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

(0) Obligation:

Clauses:

goal(A, B, C) :- ','(s2l(A, D), applast(D, B, C)).
applast(L, X, Last) :- ','(append(L, .(X, []), LX), last(Last, LX)).
last(X, .(X, [])).
last(X, .(H, T)) :- last(X, T).
append([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).

Query: goal(g,a,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: goal(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(s2l(T7, X7), applast(X7, T10, T11))\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(s2l(T7, X7), applast(X7, T10, T11)), SCOPE: 2, CLAUSE: 6 | ','(s2l(T7, X7), applast(X7, T10, T11)), SCOPE: 2, CLAUSE: 7\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(s2l(T7, X7), applast(X7, T10, T11)), SCOPE: 2, CLAUSE: 6\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(s2l(T7, X7), applast(X7, T10, T11)), SCOPE: 2, CLAUSE: 7\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(s2l(T16, X32), applast(.(X31, X32), T10, T11))\n\nT16 is ground\nX31 is free\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: s2l(T16, X32)\n\nT16 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: applast(.(X31, T18), T10, T11)\n\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: s2l(T16, X32), SCOPE: 3, CLAUSE: 6 | s2l(T16, X32), SCOPE: 3, CLAUSE: 7\n\nT16 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: s2l(T16, X32), SCOPE: 3, CLAUSE: 6\n\nT16 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: s2l(T16, X32), SCOPE: 3, CLAUSE: 7\n\nT16 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: s2l(T24, X67)\n\nT24 is ground\nX67 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: applast(.(X31, T18), T10, T11), SCOPE: 4, CLAUSE: 1\n\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(append(.(X93, T41), .(T42, []), X92), last(T43, X92))\n\nX93 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: append(.(X93, T41), .(T42, []), X92)\n\nX93 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: last(T46, T45)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: append(.(X93, T41), .(T42, []), X92), SCOPE: 5, CLAUSE: 4 | append(.(X93, T41), .(T42, []), X92), SCOPE: 5, CLAUSE: 5\n\nX93 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: append(.(X93, T41), .(T42, []), X92), SCOPE: 5, CLAUSE: 5\n\nX93 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: append(T57, .(T58, []), X118)\n\nX118 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: append(T57, .(T58, []), X118), SCOPE: 6, CLAUSE: 4 | append(T57, .(T58, []), X118), SCOPE: 6, CLAUSE: 5\n\nX118 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: append(T57, .(T58, []), X118), SCOPE: 6, CLAUSE: 4\n\nX118 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: append(T57, .(T58, []), X118), SCOPE: 6, CLAUSE: 5\n\nX118 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: append(T75, .(T76, []), X140)\n\nX140 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: last(T46, T45), SCOPE: 7, CLAUSE: 2 | last(T46, T45), SCOPE: 7, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: last(T46, T45), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: last(T46, T45), SCOPE: 7, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: last(T95, T96)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: applast([], T10, T11)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: applast([], T10, T11), SCOPE: 8, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(append([], .(T111, []), X177), last(T112, X177))\n\nX177 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: append([], .(T111, []), X177)\n\nX177 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: last(T116, T115)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: append([], .(T111, []), X177), SCOPE: 9, CLAUSE: 4 | append([], .(T111, []), X177), SCOPE: 9, CLAUSE: 5\n\nX177 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: append([], .(T111, []), X177), SCOPE: 9, CLAUSE: 4\n\nX177 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: append([], .(T111, []), X177), SCOPE: 9, CLAUSE: 5\n\nX177 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="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 54 [arrowhead = none ];
54 -> 2;

55 [label="ONLY EVAL with clause\ngoal(X4, X5, X6) :- ','(s2l(X4, X7), applast(X7, X5, X6)).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T10,\nX5 -> T10,\nT3 -> T11,\nX6 -> T11,\nT8 -> T10,\nT9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 55 [arrowhead = none ];
55 -> 3;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 56 [arrowhead = none ];
56 -> 4;

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 57 [arrowhead = none ];
57 -> 5;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 58 [arrowhead = none ];
58 -> 6;

59 [label="EVAL with clause\ns2l(s(X28), .(X29, X30)) :- s2l(X28, X30).\nand substitutionX28 -> T16,\nT7 -> s(T16),\nX29 -> X31,\nX30 -> X32,\nX7 -> .(X31, X32)", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 59 [arrowhead = none ];
59 -> 7;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 60 [arrowhead = none ];
60 -> 8;

61 [label="EVAL with clause\ns2l(0, []).\nand substitutionT7 -> 0,\nX7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 61 [arrowhead = none ];
61 -> 42;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 62 [arrowhead = none ];
62 -> 43;

63 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 63 [arrowhead = none ];
63 -> 9;

64 [label="SPLIT 2\nnew knowledge:\nT16 is ground\nreplacements:X32 -> T18", fontsize=14, style = filled, fillcolor = violet];
7 -> 64 [arrowhead = none ];
64 -> 10;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 65 [arrowhead = none ];
65 -> 11;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 66 [arrowhead = none ];
66 -> 19;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 67 [arrowhead = none ];
67 -> 12;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 68 [arrowhead = none ];
68 -> 13;

69 [label="EVAL with clause\ns2l(s(X63), .(X64, X65)) :- s2l(X63, X65).\nand substitutionX63 -> T24,\nT16 -> s(T24),\nX64 -> X66,\nX65 -> X67,\nX32 -> .(X66, X67)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 69 [arrowhead = none ];
69 -> 14;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 70 [arrowhead = none ];
70 -> 15;

71 [label="EVAL with clause\ns2l(0, []).\nand substitutionT16 -> 0,\nX32 -> []", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 71 [arrowhead = none ];
71 -> 16;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 72 [arrowhead = none ];
72 -> 17;

73 [label="INSTANCE with matching:\nT16 -> T24\nX32 -> X67", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 73 [arrowhead = none , style=dashed];
73 -> 9[style=dashed];

74 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 74 [arrowhead = none ];
74 -> 18;

75 [label="ONLY EVAL with clause\napplast(X89, X90, X91) :- ','(append(X89, .(X90, []), X92), last(X91, X92)).\nand substitutionX31 -> X93,\nT18 -> T41,\nX89 -> .(X93, T41),\nT10 -> T42,\nX90 -> T42,\nT11 -> T43,\nX91 -> T43,\nT38 -> T41,\nT39 -> T42,\nT40 -> T43", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 75 [arrowhead = none ];
75 -> 20;

76 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
20 -> 76 [arrowhead = none ];
76 -> 21;

77 [label="SPLIT 2\nreplacements:X93 -> T44,\nX92 -> T45,\nT43 -> T46", fontsize=14, style = filled, fillcolor = violet];
20 -> 77 [arrowhead = none ];
77 -> 22;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 78 [arrowhead = none ];
78 -> 23;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 79 [arrowhead = none ];
79 -> 34;

80 [label="BACKTRACK\nfor clause: append([], L, L)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 80 [arrowhead = none ];
80 -> 24;

81 [label="ONLY EVAL with clause\nappend(.(X113, X114), X115, .(X113, X116)) :- append(X114, X115, X116).\nand substitutionX93 -> X117,\nX113 -> X117,\nT41 -> T57,\nX114 -> T57,\nT42 -> T58,\nX115 -> .(T58, []),\nX116 -> X118,\nX92 -> .(X117, X118),\nT55 -> T57,\nT56 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 81 [arrowhead = none ];
81 -> 25;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 82 [arrowhead = none ];
82 -> 26;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 83 [arrowhead = none ];
83 -> 27;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 84 [arrowhead = none ];
84 -> 28;

85 [label="EVAL with clause\nappend([], X125, X125).\nand substitutionT57 -> [],\nT58 -> T65,\nX125 -> .(T65, []),\nX118 -> .(T65, [])", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 85 [arrowhead = none ];
85 -> 29;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 86 [arrowhead = none ];
86 -> 30;

87 [label="EVAL with clause\nappend(.(X136, X137), X138, .(X136, X139)) :- append(X137, X138, X139).\nand substitutionX136 -> T72,\nX137 -> T75,\nT57 -> .(T72, T75),\nT58 -> T76,\nX138 -> .(T76, []),\nX139 -> X140,\nX118 -> .(T72, X140),\nT73 -> T75,\nT74 -> T76", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 87 [arrowhead = none ];
87 -> 32;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 88 [arrowhead = none ];
88 -> 33;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 89 [arrowhead = none ];
89 -> 31;

90 [label="INSTANCE with matching:\nT57 -> T75\nT58 -> T76\nX118 -> X140", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 90 [arrowhead = none , style=dashed];
90 -> 25[style=dashed];

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

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 92 [arrowhead = none ];
92 -> 36;

93 [label="EVAL with clause\nlast(X149, .(X149, [])).\nand substitutionT46 -> T85,\nX149 -> T85,\nT45 -> .(T85, [])", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 93 [arrowhead = none ];
93 -> 37;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 94 [arrowhead = none ];
94 -> 38;

95 [label="EVAL with clause\nlast(X156, .(X157, X158)) :- last(X156, X158).\nand substitutionT46 -> T95,\nX156 -> T95,\nX157 -> T93,\nX158 -> T96,\nT45 -> .(T93, T96),\nT92 -> T95,\nT94 -> T96", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 95 [arrowhead = none ];
95 -> 40;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 96 [arrowhead = none ];
96 -> 41;

97 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 97 [arrowhead = none ];
97 -> 39;

98 [label="INSTANCE with matching:\nT46 -> T95\nT45 -> T96", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 98 [arrowhead = none , style=dashed];
98 -> 22[style=dashed];

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 99 [arrowhead = none ];
99 -> 44;

100 [label="ONLY EVAL with clause\napplast(X174, X175, X176) :- ','(append(X174, .(X175, []), X177), last(X176, X177)).\nand substitutionX174 -> [],\nT10 -> T111,\nX175 -> T111,\nT11 -> T112,\nX176 -> T112,\nT109 -> T111,\nT110 -> T112", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 100 [arrowhead = none ];
100 -> 45;

101 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
45 -> 101 [arrowhead = none ];
101 -> 46;

102 [label="SPLIT 2\nreplacements:X177 -> T115,\nT112 -> T116", fontsize=14, style = filled, fillcolor = violet];
45 -> 102 [arrowhead = none ];
102 -> 47;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 103 [arrowhead = none ];
103 -> 48;

104 [label="INSTANCE with matching:\nT46 -> T116\nT45 -> T115", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 104 [arrowhead = none , style=dashed];
104 -> 22[style=dashed];

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 105 [arrowhead = none ];
105 -> 49;

106 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 106 [arrowhead = none ];
106 -> 50;

107 [label="ONLY EVAL with clause\nappend([], X185, X185).\nand substitutionT111 -> T122,\nX185 -> .(T122, []),\nX177 -> .(T122, [])", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 107 [arrowhead = none ];
107 -> 51;

108 [label="BACKTRACK\nfor clause: append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 108 [arrowhead = none ];
108 -> 53;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 109 [arrowhead = none ];
109 -> 52;

}

(2) Obligation:

Clauses:

s2lA(s(T24), .(X66, X67)) :- s2lA(T24, X67).
s2lA(0, []).
appendB([], T65, .(T65, [])).
appendB(.(T72, T75), T76, .(T72, X140)) :- appendB(T75, T76, X140).
lastC(T85, .(T85, [])).
lastC(T95, .(T93, T96)) :- lastC(T95, T96).
appendD(X117, T57, T58, .(X117, X118)) :- appendB(T57, T58, X118).
appendE(T122, .(T122, [])).
goalF(s(T16), T10, T11) :- s2lA(T16, X32).
goalF(s(T16), T42, T43) :- ','(s2lA(T16, T41), appendD(X93, T41, T42, X92)).
goalF(s(T16), T42, T46) :- ','(s2lA(T16, T41), ','(appendD(T44, T41, T42, T45), lastC(T46, T45))).
goalF(0, T111, T112) :- appendE(T111, X177).
goalF(0, T111, T116) :- ','(appendE(T111, T115), lastC(T116, T115)).

Query: goalF(g,a,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
goalF_in: (b,f,f)
s2lA_in: (b,f)
appendD_in: (f,b,f,f)
appendB_in: (b,f,f)
lastC_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goalF_in_gaa(s(T16), T10, T11) → U5_gaa(T16, T10, T11, s2lA_in_ga(T16, X32))
s2lA_in_ga(s(T24), .(X66, X67)) → U1_ga(T24, X66, X67, s2lA_in_ga(T24, X67))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
U1_ga(T24, X66, X67, s2lA_out_ga(T24, X67)) → s2lA_out_ga(s(T24), .(X66, X67))
U5_gaa(T16, T10, T11, s2lA_out_ga(T16, X32)) → goalF_out_gaa(s(T16), T10, T11)
goalF_in_gaa(s(T16), T42, T43) → U6_gaa(T16, T42, T43, s2lA_in_ga(T16, T41))
U6_gaa(T16, T42, T43, s2lA_out_ga(T16, T41)) → U7_gaa(T16, T42, T43, appendD_in_agaa(X93, T41, T42, X92))
appendD_in_agaa(X117, T57, T58, .(X117, X118)) → U4_agaa(X117, T57, T58, X118, appendB_in_gaa(T57, T58, X118))
appendB_in_gaa([], T65, .(T65, [])) → appendB_out_gaa([], T65, .(T65, []))
appendB_in_gaa(.(T72, T75), T76, .(T72, X140)) → U2_gaa(T72, T75, T76, X140, appendB_in_gaa(T75, T76, X140))
U2_gaa(T72, T75, T76, X140, appendB_out_gaa(T75, T76, X140)) → appendB_out_gaa(.(T72, T75), T76, .(T72, X140))
U4_agaa(X117, T57, T58, X118, appendB_out_gaa(T57, T58, X118)) → appendD_out_agaa(X117, T57, T58, .(X117, X118))
U7_gaa(T16, T42, T43, appendD_out_agaa(X93, T41, T42, X92)) → goalF_out_gaa(s(T16), T42, T43)
goalF_in_gaa(s(T16), T42, T46) → U8_gaa(T16, T42, T46, s2lA_in_ga(T16, T41))
U8_gaa(T16, T42, T46, s2lA_out_ga(T16, T41)) → U9_gaa(T16, T42, T46, appendD_in_agaa(T44, T41, T42, T45))
U9_gaa(T16, T42, T46, appendD_out_agaa(T44, T41, T42, T45)) → U10_gaa(T16, T42, T46, lastC_in_ag(T46, T45))
lastC_in_ag(T85, .(T85, [])) → lastC_out_ag(T85, .(T85, []))
lastC_in_ag(T95, .(T93, T96)) → U3_ag(T95, T93, T96, lastC_in_ag(T95, T96))
U3_ag(T95, T93, T96, lastC_out_ag(T95, T96)) → lastC_out_ag(T95, .(T93, T96))
U10_gaa(T16, T42, T46, lastC_out_ag(T46, T45)) → goalF_out_gaa(s(T16), T42, T46)
goalF_in_gaa(0, T111, T112) → U11_gaa(T111, T112, appendE_in_aa(T111, X177))
appendE_in_aa(T122, .(T122, [])) → appendE_out_aa(T122, .(T122, []))
U11_gaa(T111, T112, appendE_out_aa(T111, X177)) → goalF_out_gaa(0, T111, T112)
goalF_in_gaa(0, T111, T116) → U12_gaa(T111, T116, appendE_in_aa(T111, T115))
U12_gaa(T111, T116, appendE_out_aa(T111, T115)) → U13_gaa(T111, T116, lastC_in_ag(T116, T115))
U13_gaa(T111, T116, lastC_out_ag(T116, T115)) → goalF_out_gaa(0, T111, T116)

The argument filtering Pi contains the following mapping:
goalF_in_gaa(x1, x2, x3)  =  goalF_in_gaa(x1)
s(x1)  =  s(x1)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x4)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
0  =  0
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
goalF_out_gaa(x1, x2, x3)  =  goalF_out_gaa(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
appendD_in_agaa(x1, x2, x3, x4)  =  appendD_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
appendB_in_gaa(x1, x2, x3)  =  appendB_in_gaa(x1)
[]  =  []
appendB_out_gaa(x1, x2, x3)  =  appendB_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
appendD_out_agaa(x1, x2, x3, x4)  =  appendD_out_agaa(x2, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
lastC_in_ag(x1, x2)  =  lastC_in_ag(x2)
lastC_out_ag(x1, x2)  =  lastC_out_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
U11_gaa(x1, x2, x3)  =  U11_gaa(x3)
appendE_in_aa(x1, x2)  =  appendE_in_aa
appendE_out_aa(x1, x2)  =  appendE_out_aa(x2)
U12_gaa(x1, x2, x3)  =  U12_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goalF_in_gaa(s(T16), T10, T11) → U5_gaa(T16, T10, T11, s2lA_in_ga(T16, X32))
s2lA_in_ga(s(T24), .(X66, X67)) → U1_ga(T24, X66, X67, s2lA_in_ga(T24, X67))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
U1_ga(T24, X66, X67, s2lA_out_ga(T24, X67)) → s2lA_out_ga(s(T24), .(X66, X67))
U5_gaa(T16, T10, T11, s2lA_out_ga(T16, X32)) → goalF_out_gaa(s(T16), T10, T11)
goalF_in_gaa(s(T16), T42, T43) → U6_gaa(T16, T42, T43, s2lA_in_ga(T16, T41))
U6_gaa(T16, T42, T43, s2lA_out_ga(T16, T41)) → U7_gaa(T16, T42, T43, appendD_in_agaa(X93, T41, T42, X92))
appendD_in_agaa(X117, T57, T58, .(X117, X118)) → U4_agaa(X117, T57, T58, X118, appendB_in_gaa(T57, T58, X118))
appendB_in_gaa([], T65, .(T65, [])) → appendB_out_gaa([], T65, .(T65, []))
appendB_in_gaa(.(T72, T75), T76, .(T72, X140)) → U2_gaa(T72, T75, T76, X140, appendB_in_gaa(T75, T76, X140))
U2_gaa(T72, T75, T76, X140, appendB_out_gaa(T75, T76, X140)) → appendB_out_gaa(.(T72, T75), T76, .(T72, X140))
U4_agaa(X117, T57, T58, X118, appendB_out_gaa(T57, T58, X118)) → appendD_out_agaa(X117, T57, T58, .(X117, X118))
U7_gaa(T16, T42, T43, appendD_out_agaa(X93, T41, T42, X92)) → goalF_out_gaa(s(T16), T42, T43)
goalF_in_gaa(s(T16), T42, T46) → U8_gaa(T16, T42, T46, s2lA_in_ga(T16, T41))
U8_gaa(T16, T42, T46, s2lA_out_ga(T16, T41)) → U9_gaa(T16, T42, T46, appendD_in_agaa(T44, T41, T42, T45))
U9_gaa(T16, T42, T46, appendD_out_agaa(T44, T41, T42, T45)) → U10_gaa(T16, T42, T46, lastC_in_ag(T46, T45))
lastC_in_ag(T85, .(T85, [])) → lastC_out_ag(T85, .(T85, []))
lastC_in_ag(T95, .(T93, T96)) → U3_ag(T95, T93, T96, lastC_in_ag(T95, T96))
U3_ag(T95, T93, T96, lastC_out_ag(T95, T96)) → lastC_out_ag(T95, .(T93, T96))
U10_gaa(T16, T42, T46, lastC_out_ag(T46, T45)) → goalF_out_gaa(s(T16), T42, T46)
goalF_in_gaa(0, T111, T112) → U11_gaa(T111, T112, appendE_in_aa(T111, X177))
appendE_in_aa(T122, .(T122, [])) → appendE_out_aa(T122, .(T122, []))
U11_gaa(T111, T112, appendE_out_aa(T111, X177)) → goalF_out_gaa(0, T111, T112)
goalF_in_gaa(0, T111, T116) → U12_gaa(T111, T116, appendE_in_aa(T111, T115))
U12_gaa(T111, T116, appendE_out_aa(T111, T115)) → U13_gaa(T111, T116, lastC_in_ag(T116, T115))
U13_gaa(T111, T116, lastC_out_ag(T116, T115)) → goalF_out_gaa(0, T111, T116)

The argument filtering Pi contains the following mapping:
goalF_in_gaa(x1, x2, x3)  =  goalF_in_gaa(x1)
s(x1)  =  s(x1)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x4)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
0  =  0
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
goalF_out_gaa(x1, x2, x3)  =  goalF_out_gaa(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
appendD_in_agaa(x1, x2, x3, x4)  =  appendD_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
appendB_in_gaa(x1, x2, x3)  =  appendB_in_gaa(x1)
[]  =  []
appendB_out_gaa(x1, x2, x3)  =  appendB_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
appendD_out_agaa(x1, x2, x3, x4)  =  appendD_out_agaa(x2, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
lastC_in_ag(x1, x2)  =  lastC_in_ag(x2)
lastC_out_ag(x1, x2)  =  lastC_out_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
U11_gaa(x1, x2, x3)  =  U11_gaa(x3)
appendE_in_aa(x1, x2)  =  appendE_in_aa
appendE_out_aa(x1, x2)  =  appendE_out_aa(x2)
U12_gaa(x1, x2, x3)  =  U12_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)

(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:

GOALF_IN_GAA(s(T16), T10, T11) → U5_GAA(T16, T10, T11, s2lA_in_ga(T16, X32))
GOALF_IN_GAA(s(T16), T10, T11) → S2LA_IN_GA(T16, X32)
S2LA_IN_GA(s(T24), .(X66, X67)) → U1_GA(T24, X66, X67, s2lA_in_ga(T24, X67))
S2LA_IN_GA(s(T24), .(X66, X67)) → S2LA_IN_GA(T24, X67)
GOALF_IN_GAA(s(T16), T42, T43) → U6_GAA(T16, T42, T43, s2lA_in_ga(T16, T41))
U6_GAA(T16, T42, T43, s2lA_out_ga(T16, T41)) → U7_GAA(T16, T42, T43, appendD_in_agaa(X93, T41, T42, X92))
U6_GAA(T16, T42, T43, s2lA_out_ga(T16, T41)) → APPENDD_IN_AGAA(X93, T41, T42, X92)
APPENDD_IN_AGAA(X117, T57, T58, .(X117, X118)) → U4_AGAA(X117, T57, T58, X118, appendB_in_gaa(T57, T58, X118))
APPENDD_IN_AGAA(X117, T57, T58, .(X117, X118)) → APPENDB_IN_GAA(T57, T58, X118)
APPENDB_IN_GAA(.(T72, T75), T76, .(T72, X140)) → U2_GAA(T72, T75, T76, X140, appendB_in_gaa(T75, T76, X140))
APPENDB_IN_GAA(.(T72, T75), T76, .(T72, X140)) → APPENDB_IN_GAA(T75, T76, X140)
GOALF_IN_GAA(s(T16), T42, T46) → U8_GAA(T16, T42, T46, s2lA_in_ga(T16, T41))
U8_GAA(T16, T42, T46, s2lA_out_ga(T16, T41)) → U9_GAA(T16, T42, T46, appendD_in_agaa(T44, T41, T42, T45))
U8_GAA(T16, T42, T46, s2lA_out_ga(T16, T41)) → APPENDD_IN_AGAA(T44, T41, T42, T45)
U9_GAA(T16, T42, T46, appendD_out_agaa(T44, T41, T42, T45)) → U10_GAA(T16, T42, T46, lastC_in_ag(T46, T45))
U9_GAA(T16, T42, T46, appendD_out_agaa(T44, T41, T42, T45)) → LASTC_IN_AG(T46, T45)
LASTC_IN_AG(T95, .(T93, T96)) → U3_AG(T95, T93, T96, lastC_in_ag(T95, T96))
LASTC_IN_AG(T95, .(T93, T96)) → LASTC_IN_AG(T95, T96)
GOALF_IN_GAA(0, T111, T112) → U11_GAA(T111, T112, appendE_in_aa(T111, X177))
GOALF_IN_GAA(0, T111, T112) → APPENDE_IN_AA(T111, X177)
GOALF_IN_GAA(0, T111, T116) → U12_GAA(T111, T116, appendE_in_aa(T111, T115))
U12_GAA(T111, T116, appendE_out_aa(T111, T115)) → U13_GAA(T111, T116, lastC_in_ag(T116, T115))
U12_GAA(T111, T116, appendE_out_aa(T111, T115)) → LASTC_IN_AG(T116, T115)

The TRS R consists of the following rules:

goalF_in_gaa(s(T16), T10, T11) → U5_gaa(T16, T10, T11, s2lA_in_ga(T16, X32))
s2lA_in_ga(s(T24), .(X66, X67)) → U1_ga(T24, X66, X67, s2lA_in_ga(T24, X67))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
U1_ga(T24, X66, X67, s2lA_out_ga(T24, X67)) → s2lA_out_ga(s(T24), .(X66, X67))
U5_gaa(T16, T10, T11, s2lA_out_ga(T16, X32)) → goalF_out_gaa(s(T16), T10, T11)
goalF_in_gaa(s(T16), T42, T43) → U6_gaa(T16, T42, T43, s2lA_in_ga(T16, T41))
U6_gaa(T16, T42, T43, s2lA_out_ga(T16, T41)) → U7_gaa(T16, T42, T43, appendD_in_agaa(X93, T41, T42, X92))
appendD_in_agaa(X117, T57, T58, .(X117, X118)) → U4_agaa(X117, T57, T58, X118, appendB_in_gaa(T57, T58, X118))
appendB_in_gaa([], T65, .(T65, [])) → appendB_out_gaa([], T65, .(T65, []))
appendB_in_gaa(.(T72, T75), T76, .(T72, X140)) → U2_gaa(T72, T75, T76, X140, appendB_in_gaa(T75, T76, X140))
U2_gaa(T72, T75, T76, X140, appendB_out_gaa(T75, T76, X140)) → appendB_out_gaa(.(T72, T75), T76, .(T72, X140))
U4_agaa(X117, T57, T58, X118, appendB_out_gaa(T57, T58, X118)) → appendD_out_agaa(X117, T57, T58, .(X117, X118))
U7_gaa(T16, T42, T43, appendD_out_agaa(X93, T41, T42, X92)) → goalF_out_gaa(s(T16), T42, T43)
goalF_in_gaa(s(T16), T42, T46) → U8_gaa(T16, T42, T46, s2lA_in_ga(T16, T41))
U8_gaa(T16, T42, T46, s2lA_out_ga(T16, T41)) → U9_gaa(T16, T42, T46, appendD_in_agaa(T44, T41, T42, T45))
U9_gaa(T16, T42, T46, appendD_out_agaa(T44, T41, T42, T45)) → U10_gaa(T16, T42, T46, lastC_in_ag(T46, T45))
lastC_in_ag(T85, .(T85, [])) → lastC_out_ag(T85, .(T85, []))
lastC_in_ag(T95, .(T93, T96)) → U3_ag(T95, T93, T96, lastC_in_ag(T95, T96))
U3_ag(T95, T93, T96, lastC_out_ag(T95, T96)) → lastC_out_ag(T95, .(T93, T96))
U10_gaa(T16, T42, T46, lastC_out_ag(T46, T45)) → goalF_out_gaa(s(T16), T42, T46)
goalF_in_gaa(0, T111, T112) → U11_gaa(T111, T112, appendE_in_aa(T111, X177))
appendE_in_aa(T122, .(T122, [])) → appendE_out_aa(T122, .(T122, []))
U11_gaa(T111, T112, appendE_out_aa(T111, X177)) → goalF_out_gaa(0, T111, T112)
goalF_in_gaa(0, T111, T116) → U12_gaa(T111, T116, appendE_in_aa(T111, T115))
U12_gaa(T111, T116, appendE_out_aa(T111, T115)) → U13_gaa(T111, T116, lastC_in_ag(T116, T115))
U13_gaa(T111, T116, lastC_out_ag(T116, T115)) → goalF_out_gaa(0, T111, T116)

The argument filtering Pi contains the following mapping:
goalF_in_gaa(x1, x2, x3)  =  goalF_in_gaa(x1)
s(x1)  =  s(x1)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x4)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
0  =  0
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
goalF_out_gaa(x1, x2, x3)  =  goalF_out_gaa(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
appendD_in_agaa(x1, x2, x3, x4)  =  appendD_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
appendB_in_gaa(x1, x2, x3)  =  appendB_in_gaa(x1)
[]  =  []
appendB_out_gaa(x1, x2, x3)  =  appendB_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
appendD_out_agaa(x1, x2, x3, x4)  =  appendD_out_agaa(x2, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
lastC_in_ag(x1, x2)  =  lastC_in_ag(x2)
lastC_out_ag(x1, x2)  =  lastC_out_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
U11_gaa(x1, x2, x3)  =  U11_gaa(x3)
appendE_in_aa(x1, x2)  =  appendE_in_aa
appendE_out_aa(x1, x2)  =  appendE_out_aa(x2)
U12_gaa(x1, x2, x3)  =  U12_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
GOALF_IN_GAA(x1, x2, x3)  =  GOALF_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x4)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
APPENDD_IN_AGAA(x1, x2, x3, x4)  =  APPENDD_IN_AGAA(x2)
U4_AGAA(x1, x2, x3, x4, x5)  =  U4_AGAA(x2, x5)
APPENDB_IN_GAA(x1, x2, x3)  =  APPENDB_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
LASTC_IN_AG(x1, x2)  =  LASTC_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U11_GAA(x1, x2, x3)  =  U11_GAA(x3)
APPENDE_IN_AA(x1, x2)  =  APPENDE_IN_AA
U12_GAA(x1, x2, x3)  =  U12_GAA(x3)
U13_GAA(x1, x2, x3)  =  U13_GAA(x3)

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

(6) Obligation:

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

GOALF_IN_GAA(s(T16), T10, T11) → U5_GAA(T16, T10, T11, s2lA_in_ga(T16, X32))
GOALF_IN_GAA(s(T16), T10, T11) → S2LA_IN_GA(T16, X32)
S2LA_IN_GA(s(T24), .(X66, X67)) → U1_GA(T24, X66, X67, s2lA_in_ga(T24, X67))
S2LA_IN_GA(s(T24), .(X66, X67)) → S2LA_IN_GA(T24, X67)
GOALF_IN_GAA(s(T16), T42, T43) → U6_GAA(T16, T42, T43, s2lA_in_ga(T16, T41))
U6_GAA(T16, T42, T43, s2lA_out_ga(T16, T41)) → U7_GAA(T16, T42, T43, appendD_in_agaa(X93, T41, T42, X92))
U6_GAA(T16, T42, T43, s2lA_out_ga(T16, T41)) → APPENDD_IN_AGAA(X93, T41, T42, X92)
APPENDD_IN_AGAA(X117, T57, T58, .(X117, X118)) → U4_AGAA(X117, T57, T58, X118, appendB_in_gaa(T57, T58, X118))
APPENDD_IN_AGAA(X117, T57, T58, .(X117, X118)) → APPENDB_IN_GAA(T57, T58, X118)
APPENDB_IN_GAA(.(T72, T75), T76, .(T72, X140)) → U2_GAA(T72, T75, T76, X140, appendB_in_gaa(T75, T76, X140))
APPENDB_IN_GAA(.(T72, T75), T76, .(T72, X140)) → APPENDB_IN_GAA(T75, T76, X140)
GOALF_IN_GAA(s(T16), T42, T46) → U8_GAA(T16, T42, T46, s2lA_in_ga(T16, T41))
U8_GAA(T16, T42, T46, s2lA_out_ga(T16, T41)) → U9_GAA(T16, T42, T46, appendD_in_agaa(T44, T41, T42, T45))
U8_GAA(T16, T42, T46, s2lA_out_ga(T16, T41)) → APPENDD_IN_AGAA(T44, T41, T42, T45)
U9_GAA(T16, T42, T46, appendD_out_agaa(T44, T41, T42, T45)) → U10_GAA(T16, T42, T46, lastC_in_ag(T46, T45))
U9_GAA(T16, T42, T46, appendD_out_agaa(T44, T41, T42, T45)) → LASTC_IN_AG(T46, T45)
LASTC_IN_AG(T95, .(T93, T96)) → U3_AG(T95, T93, T96, lastC_in_ag(T95, T96))
LASTC_IN_AG(T95, .(T93, T96)) → LASTC_IN_AG(T95, T96)
GOALF_IN_GAA(0, T111, T112) → U11_GAA(T111, T112, appendE_in_aa(T111, X177))
GOALF_IN_GAA(0, T111, T112) → APPENDE_IN_AA(T111, X177)
GOALF_IN_GAA(0, T111, T116) → U12_GAA(T111, T116, appendE_in_aa(T111, T115))
U12_GAA(T111, T116, appendE_out_aa(T111, T115)) → U13_GAA(T111, T116, lastC_in_ag(T116, T115))
U12_GAA(T111, T116, appendE_out_aa(T111, T115)) → LASTC_IN_AG(T116, T115)

The TRS R consists of the following rules:

goalF_in_gaa(s(T16), T10, T11) → U5_gaa(T16, T10, T11, s2lA_in_ga(T16, X32))
s2lA_in_ga(s(T24), .(X66, X67)) → U1_ga(T24, X66, X67, s2lA_in_ga(T24, X67))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
U1_ga(T24, X66, X67, s2lA_out_ga(T24, X67)) → s2lA_out_ga(s(T24), .(X66, X67))
U5_gaa(T16, T10, T11, s2lA_out_ga(T16, X32)) → goalF_out_gaa(s(T16), T10, T11)
goalF_in_gaa(s(T16), T42, T43) → U6_gaa(T16, T42, T43, s2lA_in_ga(T16, T41))
U6_gaa(T16, T42, T43, s2lA_out_ga(T16, T41)) → U7_gaa(T16, T42, T43, appendD_in_agaa(X93, T41, T42, X92))
appendD_in_agaa(X117, T57, T58, .(X117, X118)) → U4_agaa(X117, T57, T58, X118, appendB_in_gaa(T57, T58, X118))
appendB_in_gaa([], T65, .(T65, [])) → appendB_out_gaa([], T65, .(T65, []))
appendB_in_gaa(.(T72, T75), T76, .(T72, X140)) → U2_gaa(T72, T75, T76, X140, appendB_in_gaa(T75, T76, X140))
U2_gaa(T72, T75, T76, X140, appendB_out_gaa(T75, T76, X140)) → appendB_out_gaa(.(T72, T75), T76, .(T72, X140))
U4_agaa(X117, T57, T58, X118, appendB_out_gaa(T57, T58, X118)) → appendD_out_agaa(X117, T57, T58, .(X117, X118))
U7_gaa(T16, T42, T43, appendD_out_agaa(X93, T41, T42, X92)) → goalF_out_gaa(s(T16), T42, T43)
goalF_in_gaa(s(T16), T42, T46) → U8_gaa(T16, T42, T46, s2lA_in_ga(T16, T41))
U8_gaa(T16, T42, T46, s2lA_out_ga(T16, T41)) → U9_gaa(T16, T42, T46, appendD_in_agaa(T44, T41, T42, T45))
U9_gaa(T16, T42, T46, appendD_out_agaa(T44, T41, T42, T45)) → U10_gaa(T16, T42, T46, lastC_in_ag(T46, T45))
lastC_in_ag(T85, .(T85, [])) → lastC_out_ag(T85, .(T85, []))
lastC_in_ag(T95, .(T93, T96)) → U3_ag(T95, T93, T96, lastC_in_ag(T95, T96))
U3_ag(T95, T93, T96, lastC_out_ag(T95, T96)) → lastC_out_ag(T95, .(T93, T96))
U10_gaa(T16, T42, T46, lastC_out_ag(T46, T45)) → goalF_out_gaa(s(T16), T42, T46)
goalF_in_gaa(0, T111, T112) → U11_gaa(T111, T112, appendE_in_aa(T111, X177))
appendE_in_aa(T122, .(T122, [])) → appendE_out_aa(T122, .(T122, []))
U11_gaa(T111, T112, appendE_out_aa(T111, X177)) → goalF_out_gaa(0, T111, T112)
goalF_in_gaa(0, T111, T116) → U12_gaa(T111, T116, appendE_in_aa(T111, T115))
U12_gaa(T111, T116, appendE_out_aa(T111, T115)) → U13_gaa(T111, T116, lastC_in_ag(T116, T115))
U13_gaa(T111, T116, lastC_out_ag(T116, T115)) → goalF_out_gaa(0, T111, T116)

The argument filtering Pi contains the following mapping:
goalF_in_gaa(x1, x2, x3)  =  goalF_in_gaa(x1)
s(x1)  =  s(x1)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x4)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
0  =  0
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
goalF_out_gaa(x1, x2, x3)  =  goalF_out_gaa(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
appendD_in_agaa(x1, x2, x3, x4)  =  appendD_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
appendB_in_gaa(x1, x2, x3)  =  appendB_in_gaa(x1)
[]  =  []
appendB_out_gaa(x1, x2, x3)  =  appendB_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
appendD_out_agaa(x1, x2, x3, x4)  =  appendD_out_agaa(x2, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
lastC_in_ag(x1, x2)  =  lastC_in_ag(x2)
lastC_out_ag(x1, x2)  =  lastC_out_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
U11_gaa(x1, x2, x3)  =  U11_gaa(x3)
appendE_in_aa(x1, x2)  =  appendE_in_aa
appendE_out_aa(x1, x2)  =  appendE_out_aa(x2)
U12_gaa(x1, x2, x3)  =  U12_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
GOALF_IN_GAA(x1, x2, x3)  =  GOALF_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x4)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
APPENDD_IN_AGAA(x1, x2, x3, x4)  =  APPENDD_IN_AGAA(x2)
U4_AGAA(x1, x2, x3, x4, x5)  =  U4_AGAA(x2, x5)
APPENDB_IN_GAA(x1, x2, x3)  =  APPENDB_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
LASTC_IN_AG(x1, x2)  =  LASTC_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U11_GAA(x1, x2, x3)  =  U11_GAA(x3)
APPENDE_IN_AA(x1, x2)  =  APPENDE_IN_AA
U12_GAA(x1, x2, x3)  =  U12_GAA(x3)
U13_GAA(x1, x2, x3)  =  U13_GAA(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 20 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LASTC_IN_AG(T95, .(T93, T96)) → LASTC_IN_AG(T95, T96)

The TRS R consists of the following rules:

goalF_in_gaa(s(T16), T10, T11) → U5_gaa(T16, T10, T11, s2lA_in_ga(T16, X32))
s2lA_in_ga(s(T24), .(X66, X67)) → U1_ga(T24, X66, X67, s2lA_in_ga(T24, X67))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
U1_ga(T24, X66, X67, s2lA_out_ga(T24, X67)) → s2lA_out_ga(s(T24), .(X66, X67))
U5_gaa(T16, T10, T11, s2lA_out_ga(T16, X32)) → goalF_out_gaa(s(T16), T10, T11)
goalF_in_gaa(s(T16), T42, T43) → U6_gaa(T16, T42, T43, s2lA_in_ga(T16, T41))
U6_gaa(T16, T42, T43, s2lA_out_ga(T16, T41)) → U7_gaa(T16, T42, T43, appendD_in_agaa(X93, T41, T42, X92))
appendD_in_agaa(X117, T57, T58, .(X117, X118)) → U4_agaa(X117, T57, T58, X118, appendB_in_gaa(T57, T58, X118))
appendB_in_gaa([], T65, .(T65, [])) → appendB_out_gaa([], T65, .(T65, []))
appendB_in_gaa(.(T72, T75), T76, .(T72, X140)) → U2_gaa(T72, T75, T76, X140, appendB_in_gaa(T75, T76, X140))
U2_gaa(T72, T75, T76, X140, appendB_out_gaa(T75, T76, X140)) → appendB_out_gaa(.(T72, T75), T76, .(T72, X140))
U4_agaa(X117, T57, T58, X118, appendB_out_gaa(T57, T58, X118)) → appendD_out_agaa(X117, T57, T58, .(X117, X118))
U7_gaa(T16, T42, T43, appendD_out_agaa(X93, T41, T42, X92)) → goalF_out_gaa(s(T16), T42, T43)
goalF_in_gaa(s(T16), T42, T46) → U8_gaa(T16, T42, T46, s2lA_in_ga(T16, T41))
U8_gaa(T16, T42, T46, s2lA_out_ga(T16, T41)) → U9_gaa(T16, T42, T46, appendD_in_agaa(T44, T41, T42, T45))
U9_gaa(T16, T42, T46, appendD_out_agaa(T44, T41, T42, T45)) → U10_gaa(T16, T42, T46, lastC_in_ag(T46, T45))
lastC_in_ag(T85, .(T85, [])) → lastC_out_ag(T85, .(T85, []))
lastC_in_ag(T95, .(T93, T96)) → U3_ag(T95, T93, T96, lastC_in_ag(T95, T96))
U3_ag(T95, T93, T96, lastC_out_ag(T95, T96)) → lastC_out_ag(T95, .(T93, T96))
U10_gaa(T16, T42, T46, lastC_out_ag(T46, T45)) → goalF_out_gaa(s(T16), T42, T46)
goalF_in_gaa(0, T111, T112) → U11_gaa(T111, T112, appendE_in_aa(T111, X177))
appendE_in_aa(T122, .(T122, [])) → appendE_out_aa(T122, .(T122, []))
U11_gaa(T111, T112, appendE_out_aa(T111, X177)) → goalF_out_gaa(0, T111, T112)
goalF_in_gaa(0, T111, T116) → U12_gaa(T111, T116, appendE_in_aa(T111, T115))
U12_gaa(T111, T116, appendE_out_aa(T111, T115)) → U13_gaa(T111, T116, lastC_in_ag(T116, T115))
U13_gaa(T111, T116, lastC_out_ag(T116, T115)) → goalF_out_gaa(0, T111, T116)

The argument filtering Pi contains the following mapping:
goalF_in_gaa(x1, x2, x3)  =  goalF_in_gaa(x1)
s(x1)  =  s(x1)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x4)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
0  =  0
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
goalF_out_gaa(x1, x2, x3)  =  goalF_out_gaa(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
appendD_in_agaa(x1, x2, x3, x4)  =  appendD_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
appendB_in_gaa(x1, x2, x3)  =  appendB_in_gaa(x1)
[]  =  []
appendB_out_gaa(x1, x2, x3)  =  appendB_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
appendD_out_agaa(x1, x2, x3, x4)  =  appendD_out_agaa(x2, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
lastC_in_ag(x1, x2)  =  lastC_in_ag(x2)
lastC_out_ag(x1, x2)  =  lastC_out_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
U11_gaa(x1, x2, x3)  =  U11_gaa(x3)
appendE_in_aa(x1, x2)  =  appendE_in_aa
appendE_out_aa(x1, x2)  =  appendE_out_aa(x2)
U12_gaa(x1, x2, x3)  =  U12_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
LASTC_IN_AG(x1, x2)  =  LASTC_IN_AG(x2)

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

(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:

LASTC_IN_AG(T95, .(T93, T96)) → LASTC_IN_AG(T95, T96)

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

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:

LASTC_IN_AG(.(T96)) → LASTC_IN_AG(T96)

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

(14) 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:

• LASTC_IN_AG(.(T96)) → LASTC_IN_AG(T96)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

APPENDB_IN_GAA(.(T72, T75), T76, .(T72, X140)) → APPENDB_IN_GAA(T75, T76, X140)

The TRS R consists of the following rules:

goalF_in_gaa(s(T16), T10, T11) → U5_gaa(T16, T10, T11, s2lA_in_ga(T16, X32))
s2lA_in_ga(s(T24), .(X66, X67)) → U1_ga(T24, X66, X67, s2lA_in_ga(T24, X67))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
U1_ga(T24, X66, X67, s2lA_out_ga(T24, X67)) → s2lA_out_ga(s(T24), .(X66, X67))
U5_gaa(T16, T10, T11, s2lA_out_ga(T16, X32)) → goalF_out_gaa(s(T16), T10, T11)
goalF_in_gaa(s(T16), T42, T43) → U6_gaa(T16, T42, T43, s2lA_in_ga(T16, T41))
U6_gaa(T16, T42, T43, s2lA_out_ga(T16, T41)) → U7_gaa(T16, T42, T43, appendD_in_agaa(X93, T41, T42, X92))
appendD_in_agaa(X117, T57, T58, .(X117, X118)) → U4_agaa(X117, T57, T58, X118, appendB_in_gaa(T57, T58, X118))
appendB_in_gaa([], T65, .(T65, [])) → appendB_out_gaa([], T65, .(T65, []))
appendB_in_gaa(.(T72, T75), T76, .(T72, X140)) → U2_gaa(T72, T75, T76, X140, appendB_in_gaa(T75, T76, X140))
U2_gaa(T72, T75, T76, X140, appendB_out_gaa(T75, T76, X140)) → appendB_out_gaa(.(T72, T75), T76, .(T72, X140))
U4_agaa(X117, T57, T58, X118, appendB_out_gaa(T57, T58, X118)) → appendD_out_agaa(X117, T57, T58, .(X117, X118))
U7_gaa(T16, T42, T43, appendD_out_agaa(X93, T41, T42, X92)) → goalF_out_gaa(s(T16), T42, T43)
goalF_in_gaa(s(T16), T42, T46) → U8_gaa(T16, T42, T46, s2lA_in_ga(T16, T41))
U8_gaa(T16, T42, T46, s2lA_out_ga(T16, T41)) → U9_gaa(T16, T42, T46, appendD_in_agaa(T44, T41, T42, T45))
U9_gaa(T16, T42, T46, appendD_out_agaa(T44, T41, T42, T45)) → U10_gaa(T16, T42, T46, lastC_in_ag(T46, T45))
lastC_in_ag(T85, .(T85, [])) → lastC_out_ag(T85, .(T85, []))
lastC_in_ag(T95, .(T93, T96)) → U3_ag(T95, T93, T96, lastC_in_ag(T95, T96))
U3_ag(T95, T93, T96, lastC_out_ag(T95, T96)) → lastC_out_ag(T95, .(T93, T96))
U10_gaa(T16, T42, T46, lastC_out_ag(T46, T45)) → goalF_out_gaa(s(T16), T42, T46)
goalF_in_gaa(0, T111, T112) → U11_gaa(T111, T112, appendE_in_aa(T111, X177))
appendE_in_aa(T122, .(T122, [])) → appendE_out_aa(T122, .(T122, []))
U11_gaa(T111, T112, appendE_out_aa(T111, X177)) → goalF_out_gaa(0, T111, T112)
goalF_in_gaa(0, T111, T116) → U12_gaa(T111, T116, appendE_in_aa(T111, T115))
U12_gaa(T111, T116, appendE_out_aa(T111, T115)) → U13_gaa(T111, T116, lastC_in_ag(T116, T115))
U13_gaa(T111, T116, lastC_out_ag(T116, T115)) → goalF_out_gaa(0, T111, T116)

The argument filtering Pi contains the following mapping:
goalF_in_gaa(x1, x2, x3)  =  goalF_in_gaa(x1)
s(x1)  =  s(x1)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x4)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
0  =  0
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
goalF_out_gaa(x1, x2, x3)  =  goalF_out_gaa(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
appendD_in_agaa(x1, x2, x3, x4)  =  appendD_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
appendB_in_gaa(x1, x2, x3)  =  appendB_in_gaa(x1)
[]  =  []
appendB_out_gaa(x1, x2, x3)  =  appendB_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
appendD_out_agaa(x1, x2, x3, x4)  =  appendD_out_agaa(x2, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
lastC_in_ag(x1, x2)  =  lastC_in_ag(x2)
lastC_out_ag(x1, x2)  =  lastC_out_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
U11_gaa(x1, x2, x3)  =  U11_gaa(x3)
appendE_in_aa(x1, x2)  =  appendE_in_aa
appendE_out_aa(x1, x2)  =  appendE_out_aa(x2)
U12_gaa(x1, x2, x3)  =  U12_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
APPENDB_IN_GAA(x1, x2, x3)  =  APPENDB_IN_GAA(x1)

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

(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:

APPENDB_IN_GAA(.(T72, T75), T76, .(T72, X140)) → APPENDB_IN_GAA(T75, T76, X140)

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

APPENDB_IN_GAA(.(T75)) → APPENDB_IN_GAA(T75)

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:

• APPENDB_IN_GAA(.(T75)) → APPENDB_IN_GAA(T75)
  The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

S2LA_IN_GA(s(T24), .(X66, X67)) → S2LA_IN_GA(T24, X67)

The TRS R consists of the following rules:

goalF_in_gaa(s(T16), T10, T11) → U5_gaa(T16, T10, T11, s2lA_in_ga(T16, X32))
s2lA_in_ga(s(T24), .(X66, X67)) → U1_ga(T24, X66, X67, s2lA_in_ga(T24, X67))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
U1_ga(T24, X66, X67, s2lA_out_ga(T24, X67)) → s2lA_out_ga(s(T24), .(X66, X67))
U5_gaa(T16, T10, T11, s2lA_out_ga(T16, X32)) → goalF_out_gaa(s(T16), T10, T11)
goalF_in_gaa(s(T16), T42, T43) → U6_gaa(T16, T42, T43, s2lA_in_ga(T16, T41))
U6_gaa(T16, T42, T43, s2lA_out_ga(T16, T41)) → U7_gaa(T16, T42, T43, appendD_in_agaa(X93, T41, T42, X92))
appendD_in_agaa(X117, T57, T58, .(X117, X118)) → U4_agaa(X117, T57, T58, X118, appendB_in_gaa(T57, T58, X118))
appendB_in_gaa([], T65, .(T65, [])) → appendB_out_gaa([], T65, .(T65, []))
appendB_in_gaa(.(T72, T75), T76, .(T72, X140)) → U2_gaa(T72, T75, T76, X140, appendB_in_gaa(T75, T76, X140))
U2_gaa(T72, T75, T76, X140, appendB_out_gaa(T75, T76, X140)) → appendB_out_gaa(.(T72, T75), T76, .(T72, X140))
U4_agaa(X117, T57, T58, X118, appendB_out_gaa(T57, T58, X118)) → appendD_out_agaa(X117, T57, T58, .(X117, X118))
U7_gaa(T16, T42, T43, appendD_out_agaa(X93, T41, T42, X92)) → goalF_out_gaa(s(T16), T42, T43)
goalF_in_gaa(s(T16), T42, T46) → U8_gaa(T16, T42, T46, s2lA_in_ga(T16, T41))
U8_gaa(T16, T42, T46, s2lA_out_ga(T16, T41)) → U9_gaa(T16, T42, T46, appendD_in_agaa(T44, T41, T42, T45))
U9_gaa(T16, T42, T46, appendD_out_agaa(T44, T41, T42, T45)) → U10_gaa(T16, T42, T46, lastC_in_ag(T46, T45))
lastC_in_ag(T85, .(T85, [])) → lastC_out_ag(T85, .(T85, []))
lastC_in_ag(T95, .(T93, T96)) → U3_ag(T95, T93, T96, lastC_in_ag(T95, T96))
U3_ag(T95, T93, T96, lastC_out_ag(T95, T96)) → lastC_out_ag(T95, .(T93, T96))
U10_gaa(T16, T42, T46, lastC_out_ag(T46, T45)) → goalF_out_gaa(s(T16), T42, T46)
goalF_in_gaa(0, T111, T112) → U11_gaa(T111, T112, appendE_in_aa(T111, X177))
appendE_in_aa(T122, .(T122, [])) → appendE_out_aa(T122, .(T122, []))
U11_gaa(T111, T112, appendE_out_aa(T111, X177)) → goalF_out_gaa(0, T111, T112)
goalF_in_gaa(0, T111, T116) → U12_gaa(T111, T116, appendE_in_aa(T111, T115))
U12_gaa(T111, T116, appendE_out_aa(T111, T115)) → U13_gaa(T111, T116, lastC_in_ag(T116, T115))
U13_gaa(T111, T116, lastC_out_ag(T116, T115)) → goalF_out_gaa(0, T111, T116)

The argument filtering Pi contains the following mapping:
goalF_in_gaa(x1, x2, x3)  =  goalF_in_gaa(x1)
s(x1)  =  s(x1)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x4)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
0  =  0
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
goalF_out_gaa(x1, x2, x3)  =  goalF_out_gaa(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
appendD_in_agaa(x1, x2, x3, x4)  =  appendD_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
appendB_in_gaa(x1, x2, x3)  =  appendB_in_gaa(x1)
[]  =  []
appendB_out_gaa(x1, x2, x3)  =  appendB_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
appendD_out_agaa(x1, x2, x3, x4)  =  appendD_out_agaa(x2, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
lastC_in_ag(x1, x2)  =  lastC_in_ag(x2)
lastC_out_ag(x1, x2)  =  lastC_out_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
U11_gaa(x1, x2, x3)  =  U11_gaa(x3)
appendE_in_aa(x1, x2)  =  appendE_in_aa
appendE_out_aa(x1, x2)  =  appendE_out_aa(x2)
U12_gaa(x1, x2, x3)  =  U12_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)

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

(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:

S2LA_IN_GA(s(T24), .(X66, X67)) → S2LA_IN_GA(T24, X67)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)

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:

S2LA_IN_GA(s(T24)) → S2LA_IN_GA(T24)

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

(28) 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:

• S2LA_IN_GA(s(T24)) → S2LA_IN_GA(T24)
  The graph contains the following edges 1 > 1

(29) YES