Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), append(Xs, X1, X2)).
append([], Y, Z) :- ','(!, eq(Y, Z)).
append(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), append(T, Y, Z))).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X3, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X4).
head(.(H, X5), H).
tail([], []).
tail(.(X6, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: goal(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(s2l(T2, X9), append(X9, X10, X11))\n\nT2 is ground\nX9 is free; \nX10 is free; \nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: s2l(T2, X9)\n\nT2 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: append(T3, X10, X11)\n\nX10 is free\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: s2l(T2, X9), SCOPE: 2, CLAUSE: 3 | s2l(T2, X9), SCOPE: 2, CLAUSE: 4 | ?_2\n\nT2 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_2, eq(X14, [])) | s2l(0, X9), SCOPE: 2, CLAUSE: 4 | ?_2\n\nX9 is free\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: s2l(T2, X9), SCOPE: 2, CLAUSE: 4\n\nT2 is ground\nX9 is free; \nX13 is free; \ns2l(T2, X9) !=? s2l(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: eq(X14, [])\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: eq(X14, []), SCOPE: 3, CLAUSE: 11\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(T4, X24), s2l(X24, X26))\n\nT4 is ground\nX9 is free; \nX13 is free; \nX26 is free; \nX24 is free; \ns2l(T4, X9) !=? s2l(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(p(T4, X24), s2l(X24, X26)), SCOPE: 4, CLAUSE: 9 | ','(p(T4, X24), s2l(X24, X26)), SCOPE: 4, CLAUSE: 10\n\nT4 is ground\nX9 is free; \nX13 is free; \nX26 is free; \nX24 is free; \ns2l(T4, X9) !=? s2l(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T4, X24), s2l(X24, X26)), SCOPE: 4, CLAUSE: 10\n\nT4 is ground\nX9 is free; \nX13 is free; \nX26 is free; \nX24 is free; \ns2l(T4, X9) !=? s2l(0, X13)\np(T4, X24) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: s2l(T5, X26)\n\nT5 is ground\nX9 is free; \nX13 is free; \nX26 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\nX9 is free\nX13 is free; \nX24 is free; \nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: append(T3, X10, X11), SCOPE: 5, CLAUSE: 1 | append(T3, X10, X11), SCOPE: 5, CLAUSE: 2 | ?_5\n\nX10 is free\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(!_5, eq(X33, X34)) | append(T3, X10, X11), SCOPE: 5, CLAUSE: 2 | ?_5\n\nX10 is free\nX11 is free; \nX33 is free; \nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: append(T3, X10, X11), SCOPE: 5, CLAUSE: 2\n\nX10 is free\nX11 is free; \nX31 is free; \nX32 is free; \nappend(T3, X10, X11) !=? append([], X31, X32)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: eq(X33, X34)\n\nX33 is free\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: eq(X33, X34), SCOPE: 6, CLAUSE: 11\n\nX33 is free\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(head(T7, X48), ','(tail(T7, X46), append(X46, X47, X49)))\n\nX10 is free\nX11 is free; \nX31 is free; \nX32 is free; \nX47 is free; \nX48 is free; \nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(head(T7, X48), ','(tail(T7, X46), append(X46, X47, X49))), SCOPE: 7, CLAUSE: 5 | ','(head(T7, X48), ','(tail(T7, X46), append(X46, X47, X49))), SCOPE: 7, CLAUSE: 6\n\nX47 is free\nX48 is free; \nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(tail([], X46), append(X46, X47, X49)) | ','(head(T7, X48), ','(tail(T7, X46), append(X46, X47, X49))), SCOPE: 7, CLAUSE: 6\n\nX47 is free\nX48 is free; \nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(head(T7, X48), ','(tail(T7, X46), append(X46, X47, X49))), SCOPE: 7, CLAUSE: 6\n\nX47 is free\nX48 is free; \nX49 is free; \nX46 is free; \nX51 is free; \nhead(T7, X48) !=? head([], X51)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(tail([], X46), append(X46, X47, X49)), SCOPE: 8, CLAUSE: 7 | ','(tail([], X46), append(X46, X47, X49)), SCOPE: 8, CLAUSE: 8 | ','(head(T7, X48), ','(tail(T7, X46), append(X46, X47, X49))), SCOPE: 7, CLAUSE: 6\n\nX47 is free\nX48 is free; \nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: append([], X47, X49) | ','(tail([], X46), append(X46, X47, X49)), SCOPE: 8, CLAUSE: 8 | ','(head(T7, X48), ','(tail(T7, X46), append(X46, X47, X49))), SCOPE: 7, CLAUSE: 6\n\nX47 is free\nX48 is free; \nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: append([], X47, X49)\n\nX47 is free\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(tail([], X46), append(X46, X47, X49)), SCOPE: 8, CLAUSE: 8 | ','(head(T7, X48), ','(tail(T7, X46), append(X46, X47, X49))), SCOPE: 7, CLAUSE: 6\n\nX47 is free\nX48 is free; \nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: append([], X47, X49), SCOPE: 9, CLAUSE: 1 | append([], X47, X49), SCOPE: 9, CLAUSE: 2 | ?_9\n\nX47 is free\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(!_9, eq(X57, X58)) | append([], X47, X49), SCOPE: 9, CLAUSE: 2 | ?_9\n\nX47 is free\nX49 is free; \nX57 is free; \nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: eq(X57, X58)\n\nX57 is free\nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: eq(X57, X58), SCOPE: 10, CLAUSE: 11\n\nX57 is free\nX58 is free; ", 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: ','(head(T7, X48), ','(tail(T7, X46), append(X46, X47, X49))), SCOPE: 7, CLAUSE: 6\n\nX47 is free\nX48 is free; \nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(tail(.(T10, T11), X46), append(X46, X47, X49))\n\nX47 is free\nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\nX48 is free\nX66 is free; \nX67 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(tail(.(T10, T11), X46), append(X46, X47, X49)), SCOPE: 11, CLAUSE: 7 | ','(tail(.(T10, T11), X46), append(X46, X47, X49)), SCOPE: 11, CLAUSE: 8\n\nX47 is free\nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(tail(.(T10, T11), X46), append(X46, X47, X49)), SCOPE: 11, CLAUSE: 8\n\nX47 is free\nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: append(T14, X47, X49)\n\nX47 is free\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(tail(.(T17, T18), X46), append(X46, X47, X49))\n\nX47 is free\nX48 is free; \nX49 is free; \nX46 is free; \nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\nX48 is free\nX51 is free; \nX74 is free; \nX75 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(tail(.(T17, T18), X46), append(X46, X47, X49)), SCOPE: 12, CLAUSE: 7 | ','(tail(.(T17, T18), X46), append(X46, X47, X49)), SCOPE: 12, CLAUSE: 8\n\nX47 is free\nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(tail(.(T17, T18), X46), append(X46, X47, X49)), SCOPE: 12, CLAUSE: 8\n\nX47 is free\nX49 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: append(T21, X47, X49)\n\nX47 is free\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 50 [arrowhead = none ];
50 -> 2;

51 [label="EVAL with clause\ngoal(X8) :- ','(s2l(X8, X9), append(X9, X10, X11)).\nand substitution\nT1 -> T2\nX8 -> T2", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 51 [arrowhead = none ];
51 -> 3;

52 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
3 -> 52 [arrowhead = none ];
52 -> 4;

53 [label="SPLIT 2\nreplacements:\nX9 -> T3", fontsize=14, style = filled, fillcolor = violet];
3 -> 53 [arrowhead = none ];
53 -> 5;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
4 -> 54 [arrowhead = none ];
54 -> 6;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 55 [arrowhead = none ];
55 -> 18;

56 [label="EVAL with clause\ns2l(0, X13) :- ','(!, eq(X13, [])).\nand substitution\nT2 -> 0\nX9 -> X14\nX13 -> X14", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 56 [arrowhead = none ];
56 -> 7;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 57 [arrowhead = none ];
57 -> 8;

58 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 58 [arrowhead = none ];
58 -> 9;

59 [label="EVAL with clause\ns2l(X21, .(X22, X23)) :- ','(p(X21, X24), s2l(X24, X23)).\nand substitution\nT2 -> T4\nX21 -> T4\nX22 -> X25\nX23 -> X26\nX9 -> .(X25, X26)", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 59 [arrowhead = none ];
59 -> 13;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 60 [arrowhead = none ];
60 -> 10;

61 [label="EVAL with clause\neq(X16, X16).\nand substitution\nX14 -> []\nX16 -> []\nX17 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 61 [arrowhead = none ];
61 -> 11;

62 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 62 [arrowhead = none ];
62 -> 12;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 63 [arrowhead = none ];
63 -> 14;

64 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T4, X9), s2l(0, X13))", fontsize=14, style = filled, fillcolor = white];
14 -> 64 [arrowhead = none ];
64 -> 15;

65 [label="EVAL with clause\np(s(X28), X28).\nand substitution\nX28 -> T5\nT4 -> s(T5)\nX24 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 65 [arrowhead = none ];
65 -> 16;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 66 [arrowhead = none ];
66 -> 17;

67 [label="INSTANCE with matching:\nT2 -> T5\nX9 -> X26", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 67 [arrowhead = none , style=dashed];
67 -> 4[style=dashed];

68 [label="EVAL with clause\nappend([], X31, X32) :- ','(!, eq(X31, X32)).\nand substitution\nT3 -> []\nX10 -> X33\nX31 -> X33\nX11 -> X34\nX32 -> X34", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 68 [arrowhead = none ];
68 -> 19;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 69 [arrowhead = none ];
69 -> 20;

70 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 70 [arrowhead = none ];
70 -> 21;

71 [label="EVAL with clause\nappend(X42, X43, .(X44, X45)) :- ','(head(X42, X44), ','(tail(X42, X46), append(X46, X43, X45))).\nand substitution\nT3 -> T7\nX42 -> T7\nX10 -> X47\nX43 -> X47\nX44 -> X48\nX45 -> X49\nX11 -> .(X48, X49)\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 71 [arrowhead = none ];
71 -> 25;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 72 [arrowhead = none ];
72 -> 22;

73 [label="EVAL with clause\neq(X36, X36).\nand substitution\nX33 -> X37\nX36 -> X37\nX34 -> X37", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 73 [arrowhead = none ];
73 -> 23;

74 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 74 [arrowhead = none ];
74 -> 24;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 75 [arrowhead = none ];
75 -> 26;

76 [label="EVAL with clause\nhead([], X51).\nand substitution\nT7 -> []\nX48 -> X52\nX51 -> X52", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 76 [arrowhead = none ];
76 -> 27;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 77 [arrowhead = none ];
77 -> 28;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 78 [arrowhead = none ];
78 -> 29;

79 [label="EVAL with clause\nhead(.(X74, X75), X74).\nand substitution\nX74 -> T17\nX75 -> T18\nT7 -> .(T17, T18)\nX48 -> T17\nT15 -> T17\nT16 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 79 [arrowhead = none ];
79 -> 45;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 80 [arrowhead = none ];
80 -> 46;

81 [label="EVAL with clause\ntail([], []).\nand substitution\nX46 -> []", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 81 [arrowhead = none ];
81 -> 30;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 82 [arrowhead = none ];
82 -> 31;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 83 [arrowhead = none ];
83 -> 32;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 84 [arrowhead = none ];
84 -> 33;

85 [label="BACKTRACK\nfor clause: tail(.(X6, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
32 -> 85 [arrowhead = none ];
85 -> 39;

86 [label="EVAL with clause\nappend([], X55, X56) :- ','(!, eq(X55, X56)).\nand substitution\nX47 -> X57\nX55 -> X57\nX49 -> X58\nX56 -> X58", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 86 [arrowhead = none ];
86 -> 34;

87 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 87 [arrowhead = none ];
87 -> 35;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 88 [arrowhead = none ];
88 -> 36;

89 [label="EVAL with clause\neq(X60, X60).\nand substitution\nX57 -> X61\nX60 -> X61\nX58 -> X61", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 89 [arrowhead = none ];
89 -> 37;

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 90 [arrowhead = none ];
90 -> 38;

91 [label="EVAL with clause\nhead(.(X66, X67), X66).\nand substitution\nX66 -> T10\nX67 -> T11\nT7 -> .(T10, T11)\nX48 -> T10\nT8 -> T10\nT9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 91 [arrowhead = none ];
91 -> 40;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 92 [arrowhead = none ];
92 -> 41;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 93 [arrowhead = none ];
93 -> 42;

94 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
42 -> 94 [arrowhead = none ];
94 -> 43;

95 [label="EVAL with clause\ntail(.(X70, X71), X71).\nand substitution\nT10 -> T12\nX70 -> T12\nT11 -> T14\nX71 -> T14\nX46 -> T14\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 95 [arrowhead = none ];
95 -> 44;

96 [label="INSTANCE with matching:\nT3 -> T14\nX10 -> X47\nX11 -> X49", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 96 [arrowhead = none , style=dashed];
96 -> 5[style=dashed];

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 97 [arrowhead = none ];
97 -> 47;

98 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
47 -> 98 [arrowhead = none ];
98 -> 48;

99 [label="EVAL with clause\ntail(.(X78, X79), X79).\nand substitution\nT17 -> T19\nX78 -> T19\nT18 -> T21\nX79 -> T21\nX46 -> T21\nT20 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 99 [arrowhead = none ];
99 -> 49;

100 [label="INSTANCE with matching:\nT3 -> T21\nX10 -> X47\nX11 -> X49", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 100 [arrowhead = none , style=dashed];
100 -> 5[style=dashed];

}

(2) Obligation:

Clauses:

s2l4(0, []).
s2l4(s(T5), .(X25, X26)) :- s2l4(T5, X26).
append5([], X37, X37).
append5([], X61, .(X52, X61)).
append5(.(T12, T14), X47, .(T12, X49)) :- append5(T14, X47, X49).
append5(.(T19, T21), X47, .(T19, X49)) :- append5(T21, X47, X49).
goal1(T2) :- s2l4(T2, X9).
goal1(T2) :- s2l4(T2, []).
goal1(T2) :- s2l4(T2, []).
goal1(T2) :- ','(s2l4(T2, .(T12, T14)), append5(T14, X47, X49)).
goal1(T2) :- ','(s2l4(T2, .(T19, T21)), append5(T21, X47, X49)).

Queries:

goal1(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:
goal1_in: (b)
s2l4_in: (b,f) (b,b)
append5_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X25, X26)) → U1_ga(T5, X25, X26, s2l4_in_ga(T5, X26))
U1_ga(T5, X25, X26, s2l4_out_ga(T5, X26)) → s2l4_out_ga(s(T5), .(X25, X26))
U4_g(T2, s2l4_out_ga(T2, X9)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X25, X26)) → U1_gg(T5, X25, X26, s2l4_in_gg(T5, X26))
U1_gg(T5, X25, X26, s2l4_out_gg(T5, X26)) → s2l4_out_gg(s(T5), .(X25, X26))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T12, T14)))
U6_g(T2, s2l4_out_ga(T2, .(T12, T14))) → U7_g(T2, append5_in_gaa(T14, X47, X49))
append5_in_gaa([], X37, X37) → append5_out_gaa([], X37, X37)
append5_in_gaa([], X61, .(X52, X61)) → append5_out_gaa([], X61, .(X52, X61))
append5_in_gaa(.(T12, T14), X47, .(T12, X49)) → U2_gaa(T12, T14, X47, X49, append5_in_gaa(T14, X47, X49))
append5_in_gaa(.(T19, T21), X47, .(T19, X49)) → U3_gaa(T19, T21, X47, X49, append5_in_gaa(T21, X47, X49))
U3_gaa(T19, T21, X47, X49, append5_out_gaa(T21, X47, X49)) → append5_out_gaa(.(T19, T21), X47, .(T19, X49))
U2_gaa(T12, T14, X47, X49, append5_out_gaa(T14, X47, X49)) → append5_out_gaa(.(T12, T14), X47, .(T12, X49))
U7_g(T2, append5_out_gaa(T14, X47, X49)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T19, T21)))
U8_g(T2, s2l4_out_ga(T2, .(T19, T21))) → U9_g(T2, append5_in_gaa(T21, X47, X49))
U9_g(T2, append5_out_gaa(T21, X47, X49)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
append5_in_gaa(x1, x2, x3)  =  append5_in_gaa(x1)
append5_out_gaa(x1, x2, x3)  =  append5_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X25, X26)) → U1_ga(T5, X25, X26, s2l4_in_ga(T5, X26))
U1_ga(T5, X25, X26, s2l4_out_ga(T5, X26)) → s2l4_out_ga(s(T5), .(X25, X26))
U4_g(T2, s2l4_out_ga(T2, X9)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X25, X26)) → U1_gg(T5, X25, X26, s2l4_in_gg(T5, X26))
U1_gg(T5, X25, X26, s2l4_out_gg(T5, X26)) → s2l4_out_gg(s(T5), .(X25, X26))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T12, T14)))
U6_g(T2, s2l4_out_ga(T2, .(T12, T14))) → U7_g(T2, append5_in_gaa(T14, X47, X49))
append5_in_gaa([], X37, X37) → append5_out_gaa([], X37, X37)
append5_in_gaa([], X61, .(X52, X61)) → append5_out_gaa([], X61, .(X52, X61))
append5_in_gaa(.(T12, T14), X47, .(T12, X49)) → U2_gaa(T12, T14, X47, X49, append5_in_gaa(T14, X47, X49))
append5_in_gaa(.(T19, T21), X47, .(T19, X49)) → U3_gaa(T19, T21, X47, X49, append5_in_gaa(T21, X47, X49))
U3_gaa(T19, T21, X47, X49, append5_out_gaa(T21, X47, X49)) → append5_out_gaa(.(T19, T21), X47, .(T19, X49))
U2_gaa(T12, T14, X47, X49, append5_out_gaa(T14, X47, X49)) → append5_out_gaa(.(T12, T14), X47, .(T12, X49))
U7_g(T2, append5_out_gaa(T14, X47, X49)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T19, T21)))
U8_g(T2, s2l4_out_ga(T2, .(T19, T21))) → U9_g(T2, append5_in_gaa(T21, X47, X49))
U9_g(T2, append5_out_gaa(T21, X47, X49)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
append5_in_gaa(x1, x2, x3)  =  append5_in_gaa(x1)
append5_out_gaa(x1, x2, x3)  =  append5_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)

(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:

GOAL1_IN_G(T2) → U4_G(T2, s2l4_in_ga(T2, X9))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, X9)
S2L4_IN_GA(s(T5), .(X25, X26)) → U1_GA(T5, X25, X26, s2l4_in_ga(T5, X26))
S2L4_IN_GA(s(T5), .(X25, X26)) → S2L4_IN_GA(T5, X26)
GOAL1_IN_G(T2) → U5_G(T2, s2l4_in_gg(T2, []))
GOAL1_IN_G(T2) → S2L4_IN_GG(T2, [])
S2L4_IN_GG(s(T5), .(X25, X26)) → U1_GG(T5, X25, X26, s2l4_in_gg(T5, X26))
S2L4_IN_GG(s(T5), .(X25, X26)) → S2L4_IN_GG(T5, X26)
GOAL1_IN_G(T2) → U6_G(T2, s2l4_in_ga(T2, .(T12, T14)))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, .(T12, T14))
U6_G(T2, s2l4_out_ga(T2, .(T12, T14))) → U7_G(T2, append5_in_gaa(T14, X47, X49))
U6_G(T2, s2l4_out_ga(T2, .(T12, T14))) → APPEND5_IN_GAA(T14, X47, X49)
APPEND5_IN_GAA(.(T12, T14), X47, .(T12, X49)) → U2_GAA(T12, T14, X47, X49, append5_in_gaa(T14, X47, X49))
APPEND5_IN_GAA(.(T12, T14), X47, .(T12, X49)) → APPEND5_IN_GAA(T14, X47, X49)
APPEND5_IN_GAA(.(T19, T21), X47, .(T19, X49)) → U3_GAA(T19, T21, X47, X49, append5_in_gaa(T21, X47, X49))
GOAL1_IN_G(T2) → U8_G(T2, s2l4_in_ga(T2, .(T19, T21)))
U8_G(T2, s2l4_out_ga(T2, .(T19, T21))) → U9_G(T2, append5_in_gaa(T21, X47, X49))
U8_G(T2, s2l4_out_ga(T2, .(T19, T21))) → APPEND5_IN_GAA(T21, X47, X49)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X25, X26)) → U1_ga(T5, X25, X26, s2l4_in_ga(T5, X26))
U1_ga(T5, X25, X26, s2l4_out_ga(T5, X26)) → s2l4_out_ga(s(T5), .(X25, X26))
U4_g(T2, s2l4_out_ga(T2, X9)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X25, X26)) → U1_gg(T5, X25, X26, s2l4_in_gg(T5, X26))
U1_gg(T5, X25, X26, s2l4_out_gg(T5, X26)) → s2l4_out_gg(s(T5), .(X25, X26))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T12, T14)))
U6_g(T2, s2l4_out_ga(T2, .(T12, T14))) → U7_g(T2, append5_in_gaa(T14, X47, X49))
append5_in_gaa([], X37, X37) → append5_out_gaa([], X37, X37)
append5_in_gaa([], X61, .(X52, X61)) → append5_out_gaa([], X61, .(X52, X61))
append5_in_gaa(.(T12, T14), X47, .(T12, X49)) → U2_gaa(T12, T14, X47, X49, append5_in_gaa(T14, X47, X49))
append5_in_gaa(.(T19, T21), X47, .(T19, X49)) → U3_gaa(T19, T21, X47, X49, append5_in_gaa(T21, X47, X49))
U3_gaa(T19, T21, X47, X49, append5_out_gaa(T21, X47, X49)) → append5_out_gaa(.(T19, T21), X47, .(T19, X49))
U2_gaa(T12, T14, X47, X49, append5_out_gaa(T14, X47, X49)) → append5_out_gaa(.(T12, T14), X47, .(T12, X49))
U7_g(T2, append5_out_gaa(T14, X47, X49)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T19, T21)))
U8_g(T2, s2l4_out_ga(T2, .(T19, T21))) → U9_g(T2, append5_in_gaa(T21, X47, X49))
U9_g(T2, append5_out_gaa(T21, X47, X49)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
append5_in_gaa(x1, x2, x3)  =  append5_in_gaa(x1)
append5_out_gaa(x1, x2, x3)  =  append5_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U5_G(x1, x2)  =  U5_G(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
APPEND5_IN_GAA(x1, x2, x3)  =  APPEND5_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
U8_G(x1, x2)  =  U8_G(x2)
U9_G(x1, x2)  =  U9_G(x2)

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

(6) Obligation:

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

GOAL1_IN_G(T2) → U4_G(T2, s2l4_in_ga(T2, X9))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, X9)
S2L4_IN_GA(s(T5), .(X25, X26)) → U1_GA(T5, X25, X26, s2l4_in_ga(T5, X26))
S2L4_IN_GA(s(T5), .(X25, X26)) → S2L4_IN_GA(T5, X26)
GOAL1_IN_G(T2) → U5_G(T2, s2l4_in_gg(T2, []))
GOAL1_IN_G(T2) → S2L4_IN_GG(T2, [])
S2L4_IN_GG(s(T5), .(X25, X26)) → U1_GG(T5, X25, X26, s2l4_in_gg(T5, X26))
S2L4_IN_GG(s(T5), .(X25, X26)) → S2L4_IN_GG(T5, X26)
GOAL1_IN_G(T2) → U6_G(T2, s2l4_in_ga(T2, .(T12, T14)))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, .(T12, T14))
U6_G(T2, s2l4_out_ga(T2, .(T12, T14))) → U7_G(T2, append5_in_gaa(T14, X47, X49))
U6_G(T2, s2l4_out_ga(T2, .(T12, T14))) → APPEND5_IN_GAA(T14, X47, X49)
APPEND5_IN_GAA(.(T12, T14), X47, .(T12, X49)) → U2_GAA(T12, T14, X47, X49, append5_in_gaa(T14, X47, X49))
APPEND5_IN_GAA(.(T12, T14), X47, .(T12, X49)) → APPEND5_IN_GAA(T14, X47, X49)
APPEND5_IN_GAA(.(T19, T21), X47, .(T19, X49)) → U3_GAA(T19, T21, X47, X49, append5_in_gaa(T21, X47, X49))
GOAL1_IN_G(T2) → U8_G(T2, s2l4_in_ga(T2, .(T19, T21)))
U8_G(T2, s2l4_out_ga(T2, .(T19, T21))) → U9_G(T2, append5_in_gaa(T21, X47, X49))
U8_G(T2, s2l4_out_ga(T2, .(T19, T21))) → APPEND5_IN_GAA(T21, X47, X49)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X25, X26)) → U1_ga(T5, X25, X26, s2l4_in_ga(T5, X26))
U1_ga(T5, X25, X26, s2l4_out_ga(T5, X26)) → s2l4_out_ga(s(T5), .(X25, X26))
U4_g(T2, s2l4_out_ga(T2, X9)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X25, X26)) → U1_gg(T5, X25, X26, s2l4_in_gg(T5, X26))
U1_gg(T5, X25, X26, s2l4_out_gg(T5, X26)) → s2l4_out_gg(s(T5), .(X25, X26))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T12, T14)))
U6_g(T2, s2l4_out_ga(T2, .(T12, T14))) → U7_g(T2, append5_in_gaa(T14, X47, X49))
append5_in_gaa([], X37, X37) → append5_out_gaa([], X37, X37)
append5_in_gaa([], X61, .(X52, X61)) → append5_out_gaa([], X61, .(X52, X61))
append5_in_gaa(.(T12, T14), X47, .(T12, X49)) → U2_gaa(T12, T14, X47, X49, append5_in_gaa(T14, X47, X49))
append5_in_gaa(.(T19, T21), X47, .(T19, X49)) → U3_gaa(T19, T21, X47, X49, append5_in_gaa(T21, X47, X49))
U3_gaa(T19, T21, X47, X49, append5_out_gaa(T21, X47, X49)) → append5_out_gaa(.(T19, T21), X47, .(T19, X49))
U2_gaa(T12, T14, X47, X49, append5_out_gaa(T14, X47, X49)) → append5_out_gaa(.(T12, T14), X47, .(T12, X49))
U7_g(T2, append5_out_gaa(T14, X47, X49)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T19, T21)))
U8_g(T2, s2l4_out_ga(T2, .(T19, T21))) → U9_g(T2, append5_in_gaa(T21, X47, X49))
U9_g(T2, append5_out_gaa(T21, X47, X49)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
append5_in_gaa(x1, x2, x3)  =  append5_in_gaa(x1)
append5_out_gaa(x1, x2, x3)  =  append5_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U5_G(x1, x2)  =  U5_G(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
APPEND5_IN_GAA(x1, x2, x3)  =  APPEND5_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
U8_G(x1, x2)  =  U8_G(x2)
U9_G(x1, x2)  =  U9_G(x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND5_IN_GAA(.(T12, T14), X47, .(T12, X49)) → APPEND5_IN_GAA(T14, X47, X49)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X25, X26)) → U1_ga(T5, X25, X26, s2l4_in_ga(T5, X26))
U1_ga(T5, X25, X26, s2l4_out_ga(T5, X26)) → s2l4_out_ga(s(T5), .(X25, X26))
U4_g(T2, s2l4_out_ga(T2, X9)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X25, X26)) → U1_gg(T5, X25, X26, s2l4_in_gg(T5, X26))
U1_gg(T5, X25, X26, s2l4_out_gg(T5, X26)) → s2l4_out_gg(s(T5), .(X25, X26))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T12, T14)))
U6_g(T2, s2l4_out_ga(T2, .(T12, T14))) → U7_g(T2, append5_in_gaa(T14, X47, X49))
append5_in_gaa([], X37, X37) → append5_out_gaa([], X37, X37)
append5_in_gaa([], X61, .(X52, X61)) → append5_out_gaa([], X61, .(X52, X61))
append5_in_gaa(.(T12, T14), X47, .(T12, X49)) → U2_gaa(T12, T14, X47, X49, append5_in_gaa(T14, X47, X49))
append5_in_gaa(.(T19, T21), X47, .(T19, X49)) → U3_gaa(T19, T21, X47, X49, append5_in_gaa(T21, X47, X49))
U3_gaa(T19, T21, X47, X49, append5_out_gaa(T21, X47, X49)) → append5_out_gaa(.(T19, T21), X47, .(T19, X49))
U2_gaa(T12, T14, X47, X49, append5_out_gaa(T14, X47, X49)) → append5_out_gaa(.(T12, T14), X47, .(T12, X49))
U7_g(T2, append5_out_gaa(T14, X47, X49)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T19, T21)))
U8_g(T2, s2l4_out_ga(T2, .(T19, T21))) → U9_g(T2, append5_in_gaa(T21, X47, X49))
U9_g(T2, append5_out_gaa(T21, X47, X49)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
append5_in_gaa(x1, x2, x3)  =  append5_in_gaa(x1)
append5_out_gaa(x1, x2, x3)  =  append5_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
APPEND5_IN_GAA(x1, x2, x3)  =  APPEND5_IN_GAA(x1)

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:

APPEND5_IN_GAA(.(T12, T14), X47, .(T12, X49)) → APPEND5_IN_GAA(T14, X47, X49)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND5_IN_GAA(x1, x2, x3)  =  APPEND5_IN_GAA(x1)

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:

APPEND5_IN_GAA(.(T14)) → APPEND5_IN_GAA(T14)

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:

• APPEND5_IN_GAA(.(T14)) → APPEND5_IN_GAA(T14)
  The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

S2L4_IN_GG(s(T5), .(X25, X26)) → S2L4_IN_GG(T5, X26)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X25, X26)) → U1_ga(T5, X25, X26, s2l4_in_ga(T5, X26))
U1_ga(T5, X25, X26, s2l4_out_ga(T5, X26)) → s2l4_out_ga(s(T5), .(X25, X26))
U4_g(T2, s2l4_out_ga(T2, X9)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X25, X26)) → U1_gg(T5, X25, X26, s2l4_in_gg(T5, X26))
U1_gg(T5, X25, X26, s2l4_out_gg(T5, X26)) → s2l4_out_gg(s(T5), .(X25, X26))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T12, T14)))
U6_g(T2, s2l4_out_ga(T2, .(T12, T14))) → U7_g(T2, append5_in_gaa(T14, X47, X49))
append5_in_gaa([], X37, X37) → append5_out_gaa([], X37, X37)
append5_in_gaa([], X61, .(X52, X61)) → append5_out_gaa([], X61, .(X52, X61))
append5_in_gaa(.(T12, T14), X47, .(T12, X49)) → U2_gaa(T12, T14, X47, X49, append5_in_gaa(T14, X47, X49))
append5_in_gaa(.(T19, T21), X47, .(T19, X49)) → U3_gaa(T19, T21, X47, X49, append5_in_gaa(T21, X47, X49))
U3_gaa(T19, T21, X47, X49, append5_out_gaa(T21, X47, X49)) → append5_out_gaa(.(T19, T21), X47, .(T19, X49))
U2_gaa(T12, T14, X47, X49, append5_out_gaa(T14, X47, X49)) → append5_out_gaa(.(T12, T14), X47, .(T12, X49))
U7_g(T2, append5_out_gaa(T14, X47, X49)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T19, T21)))
U8_g(T2, s2l4_out_ga(T2, .(T19, T21))) → U9_g(T2, append5_in_gaa(T21, X47, X49))
U9_g(T2, append5_out_gaa(T21, X47, X49)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
append5_in_gaa(x1, x2, x3)  =  append5_in_gaa(x1)
append5_out_gaa(x1, x2, x3)  =  append5_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)

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:

S2L4_IN_GG(s(T5), .(X25, X26)) → S2L4_IN_GG(T5, X26)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)

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:

S2L4_IN_GG(s(T5), .(X26)) → S2L4_IN_GG(T5, X26)

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:

• S2L4_IN_GG(s(T5), .(X26)) → S2L4_IN_GG(T5, X26)
  The graph contains the following edges 1 > 1, 2 > 2

(22) TRUE

(23) Obligation:

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

S2L4_IN_GA(s(T5), .(X25, X26)) → S2L4_IN_GA(T5, X26)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X25, X26)) → U1_ga(T5, X25, X26, s2l4_in_ga(T5, X26))
U1_ga(T5, X25, X26, s2l4_out_ga(T5, X26)) → s2l4_out_ga(s(T5), .(X25, X26))
U4_g(T2, s2l4_out_ga(T2, X9)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X25, X26)) → U1_gg(T5, X25, X26, s2l4_in_gg(T5, X26))
U1_gg(T5, X25, X26, s2l4_out_gg(T5, X26)) → s2l4_out_gg(s(T5), .(X25, X26))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T12, T14)))
U6_g(T2, s2l4_out_ga(T2, .(T12, T14))) → U7_g(T2, append5_in_gaa(T14, X47, X49))
append5_in_gaa([], X37, X37) → append5_out_gaa([], X37, X37)
append5_in_gaa([], X61, .(X52, X61)) → append5_out_gaa([], X61, .(X52, X61))
append5_in_gaa(.(T12, T14), X47, .(T12, X49)) → U2_gaa(T12, T14, X47, X49, append5_in_gaa(T14, X47, X49))
append5_in_gaa(.(T19, T21), X47, .(T19, X49)) → U3_gaa(T19, T21, X47, X49, append5_in_gaa(T21, X47, X49))
U3_gaa(T19, T21, X47, X49, append5_out_gaa(T21, X47, X49)) → append5_out_gaa(.(T19, T21), X47, .(T19, X49))
U2_gaa(T12, T14, X47, X49, append5_out_gaa(T14, X47, X49)) → append5_out_gaa(.(T12, T14), X47, .(T12, X49))
U7_g(T2, append5_out_gaa(T14, X47, X49)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T19, T21)))
U8_g(T2, s2l4_out_ga(T2, .(T19, T21))) → U9_g(T2, append5_in_gaa(T21, X47, X49))
U9_g(T2, append5_out_gaa(T21, X47, X49)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
append5_in_gaa(x1, x2, x3)  =  append5_in_gaa(x1)
append5_out_gaa(x1, x2, x3)  =  append5_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_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:

S2L4_IN_GA(s(T5), .(X25, X26)) → S2L4_IN_GA(T5, X26)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_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:

S2L4_IN_GA(s(T5)) → S2L4_IN_GA(T5)

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:

• S2L4_IN_GA(s(T5)) → S2L4_IN_GA(T5)
  The graph contains the following edges 1 > 1

(29) TRUE