Left Termination proof of /tmp/tmpEVcsSU/palindrome.pl

Left Termination of the query pattern palindrome_in_1(g) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 AND

            ↳9 PiDP

                ↳10 UsableRulesProof (⇔)

                ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                ↳13 QDP

                ↳14 NonTerminationProof (⇔)

                ↳15 NO

            ↳16 PiDP

                ↳17 UsableRulesProof (⇔)

                ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                ↳20 QDP

                ↳21 QDPSizeChangeProof (⇔)

                ↳22 YES

            ↳23 PiDP

                ↳24 UsableRulesProof (⇔)

                ↳25 PiDP

                ↳26 PiDPToQDPProof (⇐)

                ↳27 QDP

                ↳28 MRRProof (⇔)

                ↳29 QDP

                ↳30 PisEmptyProof (⇔)

                ↳31 YES

            ↳32 PiDP

                ↳33 UsableRulesProof (⇔)

                ↳34 PiDP

                ↳35 PiDPToQDPProof (⇐)

                ↳36 QDP

                ↳37 MRRProof (⇔)

                ↳38 QDP

                ↳39 PisEmptyProof (⇔)

                ↳40 YES

    ↳41 PrologToPiTRSProof (⇐)

        ↳42 PiTRS

        ↳43 DependencyPairsProof (⇔)

        ↳44 PiDP

        ↳45 DependencyGraphProof (⇔)

        ↳46 AND

            ↳47 PiDP

                ↳48 UsableRulesProof (⇔)

                ↳49 PiDP

                ↳50 PiDPToQDPProof (⇐)

                ↳51 QDP

                ↳52 NonTerminationProof (⇔)

                ↳53 NO

            ↳54 PiDP

                ↳55 UsableRulesProof (⇔)

                ↳56 PiDP

                ↳57 PiDPToQDPProof (⇐)

                ↳58 QDP

                ↳59 QDPSizeChangeProof (⇔)

                ↳60 YES

            ↳61 PiDP

                ↳62 UsableRulesProof (⇔)

                ↳63 PiDP

                ↳64 PiDPToQDPProof (⇐)

                ↳65 QDP

                ↳66 QDPOrderProof (⇔)

                ↳67 QDP

                ↳68 DependencyGraphProof (⇔)

                ↳69 TRUE

            ↳70 PiDP

                ↳71 UsableRulesProof (⇔)

                ↳72 PiDP

                ↳73 PiDPToQDPProof (⇐)

                ↳74 QDP

                ↳75 QDPOrderProof (⇔)

                ↳76 QDP

                ↳77 DependencyGraphProof (⇔)

                ↳78 TRUE

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog 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:

Clauses:

last32(.(T49, []), T49, []).
last32(.(T54, T55), X139, .(T54, X140)) :- last32(T55, X139, X140).
last24(T34, [], T34, []).
last24(T41, T42, X113, .(T41, X114)) :- last32(T42, X113, X114).
halves41([], [], [], even).
halves41(.(T65, []), .(T65, []), [], odd).
halves41(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) :- last24(T73, T74, X185, X183).
halves41(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) :- ','(last24(T73, T74, T77, T78), halves41(T78, X184, X186, X187)).
last70(.(T107, []), T107, []).
last70(.(T114, T115), X246, .(T114, T116)) :- last70(T115, X246, T116).
palindrome1([]).
palindrome1(.(T21, .(T22, T23))) :- last24(T22, T23, X75, X73).
palindrome1(.(T21, .(T22, T23))) :- ','(last24(T22, T23, T26, T27), halves41(T27, X74, X76, X77)).
palindrome1(.(T88, .(T22, T23))) :- ','(last24(T22, T23, T88, T27), halves41(T27, T89, T89, even)).
palindrome1(T92) :- halves41(T92, X214, X215, X216).
palindrome1(T92) :- ','(halves41(T92, T93, T94, odd), last70(T93, X217, T94)).

Queries:

palindrome1(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
palindrome1_in: (b)
last24_in: (f,b,f,f)
last32_in: (b,f,f)
halves41_in: (b,f,f,f) (b,f,f,b)
last70_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

PALINDROME1_IN_G(.(T21, .(T22, T23))) → U7_G(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
PALINDROME1_IN_G(.(T21, .(T22, T23))) → LAST24_IN_AGAA(T22, T23, X75, X73)
LAST24_IN_AGAA(T41, T42, X113, .(T41, X114)) → U2_AGAA(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
LAST24_IN_AGAA(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, last24_in_agaa(T22, T23, T26, T27))
U8_G(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_G(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
U8_G(T21, T22, T23, last24_out_agaa(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_agaa(T73, T74, X185, X183))
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → LAST24_IN_AGAA(T73, T74, X185, X183)
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(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, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)
PALINDROME1_IN_G(.(T88, .(T22, T23))) → U10_G(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
PALINDROME1_IN_G(.(T88, .(T22, T23))) → LAST24_IN_AGAA(T22, T23, T88, T27)
U10_G(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_G(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
U10_G(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → HALVES41_IN_GAAG(T27, T89, T89, even)
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_GAAG(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → LAST24_IN_AGAA(T73, T74, X185, X183)
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_GAAG(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAG(T78, X184, X186, X187)
PALINDROME1_IN_G(T92) → U12_G(T92, halves41_in_gaaa(T92, X214, X215, X216))
PALINDROME1_IN_G(T92) → HALVES41_IN_GAAA(T92, X214, X215, X216)
PALINDROME1_IN_G(T92) → U13_G(T92, halves41_in_gaag(T92, T93, T94, odd))
PALINDROME1_IN_G(T92) → HALVES41_IN_GAAG(T92, T93, T94, odd)
U13_G(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_G(T92, last70_in_aaa(T93, X217, T94))
U13_G(T92, halves41_out_gaag(T92, T93, T94, odd)) → LAST70_IN_AAA(T93, X217, T94)
LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → U6_AAA(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_AAA(T115, X246, T116)

The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

The argument filtering Pi contains the following mapping:
palindrome1_in_g(x1) = palindrome1_in_g(x1)
[] = []
palindrome1_out_g(x1) = palindrome1_out_g
.(x1, x2) = .(x2)
U7_g(x1, x2, x3, x4) = U7_g(x4)
last32_in_gaa(x1, x2, x3) = last32_in_gaa(x1)
last32_out_gaa(x1, x2, x3) = last32_out_gaa(x3)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
U8_g(x1, x2, x3, x4) = U8_g(x4)
U9_g(x1, x2, x3, x4) = U9_g(x4)
halves41_in_gaaa(x1, x2, x3, x4) = halves41_in_gaaa(x1)
halves41_out_gaaa(x1, x2, x3, x4) = halves41_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U3_gaaa(x8)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U4_gaaa(x8)
U5_gaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U5_gaaa(x8)
U10_g(x1, x2, x3, x4) = U10_g(x4)
U11_g(x1, x2, x3, x4) = U11_g(x4)
halves41_in_gaag(x1, x2, x3, x4) = halves41_in_gaag(x1, x4)
even = even
halves41_out_gaag(x1, x2, x3, x4) = halves41_out_gaag
odd = odd
U3_gaag(x1, x2, x3, x4, x5, x6, x7, x8) = U3_gaag(x8)
U4_gaag(x1, x2, x3, x4, x5, x6, x7, x8) = U4_gaag(x7, x8)
U5_gaag(x1, x2, x3, x4, x5, x6, x7, x8) = U5_gaag(x8)
U12_g(x1, x2) = U12_g(x2)
U13_g(x1, x2) = U13_g(x2)
U14_g(x1, x2) = U14_g(x2)
last70_in_aaa(x1, x2, x3) = last70_in_aaa
last70_out_aaa(x1, x2, x3) = last70_out_aaa(x1, x3)
last24_in_agaa(x1, x2, x3, x4) = last24_in_agaa(x2)
last24_out_agaa(x1, x2, x3, x4) = last24_out_agaa(x4)
U2_agaa(x1, x2, x3, x4, x5) = U2_agaa(x5)
U6_aaa(x1, x2, x3, x4, x5) = U6_aaa(x5)
PALINDROME1_IN_G(x1) = PALINDROME1_IN_G(x1)
U7_G(x1, x2, x3, x4) = U7_G(x4)
LAST24_IN_AGAA(x1, x2, x3, x4) = LAST24_IN_AGAA(x2)
U2_AGAA(x1, x2, x3, x4, x5) = U2_AGAA(x5)
LAST32_IN_GAA(x1, x2, x3) = LAST32_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x5)
U8_G(x1, x2, x3, x4) = U8_G(x4)
U9_G(x1, x2, x3, x4) = U9_G(x4)
HALVES41_IN_GAAA(x1, x2, x3, x4) = HALVES41_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U3_GAAA(x8)
U4_GAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U4_GAAA(x8)
U5_GAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U5_GAAA(x8)
U10_G(x1, x2, x3, x4) = U10_G(x4)
U11_G(x1, x2, x3, x4) = U11_G(x4)
HALVES41_IN_GAAG(x1, x2, x3, x4) = HALVES41_IN_GAAG(x1, x4)
U3_GAAG(x1, x2, x3, x4, x5, x6, x7, x8) = U3_GAAG(x8)
U4_GAAG(x1, x2, x3, x4, x5, x6, x7, x8) = U4_GAAG(x7, x8)
U5_GAAG(x1, x2, x3, x4, x5, x6, x7, x8) = U5_GAAG(x8)
U12_G(x1, x2) = U12_G(x2)
U13_G(x1, x2) = U13_G(x2)
U14_G(x1, x2) = U14_G(x2)
LAST70_IN_AAA(x1, x2, x3) = LAST70_IN_AAA
U6_AAA(x1, x2, x3, x4, x5) = U6_AAA(x5)

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

(6) 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_agaa(T22, T23, X75, X73))
PALINDROME1_IN_G(.(T21, .(T22, T23))) → LAST24_IN_AGAA(T22, T23, X75, X73)
LAST24_IN_AGAA(T41, T42, X113, .(T41, X114)) → U2_AGAA(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
LAST24_IN_AGAA(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, last24_in_agaa(T22, T23, T26, T27))
U8_G(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_G(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
U8_G(T21, T22, T23, last24_out_agaa(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_agaa(T73, T74, X185, X183))
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → LAST24_IN_AGAA(T73, T74, X185, X183)
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(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, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)
PALINDROME1_IN_G(.(T88, .(T22, T23))) → U10_G(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
PALINDROME1_IN_G(.(T88, .(T22, T23))) → LAST24_IN_AGAA(T22, T23, T88, T27)
U10_G(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_G(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
U10_G(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → HALVES41_IN_GAAG(T27, T89, T89, even)
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_GAAG(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → LAST24_IN_AGAA(T73, T74, X185, X183)
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_GAAG(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAG(T78, X184, X186, X187)
PALINDROME1_IN_G(T92) → U12_G(T92, halves41_in_gaaa(T92, X214, X215, X216))
PALINDROME1_IN_G(T92) → HALVES41_IN_GAAA(T92, X214, X215, X216)
PALINDROME1_IN_G(T92) → U13_G(T92, halves41_in_gaag(T92, T93, T94, odd))
PALINDROME1_IN_G(T92) → HALVES41_IN_GAAG(T92, T93, T94, odd)
U13_G(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_G(T92, last70_in_aaa(T93, X217, T94))
U13_G(T92, halves41_out_gaag(T92, T93, T94, odd)) → LAST70_IN_AAA(T93, X217, T94)
LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → U6_AAA(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_AAA(T115, X246, T116)

The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 25 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_AAA(T115, X246, T116)

The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_AAA(T115, X246, T116)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
LAST70_IN_AAA(x1, x2, x3) = LAST70_IN_AAA

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

LAST70_IN_AAA → LAST70_IN_AAA

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

(14) NonTerminationProof (EQUIVALENT transformation)

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

s = LAST70_IN_AAA evaluates to t =LAST70_IN_AAA

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LAST70_IN_AAA to LAST70_IN_AAA.

(15) NO

(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)

The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

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) = .(x2)
LAST32_IN_GAA(x1, x2, x3) = LAST32_IN_GAA(x1)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

LAST32_IN_GAA(.(T55)) → LAST32_IN_GAA(T55)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(22) YES

(23) Obligation:

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

HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAG(T78, X184, X186, X187)

The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAG(T78, X184, X186, X187)

The TRS R consists of the following rules:

last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
last32_in_gaa(x1, x2, x3) = last32_in_gaa(x1)
last32_out_gaa(x1, x2, x3) = last32_out_gaa(x3)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
last24_in_agaa(x1, x2, x3, x4) = last24_in_agaa(x2)
last24_out_agaa(x1, x2, x3, x4) = last24_out_agaa(x4)
U2_agaa(x1, x2, x3, x4, x5) = U2_agaa(x5)
HALVES41_IN_GAAG(x1, x2, x3, x4) = HALVES41_IN_GAAG(x1, x4)
U4_GAAG(x1, x2, x3, x4, x5, x6, x7, x8) = U4_GAAG(x7, x8)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

HALVES41_IN_GAAG(.(.(T74)), X187) → U4_GAAG(X187, last24_in_agaa(T74))
U4_GAAG(X187, last24_out_agaa(T78)) → HALVES41_IN_GAAG(T78, X187)

The TRS R consists of the following rules:

last24_in_agaa([]) → last24_out_agaa([])
last24_in_agaa(T42) → U2_agaa(last32_in_gaa(T42))
U2_agaa(last32_out_gaa(X114)) → last24_out_agaa(.(X114))
last32_in_gaa(.([])) → last32_out_gaa([])
last32_in_gaa(.(T55)) → U1_gaa(last32_in_gaa(T55))
U1_gaa(last32_out_gaa(X140)) → last32_out_gaa(.(X140))

The set Q consists of the following terms:

last24_in_agaa(x0)
U2_agaa(x0)
last32_in_gaa(x0)
U1_gaa(x0)

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

(28) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

HALVES41_IN_GAAG(.(.(T74)), X187) → U4_GAAG(X187, last24_in_agaa(T74))
U4_GAAG(X187, last24_out_agaa(T78)) → HALVES41_IN_GAAG(T78, X187)

Strictly oriented rules of the TRS R:

last24_in_agaa([]) → last24_out_agaa([])
last24_in_agaa(T42) → U2_agaa(last32_in_gaa(T42))
U2_agaa(last32_out_gaa(X114)) → last24_out_agaa(.(X114))
last32_in_gaa(.([])) → last32_out_gaa([])

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁)) = 3 + x₁
POL(HALVES41_IN_GAAG(x₁, x₂)) = x₁ + x₂
POL(U1_gaa(x₁)) = 3 + x₁
POL(U2_agaa(x₁)) = x₁
POL(U4_GAAG(x₁, x₂)) = 1 + x₁ + x₂
POL([]) = 0
POL(last24_in_agaa(x₁)) = 3 + x₁
POL(last24_out_agaa(x₁)) = x₁
POL(last32_in_gaa(x₁)) = 2 + x₁
POL(last32_out_gaa(x₁)) = 4 + x₁

(29) Obligation:

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

last32_in_gaa(.(T55)) → U1_gaa(last32_in_gaa(T55))
U1_gaa(last32_out_gaa(X140)) → last32_out_gaa(.(X140))

The set Q consists of the following terms:

last24_in_agaa(x0)
U2_agaa(x0)
last32_in_gaa(x0)
U1_gaa(x0)

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

(30) PisEmptyProof (EQUIVALENT transformation)

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

(31) YES

(32) 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, last24_in_agaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)

The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(33) UsableRulesProof (EQUIVALENT transformation)

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

(34) 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, last24_in_agaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)

The TRS R consists of the following rules:

last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
last32_in_gaa(x1, x2, x3) = last32_in_gaa(x1)
last32_out_gaa(x1, x2, x3) = last32_out_gaa(x3)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
last24_in_agaa(x1, x2, x3, x4) = last24_in_agaa(x2)
last24_out_agaa(x1, x2, x3, x4) = last24_out_agaa(x4)
U2_agaa(x1, x2, x3, x4, x5) = U2_agaa(x5)
HALVES41_IN_GAAA(x1, x2, x3, x4) = HALVES41_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U4_GAAA(x8)

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

(35) PiDPToQDPProof (SOUND transformation)

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

(36) Obligation:

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

HALVES41_IN_GAAA(.(.(T74))) → U4_GAAA(last24_in_agaa(T74))
U4_GAAA(last24_out_agaa(T78)) → HALVES41_IN_GAAA(T78)

The TRS R consists of the following rules:

last24_in_agaa([]) → last24_out_agaa([])
last24_in_agaa(T42) → U2_agaa(last32_in_gaa(T42))
U2_agaa(last32_out_gaa(X114)) → last24_out_agaa(.(X114))
last32_in_gaa(.([])) → last32_out_gaa([])
last32_in_gaa(.(T55)) → U1_gaa(last32_in_gaa(T55))
U1_gaa(last32_out_gaa(X140)) → last32_out_gaa(.(X140))

The set Q consists of the following terms:

last24_in_agaa(x0)
U2_agaa(x0)
last32_in_gaa(x0)
U1_gaa(x0)

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

(37) MRRProof (EQUIVALENT transformation)

HALVES41_IN_GAAA(.(.(T74))) → U4_GAAA(last24_in_agaa(T74))
U4_GAAA(last24_out_agaa(T78)) → HALVES41_IN_GAAA(T78)

Strictly oriented rules of the TRS R:

last24_in_agaa([]) → last24_out_agaa([])
last24_in_agaa(T42) → U2_agaa(last32_in_gaa(T42))
U2_agaa(last32_out_gaa(X114)) → last24_out_agaa(.(X114))
last32_in_gaa(.([])) → last32_out_gaa([])

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁)) = 3 + x₁
POL(HALVES41_IN_GAAA(x₁)) = x₁
POL(U1_gaa(x₁)) = 3 + x₁
POL(U2_agaa(x₁)) = x₁
POL(U4_GAAA(x₁)) = 1 + x₁
POL([]) = 0
POL(last24_in_agaa(x₁)) = 3 + x₁
POL(last24_out_agaa(x₁)) = x₁
POL(last32_in_gaa(x₁)) = 2 + x₁
POL(last32_out_gaa(x₁)) = 4 + x₁

(38) Obligation:

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

last32_in_gaa(.(T55)) → U1_gaa(last32_in_gaa(T55))
U1_gaa(last32_out_gaa(X140)) → last32_out_gaa(.(X140))

The set Q consists of the following terms:

last24_in_agaa(x0)
U2_agaa(x0)
last32_in_gaa(x0)
U1_gaa(x0)

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

(39) PisEmptyProof (EQUIVALENT transformation)

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

(40) YES

(41) PrologToPiTRSProof (SOUND transformation)

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(42) Obligation:

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

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(43) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

PALINDROME1_IN_G(.(T21, .(T22, T23))) → U7_G(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
PALINDROME1_IN_G(.(T21, .(T22, T23))) → LAST24_IN_AGAA(T22, T23, X75, X73)
LAST24_IN_AGAA(T41, T42, X113, .(T41, X114)) → U2_AGAA(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
LAST24_IN_AGAA(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, last24_in_agaa(T22, T23, T26, T27))
U8_G(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_G(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
U8_G(T21, T22, T23, last24_out_agaa(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_agaa(T73, T74, X185, X183))
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → LAST24_IN_AGAA(T73, T74, X185, X183)
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(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, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)
PALINDROME1_IN_G(.(T88, .(T22, T23))) → U10_G(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
PALINDROME1_IN_G(.(T88, .(T22, T23))) → LAST24_IN_AGAA(T22, T23, T88, T27)
U10_G(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_G(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
U10_G(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → HALVES41_IN_GAAG(T27, T89, T89, even)
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_GAAG(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → LAST24_IN_AGAA(T73, T74, X185, X183)
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_GAAG(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAG(T78, X184, X186, X187)
PALINDROME1_IN_G(T92) → U12_G(T92, halves41_in_gaaa(T92, X214, X215, X216))
PALINDROME1_IN_G(T92) → HALVES41_IN_GAAA(T92, X214, X215, X216)
PALINDROME1_IN_G(T92) → U13_G(T92, halves41_in_gaag(T92, T93, T94, odd))
PALINDROME1_IN_G(T92) → HALVES41_IN_GAAG(T92, T93, T94, odd)
U13_G(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_G(T92, last70_in_aaa(T93, X217, T94))
U13_G(T92, halves41_out_gaag(T92, T93, T94, odd)) → LAST70_IN_AAA(T93, X217, T94)
LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → U6_AAA(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_AAA(T115, X246, T116)

The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

The argument filtering Pi contains the following mapping:
palindrome1_in_g(x1) = palindrome1_in_g(x1)
[] = []
palindrome1_out_g(x1) = palindrome1_out_g(x1)
.(x1, x2) = .(x2)
U7_g(x1, x2, x3, x4) = U7_g(x3, x4)
last32_in_gaa(x1, x2, x3) = last32_in_gaa(x1)
last32_out_gaa(x1, x2, x3) = last32_out_gaa(x1, x3)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x2, x5)
U8_g(x1, x2, x3, x4) = U8_g(x3, x4)
U9_g(x1, x2, x3, x4) = U9_g(x3, x4)
halves41_in_gaaa(x1, x2, x3, x4) = halves41_in_gaaa(x1)
halves41_out_gaaa(x1, x2, x3, x4) = halves41_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U3_gaaa(x3, x8)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U4_gaaa(x3, x8)
U5_gaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U5_gaaa(x3, x8)
U10_g(x1, x2, x3, x4) = U10_g(x3, x4)
U11_g(x1, x2, x3, x4) = U11_g(x3, x4)
halves41_in_gaag(x1, x2, x3, x4) = halves41_in_gaag(x1, x4)
even = even
halves41_out_gaag(x1, x2, x3, x4) = halves41_out_gaag(x1, x4)
odd = odd
U3_gaag(x1, x2, x3, x4, x5, x6, x7, x8) = U3_gaag(x3, x7, x8)
U4_gaag(x1, x2, x3, x4, x5, x6, x7, x8) = U4_gaag(x3, x7, x8)
U5_gaag(x1, x2, x3, x4, x5, x6, x7, x8) = U5_gaag(x3, x7, x8)
U12_g(x1, x2) = U12_g(x1, x2)
U13_g(x1, x2) = U13_g(x1, x2)
U14_g(x1, x2) = U14_g(x1, x2)
last70_in_aaa(x1, x2, x3) = last70_in_aaa
last70_out_aaa(x1, x2, x3) = last70_out_aaa(x1, x3)
last24_in_agaa(x1, x2, x3, x4) = last24_in_agaa(x2)
last24_out_agaa(x1, x2, x3, x4) = last24_out_agaa(x2, x4)
U2_agaa(x1, x2, x3, x4, x5) = U2_agaa(x2, x5)
U6_aaa(x1, x2, x3, x4, x5) = U6_aaa(x5)
PALINDROME1_IN_G(x1) = PALINDROME1_IN_G(x1)
U7_G(x1, x2, x3, x4) = U7_G(x3, x4)
LAST24_IN_AGAA(x1, x2, x3, x4) = LAST24_IN_AGAA(x2)
U2_AGAA(x1, x2, x3, x4, x5) = U2_AGAA(x2, x5)
LAST32_IN_GAA(x1, x2, x3) = LAST32_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x2, x5)
U8_G(x1, x2, x3, x4) = U8_G(x3, x4)
U9_G(x1, x2, x3, x4) = U9_G(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(x3, x8)
U4_GAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U4_GAAA(x3, x8)
U5_GAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U5_GAAA(x3, x8)
U10_G(x1, x2, x3, x4) = U10_G(x3, x4)
U11_G(x1, x2, x3, x4) = U11_G(x3, x4)
HALVES41_IN_GAAG(x1, x2, x3, x4) = HALVES41_IN_GAAG(x1, x4)
U3_GAAG(x1, x2, x3, x4, x5, x6, x7, x8) = U3_GAAG(x3, x7, x8)
U4_GAAG(x1, x2, x3, x4, x5, x6, x7, x8) = U4_GAAG(x3, x7, x8)
U5_GAAG(x1, x2, x3, x4, x5, x6, x7, x8) = U5_GAAG(x3, x7, x8)
U12_G(x1, x2) = U12_G(x1, x2)
U13_G(x1, x2) = U13_G(x1, x2)
U14_G(x1, x2) = U14_G(x1, x2)
LAST70_IN_AAA(x1, x2, x3) = LAST70_IN_AAA
U6_AAA(x1, x2, x3, x4, x5) = U6_AAA(x5)

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

(44) 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_agaa(T22, T23, X75, X73))
PALINDROME1_IN_G(.(T21, .(T22, T23))) → LAST24_IN_AGAA(T22, T23, X75, X73)
LAST24_IN_AGAA(T41, T42, X113, .(T41, X114)) → U2_AGAA(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
LAST24_IN_AGAA(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, last24_in_agaa(T22, T23, T26, T27))
U8_G(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_G(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
U8_G(T21, T22, T23, last24_out_agaa(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_agaa(T73, T74, X185, X183))
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → LAST24_IN_AGAA(T73, T74, X185, X183)
HALVES41_IN_GAAA(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(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, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)
PALINDROME1_IN_G(.(T88, .(T22, T23))) → U10_G(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
PALINDROME1_IN_G(.(T88, .(T22, T23))) → LAST24_IN_AGAA(T22, T23, T88, T27)
U10_G(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_G(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
U10_G(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → HALVES41_IN_GAAG(T27, T89, T89, even)
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_GAAG(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → LAST24_IN_AGAA(T73, T74, X185, X183)
HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_GAAG(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAG(T78, X184, X186, X187)
PALINDROME1_IN_G(T92) → U12_G(T92, halves41_in_gaaa(T92, X214, X215, X216))
PALINDROME1_IN_G(T92) → HALVES41_IN_GAAA(T92, X214, X215, X216)
PALINDROME1_IN_G(T92) → U13_G(T92, halves41_in_gaag(T92, T93, T94, odd))
PALINDROME1_IN_G(T92) → HALVES41_IN_GAAG(T92, T93, T94, odd)
U13_G(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_G(T92, last70_in_aaa(T93, X217, T94))
U13_G(T92, halves41_out_gaag(T92, T93, T94, odd)) → LAST70_IN_AAA(T93, X217, T94)
LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → U6_AAA(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_AAA(T115, X246, T116)

The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(45) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 25 less nodes.

(46) Complex Obligation (AND)

(47) Obligation:

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

LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_AAA(T115, X246, T116)

The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(48) UsableRulesProof (EQUIVALENT transformation)

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

(49) Obligation:

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

LAST70_IN_AAA(.(T114, T115), X246, .(T114, T116)) → LAST70_IN_AAA(T115, X246, T116)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
LAST70_IN_AAA(x1, x2, x3) = LAST70_IN_AAA

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

(50) PiDPToQDPProof (SOUND transformation)

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

(51) Obligation:

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

LAST70_IN_AAA → LAST70_IN_AAA

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

(52) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LAST70_IN_AAA to LAST70_IN_AAA.

(53) NO

(54) 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:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(55) UsableRulesProof (EQUIVALENT transformation)

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

(56) 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) = .(x2)
LAST32_IN_GAA(x1, x2, x3) = LAST32_IN_GAA(x1)

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

(57) PiDPToQDPProof (SOUND transformation)

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

(58) Obligation:

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

LAST32_IN_GAA(.(T55)) → LAST32_IN_GAA(T55)

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

(59) 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(.(T55)) → LAST32_IN_GAA(T55)
The graph contains the following edges 1 > 1

(60) YES

(61) Obligation:

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

HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAG(T78, X184, X186, X187)

The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(62) UsableRulesProof (EQUIVALENT transformation)

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

(63) Obligation:

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

HALVES41_IN_GAAG(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_GAAG(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAG(T78, X184, X186, X187)

The TRS R consists of the following rules:

last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
last32_in_gaa(x1, x2, x3) = last32_in_gaa(x1)
last32_out_gaa(x1, x2, x3) = last32_out_gaa(x1, x3)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x2, x5)
last24_in_agaa(x1, x2, x3, x4) = last24_in_agaa(x2)
last24_out_agaa(x1, x2, x3, x4) = last24_out_agaa(x2, x4)
U2_agaa(x1, x2, x3, x4, x5) = U2_agaa(x2, x5)
HALVES41_IN_GAAG(x1, x2, x3, x4) = HALVES41_IN_GAAG(x1, x4)
U4_GAAG(x1, x2, x3, x4, x5, x6, x7, x8) = U4_GAAG(x3, x7, x8)

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

(64) PiDPToQDPProof (SOUND transformation)

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

(65) Obligation:

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

HALVES41_IN_GAAG(.(.(T74)), X187) → U4_GAAG(T74, X187, last24_in_agaa(T74))
U4_GAAG(T74, X187, last24_out_agaa(T74, T78)) → HALVES41_IN_GAAG(T78, X187)

The TRS R consists of the following rules:

last24_in_agaa([]) → last24_out_agaa([], [])
last24_in_agaa(T42) → U2_agaa(T42, last32_in_gaa(T42))
U2_agaa(T42, last32_out_gaa(T42, X114)) → last24_out_agaa(T42, .(X114))
last32_in_gaa(.([])) → last32_out_gaa(.([]), [])
last32_in_gaa(.(T55)) → U1_gaa(T55, last32_in_gaa(T55))
U1_gaa(T55, last32_out_gaa(T55, X140)) → last32_out_gaa(.(T55), .(X140))

The set Q consists of the following terms:

last24_in_agaa(x0)
U2_agaa(x0, x1)
last32_in_gaa(x0)
U1_gaa(x0, x1)

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

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HALVES41_IN_GAAG(.(.(T74)), X187) → U4_GAAG(T74, X187, last24_in_agaa(T74))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U4_GAAG(x₁, ..., x₃) ) = 2x₂ + 2x₃ + 2

POL( last24_in_agaa(x₁) ) = 2x₁

POL( [] ) = 0

POL( last24_out_agaa(x₁, x₂) ) = max{0, x₂ - 1}

POL( U2_agaa(x₁, x₂) ) = max{0, 2x₂ - 2}

POL( last32_in_gaa(x₁) ) = x₁

POL( .(x₁) ) = 2x₁ + 2

POL( last32_out_gaa(x₁, x₂) ) = x₂ + 2

POL( U1_gaa(x₁, x₂) ) = 2x₂ + 2

POL( HALVES41_IN_GAAG(x₁, x₂) ) = x₁ + 2x₂

The following usable rules [FROCOS05] were oriented:

last24_in_agaa([]) → last24_out_agaa([], [])
last24_in_agaa(T42) → U2_agaa(T42, last32_in_gaa(T42))
last32_in_gaa(.([])) → last32_out_gaa(.([]), [])
last32_in_gaa(.(T55)) → U1_gaa(T55, last32_in_gaa(T55))
U2_agaa(T42, last32_out_gaa(T42, X114)) → last24_out_agaa(T42, .(X114))
U1_gaa(T55, last32_out_gaa(T55, X140)) → last32_out_gaa(.(T55), .(X140))

(67) Obligation:

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

U4_GAAG(T74, X187, last24_out_agaa(T74, T78)) → HALVES41_IN_GAAG(T78, X187)

The TRS R consists of the following rules:

last24_in_agaa([]) → last24_out_agaa([], [])
last24_in_agaa(T42) → U2_agaa(T42, last32_in_gaa(T42))
U2_agaa(T42, last32_out_gaa(T42, X114)) → last24_out_agaa(T42, .(X114))
last32_in_gaa(.([])) → last32_out_gaa(.([]), [])
last32_in_gaa(.(T55)) → U1_gaa(T55, last32_in_gaa(T55))
U1_gaa(T55, last32_out_gaa(T55, X140)) → last32_out_gaa(.(T55), .(X140))

The set Q consists of the following terms:

last24_in_agaa(x0)
U2_agaa(x0, x1)
last32_in_gaa(x0)
U1_gaa(x0, x1)

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

(68) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(69) TRUE

(70) 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, last24_in_agaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)

The TRS R consists of the following rules:

palindrome1_in_g([]) → palindrome1_out_g([])
palindrome1_in_g(.(T21, .(T22, T23))) → U7_g(T21, T22, T23, last24_in_agaa(T22, T23, X75, X73))
last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
U7_g(T21, T22, T23, last24_out_agaa(T22, T23, X75, X73)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T21, .(T22, T23))) → U8_g(T21, T22, T23, last24_in_agaa(T22, T23, T26, T27))
U8_g(T21, T22, T23, last24_out_agaa(T22, T23, T26, T27)) → U9_g(T21, T22, T23, halves41_in_gaaa(T27, X74, X76, X77))
halves41_in_gaaa([], [], [], even) → halves41_out_gaaa([], [], [], even)
halves41_in_gaaa(.(T65, []), .(T65, []), [], odd) → halves41_out_gaaa(.(T65, []), .(T65, []), [], odd)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaaa(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaaa(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaaa(T78, X184, X186, X187))
U5_gaaa(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaaa(T78, X184, X186, X187)) → halves41_out_gaaa(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U9_g(T21, T22, T23, halves41_out_gaaa(T27, X74, X76, X77)) → palindrome1_out_g(.(T21, .(T22, T23)))
palindrome1_in_g(.(T88, .(T22, T23))) → U10_g(T88, T22, T23, last24_in_agaa(T22, T23, T88, T27))
U10_g(T88, T22, T23, last24_out_agaa(T22, T23, T88, T27)) → U11_g(T88, T22, T23, halves41_in_gaag(T27, T89, T89, even))
halves41_in_gaag([], [], [], even) → halves41_out_gaag([], [], [], even)
halves41_in_gaag(.(T65, []), .(T65, []), [], odd) → halves41_out_gaag(.(T65, []), .(T65, []), [], odd)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187) → U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_in_agaa(T73, T74, X185, X183))
U3_gaag(T72, T73, T74, X184, X185, X186, X187, last24_out_agaa(T73, T74, X185, X183)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(X185, X186), X187)
halves41_in_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187) → U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_in_agaa(T73, T74, T77, T78))
U4_gaag(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_in_gaag(T78, X184, X186, X187))
U5_gaag(T72, T73, T74, X184, T77, X186, X187, halves41_out_gaag(T78, X184, X186, X187)) → halves41_out_gaag(.(T72, .(T73, T74)), .(T72, X184), .(T77, X186), X187)
U11_g(T88, T22, T23, halves41_out_gaag(T27, T89, T89, even)) → palindrome1_out_g(.(T88, .(T22, T23)))
palindrome1_in_g(T92) → U12_g(T92, halves41_in_gaaa(T92, X214, X215, X216))
U12_g(T92, halves41_out_gaaa(T92, X214, X215, X216)) → palindrome1_out_g(T92)
palindrome1_in_g(T92) → U13_g(T92, halves41_in_gaag(T92, T93, T94, odd))
U13_g(T92, halves41_out_gaag(T92, T93, T94, odd)) → U14_g(T92, last70_in_aaa(T93, X217, T94))
last70_in_aaa(.(T107, []), T107, []) → last70_out_aaa(.(T107, []), T107, [])
last70_in_aaa(.(T114, T115), X246, .(T114, T116)) → U6_aaa(T114, T115, X246, T116, last70_in_aaa(T115, X246, T116))
U6_aaa(T114, T115, X246, T116, last70_out_aaa(T115, X246, T116)) → last70_out_aaa(.(T114, T115), X246, .(T114, T116))
U14_g(T92, last70_out_aaa(T93, X217, T94)) → palindrome1_out_g(T92)

(71) UsableRulesProof (EQUIVALENT transformation)

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

(72) 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, last24_in_agaa(T73, T74, T77, T78))
U4_GAAA(T72, T73, T74, X184, T77, X186, X187, last24_out_agaa(T73, T74, T77, T78)) → HALVES41_IN_GAAA(T78, X184, X186, X187)

The TRS R consists of the following rules:

last24_in_agaa(T34, [], T34, []) → last24_out_agaa(T34, [], T34, [])
last24_in_agaa(T41, T42, X113, .(T41, X114)) → U2_agaa(T41, T42, X113, X114, last32_in_gaa(T42, X113, X114))
U2_agaa(T41, T42, X113, X114, last32_out_gaa(T42, X113, X114)) → last24_out_agaa(T41, T42, X113, .(T41, X114))
last32_in_gaa(.(T49, []), T49, []) → last32_out_gaa(.(T49, []), T49, [])
last32_in_gaa(.(T54, T55), X139, .(T54, X140)) → U1_gaa(T54, T55, X139, X140, last32_in_gaa(T55, X139, X140))
U1_gaa(T54, T55, X139, X140, last32_out_gaa(T55, X139, X140)) → last32_out_gaa(.(T54, T55), X139, .(T54, X140))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
last32_in_gaa(x1, x2, x3) = last32_in_gaa(x1)
last32_out_gaa(x1, x2, x3) = last32_out_gaa(x1, x3)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x2, x5)
last24_in_agaa(x1, x2, x3, x4) = last24_in_agaa(x2)
last24_out_agaa(x1, x2, x3, x4) = last24_out_agaa(x2, x4)
U2_agaa(x1, x2, x3, x4, x5) = U2_agaa(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(x3, x8)

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

(73) PiDPToQDPProof (SOUND transformation)

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

(74) Obligation:

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

HALVES41_IN_GAAA(.(.(T74))) → U4_GAAA(T74, last24_in_agaa(T74))
U4_GAAA(T74, last24_out_agaa(T74, T78)) → HALVES41_IN_GAAA(T78)

The TRS R consists of the following rules:

last24_in_agaa([]) → last24_out_agaa([], [])
last24_in_agaa(T42) → U2_agaa(T42, last32_in_gaa(T42))
U2_agaa(T42, last32_out_gaa(T42, X114)) → last24_out_agaa(T42, .(X114))
last32_in_gaa(.([])) → last32_out_gaa(.([]), [])
last32_in_gaa(.(T55)) → U1_gaa(T55, last32_in_gaa(T55))
U1_gaa(T55, last32_out_gaa(T55, X140)) → last32_out_gaa(.(T55), .(X140))

The set Q consists of the following terms:

last24_in_agaa(x0)
U2_agaa(x0, x1)
last32_in_gaa(x0)
U1_gaa(x0, x1)

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

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HALVES41_IN_GAAA(.(.(T74))) → U4_GAAA(T74, last24_in_agaa(T74))

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

POL(.(x₁)) = 1 + x₁
POL(HALVES41_IN_GAAA(x₁)) = x₁
POL(U1_gaa(x₁, x₂)) = 1 + x₂
POL(U2_agaa(x₁, x₂)) = 1 + x₂
POL(U4_GAAA(x₁, x₂)) = x₂
POL([]) = 0
POL(last24_in_agaa(x₁)) = 1 + x₁
POL(last24_out_agaa(x₁, x₂)) = x₂
POL(last32_in_gaa(x₁)) = x₁
POL(last32_out_gaa(x₁, x₂)) = x₂

The following usable rules [FROCOS05] were oriented:

last24_in_agaa([]) → last24_out_agaa([], [])
last24_in_agaa(T42) → U2_agaa(T42, last32_in_gaa(T42))
last32_in_gaa(.([])) → last32_out_gaa(.([]), [])
last32_in_gaa(.(T55)) → U1_gaa(T55, last32_in_gaa(T55))
U2_agaa(T42, last32_out_gaa(T42, X114)) → last24_out_agaa(T42, .(X114))
U1_gaa(T55, last32_out_gaa(T55, X140)) → last32_out_gaa(.(T55), .(X140))

(76) Obligation:

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

U4_GAAA(T74, last24_out_agaa(T74, T78)) → HALVES41_IN_GAAA(T78)

The TRS R consists of the following rules:

last24_in_agaa([]) → last24_out_agaa([], [])
last24_in_agaa(T42) → U2_agaa(T42, last32_in_gaa(T42))
U2_agaa(T42, last32_out_gaa(T42, X114)) → last24_out_agaa(T42, .(X114))
last32_in_gaa(.([])) → last32_out_gaa(.([]), [])
last32_in_gaa(.(T55)) → U1_gaa(T55, last32_in_gaa(T55))
U1_gaa(T55, last32_out_gaa(T55, X140)) → last32_out_gaa(.(T55), .(X140))

The set Q consists of the following terms:

last24_in_agaa(x0)
U2_agaa(x0, x1)
last32_in_gaa(x0)
U1_gaa(x0, x1)

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

(77) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) NonTerminationProof (EQUIVALENT transformation)

(15) NO

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) MRRProof (EQUIVALENT transformation)

(29) Obligation:

(30) PisEmptyProof (EQUIVALENT transformation)

(31) YES

(32) Obligation:

(33) UsableRulesProof (EQUIVALENT transformation)

(34) Obligation:

(35) PiDPToQDPProof (SOUND transformation)

(36) Obligation:

(37) MRRProof (EQUIVALENT transformation)

(38) Obligation:

(39) PisEmptyProof (EQUIVALENT transformation)

(40) YES

(41) PrologToPiTRSProof (SOUND transformation)

(42) Obligation:

(43) DependencyPairsProof (EQUIVALENT transformation)

(44) Obligation:

(45) DependencyGraphProof (EQUIVALENT transformation)

(46) Complex Obligation (AND)

(47) Obligation:

(48) UsableRulesProof (EQUIVALENT transformation)

(49) Obligation:

(50) PiDPToQDPProof (SOUND transformation)

(51) Obligation:

(52) NonTerminationProof (EQUIVALENT transformation)

(53) NO

(54) Obligation:

(55) UsableRulesProof (EQUIVALENT transformation)

(56) Obligation:

(57) PiDPToQDPProof (SOUND transformation)

(58) Obligation:

(59) QDPSizeChangeProof (EQUIVALENT transformation)

(60) YES

(61) Obligation:

(62) UsableRulesProof (EQUIVALENT transformation)

(63) Obligation:

(64) PiDPToQDPProof (SOUND transformation)

(65) Obligation:

(66) QDPOrderProof (EQUIVALENT transformation)

(67) Obligation:

(68) DependencyGraphProof (EQUIVALENT transformation)

(69) TRUE

(70) Obligation:

(71) UsableRulesProof (EQUIVALENT transformation)

(72) Obligation:

(73) PiDPToQDPProof (SOUND transformation)

(74) Obligation:

(75) QDPOrderProof (EQUIVALENT transformation)

(76) Obligation:

(77) DependencyGraphProof (EQUIVALENT transformation)

(78) TRUE