -> 99 [arrowhead = none ];
99 -> 19;

100 [label="EVAL with clause\nhalves(.(X66, .(X67, X68)), .(X66, X69), .(X70, X71), X72) :- ','(last(.(X67, X68), X70, X73), halves(X73, X69, X71, X72)).\nand substitution\nX66 -> T21\nX67 -> T22\nX68 -> T23\nT3 -> .(T21, .(T22, T23))\nX69 -> X74\nX4 -> .(T21, X74)\nX70 -> X75\nX71 -> X76\nX5 -> .(X75, X76)\nX6 -> X77\nX72 -> X77", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 100 [arrowhead = none ];
100 -> 22;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 101 [arrowhead = none ];
101 -> 23;

102 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 102 [arrowhead = none ];
102 -> 65;

103 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 103 [arrowhead = none ];
103 -> 24;

104 [label="SPLIT 2\nnew knowledge:\nT26 is ground\nT27 is ground\nreplacements:\nX75 -> T26\nX73 -> T27", fontsize=14, style = filled, fillcolor = violet];
22 -> 104 [arrowhead = none ];
104 -> 25;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 105 [arrowhead = none ];
105 -> 26;

106 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 106 [arrowhead = none ];
106 -> 41;

107 [label="SPLIT 2\nnew knowledge:\nT58 is ground\nT59 is ground\nT60 is ground\nreplacements:\nX74 -> T58\nX76 -> T59\nX77 -> T60", fontsize=14, style = filled, fillcolor = violet];
25 -> 107 [arrowhead = none ];
107 -> 42;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 108 [arrowhead = none ];
108 -> 27;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 109 [arrowhead = none ];
109 -> 28;

110 [label="EVAL with clause\nlast(.(X93, []), X93, []).\nand substitution\nT22 -> T34\nX93 -> T34\nT23 -> []\nX75 -> T34\nX73 -> []", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 110 [arrowhead = none ];
110 -> 29;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 111 [arrowhead = none ];
111 -> 30;

112 [label="ONLY EVAL with clause\nlast(.(X109, X110), X111, .(X109, X112)) :- last(X110, X111, X112).\nand substitution\nT22 -> T41\nX109 -> T41\nT23 -> T42\nX110 -> T42\nX75 -> X113\nX111 -> X113\nX112 -> X114\nX73 -> .(T41, X114)", fontsize=14, style = filled, fillcolor = white];
28 -> 112 [arrowhead = none ];
112 -> 32;

113 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 113 [arrowhead = none ];
113 -> 31;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 114 [arrowhead = none ];
114 -> 33;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 115 [arrowhead = none ];
115 -> 34;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 116 [arrowhead = none ];
116 -> 35;

117 [label="EVAL with clause\nlast(.(X121, []), X121, []).\nand substitution\nX121 -> T49\nT42 -> .(T49, [])\nX113 -> T49\nX114 -> []", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 117 [arrowhead = none ];
117 -> 36;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 118 [arrowhead = none ];
118 -> 37;

119 [label="EVAL with clause\nlast(.(X135, X136), X137, .(X135, X138)) :- last(X136, X137, X138).\nand substitution\nX135 -> T54\nX136 -> T55\nT42 -> .(T54, T55)\nX113 -> X139\nX137 -> X139\nX138 -> X140\nX114 -> .(T54, X140)", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 119 [arrowhead = none ];
119 -> 39;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 120 [arrowhead = none ];
120 -> 40;

121 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 121 [arrowhead = none ];
121 -> 38;

122 [label="INSTANCE with matching:\nT42 -> T55\nX113 -> X139\nX114 -> X140", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 122 [arrowhead = none , style=dashed];
122 -> 32[style=dashed];

123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 123 [arrowhead = none ];
123 -> 43;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 124 [arrowhead = none ];
124 -> 58;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 125 [arrowhead = none ];
125 -> 44;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 126 [arrowhead = none ];
126 -> 45;

127 [label="EVAL with clause\nhalves([], [], [], even).\nand substitution\nT27 -> []\nX74 -> []\nX76 -> []\nX77 -> even", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 127 [arrowhead = none ];
127 -> 46;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 128 [arrowhead = none ];
128 -> 47;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 129 [arrowhead = none ];
129 -> 49;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 130 [arrowhead = none ];
130 -> 50;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 131 [arrowhead = none ];
131 -> 48;

132 [label="EVAL with clause\nhalves(.(X151, []), .(X151, []), [], odd).\nand substitution\nX151 -> T65\nT27 -> .(T65, [])\nX74 -> .(T65, [])\nX76 -> []\nX77 -> odd", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 132 [arrowhead = none ];
132 -> 51;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 133 [arrowhead = none ];
133 -> 52;

134 [label="EVAL with clause\nhalves(.(X176, .(X177, X178)), .(X176, X179), .(X180, X181), X182) :- ','(last(.(X177, X178), X180, X183), halves(X183, X179, X181, X182)).\nand substitution\nX176 -> T72\nX177 -> T73\nX178 -> T74\nT27 -> .(T72, .(T73, T74))\nX179 -> X184\nX74 -> .(T72, X184)\nX180 -> X185\nX181 -> X186\nX76 -> .(X185, X186)\nX77 -> X187\nX182 -> X187", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 134 [arrowhead = none ];
134 -> 54;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 135 [arrowhead = none ];
135 -> 55;

136 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 136 [arrowhead = none ];
136 -> 53;

137 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
54 -> 137 [arrowhead = none ];
137 -> 56;

138 [label="SPLIT 2\nnew knowledge:\nT77 is ground\nT78 is ground\nreplacements:\nX185 -> T77\nX183 -> T78", fontsize=14, style = filled, fillcolor = violet];
54 -> 138 [arrowhead = none ];
138 -> 57;

139 [label="INSTANCE with matching:\nT22 -> T73\nT23 -> T74\nX75 -> X185\nX73 -> X183", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 139 [arrowhead = none , style=dashed];
139 -> 24[style=dashed];

140 [label="INSTANCE with matching:\nT27 -> T78\nX74 -> X184\nX76 -> X186\nX77 -> X187", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 140 [arrowhead = none , style=dashed];
140 -> 41[style=dashed];

141 [label="EVAL with clause\neq(X201, X201).\nand substitution\nT60 -> even\nX201 -> even\nT83 -> even", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 141 [arrowhead = none ];
141 -> 59;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 142 [arrowhead = none ];
142 -> 60;

143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 143 [arrowhead = none ];
143 -> 61;

144 [label="EVAL with clause\neq(X205, X205).\nand substitution\nT21 -> T88\nT58 -> T89\nX205 -> .(T88, T89)\nT26 -> T88\nT59 -> T89", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 144 [arrowhead = none ];
144 -> 62;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 145 [arrowhead = none ];
145 -> 63;

146 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 146 [arrowhead = none ];
146 -> 64;

147 [label="ONLY EVAL with clause\npalindrome(X213) :- ','(halves(X213, X214, X215, X216), ','(eq(X216, odd), last(X214, X217, X215))).\nand substitution\nT3 -> T92\nX213 -> T92", fontsize=14, style = filled, fillcolor = white];
65 -> 147 [arrowhead = none ];
147 -> 66;

148 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
66 -> 148 [arrowhead = none ];
148 -> 67;

149 [label="SPLIT 2\nnew knowledge:\nT93 is ground\nT94 is ground\nT95 is ground\nreplacements:\nX214 -> T93\nX215 -> T94\nX216 -> T95", fontsize=14, style = filled, fillcolor = violet];
66 -> 149 [arrowhead = none ];
149 -> 68;

150 [label="INSTANCE with matching:\nT27 -> T92\nX74 -> X214\nX76 -> X215\nX77 -> X216", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 150 [arrowhead = none , style=dashed];
150 -> 41[style=dashed];

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 151 [arrowhead = none ];
151 -> 69;

152 [label="EVAL with clause\neq(X222, X222).\nand substitution\nT95 -> odd\nX222 -> odd\nT100 -> odd", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 152 [arrowhead = none ];
152 -> 70;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 153 [arrowhead = none ];
153 -> 71;

154 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 154 [arrowhead = none ];
154 -> 72;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 155 [arrowhead = none ];
155 -> 73;

156 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 156 [arrowhead = none ];
156 -> 74;

157 [label="EVAL with clause\nlast(.(X230, []), X230, []).\nand substitution\nX230 -> T107\nT93 -> .(T107, [])\nX217 -> T107\nT94 -> []", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 157 [arrowhead = none ];
157 -> 75;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 158 [arrowhead = none ];
158 -> 76;

159 [label="EVAL with clause\nlast(.(X242, X243), X244, .(X242, X245)) :- last(X243, X244, X245).\nand substitution\nX242 -> T114\nX243 -> T115\nT93 -> .(T114, T115)\nX217 -> X246\nX244 -> X246\nX245 -> T116\nT94 -> .(T114, T116)", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 159 [arrowhead = none ];
159 -> 78;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 160 [arrowhead = none ];
160 -> 79;

161 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 161 [arrowhead = none ];
161 -> 77;

162 [label="INSTANCE with matching:\nT93 -> T115\nX217 -> X246\nT94 -> T116", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 162 [arrowhead = none , style=dashed];
162 -> 70[style=dashed];

}

(2) Obligation:

Triples:

last32(.(T54, T55), X139, .(T54, X140)) :- last32(T55, X139, X140).
last24(T41, T42, X113, .(T41, X114)) :- last32(T42, X113, X114).
halves41(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) :- last24(T73, T74, X185, X183).
halves41(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) :- ','(lastc24(T73, T74, T77, T78), halves41(T78, X184, X186, X187)).
last70(.(T114, T115), X246, .(T114, T116)) :- last70(T115, X246, T116).
palindrome1(.(T21, .(T22, T23))) :- last24(T22, T23, X75, X73).
palindrome1(.(T21, .(T22, T23))) :- ','(lastc24(T22, T23, T26, T27), halves41(T27, X74, X76, X77)).
palindrome1(T92) :- halves41(T92, X214, X215, X216).
palindrome1(T92) :- ','(halvesc41(T92, T93, T94, odd), last70(T93, X217, T94)).

Clauses:

lastc32(.(T49, []), T49, []).
lastc32(.(T54, T55), X139, .(T54, X140)) :- lastc32(T55, X139, X140).
lastc24(T34, [], T34, []).
lastc24(T41, T42, X113, .(T41, X114)) :- lastc32(T42, X113, X114).
halvesc41([], [], [], even).
halvesc41(.(T65, []), .(T65, []), [], odd).
halvesc41(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) :- ','(lastc24(T73, T74, T77, T78), halvesc41(T78, X184, X186, X187)).
lastc70(.(T107, []), T107, []).
lastc70(.(T114, T115), X246, .(T114, T116)) :- lastc70(T115, X246, T116).

Afs:

palindrome1(x1)  =  palindrome1(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:
palindrome1_in: (b)
last24_in: (b,b,f,f)
last32_in: (b,f,f)
lastc24_in: (b,b,f,f)
lastc32_in: (b,f,f)
halves41_in: (b,f,f,f)
halvesc41_in: (b,f,f,b)
last70_in: (b,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PALINDROME1_IN_G(.(T21, .(T22, T23))) → U7_G(T21, T22, T23, last24_in_ggaa(T22, T23, X75, X73))
PALINDROME1_IN_G(.(T21, .(T22, T23))) → LAST24_IN_GGAA(T22, T23, X75, X73)
LAST24_IN_GGAA(T41, T42, X113, .(T41, X114)) → U2_GGAA(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
LAST24_IN_GGAA(T41, T42, X113, .(T41, X114)) → LAST32_IN_GAA(T42, X113, X114)
LAST32_IN_GAA(.(T54, T55), X139, .(T54, X140)) → U1_GAA(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
LAST32_IN_GAA(.(T54, T55), X139, .(T54, X140)) → LAST32_IN_GAA(T55, X139, X140)
PALINDROME1_IN_G(.(T21, .(T22, T23))) → U8_G(T21, T22, T23, lastc24_in_ggaa(T22, T23, T26, T27))
U8_G(T21, T22, T23, lastc24_out_ggaa(T22, T23, T26, T27)) → U9_G(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
U8_G(T21, T22, T23, lastc24_out_ggaa(T22, T23, T26, T27)) → HALVES41_IN_GAAA(T27, X74, X76, X77)
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_GAAA(T72, T73, T74, X184, X185, X186, X187, last24_in_ggaa(T73, T74, X185, X183))
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → LAST24_IN_GGAA(T73, T74, X185, X183)
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAA(T72, T73, T74, X184, T77, X186, X187, lastc24_in_ggaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, lastc24_out_ggaa(T73, T74, T77, T78)) → U5_GAAA(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, lastc24_out_ggaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)
PALINDROME1_IN_G(T92) → U10_G(T92, halves41_in_gaaa(T92, X214, X215, X216))
PALINDROME1_IN_G(T92) → HALVES41_IN_GAAA(T92, X214, X215, X216)
PALINDROME1_IN_G(T92) → U11_G(T92, halvesc41_in_gaag(T92, T93, T94, odd))
U11_G(T92, halvesc41_out_gaag(T92, T93, T94, odd)) → U12_G(T92, last70_in_gag(T93, X217, T94))
U11_G(T92, halvesc41_out_gaag(T92, T93, T94, odd)) → LAST70_IN_GAG(T93, X217, T94)
LAST70_IN_GAG(.(T114, T115), X246, .(T114, T116)) → U6_GAG(T114, T115, X246, T116, last70_in_gag(T115, X246, T116))
LAST70_IN_GAG(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_GAG(T115, X246, T116)

The TRS R consists of the following rules:

lastc24_in_ggaa(T34, [], T34, []) → lastc24_out_ggaa(T34, [], T34, [])
lastc24_in_ggaa(T41, T42, X113, .(T41, X114)) → U15_ggaa(T41, T42, X113, X114, lastc32_in_gaa(T42, X113, X114))
lastc32_in_gaa(.(T49, []), T49, []) → lastc32_out_gaa(.(T49, []), T49, [])
lastc32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U14_gaa(T54, T55, X139, X140, lastc32_in_gaa(T55, X139, X140))
U14_gaa(T54, T55, X139, X140, lastc32_out_gaa(T55, X139, X140)) → lastc32_out_gaa(.(T54, T55), X139, .(T54, X140))
U15_ggaa(T41, T42, X113, X114, lastc32_out_gaa(T42, X113, X114)) → lastc24_out_ggaa(T41, T42, X113, .(T41, X114))
halvesc41_in_gaag([], [], [], even) → halvesc41_out_gaag([], [], [], even)
halvesc41_in_gaag(.(T65, []), .(T65, []), [], odd) → halvesc41_out_gaag(.(T65, []), .(T65, []), [], odd)
halvesc41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U16_gaag(T72, T73, T74, X184, T77, X186, X187, lastc24_in_ggaa(T73, T74, T77, T78))
U16_gaag(T72, T73, T74, X184, T77, X186, X187, lastc24_out_ggaa(T73, T74, T77, T78)) → U17_gaag(T72, T73, T74, X184, T77, X186, X187, halvesc41_in_gaag(T78, X184, X186, X187))
U17_gaag(T72, T73, T74, X184, T77, X186, X187, halvesc41_out_gaag(T78, X184, X186, X187)) → halvesc41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
last24_in_ggaa(x1, x2, x3, x4)  =  last24_in_ggaa(x1, x2)
last32_in_gaa(x1, x2, x3)  =  last32_in_gaa(x1)
lastc24_in_ggaa(x1, x2, x3, x4)  =  lastc24_in_ggaa(x1, x2)
[]  =  []
lastc24_out_ggaa(x1, x2, x3, x4)  =  lastc24_out_ggaa(x1, x2, x3, x4)
U15_ggaa(x1, x2, x3, x4, x5)  =  U15_ggaa(x1, x2, x5)
lastc32_in_gaa(x1, x2, x3)  =  lastc32_in_gaa(x1)
lastc32_out_gaa(x1, x2, x3)  =  lastc32_out_gaa(x1, x2, x3)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x2, x5)
halves41_in_gaaa(x1, x2, x3, x4)  =  halves41_in_gaaa(x1)
halvesc41_in_gaag(x1, x2, x3, x4)  =  halvesc41_in_gaag(x1, x4)
even  =  even
halvesc41_out_gaag(x1, x2, x3, x4)  =  halvesc41_out_gaag(x1, x2, x3, x4)
odd  =  odd
U16_gaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaag(x1, x2, x3, x7, x8)
U17_gaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaag(x1, x2, x3, x5, x7, x8)
last70_in_gag(x1, x2, x3)  =  last70_in_gag(x1, x3)
PALINDROME1_IN_G(x1)  =  PALINDROME1_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
LAST24_IN_GGAA(x1, x2, x3, x4)  =  LAST24_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5)  =  U2_GGAA(x1, x2, x5)
LAST32_IN_GAA(x1, x2, x3)  =  LAST32_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x2, x5)
U8_G(x1, x2, x3, x4)  =  U8_G(x1, x2, x3, x4)
U9_G(x1, x2, x3, x4)  =  U9_G(x1, x2, x3, x4)
HALVES41_IN_GAAA(x1, x2, x3, x4)  =  HALVES41_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U3_GAAA(x1, x2, x3, x8)
U4_GAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_GAAA(x1, x2, x3, x8)
U5_GAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_GAAA(x1, x2, x3, x8)
U10_G(x1, x2)  =  U10_G(x1, x2)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
LAST70_IN_GAG(x1, x2, x3)  =  LAST70_IN_GAG(x1, x3)
U6_GAG(x1, x2, x3, x4, x5)  =  U6_GAG(x1, x2, x4, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

PALINDROME1_IN_G(.(T21, .(T22, T23))) → U7_G(T21, T22, T23, last24_in_ggaa(T22, T23, X75, X73))
PALINDROME1_IN_G(.(T21, .(T22, T23))) → LAST24_IN_GGAA(T22, T23, X75, X73)
LAST24_IN_GGAA(T41, T42, X113, .(T41, X114)) → U2_GGAA(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
LAST24_IN_GGAA(T41, T42, X113, .(T41, X114)) → LAST32_IN_GAA(T42, X113, X114)
LAST32_IN_GAA(.(T54, T55), X139, .(T54, X140)) → U1_GAA(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
LAST32_IN_GAA(.(T54, T55), X139, .(T54, X140)) → LAST32_IN_GAA(T55, X139, X140)
PALINDROME1_IN_G(.(T21, .(T22, T23))) → U8_G(T21, T22, T23, lastc24_in_ggaa(T22, T23, T26, T27))
U8_G(T21, T22, T23, lastc24_out_ggaa(T22, T23, T26, T27)) → U9_G(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
U8_G(T21, T22, T23, lastc24_out_ggaa(T22, T23, T26, T27)) → HALVES41_IN_GAAA(T27, X74, X76, X77)
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_GAAA(T72, T73, T74, X184, X185, X186, X187, last24_in_ggaa(T73, T74, X185, X183))
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → LAST24_IN_GGAA(T73, T74, X185, X183)
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAA(T72, T73, T74, X184, T77, X186, X187, lastc24_in_ggaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, lastc24_out_ggaa(T73, T74, T77, T78)) → U5_GAAA(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, lastc24_out_ggaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)
PALINDROME1_IN_G(T92) → U10_G(T92, halves41_in_gaaa(T92, X214, X215, X216))
PALINDROME1_IN_G(T92) → HALVES41_IN_GAAA(T92, X214, X215, X216)
PALINDROME1_IN_G(T92) → U11_G(T92, halvesc41_in_gaag(T92, T93, T94, odd))
U11_G(T92, halvesc41_out_gaag(T92, T93, T94, odd)) → U12_G(T92, last70_in_gag(T93, X217, T94))
U11_G(T92, halvesc41_out_gaag(T92, T93, T94, odd)) → LAST70_IN_GAG(T93, X217, T94)
LAST70_IN_GAG(.(T114, T115), X246, .(T114, T116)) → U6_GAG(T114, T115, X246, T116, last70_in_gag(T115, X246, T116))
LAST70_IN_GAG(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_GAG(T115, X246, T116)

The TRS R consists of the following rules:

lastc24_in_ggaa(T34, [], T34, []) → lastc24_out_ggaa(T34, [], T34, [])
lastc24_in_ggaa(T41, T42, X113, .(T41, X114)) → U15_ggaa(T41, T42, X113, X114, lastc32_in_gaa(T42, X113, X114))
lastc32_in_gaa(.(T49, []), T49, []) → lastc32_out_gaa(.(T49, []), T49, [])
lastc32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U14_gaa(T54, T55, X139, X140, lastc32_in_gaa(T55, X139, X140))
U14_gaa(T54, T55, X139, X140, lastc32_out_gaa(T55, X139, X140)) → lastc32_out_gaa(.(T54, T55), X139, .(T54, X140))
U15_ggaa(T41, T42, X113, X114, lastc32_out_gaa(T42, X113, X114)) → lastc24_out_ggaa(T41, T42, X113, .(T41, X114))
halvesc41_in_gaag([], [], [], even) → halvesc41_out_gaag([], [], [], even)
halvesc41_in_gaag(.(T65, []), .(T65, []), [], odd) → halvesc41_out_gaag(.(T65, []), .(T65, []), [], odd)
halvesc41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U16_gaag(T72, T73, T74, X184, T77, X186, X187, lastc24_in_ggaa(T73, T74, T77, T78))
U16_gaag(T72, T73, T74, X184, T77, X186, X187, lastc24_out_ggaa(T73, T74, T77, T78)) → U17_gaag(T72, T73, T74, X184, T77, X186, X187, halvesc41_in_gaag(T78, X184, X186, X187))
U17_gaag(T72, T73, T74, X184, T77, X186, X187, halvesc41_out_gaag(T78, X184, X186, X187)) → halvesc41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
last24_in_ggaa(x1, x2, x3, x4)  =  last24_in_ggaa(x1, x2)
last32_in_gaa(x1, x2, x3)  =  last32_in_gaa(x1)
lastc24_in_ggaa(x1, x2, x3, x4)  =  lastc24_in_ggaa(x1, x2)
[]  =  []
lastc24_out_ggaa(x1, x2, x3, x4)  =  lastc24_out_ggaa(x1, x2, x3, x4)
U15_ggaa(x1, x2, x3, x4, x5)  =  U15_ggaa(x1, x2, x5)
lastc32_in_gaa(x1, x2, x3)  =  lastc32_in_gaa(x1)
lastc32_out_gaa(x1, x2, x3)  =  lastc32_out_gaa(x1, x2, x3)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x2, x5)
halves41_in_gaaa(x1, x2, x3, x4)  =  halves41_in_gaaa(x1)
halvesc41_in_gaag(x1, x2, x3, x4)  =  halvesc41_in_gaag(x1, x4)
even  =  even
halvesc41_out_gaag(x1, x2, x3, x4)  =  halvesc41_out_gaag(x1, x2, x3, x4)
odd  =  odd
U16_gaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaag(x1, x2, x3, x7, x8)
U17_gaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaag(x1, x2, x3, x5, x7, x8)
last70_in_gag(x1, x2, x3)  =  last70_in_gag(x1, x3)
PALINDROME1_IN_G(x1)  =  PALINDROME1_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
LAST24_IN_GGAA(x1, x2, x3, x4)  =  LAST24_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5)  =  U2_GGAA(x1, x2, x5)
LAST32_IN_GAA(x1, x2, x3)  =  LAST32_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x2, x5)
U8_G(x1, x2, x3, x4)  =  U8_G(x1, x2, x3, x4)
U9_G(x1, x2, x3, x4)  =  U9_G(x1, x2, x3, x4)
HALVES41_IN_GAAA(x1, x2, x3, x4)  =  HALVES41_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U3_GAAA(x1, x2, x3, x8)
U4_GAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_GAAA(x1, x2, x3, x8)
U5_GAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_GAAA(x1, x2, x3, x8)
U10_G(x1, x2)  =  U10_G(x1, x2)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
LAST70_IN_GAG(x1, x2, x3)  =  LAST70_IN_GAG(x1, x3)
U6_GAG(x1, x2, x3, x4, x5)  =  U6_GAG(x1, x2, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 17 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LAST70_IN_GAG(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_GAG(T115, X246, T116)

The TRS R consists of the following rules:

lastc24_in_ggaa(T34, [], T34, []) → lastc24_out_ggaa(T34, [], T34, [])
lastc24_in_ggaa(T41, T42, X113, .(T41, X114)) → U15_ggaa(T41, T42, X113, X114, lastc32_in_gaa(T42, X113, X114))
lastc32_in_gaa(.(T49, []), T49, []) → lastc32_out_gaa(.(T49, []), T49, [])
lastc32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U14_gaa(T54, T55, X139, X140, lastc32_in_gaa(T55, X139, X140))
U14_gaa(T54, T55, X139, X140, lastc32_out_gaa(T55, X139, X140)) → lastc32_out_gaa(.(T54, T55), X139, .(T54, X140))
U15_ggaa(T41, T42, X113, X114, lastc32_out_gaa(T42, X113, X114)) → lastc24_out_ggaa(T41, T42, X113, .(T41, X114))
halvesc41_in_gaag([], [], [], even) → halvesc41_out_gaag([], [], [], even)
halvesc41_in_gaag(.(T65, []), .(T65, []), [], odd) → halvesc41_out_gaag(.(T65, []), .(T65, []), [], odd)
halvesc41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U16_gaag(T72, T73, T74, X184, T77, X186, X187, lastc24_in_ggaa(T73, T74, T77, T78))
U16_gaag(T72, T73, T74, X184, T77, X186, X187, lastc24_out_ggaa(T73, T74, T77, T78)) → U17_gaag(T72, T73, T74, X184, T77, X186, X187, halvesc41_in_gaag(T78, X184, X186, X187))
U17_gaag(T72, T73, T74, X184, T77, X186, X187, halvesc41_out_gaag(T78, X184, X186, X187)) → halvesc41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
lastc24_in_ggaa(x1, x2, x3, x4)  =  lastc24_in_ggaa(x1, x2)
[]  =  []
lastc24_out_ggaa(x1, x2, x3, x4)  =  lastc24_out_ggaa(x1, x2, x3, x4)
U15_ggaa(x1, x2, x3, x4, x5)  =  U15_ggaa(x1, x2, x5)
lastc32_in_gaa(x1, x2, x3)  =  lastc32_in_gaa(x1)
lastc32_out_gaa(x1, x2, x3)  =  lastc32_out_gaa(x1, x2, x3)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x2, x5)
halvesc41_in_gaag(x1, x2, x3, x4)  =  halvesc41_in_gaag(x1, x4)
even  =  even
halvesc41_out_gaag(x1, x2, x3, x4)  =  halvesc41_out_gaag(x1, x2, x3, x4)
odd  =  odd
U16_gaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaag(x1, x2, x3, x7, x8)
U17_gaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaag(x1, x2, x3, x5, x7, x8)
LAST70_IN_GAG(x1, x2, x3)  =  LAST70_IN_GAG(x1, 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:

LAST70_IN_GAG(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_GAG(T115, X246, T116)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
LAST70_IN_GAG(x1, x2, x3)  =  LAST70_IN_GAG(x1, 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:

LAST70_IN_GAG(.(T114, T115), .(T114, T116)) → LAST70_IN_GAG(T115, T116)

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:

• LAST70_IN_GAG(.(T114, T115), .(T114, T116)) → LAST70_IN_GAG(T115, T116)
  The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

LAST32_IN_GAA(.(T54, T55), X139, .(T54, X140)) → LAST32_IN_GAA(T55, X139, X140)

The TRS R consists of the following rules:

lastc24_in_ggaa(T34, [], T34, []) → lastc24_out_ggaa(T34, [], T34, [])
lastc24_in_ggaa(T41, T42, X113, .(T41, X114)) → U15_ggaa(T41, T42, X113, X114, lastc32_in_gaa(T42, X113, X114))
lastc32_in_gaa(.(T49, []), T49, []) → lastc32_out_gaa(.(T49, []), T49, [])
lastc32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U14_gaa(T54, T55, X139, X140, lastc32_in_gaa(T55, X139, X140))
U14_gaa(T54, T55, X139, X140, lastc32_out_gaa(T55, X139, X140)) → lastc32_out_gaa(.(T54, T55), X139, .(T54, X140))
U15_ggaa(T41, T42, X113, X114, lastc32_out_gaa(T42, X113, X114)) → lastc24_out_ggaa(T41, T42, X113, .(T41, X114))
halvesc41_in_gaag([], [], [], even) → halvesc41_out_gaag([], [], [], even)
halvesc41_in_gaag(.(T65, []), .(T65, []), [], odd) → halvesc41_out_gaag(.(T65, []), .(T65, []), [], odd)
halvesc41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U16_gaag(T72, T73, T74, X184, T77, X186, X187, lastc24_in_ggaa(T73, T74, T77, T78))
U16_gaag(T72, T73, T74, X184, T77, X186, X187, lastc24_out_ggaa(T73, T74, T77, T78)) → U17_gaag(T72, T73, T74, X184, T77, X186, X187, halvesc41_in_gaag(T78, X184, X186, X187))
U17_gaag(T72, T73, T74, X184, T77, X186, X187, halvesc41_out_gaag(T78, X184, X186, X187)) → halvesc41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
lastc24_in_ggaa(x1, x2, x3, x4)  =  lastc24_in_ggaa(x1, x2)
[]  =  []
lastc24_out_ggaa(x1, x2, x3, x4)  =  lastc24_out_ggaa(x1, x2, x3, x4)
U15_ggaa(x1, x2, x3, x4, x5)  =  U15_ggaa(x1, x2, x5)
lastc32_in_gaa(x1, x2, x3)  =  lastc32_in_gaa(x1)
lastc32_out_gaa(x1, x2, x3)  =  lastc32_out_gaa(x1, x2, x3)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x2, x5)
halvesc41_in_gaag(x1, x2, x3, x4)  =  halvesc41_in_gaag(x1, x4)
even  =  even
halvesc41_out_gaag(x1, x2, x3, x4)  =  halvesc41_out_gaag(x1, x2, x3, x4)
odd  =  odd
U16_gaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaag(x1, x2, x3, x7, x8)
U17_gaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaag(x1, x2, x3, x5, x7, x8)
LAST32_IN_GAA(x1, x2, x3)  =  LAST32_IN_GAA(x1)

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:

LAST32_IN_GAA(.(T54, T55), X139, .(T54, X140)) → LAST32_IN_GAA(T55, X139, X140)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
LAST32_IN_GAA(x1, x2, x3)  =  LAST32_IN_GAA(x1)

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:

LAST32_IN_GAA(.(T54, T55)) → LAST32_IN_GAA(T55)

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:

• LAST32_IN_GAA(.(T54, T55)) → LAST32_IN_GAA(T55)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAA(T72, T73, T74, X184, T77, X186, X187, lastc24_in_ggaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, lastc24_out_ggaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)

The TRS R consists of the following rules:

lastc24_in_ggaa(T34, [], T34, []) → lastc24_out_ggaa(T34, [], T34, [])
lastc24_in_ggaa(T41, T42, X113, .(T41, X114)) → U15_ggaa(T41, T42, X113, X114, lastc32_in_gaa(T42, X113, X114))
lastc32_in_gaa(.(T49, []), T49, []) → lastc32_out_gaa(.(T49, []), T49, [])
lastc32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U14_gaa(T54, T55, X139, X140, lastc32_in_gaa(T55, X139, X140))
U14_gaa(T54, T55, X139, X140, lastc32_out_gaa(T55, X139, X140)) → lastc32_out_gaa(.(T54, T55), X139, .(T54, X140))
U15_ggaa(T41, T42, X113, X114, lastc32_out_gaa(T42, X113, X114)) → lastc24_out_ggaa(T41, T42, X113, .(T41, X114))
halvesc41_in_gaag([], [], [], even) → halvesc41_out_gaag([], [], [], even)
halvesc41_in_gaag(.(T65, []), .(T65, []), [], odd) → halvesc41_out_gaag(.(T65, []), .(T65, []), [], odd)
halvesc41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U16_gaag(T72, T73, T74, X184, T77, X186, X187, lastc24_in_ggaa(T73, T74, T77, T78))
U16_gaag(T72, T73, T74, X184, T77, X186, X187, lastc24_out_ggaa(T73, T74, T77, T78)) → U17_gaag(T72, T73, T74, X184, T77, X186, X187, halvesc41_in_gaag(T78, X184, X186, X187))
U17_gaag(T72, T73, T74, X184, T77, X186, X187, halvesc41_out_gaag(T78, X184, X186, X187)) → halvesc41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
lastc24_in_ggaa(x1, x2, x3, x4)  =  lastc24_in_ggaa(x1, x2)
[]  =  []
lastc24_out_ggaa(x1, x2, x3, x4)  =  lastc24_out_ggaa(x1, x2, x3, x4)
U15_ggaa(x1, x2, x3, x4, x5)  =  U15_ggaa(x1, x2, x5)
lastc32_in_gaa(x1, x2, x3)  =  lastc32_in_gaa(x1)
lastc32_out_gaa(x1, x2, x3)  =  lastc32_out_gaa(x1, x2, x3)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x2, x5)
halvesc41_in_gaag(x1, x2, x3, x4)  =  halvesc41_in_gaag(x1, x4)
even  =  even
halvesc41_out_gaag(x1, x2, x3, x4)  =  halvesc41_out_gaag(x1, x2, x3, x4)
odd  =  odd
U16_gaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaag(x1, x2, x3, x7, x8)
U17_gaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaag(x1, x2, x3, x5, x7, x8)
HALVES41_IN_GAAA(x1, x2, x3, x4)  =  HALVES41_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_GAAA(x1, x2, x3, x8)

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:

HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAA(T72, T73, T74, X184, T77, X186, X187, lastc24_in_ggaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, lastc24_out_ggaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)

The TRS R consists of the following rules:

lastc24_in_ggaa(T34, [], T34, []) → lastc24_out_ggaa(T34, [], T34, [])
lastc24_in_ggaa(T41, T42, X113, .(T41, X114)) → U15_ggaa(T41, T42, X113, X114, lastc32_in_gaa(T42, X113, X114))
U15_ggaa(T41, T42, X113, X114, lastc32_out_gaa(T42, X113, X114)) → lastc24_out_ggaa(T41, T42, X113, .(T41, X114))
lastc32_in_gaa(.(T49, []), T49, []) → lastc32_out_gaa(.(T49, []), T49, [])
lastc32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U14_gaa(T54, T55, X139, X140, lastc32_in_gaa(T55, X139, X140))
U14_gaa(T54, T55, X139, X140, lastc32_out_gaa(T55, X139, X140)) → lastc32_out_gaa(.(T54, T55), X139, .(T54, X140))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
lastc24_in_ggaa(x1, x2, x3, x4)  =  lastc24_in_ggaa(x1, x2)
[]  =  []
lastc24_out_ggaa(x1, x2, x3, x4)  =  lastc24_out_ggaa(x1, x2, x3, x4)
U15_ggaa(x1, x2, x3, x4, x5)  =  U15_ggaa(x1, x2, x5)
lastc32_in_gaa(x1, x2, x3)  =  lastc32_in_gaa(x1)
lastc32_out_gaa(x1, x2, x3)  =  lastc32_out_gaa(x1, x2, x3)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x2, x5)
HALVES41_IN_GAAA(x1, x2, x3, x4)  =  HALVES41_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_GAAA(x1, x2, x3, x8)

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:

HALVES41_IN_GAAA(.(T72, .(T73, T74))) → U4_GAAA(T72, T73, T74, lastc24_in_ggaa(T73, T74))
U4_GAAA(T72, T73, T74, lastc24_out_ggaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78)

The TRS R consists of the following rules:

lastc24_in_ggaa(T34, []) → lastc24_out_ggaa(T34, [], T34, [])
lastc24_in_ggaa(T41, T42) → U15_ggaa(T41, T42, lastc32_in_gaa(T42))
U15_ggaa(T41, T42, lastc32_out_gaa(T42, X113, X114)) → lastc24_out_ggaa(T41, T42, X113, .(T41, X114))
lastc32_in_gaa(.(T49, [])) → lastc32_out_gaa(.(T49, []), T49, [])
lastc32_in_gaa(.(T54, T55)) → U14_gaa(T54, T55, lastc32_in_gaa(T55))
U14_gaa(T54, T55, lastc32_out_gaa(T55, X139, X140)) → lastc32_out_gaa(.(T54, T55), X139, .(T54, X140))

The set Q consists of the following terms:

lastc24_in_ggaa(x0, x1)
U15_ggaa(x0, x1, x2)
lastc32_in_gaa(x0)
U14_gaa(x0, x1, x2)

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.

HALVES41_IN_GAAA(.(T72, .(T73, T74))) → U4_GAAA(T72, T73, T74, lastc24_in_ggaa(T73, T74))
U4_GAAA(T72, T73, T74, lastc24_out_ggaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78)

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(HALVES41_IN_GAAA(x₁)) = x₁   
POL(U14_gaa(x₁, x₂, x₃)) = 1 + x₃   
POL(U15_ggaa(x₁, x₂, x₃)) = 1 + x₃   
POL(U4_GAAA(x₁, x₂, x₃, x₄)) = x₄   
POL([]) = 1   
POL(lastc24_in_ggaa(x₁, x₂)) = 1 + x₂   
POL(lastc24_out_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(lastc32_in_gaa(x₁)) = x₁   
POL(lastc32_out_gaa(x₁, x₂, x₃)) = 1 + x₃   

The following usable rules [FROCOS05] were oriented:

lastc24_in_ggaa(T34, []) → lastc24_out_ggaa(T34, [], T34, [])
lastc24_in_ggaa(T41, T42) → U15_ggaa(T41, T42, lastc32_in_gaa(T42))
lastc32_in_gaa(.(T49, [])) → lastc32_out_gaa(.(T49, []), T49, [])
lastc32_in_gaa(.(T54, T55)) → U14_gaa(T54, T55, lastc32_in_gaa(T55))
U15_ggaa(T41, T42, lastc32_out_gaa(T42, X113, X114)) → lastc24_out_ggaa(T41, T42, X113, .(T41, X114))
U14_gaa(T54, T55, lastc32_out_gaa(T55, X139, X140)) → lastc32_out_gaa(.(T54, T55), X139, .(T54, X140))

(27) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

lastc24_in_ggaa(T34, []) → lastc24_out_ggaa(T34, [], T34, [])
lastc24_in_ggaa(T41, T42) → U15_ggaa(T41, T42, lastc32_in_gaa(T42))
U15_ggaa(T41, T42, lastc32_out_gaa(T42, X113, X114)) → lastc24_out_ggaa(T41, T42, X113, .(T41, X114))
lastc32_in_gaa(.(T49, [])) → lastc32_out_gaa(.(T49, []), T49, [])
lastc32_in_gaa(.(T54, T55)) → U14_gaa(T54, T55, lastc32_in_gaa(T55))
U14_gaa(T54, T55, lastc32_out_gaa(T55, X139, X140)) → lastc32_out_gaa(.(T54, T55), X139, .(T54, X140))

The set Q consists of the following terms:

lastc24_in_ggaa(x0, x1)
U15_ggaa(x0, x1, x2)
lastc32_in_gaa(x0)
U14_gaa(x0, x1, x2)

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

(28) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(29) YES