Left Termination of the query pattern goal_in_3(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, []).

Queries:

goal(g,a,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

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, X70)\n\nT24 is ground\nX70 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(.(X105, T43), .(T44, []), X104), last(T45, X104))\n\nX105 is free\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: append(.(X105, T43), .(T44, []), X104)\n\nX105 is free\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: last(T50, T49)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: append(.(X105, T43), .(T44, []), X104), SCOPE: 5, CLAUSE: 4 | append(.(X105, T43), .(T44, []), X104), SCOPE: 5, CLAUSE: 5\n\nX105 is free\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: append(.(X105, T43), .(T44, []), X104), SCOPE: 5, CLAUSE: 5\n\nX105 is free\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: append(T63, .(T64, []), X142)\n\nX142 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: append(T63, .(T64, []), X142), SCOPE: 6, CLAUSE: 4 | append(T63, .(T64, []), X142), SCOPE: 6, CLAUSE: 5\n\nX142 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: append(T63, .(T64, []), X142), SCOPE: 6, CLAUSE: 4\n\nX142 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: append(T63, .(T64, []), X142), SCOPE: 6, CLAUSE: 5\n\nX142 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(T81, .(T82, []), X165)\n\nX165 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: last(T50, T49), SCOPE: 7, CLAUSE: 2 | last(T50, T49), SCOPE: 7, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: last(T50, T49), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: last(T50, T49), 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(T101, T102)\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([], .(T117, []), X210), last(T118, X210))\n\nX210 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: append([], .(T117, []), X210)\n\nX210 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: last(T122, T121)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: append([], .(T117, []), X210), SCOPE: 9, CLAUSE: 4 | append([], .(T117, []), X210), SCOPE: 9, CLAUSE: 5\n\nX210 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: append([], .(T117, []), X210), SCOPE: 9, CLAUSE: 4\n\nX210 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: append([], .(T117, []), X210), SCOPE: 9, CLAUSE: 5\n\nX210 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 substitution\nT1 -> T7\nX4 -> T7\nT2 -> T10\nX5 -> T10\nT3 -> T11\nX6 -> T11\nT8 -> T10\nT9 -> T11", fontsize=14, style = filled, fillcolor = white];
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 substitution\nX28 -> 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 = lightblue];
5 -> 60 [arrowhead = none ];
60 -> 8;

61 [label="EVAL with clause\ns2l(0, []).\nand substitution\nT7 -> 0\nX7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 61 [arrowhead = none ];
61 -> 42;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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\nreplacements:\nX32 -> 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(X66), .(X67, X68)) :- s2l(X66, X68).\nand substitution\nX66 -> T24\nT16 -> s(T24)\nX67 -> X69\nX68 -> X70\nX32 -> .(X69, X70)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 69 [arrowhead = none ];
69 -> 14;

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

71 [label="EVAL with clause\ns2l(0, []).\nand substitution\nT16 -> 0\nX32 -> []", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 71 [arrowhead = none ];
71 -> 16;

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

73 [label="INSTANCE with matching:\nT16 -> T24\nX32 -> X70", 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(X101, X102, X103) :- ','(append(X101, .(X102, []), X104), last(X103, X104)).\nand substitution\nX31 -> X105\nT18 -> T43\nX101 -> .(X105, T43)\nT10 -> T44\nX102 -> T44\nT11 -> T45\nX103 -> T45\nT40 -> T43\nT41 -> T44\nT42 -> T45", fontsize=14, style = filled, fillcolor = white];
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:\nX105 -> T48\nX104 -> T49\nT45 -> T50", 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 = white];
23 -> 80 [arrowhead = none ];
80 -> 24;

81 [label="ONLY EVAL with clause\nappend(.(X137, X138), X139, .(X137, X140)) :- append(X138, X139, X140).\nand substitution\nX105 -> X141\nX137 -> X141\nT43 -> T63\nX138 -> T63\nT44 -> T64\nX139 -> .(T64, [])\nX140 -> X142\nX104 -> .(X141, X142)\nT61 -> T63\nT62 -> T64", fontsize=14, style = filled, fillcolor = white];
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([], X149, X149).\nand substitution\nT63 -> []\nT64 -> T71\nX149 -> .(T71, [])\nX142 -> .(T71, [])", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 85 [arrowhead = none ];
85 -> 29;

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

87 [label="EVAL with clause\nappend(.(X161, X162), X163, .(X161, X164)) :- append(X162, X163, X164).\nand substitution\nX161 -> T78\nX162 -> T81\nT63 -> .(T78, T81)\nT64 -> T82\nX163 -> .(T82, [])\nX164 -> X165\nX142 -> .(T78, X165)\nT79 -> T81\nT80 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 87 [arrowhead = none ];
87 -> 32;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT63 -> T81\nT64 -> T82\nX142 -> X165", 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(X178, .(X178, [])).\nand substitution\nT50 -> T91\nX178 -> T91\nT49 -> .(T91, [])", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 93 [arrowhead = none ];
93 -> 37;

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

95 [label="EVAL with clause\nlast(X186, .(X187, X188)) :- last(X186, X188).\nand substitution\nT50 -> T101\nX186 -> T101\nX187 -> T99\nX188 -> T102\nT49 -> .(T99, T102)\nT98 -> T101\nT100 -> T102", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 95 [arrowhead = none ];
95 -> 40;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT50 -> T101\nT49 -> T102", 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(X207, X208, X209) :- ','(append(X207, .(X208, []), X210), last(X209, X210)).\nand substitution\nX207 -> []\nT10 -> T117\nX208 -> T117\nT11 -> T118\nX209 -> T118\nT115 -> T117\nT116 -> T118", fontsize=14, style = filled, fillcolor = white];
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:\nX210 -> T121\nT118 -> T122", 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:\nT50 -> T122\nT49 -> T121", 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([], X218, X218).\nand substitution\nT117 -> T128\nX218 -> .(T128, [])\nX210 -> .(T128, [])", fontsize=14, style = filled, fillcolor = white];
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 = white];
50 -> 108 [arrowhead = none ];
108 -> 53;

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

}

(2) Obligation:

Triples:

s2l9(s(T24), .(X69, X70)) :- s2l9(T24, X70).
append25(.(T78, T81), T82, .(T78, X165)) :- append25(T81, T82, X165).
last22(T101, .(T99, T102)) :- last22(T101, T102).
goal1(s(T16), T10, T11) :- s2l9(T16, X32).
goal1(s(T16), T64, T45) :- ','(s2lc9(T16, T63), append25(T63, T64, X142)).
goal1(s(T16), T44, T50) :- ','(s2lc9(T16, T43), ','(appendc21(T48, T43, T44, T49), last22(T50, T49))).
goal1(0, T117, T122) :- ','(appendc46(T117, T121), last22(T122, T121)).

Clauses:

s2lc9(s(T24), .(X69, X70)) :- s2lc9(T24, X70).
s2lc9(0, []).
appendc25([], T71, .(T71, [])).
appendc25(.(T78, T81), T82, .(T78, X165)) :- appendc25(T81, T82, X165).
lastc22(T91, .(T91, [])).
lastc22(T101, .(T99, T102)) :- lastc22(T101, T102).
appendc21(X141, T63, T64, .(X141, X142)) :- appendc25(T63, T64, X142).
appendc46(T128, .(T128, [])).

Afs:

goal1(x1, x2, x3)  =  goal1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal1_in: (b,f,f)
s2l9_in: (b,f)
s2lc9_in: (b,f)
append25_in: (b,f,f)
appendc21_in: (f,b,f,f)
appendc25_in: (b,f,f)
last22_in: (f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_GAA(s(T16), T10, T11) → U4_GAA(T16, T10, T11, s2l9_in_ga(T16, X32))
GOAL1_IN_GAA(s(T16), T10, T11) → S2L9_IN_GA(T16, X32)
S2L9_IN_GA(s(T24), .(X69, X70)) → U1_GA(T24, X69, X70, s2l9_in_ga(T24, X70))
S2L9_IN_GA(s(T24), .(X69, X70)) → S2L9_IN_GA(T24, X70)
GOAL1_IN_GAA(s(T16), T64, T45) → U5_GAA(T16, T64, T45, s2lc9_in_ga(T16, T63))
U5_GAA(T16, T64, T45, s2lc9_out_ga(T16, T63)) → U6_GAA(T16, T64, T45, append25_in_gaa(T63, T64, X142))
U5_GAA(T16, T64, T45, s2lc9_out_ga(T16, T63)) → APPEND25_IN_GAA(T63, T64, X142)
APPEND25_IN_GAA(.(T78, T81), T82, .(T78, X165)) → U2_GAA(T78, T81, T82, X165, append25_in_gaa(T81, T82, X165))
APPEND25_IN_GAA(.(T78, T81), T82, .(T78, X165)) → APPEND25_IN_GAA(T81, T82, X165)
GOAL1_IN_GAA(s(T16), T44, T50) → U7_GAA(T16, T44, T50, s2lc9_in_ga(T16, T43))
U7_GAA(T16, T44, T50, s2lc9_out_ga(T16, T43)) → U8_GAA(T16, T44, T50, appendc21_in_agaa(T48, T43, T44, T49))
U8_GAA(T16, T44, T50, appendc21_out_agaa(T48, T43, T44, T49)) → U9_GAA(T16, T44, T50, last22_in_ag(T50, T49))
U8_GAA(T16, T44, T50, appendc21_out_agaa(T48, T43, T44, T49)) → LAST22_IN_AG(T50, T49)
LAST22_IN_AG(T101, .(T99, T102)) → U3_AG(T101, T99, T102, last22_in_ag(T101, T102))
LAST22_IN_AG(T101, .(T99, T102)) → LAST22_IN_AG(T101, T102)
GOAL1_IN_GAA(0, T117, T122) → U10_GAA(T117, T122, appendc46_in_aa(T117, T121))
U10_GAA(T117, T122, appendc46_out_aa(T117, T121)) → U11_GAA(T117, T122, last22_in_ag(T122, T121))
U10_GAA(T117, T122, appendc46_out_aa(T117, T121)) → LAST22_IN_AG(T122, T121)

The TRS R consists of the following rules:

s2lc9_in_ga(s(T24), .(X69, X70)) → U13_ga(T24, X69, X70, s2lc9_in_ga(T24, X70))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U13_ga(T24, X69, X70, s2lc9_out_ga(T24, X70)) → s2lc9_out_ga(s(T24), .(X69, X70))
appendc21_in_agaa(X141, T63, T64, .(X141, X142)) → U16_agaa(X141, T63, T64, X142, appendc25_in_gaa(T63, T64, X142))
appendc25_in_gaa([], T71, .(T71, [])) → appendc25_out_gaa([], T71, .(T71, []))
appendc25_in_gaa(.(T78, T81), T82, .(T78, X165)) → U14_gaa(T78, T81, T82, X165, appendc25_in_gaa(T81, T82, X165))
U14_gaa(T78, T81, T82, X165, appendc25_out_gaa(T81, T82, X165)) → appendc25_out_gaa(.(T78, T81), T82, .(T78, X165))
U16_agaa(X141, T63, T64, X142, appendc25_out_gaa(T63, T64, X142)) → appendc21_out_agaa(X141, T63, T64, .(X141, X142))
appendc46_in_aa(T128, .(T128, [])) → appendc46_out_aa(T128, .(T128, []))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
.(x1, x2)  =  .(x2)
s2lc9_in_ga(x1, x2)  =  s2lc9_in_ga(x1)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
0  =  0
s2lc9_out_ga(x1, x2)  =  s2lc9_out_ga(x1, x2)
append25_in_gaa(x1, x2, x3)  =  append25_in_gaa(x1)
appendc21_in_agaa(x1, x2, x3, x4)  =  appendc21_in_agaa(x2)
U16_agaa(x1, x2, x3, x4, x5)  =  U16_agaa(x2, x5)
appendc25_in_gaa(x1, x2, x3)  =  appendc25_in_gaa(x1)
[]  =  []
appendc25_out_gaa(x1, x2, x3)  =  appendc25_out_gaa(x1, x3)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x2, x5)
appendc21_out_agaa(x1, x2, x3, x4)  =  appendc21_out_agaa(x2, x4)
last22_in_ag(x1, x2)  =  last22_in_ag(x2)
appendc46_in_aa(x1, x2)  =  appendc46_in_aa
appendc46_out_aa(x1, x2)  =  appendc46_out_aa(x2)
GOAL1_IN_GAA(x1, x2, x3)  =  GOAL1_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x1, x4)
S2L9_IN_GA(x1, x2)  =  S2L9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x4)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
APPEND25_IN_GAA(x1, x2, x3)  =  APPEND25_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
LAST22_IN_AG(x1, x2)  =  LAST22_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U10_GAA(x1, x2, x3)  =  U10_GAA(x3)
U11_GAA(x1, x2, x3)  =  U11_GAA(x3)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

GOAL1_IN_GAA(s(T16), T10, T11) → U4_GAA(T16, T10, T11, s2l9_in_ga(T16, X32))
GOAL1_IN_GAA(s(T16), T10, T11) → S2L9_IN_GA(T16, X32)
S2L9_IN_GA(s(T24), .(X69, X70)) → U1_GA(T24, X69, X70, s2l9_in_ga(T24, X70))
S2L9_IN_GA(s(T24), .(X69, X70)) → S2L9_IN_GA(T24, X70)
GOAL1_IN_GAA(s(T16), T64, T45) → U5_GAA(T16, T64, T45, s2lc9_in_ga(T16, T63))
U5_GAA(T16, T64, T45, s2lc9_out_ga(T16, T63)) → U6_GAA(T16, T64, T45, append25_in_gaa(T63, T64, X142))
U5_GAA(T16, T64, T45, s2lc9_out_ga(T16, T63)) → APPEND25_IN_GAA(T63, T64, X142)
APPEND25_IN_GAA(.(T78, T81), T82, .(T78, X165)) → U2_GAA(T78, T81, T82, X165, append25_in_gaa(T81, T82, X165))
APPEND25_IN_GAA(.(T78, T81), T82, .(T78, X165)) → APPEND25_IN_GAA(T81, T82, X165)
GOAL1_IN_GAA(s(T16), T44, T50) → U7_GAA(T16, T44, T50, s2lc9_in_ga(T16, T43))
U7_GAA(T16, T44, T50, s2lc9_out_ga(T16, T43)) → U8_GAA(T16, T44, T50, appendc21_in_agaa(T48, T43, T44, T49))
U8_GAA(T16, T44, T50, appendc21_out_agaa(T48, T43, T44, T49)) → U9_GAA(T16, T44, T50, last22_in_ag(T50, T49))
U8_GAA(T16, T44, T50, appendc21_out_agaa(T48, T43, T44, T49)) → LAST22_IN_AG(T50, T49)
LAST22_IN_AG(T101, .(T99, T102)) → U3_AG(T101, T99, T102, last22_in_ag(T101, T102))
LAST22_IN_AG(T101, .(T99, T102)) → LAST22_IN_AG(T101, T102)
GOAL1_IN_GAA(0, T117, T122) → U10_GAA(T117, T122, appendc46_in_aa(T117, T121))
U10_GAA(T117, T122, appendc46_out_aa(T117, T121)) → U11_GAA(T117, T122, last22_in_ag(T122, T121))
U10_GAA(T117, T122, appendc46_out_aa(T117, T121)) → LAST22_IN_AG(T122, T121)

The TRS R consists of the following rules:

s2lc9_in_ga(s(T24), .(X69, X70)) → U13_ga(T24, X69, X70, s2lc9_in_ga(T24, X70))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U13_ga(T24, X69, X70, s2lc9_out_ga(T24, X70)) → s2lc9_out_ga(s(T24), .(X69, X70))
appendc21_in_agaa(X141, T63, T64, .(X141, X142)) → U16_agaa(X141, T63, T64, X142, appendc25_in_gaa(T63, T64, X142))
appendc25_in_gaa([], T71, .(T71, [])) → appendc25_out_gaa([], T71, .(T71, []))
appendc25_in_gaa(.(T78, T81), T82, .(T78, X165)) → U14_gaa(T78, T81, T82, X165, appendc25_in_gaa(T81, T82, X165))
U14_gaa(T78, T81, T82, X165, appendc25_out_gaa(T81, T82, X165)) → appendc25_out_gaa(.(T78, T81), T82, .(T78, X165))
U16_agaa(X141, T63, T64, X142, appendc25_out_gaa(T63, T64, X142)) → appendc21_out_agaa(X141, T63, T64, .(X141, X142))
appendc46_in_aa(T128, .(T128, [])) → appendc46_out_aa(T128, .(T128, []))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
.(x1, x2)  =  .(x2)
s2lc9_in_ga(x1, x2)  =  s2lc9_in_ga(x1)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
0  =  0
s2lc9_out_ga(x1, x2)  =  s2lc9_out_ga(x1, x2)
append25_in_gaa(x1, x2, x3)  =  append25_in_gaa(x1)
appendc21_in_agaa(x1, x2, x3, x4)  =  appendc21_in_agaa(x2)
U16_agaa(x1, x2, x3, x4, x5)  =  U16_agaa(x2, x5)
appendc25_in_gaa(x1, x2, x3)  =  appendc25_in_gaa(x1)
[]  =  []
appendc25_out_gaa(x1, x2, x3)  =  appendc25_out_gaa(x1, x3)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x2, x5)
appendc21_out_agaa(x1, x2, x3, x4)  =  appendc21_out_agaa(x2, x4)
last22_in_ag(x1, x2)  =  last22_in_ag(x2)
appendc46_in_aa(x1, x2)  =  appendc46_in_aa
appendc46_out_aa(x1, x2)  =  appendc46_out_aa(x2)
GOAL1_IN_GAA(x1, x2, x3)  =  GOAL1_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x1, x4)
S2L9_IN_GA(x1, x2)  =  S2L9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x4)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
APPEND25_IN_GAA(x1, x2, x3)  =  APPEND25_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
LAST22_IN_AG(x1, x2)  =  LAST22_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U10_GAA(x1, x2, x3)  =  U10_GAA(x3)
U11_GAA(x1, x2, x3)  =  U11_GAA(x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 15 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LAST22_IN_AG(T101, .(T99, T102)) → LAST22_IN_AG(T101, T102)

The TRS R consists of the following rules:

s2lc9_in_ga(s(T24), .(X69, X70)) → U13_ga(T24, X69, X70, s2lc9_in_ga(T24, X70))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U13_ga(T24, X69, X70, s2lc9_out_ga(T24, X70)) → s2lc9_out_ga(s(T24), .(X69, X70))
appendc21_in_agaa(X141, T63, T64, .(X141, X142)) → U16_agaa(X141, T63, T64, X142, appendc25_in_gaa(T63, T64, X142))
appendc25_in_gaa([], T71, .(T71, [])) → appendc25_out_gaa([], T71, .(T71, []))
appendc25_in_gaa(.(T78, T81), T82, .(T78, X165)) → U14_gaa(T78, T81, T82, X165, appendc25_in_gaa(T81, T82, X165))
U14_gaa(T78, T81, T82, X165, appendc25_out_gaa(T81, T82, X165)) → appendc25_out_gaa(.(T78, T81), T82, .(T78, X165))
U16_agaa(X141, T63, T64, X142, appendc25_out_gaa(T63, T64, X142)) → appendc21_out_agaa(X141, T63, T64, .(X141, X142))
appendc46_in_aa(T128, .(T128, [])) → appendc46_out_aa(T128, .(T128, []))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lc9_in_ga(x1, x2)  =  s2lc9_in_ga(x1)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
0  =  0
s2lc9_out_ga(x1, x2)  =  s2lc9_out_ga(x1, x2)
appendc21_in_agaa(x1, x2, x3, x4)  =  appendc21_in_agaa(x2)
U16_agaa(x1, x2, x3, x4, x5)  =  U16_agaa(x2, x5)
appendc25_in_gaa(x1, x2, x3)  =  appendc25_in_gaa(x1)
[]  =  []
appendc25_out_gaa(x1, x2, x3)  =  appendc25_out_gaa(x1, x3)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x2, x5)
appendc21_out_agaa(x1, x2, x3, x4)  =  appendc21_out_agaa(x2, x4)
appendc46_in_aa(x1, x2)  =  appendc46_in_aa
appendc46_out_aa(x1, x2)  =  appendc46_out_aa(x2)
LAST22_IN_AG(x1, x2)  =  LAST22_IN_AG(x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

LAST22_IN_AG(T101, .(T99, T102)) → LAST22_IN_AG(T101, T102)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
LAST22_IN_AG(x1, x2)  =  LAST22_IN_AG(x2)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

LAST22_IN_AG(.(T102)) → LAST22_IN_AG(T102)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LAST22_IN_AG(.(T102)) → LAST22_IN_AG(T102)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

APPEND25_IN_GAA(.(T78, T81), T82, .(T78, X165)) → APPEND25_IN_GAA(T81, T82, X165)

The TRS R consists of the following rules:

s2lc9_in_ga(s(T24), .(X69, X70)) → U13_ga(T24, X69, X70, s2lc9_in_ga(T24, X70))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U13_ga(T24, X69, X70, s2lc9_out_ga(T24, X70)) → s2lc9_out_ga(s(T24), .(X69, X70))
appendc21_in_agaa(X141, T63, T64, .(X141, X142)) → U16_agaa(X141, T63, T64, X142, appendc25_in_gaa(T63, T64, X142))
appendc25_in_gaa([], T71, .(T71, [])) → appendc25_out_gaa([], T71, .(T71, []))
appendc25_in_gaa(.(T78, T81), T82, .(T78, X165)) → U14_gaa(T78, T81, T82, X165, appendc25_in_gaa(T81, T82, X165))
U14_gaa(T78, T81, T82, X165, appendc25_out_gaa(T81, T82, X165)) → appendc25_out_gaa(.(T78, T81), T82, .(T78, X165))
U16_agaa(X141, T63, T64, X142, appendc25_out_gaa(T63, T64, X142)) → appendc21_out_agaa(X141, T63, T64, .(X141, X142))
appendc46_in_aa(T128, .(T128, [])) → appendc46_out_aa(T128, .(T128, []))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lc9_in_ga(x1, x2)  =  s2lc9_in_ga(x1)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
0  =  0
s2lc9_out_ga(x1, x2)  =  s2lc9_out_ga(x1, x2)
appendc21_in_agaa(x1, x2, x3, x4)  =  appendc21_in_agaa(x2)
U16_agaa(x1, x2, x3, x4, x5)  =  U16_agaa(x2, x5)
appendc25_in_gaa(x1, x2, x3)  =  appendc25_in_gaa(x1)
[]  =  []
appendc25_out_gaa(x1, x2, x3)  =  appendc25_out_gaa(x1, x3)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x2, x5)
appendc21_out_agaa(x1, x2, x3, x4)  =  appendc21_out_agaa(x2, x4)
appendc46_in_aa(x1, x2)  =  appendc46_in_aa
appendc46_out_aa(x1, x2)  =  appendc46_out_aa(x2)
APPEND25_IN_GAA(x1, x2, x3)  =  APPEND25_IN_GAA(x1)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

APPEND25_IN_GAA(.(T78, T81), T82, .(T78, X165)) → APPEND25_IN_GAA(T81, T82, X165)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND25_IN_GAA(x1, x2, x3)  =  APPEND25_IN_GAA(x1)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APPEND25_IN_GAA(.(T81)) → APPEND25_IN_GAA(T81)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• APPEND25_IN_GAA(.(T81)) → APPEND25_IN_GAA(T81)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

S2L9_IN_GA(s(T24), .(X69, X70)) → S2L9_IN_GA(T24, X70)

The TRS R consists of the following rules:

s2lc9_in_ga(s(T24), .(X69, X70)) → U13_ga(T24, X69, X70, s2lc9_in_ga(T24, X70))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U13_ga(T24, X69, X70, s2lc9_out_ga(T24, X70)) → s2lc9_out_ga(s(T24), .(X69, X70))
appendc21_in_agaa(X141, T63, T64, .(X141, X142)) → U16_agaa(X141, T63, T64, X142, appendc25_in_gaa(T63, T64, X142))
appendc25_in_gaa([], T71, .(T71, [])) → appendc25_out_gaa([], T71, .(T71, []))
appendc25_in_gaa(.(T78, T81), T82, .(T78, X165)) → U14_gaa(T78, T81, T82, X165, appendc25_in_gaa(T81, T82, X165))
U14_gaa(T78, T81, T82, X165, appendc25_out_gaa(T81, T82, X165)) → appendc25_out_gaa(.(T78, T81), T82, .(T78, X165))
U16_agaa(X141, T63, T64, X142, appendc25_out_gaa(T63, T64, X142)) → appendc21_out_agaa(X141, T63, T64, .(X141, X142))
appendc46_in_aa(T128, .(T128, [])) → appendc46_out_aa(T128, .(T128, []))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lc9_in_ga(x1, x2)  =  s2lc9_in_ga(x1)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
0  =  0
s2lc9_out_ga(x1, x2)  =  s2lc9_out_ga(x1, x2)
appendc21_in_agaa(x1, x2, x3, x4)  =  appendc21_in_agaa(x2)
U16_agaa(x1, x2, x3, x4, x5)  =  U16_agaa(x2, x5)
appendc25_in_gaa(x1, x2, x3)  =  appendc25_in_gaa(x1)
[]  =  []
appendc25_out_gaa(x1, x2, x3)  =  appendc25_out_gaa(x1, x3)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x2, x5)
appendc21_out_agaa(x1, x2, x3, x4)  =  appendc21_out_agaa(x2, x4)
appendc46_in_aa(x1, x2)  =  appendc46_in_aa
appendc46_out_aa(x1, x2)  =  appendc46_out_aa(x2)
S2L9_IN_GA(x1, x2)  =  S2L9_IN_GA(x1)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

S2L9_IN_GA(s(T24), .(X69, X70)) → S2L9_IN_GA(T24, X70)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L9_IN_GA(x1, x2)  =  S2L9_IN_GA(x1)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

S2L9_IN_GA(s(T24)) → S2L9_IN_GA(T24)

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

(26) 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:

• S2L9_IN_GA(s(T24)) → S2L9_IN_GA(T24)
  The graph contains the following edges 1 > 1

(27) YES