Left Termination of the query pattern goal_in_3(g, a, a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

goal(A, B, C) :- ','(s2l(A, D), applast(D, B, C)).
applast(L, X, Last) :- ','(app(L, .(X, []), LX), last(Last, LX)).
last(X, .(X, [])) :- !.
last(X, Y) :- ','(tail(Y, T), last(X, T)).
app([], Y, Z) :- ','(!, eq(Y, Z)).
app(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), app(T, Y, Z))).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X1, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X2).
head(.(X, X3), X).
tail([], []).
tail(.(X4, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g,a,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog 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(T4, X11), applast(X11, T7, T8))\n\nT4 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: s2l(T4, X11)\n\nT4 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: applast(T9, T7, T8)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: s2l(T4, X11), SCOPE: 2, CLAUSE: 6 | s2l(T4, X11), SCOPE: 2, CLAUSE: 7 | ?_2\n\nT4 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_2, eq(X14, [])) | s2l(0, X11), SCOPE: 2, CLAUSE: 7 | ?_2\n\nX11 is free\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: s2l(T4, X11), SCOPE: 2, CLAUSE: 7\n\nT4 is ground\nX11 is free; \nX13 is free; \ns2l(T4, X11) !=? 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: 14\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(T10, X24), s2l(X24, X26))\n\nT10 is ground\nX11 is free; \nX13 is free; \nX26 is free; \nX24 is free; \ns2l(T10, X11) !=? s2l(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(p(T10, X24), s2l(X24, X26)), SCOPE: 4, CLAUSE: 12 | ','(p(T10, X24), s2l(X24, X26)), SCOPE: 4, CLAUSE: 13\n\nT10 is ground\nX11 is free; \nX13 is free; \nX26 is free; \nX24 is free; \ns2l(T10, X11) !=? s2l(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T10, X24), s2l(X24, X26)), SCOPE: 4, CLAUSE: 13\n\nT10 is ground\nX11 is free; \nX13 is free; \nX26 is free; \nX24 is free; \ns2l(T10, X11) !=? s2l(0, X13)\np(T10, X24) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: s2l(T11, X26)\n\nT11 is ground\nX11 is free; \nX13 is free; \nX26 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\nX11 is free\nX13 is free; \nX24 is free; \nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: applast(T9, T7, T8), SCOPE: 5, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(app(T15, .(T16, []), X35), last(T17, X35))\n\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(T15, .(T16, []), X35)\n\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: last(T19, T18)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(T15, .(T16, []), X35), SCOPE: 6, CLAUSE: 4 | app(T15, .(T16, []), X35), SCOPE: 6, CLAUSE: 5 | ?_6\n\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(!_6, eq(.(T21, []), X40)) | app(T15, .(T16, []), X35), SCOPE: 6, CLAUSE: 5 | ?_6\n\nX35 is free\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app(T15, .(T16, []), X35), SCOPE: 6, CLAUSE: 5\n\nX35 is free\nX38 is free; \nX39 is free; \napp(T15, .(T16, []), X35) !=? app([], X38, X39)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: eq(.(T21, []), X40)\n\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: eq(.(T21, []), X40), SCOPE: 7, CLAUSE: 14\n\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(head(T25, X52), ','(tail(T25, X51), app(X51, .(T26, []), X53)))\n\nX35 is free\nX38 is free; \nX39 is free; \nX52 is free; \nX53 is free; \nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(head(T25, X52), ','(tail(T25, X51), app(X51, .(T26, []), X53))), SCOPE: 8, CLAUSE: 8 | ','(head(T25, X52), ','(tail(T25, X51), app(X51, .(T26, []), X53))), SCOPE: 8, CLAUSE: 9\n\nX52 is free\nX53 is free; \nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(tail([], X51), app(X51, .(T27, []), X53)) | ','(head(T25, X52), ','(tail(T25, X51), app(X51, .(T26, []), X53))), SCOPE: 8, CLAUSE: 9\n\nX52 is free\nX53 is free; \nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(head(T25, X52), ','(tail(T25, X51), app(X51, .(T26, []), X53))), SCOPE: 8, CLAUSE: 9\n\nX52 is free\nX53 is free; \nX51 is free; \nX55 is free; \nhead(T25, X52) !=? head([], X55)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(tail([], X51), app(X51, .(T27, []), X53)), SCOPE: 9, CLAUSE: 10 | ','(tail([], X51), app(X51, .(T27, []), X53)), SCOPE: 9, CLAUSE: 11 | ','(head(T25, X52), ','(tail(T25, X51), app(X51, .(T26, []), X53))), SCOPE: 8, CLAUSE: 9\n\nX52 is free\nX53 is free; \nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app([], .(T27, []), X53) | ','(tail([], X51), app(X51, .(T27, []), X53)), SCOPE: 9, CLAUSE: 11 | ','(head(T25, X52), ','(tail(T25, X51), app(X51, .(T26, []), X53))), SCOPE: 8, CLAUSE: 9\n\nX52 is free\nX53 is free; \nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: app([], .(T27, []), X53)\n\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(tail([], X51), app(X51, .(T27, []), X53)), SCOPE: 9, CLAUSE: 11 | ','(head(T25, X52), ','(tail(T25, X51), app(X51, .(T26, []), X53))), SCOPE: 8, CLAUSE: 9\n\nX52 is free\nX53 is free; \nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: app([], .(T27, []), X53), SCOPE: 10, CLAUSE: 4 | app([], .(T27, []), X53), SCOPE: 10, CLAUSE: 5 | ?_10\n\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(!_10, eq(.(T29, []), X61)) | app([], .(T27, []), X53), SCOPE: 10, CLAUSE: 5 | ?_10\n\nX53 is free\nX61 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: eq(.(T29, []), X61)\n\nX61 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: eq(.(T29, []), X61), SCOPE: 11, CLAUSE: 14\n\nX61 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(head(T25, X52), ','(tail(T25, X51), app(X51, .(T26, []), X53))), SCOPE: 8, CLAUSE: 9\n\nX52 is free\nX53 is free; \nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(tail(.(T33, T34), X51), app(X51, .(T35, []), X53))\n\nX53 is free\nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\nX52 is free\nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(tail(.(T33, T34), X51), app(X51, .(T35, []), X53)), SCOPE: 12, CLAUSE: 10 | ','(tail(.(T33, T34), X51), app(X51, .(T35, []), X53)), SCOPE: 12, CLAUSE: 11\n\nX53 is free\nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(tail(.(T33, T34), X51), app(X51, .(T35, []), X53)), SCOPE: 12, CLAUSE: 11\n\nX53 is free\nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: app(T38, .(T39, []), X53)\n\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(tail(.(T42, T43), X51), app(X51, .(T44, []), X53))\n\nX52 is free\nX53 is free; \nX51 is free; \nX55 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\nX52 is free\nX55 is free; \nX76 is free; \nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(tail(.(T42, T43), X51), app(X51, .(T44, []), X53)), SCOPE: 13, CLAUSE: 10 | ','(tail(.(T42, T43), X51), app(X51, .(T44, []), X53)), SCOPE: 13, CLAUSE: 11\n\nX53 is free\nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(tail(.(T42, T43), X51), app(X51, .(T44, []), X53)), SCOPE: 13, CLAUSE: 11\n\nX53 is free\nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: app(T47, .(T48, []), X53)\n\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: last(T19, T18), SCOPE: 14, CLAUSE: 2 | last(T19, T18), SCOPE: 14, CLAUSE: 3 | ?_14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: !_14 | last(T19, T18), SCOPE: 14, CLAUSE: 3 | ?_14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: last(T19, T18), SCOPE: 14, CLAUSE: 3\n\nX83 is free\nlast(T19, T18) !=? last(X83, .(X83, []))", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(tail(T52, X88), last(T53, X88))\n\nX83 is free\nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(tail(T52, X88), last(T53, X88)), SCOPE: 15, CLAUSE: 10 | ','(tail(T52, X88), last(T53, X88)), SCOPE: 15, CLAUSE: 11\n\nX88 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: last(T54, []) | ','(tail(T52, X88), last(T53, X88)), SCOPE: 15, CLAUSE: 11\n\nX88 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(tail(T52, X88), last(T53, X88)), SCOPE: 15, CLAUSE: 11\n\nX88 is free\ntail(T52, X88) !=? tail([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: last(T54, [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(tail(T52, X88), last(T53, X88)), SCOPE: 15, CLAUSE: 11\n\nX88 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: last(T54, []), SCOPE: 16, CLAUSE: 2 | last(T54, []), SCOPE: 16, CLAUSE: 3 | ?_16\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: last(T54, []), SCOPE: 16, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(tail([], X94), last(T56, X94))\n\nX94 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(tail([], X94), last(T56, X94)), SCOPE: 17, CLAUSE: 10 | ','(tail([], X94), last(T56, X94)), SCOPE: 17, CLAUSE: 11\n\nX94 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: last(T56, []) | ','(tail([], X94), last(T56, X94)), SCOPE: 17, CLAUSE: 11\n\nX94 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: last(T56, [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(tail([], X94), last(T56, X94)), SCOPE: 17, CLAUSE: 11\n\nX94 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: last(T59, T60)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\nX88 is free\nX99 is free; \nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: last(T63, T64)\n\nX88 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\nX88 is free\nX103 is free; \nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 77 [arrowhead = none ];
77 -> 2;

78 [label="EVAL with clause\ngoal(X8, X9, X10) :- ','(s2l(X8, X11), applast(X11, X9, X10)).\nand substitution\nT1 -> T4\nX8 -> T4\nT2 -> T7\nX9 -> T7\nT3 -> T8\nX10 -> T8\nT5 -> T7\nT6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 78 [arrowhead = none ];
78 -> 3;

79 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
3 -> 79 [arrowhead = none ];
79 -> 4;

80 [label="SPLIT 2\nreplacements:\nX11 -> T9", fontsize=14, style = filled, fillcolor = violet];
3 -> 80 [arrowhead = none ];
80 -> 5;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
4 -> 81 [arrowhead = none ];
81 -> 6;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 82 [arrowhead = none ];
82 -> 18;

83 [label="EVAL with clause\ns2l(0, X13) :- ','(!, eq(X13, [])).\nand substitution\nT4 -> 0\nX11 -> X14\nX13 -> X14", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 83 [arrowhead = none ];
83 -> 7;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 84 [arrowhead = none ];
84 -> 8;

85 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 85 [arrowhead = none ];
85 -> 9;

86 [label="EVAL with clause\ns2l(X21, .(X22, X23)) :- ','(p(X21, X24), s2l(X24, X23)).\nand substitution\nT4 -> T10\nX21 -> T10\nX22 -> X25\nX23 -> X26\nX11 -> .(X25, X26)", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 86 [arrowhead = none ];
86 -> 13;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 87 [arrowhead = none ];
87 -> 10;

88 [label="EVAL with clause\neq(X16, X16).\nand substitution\nX14 -> []\nX16 -> []\nX17 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 88 [arrowhead = none ];
88 -> 11;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 89 [arrowhead = none ];
89 -> 12;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 90 [arrowhead = none ];
90 -> 14;

91 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T10, X11), s2l(0, X13))", fontsize=14, style = filled, fillcolor = white];
14 -> 91 [arrowhead = none ];
91 -> 15;

92 [label="EVAL with clause\np(s(X28), X28).\nand substitution\nX28 -> T11\nT10 -> s(T11)\nX24 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 92 [arrowhead = none ];
92 -> 16;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 93 [arrowhead = none ];
93 -> 17;

94 [label="INSTANCE with matching:\nT4 -> T11\nX11 -> X26", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 94 [arrowhead = none , style=dashed];
94 -> 4[style=dashed];

95 [label="EVAL with clause\napplast(X32, X33, X34) :- ','(app(X32, .(X33, []), X35), last(X34, X35)).\nand substitution\nT9 -> T15\nX32 -> T15\nT7 -> T16\nX33 -> T16\nT8 -> T17\nX34 -> T17\nT12 -> T15\nT13 -> T16\nT14 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 95 [arrowhead = none ];
95 -> 19;

96 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 96 [arrowhead = none ];
96 -> 20;

97 [label="SPLIT 2\nreplacements:\nX35 -> T18\nT17 -> T19", fontsize=14, style = filled, fillcolor = violet];
19 -> 97 [arrowhead = none ];
97 -> 21;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 98 [arrowhead = none ];
98 -> 22;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 99 [arrowhead = none ];
99 -> 54;

100 [label="EVAL with clause\napp([], X38, X39) :- ','(!, eq(X38, X39)).\nand substitution\nT15 -> []\nT16 -> T21\nX38 -> .(T21, [])\nX35 -> X40\nX39 -> X40\nT20 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 100 [arrowhead = none ];
100 -> 23;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 101 [arrowhead = none ];
101 -> 24;

102 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 102 [arrowhead = none ];
102 -> 25;

103 [label="EVAL with clause\napp(X47, X48, .(X49, X50)) :- ','(head(X47, X49), ','(tail(X47, X51), app(X51, X48, X50))).\nand substitution\nT15 -> T25\nX47 -> T25\nT16 -> T26\nX48 -> .(T26, [])\nX49 -> X52\nX50 -> X53\nX35 -> .(X52, X53)\nT23 -> T25\nT24 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 103 [arrowhead = none ];
103 -> 29;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 104 [arrowhead = none ];
104 -> 26;

105 [label="EVAL with clause\neq(X42, X42).\nand substitution\nT21 -> T22\nX42 -> .(T22, [])\nX40 -> .(T22, [])", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 105 [arrowhead = none ];
105 -> 27;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 106 [arrowhead = none ];
106 -> 28;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 107 [arrowhead = none ];
107 -> 30;

108 [label="EVAL with clause\nhead([], X55).\nand substitution\nT25 -> []\nX52 -> X56\nX55 -> X56\nT26 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 108 [arrowhead = none ];
108 -> 31;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 109 [arrowhead = none ];
109 -> 32;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 110 [arrowhead = none ];
110 -> 33;

111 [label="EVAL with clause\nhead(.(X76, X77), X76).\nand substitution\nX76 -> T42\nX77 -> T43\nT25 -> .(T42, T43)\nX52 -> T42\nT40 -> T42\nT41 -> T43\nT26 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 111 [arrowhead = none ];
111 -> 49;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 112 [arrowhead = none ];
112 -> 50;

113 [label="EVAL with clause\ntail([], []).\nand substitution\nX51 -> []", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 113 [arrowhead = none ];
113 -> 34;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 114 [arrowhead = none ];
114 -> 35;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 115 [arrowhead = none ];
115 -> 36;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 116 [arrowhead = none ];
116 -> 37;

117 [label="BACKTRACK\nfor clause: tail(.(X4, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
36 -> 117 [arrowhead = none ];
117 -> 43;

118 [label="EVAL with clause\napp([], X59, X60) :- ','(!, eq(X59, X60)).\nand substitution\nT27 -> T29\nX59 -> .(T29, [])\nX53 -> X61\nX60 -> X61\nT28 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 118 [arrowhead = none ];
118 -> 38;

119 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 119 [arrowhead = none ];
119 -> 39;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 120 [arrowhead = none ];
120 -> 40;

121 [label="EVAL with clause\neq(X63, X63).\nand substitution\nT29 -> T30\nX63 -> .(T30, [])\nX61 -> .(T30, [])", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 121 [arrowhead = none ];
121 -> 41;

122 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 122 [arrowhead = none ];
122 -> 42;

123 [label="EVAL with clause\nhead(.(X68, X69), X68).\nand substitution\nX68 -> T33\nX69 -> T34\nT25 -> .(T33, T34)\nX52 -> T33\nT31 -> T33\nT32 -> T34\nT26 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 123 [arrowhead = none ];
123 -> 44;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 124 [arrowhead = none ];
124 -> 45;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 125 [arrowhead = none ];
125 -> 46;

126 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
46 -> 126 [arrowhead = none ];
126 -> 47;

127 [label="EVAL with clause\ntail(.(X72, X73), X73).\nand substitution\nT33 -> T36\nX72 -> T36\nT34 -> T38\nX73 -> T38\nX51 -> T38\nT37 -> T38\nT35 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 127 [arrowhead = none ];
127 -> 48;

128 [label="INSTANCE with matching:\nT15 -> T38\nT16 -> T39\nX35 -> X53", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 128 [arrowhead = none , style=dashed];
128 -> 20[style=dashed];

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 129 [arrowhead = none ];
129 -> 51;

130 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
51 -> 130 [arrowhead = none ];
130 -> 52;

131 [label="EVAL with clause\ntail(.(X80, X81), X81).\nand substitution\nT42 -> T45\nX80 -> T45\nT43 -> T47\nX81 -> T47\nX51 -> T47\nT46 -> T47\nT44 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 131 [arrowhead = none ];
131 -> 53;

132 [label="INSTANCE with matching:\nT15 -> T47\nT16 -> T48\nX35 -> X53", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 132 [arrowhead = none , style=dashed];
132 -> 20[style=dashed];

133 [label="EVAL with clause\nlast(X83, .(X83, [])) :- !.\nand substitution\nT19 -> T49\nX83 -> T49\nT18 -> .(T49, [])", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 133 [arrowhead = none ];
133 -> 55;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 134 [arrowhead = none ];
134 -> 56;

135 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 135 [arrowhead = none ];
135 -> 57;

136 [label="EVAL with clause\nlast(X86, X87) :- ','(tail(X87, X88), last(X86, X88)).\nand substitution\nT19 -> T53\nX86 -> T53\nT18 -> T52\nX87 -> T52\nT51 -> T52\nT50 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 136 [arrowhead = none ];
136 -> 59;

137 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 137 [arrowhead = none ];
137 -> 58;

138 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 138 [arrowhead = none ];
138 -> 60;

139 [label="EVAL with clause\ntail([], []).\nand substitution\nT52 -> []\nX88 -> []\nT53 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 139 [arrowhead = none ];
139 -> 61;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 140 [arrowhead = none ];
140 -> 62;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 141 [arrowhead = none ];
141 -> 63;

142 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 142 [arrowhead = none ];
142 -> 64;

143 [label="EVAL with clause\ntail(.(X103, X104), X104).\nand substitution\nX103 -> T61\nX104 -> T64\nT52 -> .(T61, T64)\nX88 -> T64\nT53 -> T63\nT62 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 143 [arrowhead = none ];
143 -> 75;

144 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 144 [arrowhead = none ];
144 -> 76;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 145 [arrowhead = none ];
145 -> 65;

146 [label="EVAL with clause\ntail(.(X99, X100), X100).\nand substitution\nX99 -> T57\nX100 -> T60\nT52 -> .(T57, T60)\nX88 -> T60\nT53 -> T59\nT58 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 146 [arrowhead = none ];
146 -> 73;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 147 [arrowhead = none ];
147 -> 74;

148 [label="BACKTRACK\nfor clause: last(X, .(X, [])) :- !because of non-unification", fontsize=14, style = filled, fillcolor = white];
65 -> 148 [arrowhead = none ];
148 -> 66;

149 [label="EVAL with clause\nlast(X92, X93) :- ','(tail(X93, X94), last(X92, X94)).\nand substitution\nT54 -> T56\nX92 -> T56\nX93 -> []\nT55 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 149 [arrowhead = none ];
149 -> 67;

150 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 150 [arrowhead = none ];
150 -> 68;

151 [label="EVAL with clause\ntail([], []).\nand substitution\nX94 -> []", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 151 [arrowhead = none ];
151 -> 69;

152 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 152 [arrowhead = none ];
152 -> 70;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 153 [arrowhead = none ];
153 -> 71;

154 [label="INSTANCE with matching:\nT54 -> T56", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 154 [arrowhead = none , style=dashed];
154 -> 63[style=dashed];

155 [label="BACKTRACK\nfor clause: tail(.(X4, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
71 -> 155 [arrowhead = none ];
155 -> 72;

156 [label="INSTANCE with matching:\nT19 -> T59\nT18 -> T60", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 156 [arrowhead = none , style=dashed];
156 -> 21[style=dashed];

157 [label="INSTANCE with matching:\nT19 -> T63\nT18 -> T64", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 157 [arrowhead = none , style=dashed];
157 -> 21[style=dashed];

}

(2) Obligation:

Clauses:

s2l4(0, []).
s2l4(s(T11), .(X25, X26)) :- s2l4(T11, X26).
app20([], T22, .(T22, [])).
app20([], T30, .(X56, .(T30, []))).
app20(.(T36, T38), T39, .(T36, X53)) :- app20(T38, T39, X53).
app20(.(T45, T47), T48, .(T45, X53)) :- app20(T47, T48, X53).
last63(T56) :- last63(T56).
last21(T49, .(T49, [])).
last21(T56, []) :- last63(T56).
last21(T59, .(T57, T60)) :- last21(T59, T60).
last21(T63, .(T61, T64)) :- last21(T63, T64).
goal1(T4, T7, T8) :- s2l4(T4, X11).
goal1(T4, T16, T17) :- ','(s2l4(T4, T15), app20(T15, T16, X35)).
goal1(T4, T16, T49) :- ','(s2l4(T4, T15), app20(T15, T16, .(T49, []))).
goal1(T4, T16, T56) :- ','(s2l4(T4, T15), ','(app20(T15, T16, []), last63(T56))).
goal1(T4, T16, T59) :- ','(s2l4(T4, T15), ','(app20(T15, T16, .(T57, T60)), last21(T59, T60))).
goal1(T4, T16, T63) :- ','(s2l4(T4, T15), ','(app20(T15, T16, .(T61, T64)), last21(T63, T64))).

Queries:

goal1(g,a,a).

(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,f,f)
s2l4_in: (b,f)
app20_in: (b,f,f) (b,f,b)
last63_in: (f)
last21_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
goal1_out_gaa(x1, x2, x3)  =  goal1_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag(x1, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x2, x4, x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x1, x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x1, x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x3, x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x3, x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x1, x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x1, x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x1, x4)

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_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
goal1_out_gaa(x1, x2, x3)  =  goal1_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag(x1, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x2, x4, x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x1, x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x1, x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x3, x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x3, x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x1, x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x1, x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x1, x4)

(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_GAA(T4, T7, T8) → U8_GAA(T4, T7, T8, s2l4_in_ga(T4, X11))
GOAL1_IN_GAA(T4, T7, T8) → S2L4_IN_GA(T4, X11)
S2L4_IN_GA(s(T11), .(X25, X26)) → U1_GA(T11, X25, X26, s2l4_in_ga(T11, X26))
S2L4_IN_GA(s(T11), .(X25, X26)) → S2L4_IN_GA(T11, X26)
GOAL1_IN_GAA(T4, T16, T17) → U9_GAA(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_GAA(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_GAA(T4, T16, T17, app20_in_gaa(T15, T16, X35))
U9_GAA(T4, T16, T17, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, X35)
APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → U2_GAA(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAA(T38, T39, X53)
APP20_IN_GAA(.(T45, T47), T48, .(T45, X53)) → U3_GAA(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
GOAL1_IN_GAA(T4, T16, T49) → U11_GAA(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_GAA(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_GAA(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
U11_GAA(T4, T16, T49, s2l4_out_ga(T4, T15)) → APP20_IN_GAG(T15, T16, .(T49, []))
APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → U2_GAG(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAG(T38, T39, X53)
APP20_IN_GAG(.(T45, T47), T48, .(T45, X53)) → U3_GAG(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
GOAL1_IN_GAA(T4, T16, T56) → U13_GAA(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_GAA(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_GAA(T4, T16, T56, app20_in_gag(T15, T16, []))
U13_GAA(T4, T16, T56, s2l4_out_ga(T4, T15)) → APP20_IN_GAG(T15, T16, [])
U14_GAA(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_GAA(T4, T16, T56, last63_in_a(T56))
U14_GAA(T4, T16, T56, app20_out_gag(T15, T16, [])) → LAST63_IN_A(T56)
LAST63_IN_A(T56) → U4_A(T56, last63_in_a(T56))
LAST63_IN_A(T56) → LAST63_IN_A(T56)
GOAL1_IN_GAA(T4, T16, T59) → U16_GAA(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_GAA(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_GAA(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U16_GAA(T4, T16, T59, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, .(T57, T60))
U17_GAA(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_GAA(T4, T16, T59, last21_in_ag(T59, T60))
U17_GAA(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → LAST21_IN_AG(T59, T60)
LAST21_IN_AG(T56, []) → U5_AG(T56, last63_in_a(T56))
LAST21_IN_AG(T56, []) → LAST63_IN_A(T56)
LAST21_IN_AG(T59, .(T57, T60)) → U6_AG(T59, T57, T60, last21_in_ag(T59, T60))
LAST21_IN_AG(T59, .(T57, T60)) → LAST21_IN_AG(T59, T60)
LAST21_IN_AG(T63, .(T61, T64)) → U7_AG(T63, T61, T64, last21_in_ag(T63, T64))
GOAL1_IN_GAA(T4, T16, T63) → U19_GAA(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_GAA(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_GAA(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U19_GAA(T4, T16, T63, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, .(T61, T64))
U20_GAA(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_GAA(T4, T16, T63, last21_in_ag(T63, T64))
U20_GAA(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → LAST21_IN_AG(T63, T64)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
goal1_out_gaa(x1, x2, x3)  =  goal1_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag(x1, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x2, x4, x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x1, x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x1, x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x3, x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x3, x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x1, x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x1, x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x1, x4)
GOAL1_IN_GAA(x1, x2, x3)  =  GOAL1_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
APP20_IN_GAA(x1, x2, x3)  =  APP20_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, x5)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x4)
U12_GAA(x1, x2, x3, x4)  =  U12_GAA(x1, x4)
APP20_IN_GAG(x1, x2, x3)  =  APP20_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x2, x4, x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x2, x4, x5)
U13_GAA(x1, x2, x3, x4)  =  U13_GAA(x1, x4)
U14_GAA(x1, x2, x3, x4)  =  U14_GAA(x1, x4)
U15_GAA(x1, x2, x3, x4)  =  U15_GAA(x1, x4)
LAST63_IN_A(x1)  =  LAST63_IN_A
U4_A(x1, x2)  =  U4_A(x2)
U16_GAA(x1, x2, x3, x4)  =  U16_GAA(x1, x4)
U17_GAA(x1, x2, x3, x4)  =  U17_GAA(x1, x4)
U18_GAA(x1, x2, x3, x4)  =  U18_GAA(x1, x4)
LAST21_IN_AG(x1, x2)  =  LAST21_IN_AG(x2)
U5_AG(x1, x2)  =  U5_AG(x2)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x3, x4)
U7_AG(x1, x2, x3, x4)  =  U7_AG(x3, x4)
U19_GAA(x1, x2, x3, x4)  =  U19_GAA(x1, x4)
U20_GAA(x1, x2, x3, x4)  =  U20_GAA(x1, x4)
U21_GAA(x1, x2, x3, x4)  =  U21_GAA(x1, x4)

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

(6) Obligation:

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

GOAL1_IN_GAA(T4, T7, T8) → U8_GAA(T4, T7, T8, s2l4_in_ga(T4, X11))
GOAL1_IN_GAA(T4, T7, T8) → S2L4_IN_GA(T4, X11)
S2L4_IN_GA(s(T11), .(X25, X26)) → U1_GA(T11, X25, X26, s2l4_in_ga(T11, X26))
S2L4_IN_GA(s(T11), .(X25, X26)) → S2L4_IN_GA(T11, X26)
GOAL1_IN_GAA(T4, T16, T17) → U9_GAA(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_GAA(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_GAA(T4, T16, T17, app20_in_gaa(T15, T16, X35))
U9_GAA(T4, T16, T17, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, X35)
APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → U2_GAA(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAA(T38, T39, X53)
APP20_IN_GAA(.(T45, T47), T48, .(T45, X53)) → U3_GAA(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
GOAL1_IN_GAA(T4, T16, T49) → U11_GAA(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_GAA(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_GAA(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
U11_GAA(T4, T16, T49, s2l4_out_ga(T4, T15)) → APP20_IN_GAG(T15, T16, .(T49, []))
APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → U2_GAG(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAG(T38, T39, X53)
APP20_IN_GAG(.(T45, T47), T48, .(T45, X53)) → U3_GAG(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
GOAL1_IN_GAA(T4, T16, T56) → U13_GAA(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_GAA(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_GAA(T4, T16, T56, app20_in_gag(T15, T16, []))
U13_GAA(T4, T16, T56, s2l4_out_ga(T4, T15)) → APP20_IN_GAG(T15, T16, [])
U14_GAA(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_GAA(T4, T16, T56, last63_in_a(T56))
U14_GAA(T4, T16, T56, app20_out_gag(T15, T16, [])) → LAST63_IN_A(T56)
LAST63_IN_A(T56) → U4_A(T56, last63_in_a(T56))
LAST63_IN_A(T56) → LAST63_IN_A(T56)
GOAL1_IN_GAA(T4, T16, T59) → U16_GAA(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_GAA(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_GAA(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U16_GAA(T4, T16, T59, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, .(T57, T60))
U17_GAA(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_GAA(T4, T16, T59, last21_in_ag(T59, T60))
U17_GAA(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → LAST21_IN_AG(T59, T60)
LAST21_IN_AG(T56, []) → U5_AG(T56, last63_in_a(T56))
LAST21_IN_AG(T56, []) → LAST63_IN_A(T56)
LAST21_IN_AG(T59, .(T57, T60)) → U6_AG(T59, T57, T60, last21_in_ag(T59, T60))
LAST21_IN_AG(T59, .(T57, T60)) → LAST21_IN_AG(T59, T60)
LAST21_IN_AG(T63, .(T61, T64)) → U7_AG(T63, T61, T64, last21_in_ag(T63, T64))
GOAL1_IN_GAA(T4, T16, T63) → U19_GAA(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_GAA(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_GAA(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U19_GAA(T4, T16, T63, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, .(T61, T64))
U20_GAA(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_GAA(T4, T16, T63, last21_in_ag(T63, T64))
U20_GAA(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → LAST21_IN_AG(T63, T64)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
goal1_out_gaa(x1, x2, x3)  =  goal1_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag(x1, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x2, x4, x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x1, x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x1, x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x3, x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x3, x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x1, x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x1, x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x1, x4)
GOAL1_IN_GAA(x1, x2, x3)  =  GOAL1_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
APP20_IN_GAA(x1, x2, x3)  =  APP20_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, x5)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x4)
U12_GAA(x1, x2, x3, x4)  =  U12_GAA(x1, x4)
APP20_IN_GAG(x1, x2, x3)  =  APP20_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x2, x4, x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x2, x4, x5)
U13_GAA(x1, x2, x3, x4)  =  U13_GAA(x1, x4)
U14_GAA(x1, x2, x3, x4)  =  U14_GAA(x1, x4)
U15_GAA(x1, x2, x3, x4)  =  U15_GAA(x1, x4)
LAST63_IN_A(x1)  =  LAST63_IN_A
U4_A(x1, x2)  =  U4_A(x2)
U16_GAA(x1, x2, x3, x4)  =  U16_GAA(x1, x4)
U17_GAA(x1, x2, x3, x4)  =  U17_GAA(x1, x4)
U18_GAA(x1, x2, x3, x4)  =  U18_GAA(x1, x4)
LAST21_IN_AG(x1, x2)  =  LAST21_IN_AG(x2)
U5_AG(x1, x2)  =  U5_AG(x2)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x3, x4)
U7_AG(x1, x2, x3, x4)  =  U7_AG(x3, x4)
U19_GAA(x1, x2, x3, x4)  =  U19_GAA(x1, x4)
U20_GAA(x1, x2, x3, x4)  =  U20_GAA(x1, x4)
U21_GAA(x1, x2, x3, x4)  =  U21_GAA(x1, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 33 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LAST63_IN_A(T56) → LAST63_IN_A(T56)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
goal1_out_gaa(x1, x2, x3)  =  goal1_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag(x1, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x2, x4, x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x1, x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x1, x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x3, x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x3, x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x1, x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x1, x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x1, x4)
LAST63_IN_A(x1)  =  LAST63_IN_A

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:

LAST63_IN_A(T56) → LAST63_IN_A(T56)

R is empty.
The argument filtering Pi contains the following mapping:
LAST63_IN_A(x1)  =  LAST63_IN_A

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:

LAST63_IN_A → LAST63_IN_A

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

(14) NonTerminationProof (EQUIVALENT transformation)

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

s = LAST63_IN_A evaluates to t =LAST63_IN_A

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LAST63_IN_A to LAST63_IN_A.

(15) FALSE

(16) Obligation:

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

LAST21_IN_AG(T59, .(T57, T60)) → LAST21_IN_AG(T59, T60)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
goal1_out_gaa(x1, x2, x3)  =  goal1_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag(x1, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x2, x4, x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x1, x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x1, x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x3, x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x3, x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x1, x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x1, x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x1, x4)
LAST21_IN_AG(x1, x2)  =  LAST21_IN_AG(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:

LAST21_IN_AG(T59, .(T57, T60)) → LAST21_IN_AG(T59, T60)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
LAST21_IN_AG(x1, x2)  =  LAST21_IN_AG(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:

LAST21_IN_AG(.(T60)) → LAST21_IN_AG(T60)

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:

• LAST21_IN_AG(.(T60)) → LAST21_IN_AG(T60)
  The graph contains the following edges 1 > 1

(22) TRUE

(23) Obligation:

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

APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAG(T38, T39, X53)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
goal1_out_gaa(x1, x2, x3)  =  goal1_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag(x1, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x2, x4, x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x1, x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x1, x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x3, x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x3, x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x1, x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x1, x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x1, x4)
APP20_IN_GAG(x1, x2, x3)  =  APP20_IN_GAG(x1, x3)

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:

APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAG(T38, T39, X53)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP20_IN_GAG(x1, x2, x3)  =  APP20_IN_GAG(x1, x3)

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:

APP20_IN_GAG(.(T38), .(X53)) → APP20_IN_GAG(T38, X53)

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:

• APP20_IN_GAG(.(T38), .(X53)) → APP20_IN_GAG(T38, X53)
  The graph contains the following edges 1 > 1, 2 > 2

(29) TRUE

(30) Obligation:

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

APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAA(T38, T39, X53)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
goal1_out_gaa(x1, x2, x3)  =  goal1_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag(x1, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x2, x4, x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x1, x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x1, x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x3, x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x3, x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x1, x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x1, x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x1, x4)
APP20_IN_GAA(x1, x2, x3)  =  APP20_IN_GAA(x1)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAA(T38, T39, X53)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP20_IN_GAA(x1, x2, x3)  =  APP20_IN_GAA(x1)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

APP20_IN_GAA(.(T38)) → APP20_IN_GAA(T38)

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

(35) 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:

• APP20_IN_GAA(.(T38)) → APP20_IN_GAA(T38)
  The graph contains the following edges 1 > 1

(36) TRUE

(37) Obligation:

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

S2L4_IN_GA(s(T11), .(X25, X26)) → S2L4_IN_GA(T11, X26)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
goal1_out_gaa(x1, x2, x3)  =  goal1_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x1, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag(x1, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x2, x4, x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x1, x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x1, x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x1, x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x3, x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x3, x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x1, x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x1, x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x1, x4)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

S2L4_IN_GA(s(T11), .(X25, X26)) → S2L4_IN_GA(T11, 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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

S2L4_IN_GA(s(T11)) → S2L4_IN_GA(T11)

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

(42) 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(T11)) → S2L4_IN_GA(T11)
  The graph contains the following edges 1 > 1

(43) TRUE

(44) 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,f,f)
s2l4_in: (b,f)
app20_in: (b,f,f) (b,f,b)
last63_in: (f)
last21_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
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_gaa(x1, x2, x3)  =  goal1_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(45) Obligation:

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

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
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_gaa(x1, x2, x3)  =  goal1_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x4)

(46) 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_GAA(T4, T7, T8) → U8_GAA(T4, T7, T8, s2l4_in_ga(T4, X11))
GOAL1_IN_GAA(T4, T7, T8) → S2L4_IN_GA(T4, X11)
S2L4_IN_GA(s(T11), .(X25, X26)) → U1_GA(T11, X25, X26, s2l4_in_ga(T11, X26))
S2L4_IN_GA(s(T11), .(X25, X26)) → S2L4_IN_GA(T11, X26)
GOAL1_IN_GAA(T4, T16, T17) → U9_GAA(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_GAA(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_GAA(T4, T16, T17, app20_in_gaa(T15, T16, X35))
U9_GAA(T4, T16, T17, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, X35)
APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → U2_GAA(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAA(T38, T39, X53)
APP20_IN_GAA(.(T45, T47), T48, .(T45, X53)) → U3_GAA(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
GOAL1_IN_GAA(T4, T16, T49) → U11_GAA(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_GAA(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_GAA(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
U11_GAA(T4, T16, T49, s2l4_out_ga(T4, T15)) → APP20_IN_GAG(T15, T16, .(T49, []))
APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → U2_GAG(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAG(T38, T39, X53)
APP20_IN_GAG(.(T45, T47), T48, .(T45, X53)) → U3_GAG(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
GOAL1_IN_GAA(T4, T16, T56) → U13_GAA(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_GAA(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_GAA(T4, T16, T56, app20_in_gag(T15, T16, []))
U13_GAA(T4, T16, T56, s2l4_out_ga(T4, T15)) → APP20_IN_GAG(T15, T16, [])
U14_GAA(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_GAA(T4, T16, T56, last63_in_a(T56))
U14_GAA(T4, T16, T56, app20_out_gag(T15, T16, [])) → LAST63_IN_A(T56)
LAST63_IN_A(T56) → U4_A(T56, last63_in_a(T56))
LAST63_IN_A(T56) → LAST63_IN_A(T56)
GOAL1_IN_GAA(T4, T16, T59) → U16_GAA(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_GAA(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_GAA(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U16_GAA(T4, T16, T59, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, .(T57, T60))
U17_GAA(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_GAA(T4, T16, T59, last21_in_ag(T59, T60))
U17_GAA(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → LAST21_IN_AG(T59, T60)
LAST21_IN_AG(T56, []) → U5_AG(T56, last63_in_a(T56))
LAST21_IN_AG(T56, []) → LAST63_IN_A(T56)
LAST21_IN_AG(T59, .(T57, T60)) → U6_AG(T59, T57, T60, last21_in_ag(T59, T60))
LAST21_IN_AG(T59, .(T57, T60)) → LAST21_IN_AG(T59, T60)
LAST21_IN_AG(T63, .(T61, T64)) → U7_AG(T63, T61, T64, last21_in_ag(T63, T64))
GOAL1_IN_GAA(T4, T16, T63) → U19_GAA(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_GAA(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_GAA(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U19_GAA(T4, T16, T63, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, .(T61, T64))
U20_GAA(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_GAA(T4, T16, T63, last21_in_ag(T63, T64))
U20_GAA(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → LAST21_IN_AG(T63, T64)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
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_gaa(x1, x2, x3)  =  goal1_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x4)
GOAL1_IN_GAA(x1, x2, x3)  =  GOAL1_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x4)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x4)
APP20_IN_GAA(x1, x2, x3)  =  APP20_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x4)
U12_GAA(x1, x2, x3, x4)  =  U12_GAA(x4)
APP20_IN_GAG(x1, x2, x3)  =  APP20_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x5)
U13_GAA(x1, x2, x3, x4)  =  U13_GAA(x4)
U14_GAA(x1, x2, x3, x4)  =  U14_GAA(x4)
U15_GAA(x1, x2, x3, x4)  =  U15_GAA(x4)
LAST63_IN_A(x1)  =  LAST63_IN_A
U4_A(x1, x2)  =  U4_A(x2)
U16_GAA(x1, x2, x3, x4)  =  U16_GAA(x4)
U17_GAA(x1, x2, x3, x4)  =  U17_GAA(x4)
U18_GAA(x1, x2, x3, x4)  =  U18_GAA(x4)
LAST21_IN_AG(x1, x2)  =  LAST21_IN_AG(x2)
U5_AG(x1, x2)  =  U5_AG(x2)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x4)
U7_AG(x1, x2, x3, x4)  =  U7_AG(x4)
U19_GAA(x1, x2, x3, x4)  =  U19_GAA(x4)
U20_GAA(x1, x2, x3, x4)  =  U20_GAA(x4)
U21_GAA(x1, x2, x3, x4)  =  U21_GAA(x4)

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

(47) Obligation:

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

GOAL1_IN_GAA(T4, T7, T8) → U8_GAA(T4, T7, T8, s2l4_in_ga(T4, X11))
GOAL1_IN_GAA(T4, T7, T8) → S2L4_IN_GA(T4, X11)
S2L4_IN_GA(s(T11), .(X25, X26)) → U1_GA(T11, X25, X26, s2l4_in_ga(T11, X26))
S2L4_IN_GA(s(T11), .(X25, X26)) → S2L4_IN_GA(T11, X26)
GOAL1_IN_GAA(T4, T16, T17) → U9_GAA(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_GAA(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_GAA(T4, T16, T17, app20_in_gaa(T15, T16, X35))
U9_GAA(T4, T16, T17, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, X35)
APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → U2_GAA(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAA(T38, T39, X53)
APP20_IN_GAA(.(T45, T47), T48, .(T45, X53)) → U3_GAA(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
GOAL1_IN_GAA(T4, T16, T49) → U11_GAA(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_GAA(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_GAA(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
U11_GAA(T4, T16, T49, s2l4_out_ga(T4, T15)) → APP20_IN_GAG(T15, T16, .(T49, []))
APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → U2_GAG(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAG(T38, T39, X53)
APP20_IN_GAG(.(T45, T47), T48, .(T45, X53)) → U3_GAG(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
GOAL1_IN_GAA(T4, T16, T56) → U13_GAA(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_GAA(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_GAA(T4, T16, T56, app20_in_gag(T15, T16, []))
U13_GAA(T4, T16, T56, s2l4_out_ga(T4, T15)) → APP20_IN_GAG(T15, T16, [])
U14_GAA(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_GAA(T4, T16, T56, last63_in_a(T56))
U14_GAA(T4, T16, T56, app20_out_gag(T15, T16, [])) → LAST63_IN_A(T56)
LAST63_IN_A(T56) → U4_A(T56, last63_in_a(T56))
LAST63_IN_A(T56) → LAST63_IN_A(T56)
GOAL1_IN_GAA(T4, T16, T59) → U16_GAA(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_GAA(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_GAA(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U16_GAA(T4, T16, T59, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, .(T57, T60))
U17_GAA(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_GAA(T4, T16, T59, last21_in_ag(T59, T60))
U17_GAA(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → LAST21_IN_AG(T59, T60)
LAST21_IN_AG(T56, []) → U5_AG(T56, last63_in_a(T56))
LAST21_IN_AG(T56, []) → LAST63_IN_A(T56)
LAST21_IN_AG(T59, .(T57, T60)) → U6_AG(T59, T57, T60, last21_in_ag(T59, T60))
LAST21_IN_AG(T59, .(T57, T60)) → LAST21_IN_AG(T59, T60)
LAST21_IN_AG(T63, .(T61, T64)) → U7_AG(T63, T61, T64, last21_in_ag(T63, T64))
GOAL1_IN_GAA(T4, T16, T63) → U19_GAA(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_GAA(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_GAA(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U19_GAA(T4, T16, T63, s2l4_out_ga(T4, T15)) → APP20_IN_GAA(T15, T16, .(T61, T64))
U20_GAA(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_GAA(T4, T16, T63, last21_in_ag(T63, T64))
U20_GAA(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → LAST21_IN_AG(T63, T64)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
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_gaa(x1, x2, x3)  =  goal1_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x4)
GOAL1_IN_GAA(x1, x2, x3)  =  GOAL1_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x4)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x4)
APP20_IN_GAA(x1, x2, x3)  =  APP20_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x4)
U12_GAA(x1, x2, x3, x4)  =  U12_GAA(x4)
APP20_IN_GAG(x1, x2, x3)  =  APP20_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x5)
U13_GAA(x1, x2, x3, x4)  =  U13_GAA(x4)
U14_GAA(x1, x2, x3, x4)  =  U14_GAA(x4)
U15_GAA(x1, x2, x3, x4)  =  U15_GAA(x4)
LAST63_IN_A(x1)  =  LAST63_IN_A
U4_A(x1, x2)  =  U4_A(x2)
U16_GAA(x1, x2, x3, x4)  =  U16_GAA(x4)
U17_GAA(x1, x2, x3, x4)  =  U17_GAA(x4)
U18_GAA(x1, x2, x3, x4)  =  U18_GAA(x4)
LAST21_IN_AG(x1, x2)  =  LAST21_IN_AG(x2)
U5_AG(x1, x2)  =  U5_AG(x2)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x4)
U7_AG(x1, x2, x3, x4)  =  U7_AG(x4)
U19_GAA(x1, x2, x3, x4)  =  U19_GAA(x4)
U20_GAA(x1, x2, x3, x4)  =  U20_GAA(x4)
U21_GAA(x1, x2, x3, x4)  =  U21_GAA(x4)

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

(48) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 33 less nodes.

(49) Complex Obligation (AND)

(50) Obligation:

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

LAST63_IN_A(T56) → LAST63_IN_A(T56)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
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_gaa(x1, x2, x3)  =  goal1_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x4)
LAST63_IN_A(x1)  =  LAST63_IN_A

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

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

LAST63_IN_A(T56) → LAST63_IN_A(T56)

R is empty.
The argument filtering Pi contains the following mapping:
LAST63_IN_A(x1)  =  LAST63_IN_A

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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) Obligation:

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

LAST63_IN_A → LAST63_IN_A

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

(55) NonTerminationProof (EQUIVALENT transformation)

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

s = LAST63_IN_A evaluates to t =LAST63_IN_A

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LAST63_IN_A to LAST63_IN_A.

(56) FALSE

(57) Obligation:

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

LAST21_IN_AG(T59, .(T57, T60)) → LAST21_IN_AG(T59, T60)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
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_gaa(x1, x2, x3)  =  goal1_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x4)
LAST21_IN_AG(x1, x2)  =  LAST21_IN_AG(x2)

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

(58) UsableRulesProof (EQUIVALENT transformation)

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

(59) Obligation:

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

LAST21_IN_AG(T59, .(T57, T60)) → LAST21_IN_AG(T59, T60)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
LAST21_IN_AG(x1, x2)  =  LAST21_IN_AG(x2)

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

(60) PiDPToQDPProof (SOUND transformation)

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

(61) Obligation:

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

LAST21_IN_AG(.(T60)) → LAST21_IN_AG(T60)

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

(62) 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:

• LAST21_IN_AG(.(T60)) → LAST21_IN_AG(T60)
  The graph contains the following edges 1 > 1

(63) TRUE

(64) Obligation:

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

APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAG(T38, T39, X53)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
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_gaa(x1, x2, x3)  =  goal1_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x4)
APP20_IN_GAG(x1, x2, x3)  =  APP20_IN_GAG(x1, x3)

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

(65) UsableRulesProof (EQUIVALENT transformation)

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

(66) Obligation:

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

APP20_IN_GAG(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAG(T38, T39, X53)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP20_IN_GAG(x1, x2, x3)  =  APP20_IN_GAG(x1, x3)

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

(67) PiDPToQDPProof (SOUND transformation)

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

(68) Obligation:

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

APP20_IN_GAG(.(T38), .(X53)) → APP20_IN_GAG(T38, X53)

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

(69) 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:

• APP20_IN_GAG(.(T38), .(X53)) → APP20_IN_GAG(T38, X53)
  The graph contains the following edges 1 > 1, 2 > 2

(70) TRUE

(71) Obligation:

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

APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAA(T38, T39, X53)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
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_gaa(x1, x2, x3)  =  goal1_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x4)
APP20_IN_GAA(x1, x2, x3)  =  APP20_IN_GAA(x1)

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

(72) UsableRulesProof (EQUIVALENT transformation)

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

(73) Obligation:

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

APP20_IN_GAA(.(T36, T38), T39, .(T36, X53)) → APP20_IN_GAA(T38, T39, X53)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP20_IN_GAA(x1, x2, x3)  =  APP20_IN_GAA(x1)

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

(74) PiDPToQDPProof (SOUND transformation)

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

(75) Obligation:

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

APP20_IN_GAA(.(T38)) → APP20_IN_GAA(T38)

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

(76) 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:

• APP20_IN_GAA(.(T38)) → APP20_IN_GAA(T38)
  The graph contains the following edges 1 > 1

(77) TRUE

(78) Obligation:

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

S2L4_IN_GA(s(T11), .(X25, X26)) → S2L4_IN_GA(T11, X26)

The TRS R consists of the following rules:

goal1_in_gaa(T4, T7, T8) → U8_gaa(T4, T7, T8, s2l4_in_ga(T4, X11))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T11), .(X25, X26)) → U1_ga(T11, X25, X26, s2l4_in_ga(T11, X26))
U1_ga(T11, X25, X26, s2l4_out_ga(T11, X26)) → s2l4_out_ga(s(T11), .(X25, X26))
U8_gaa(T4, T7, T8, s2l4_out_ga(T4, X11)) → goal1_out_gaa(T4, T7, T8)
goal1_in_gaa(T4, T16, T17) → U9_gaa(T4, T16, T17, s2l4_in_ga(T4, T15))
U9_gaa(T4, T16, T17, s2l4_out_ga(T4, T15)) → U10_gaa(T4, T16, T17, app20_in_gaa(T15, T16, X35))
app20_in_gaa([], T22, .(T22, [])) → app20_out_gaa([], T22, .(T22, []))
app20_in_gaa([], T30, .(X56, .(T30, []))) → app20_out_gaa([], T30, .(X56, .(T30, [])))
app20_in_gaa(.(T36, T38), T39, .(T36, X53)) → U2_gaa(T36, T38, T39, X53, app20_in_gaa(T38, T39, X53))
app20_in_gaa(.(T45, T47), T48, .(T45, X53)) → U3_gaa(T45, T47, T48, X53, app20_in_gaa(T47, T48, X53))
U3_gaa(T45, T47, T48, X53, app20_out_gaa(T47, T48, X53)) → app20_out_gaa(.(T45, T47), T48, .(T45, X53))
U2_gaa(T36, T38, T39, X53, app20_out_gaa(T38, T39, X53)) → app20_out_gaa(.(T36, T38), T39, .(T36, X53))
U10_gaa(T4, T16, T17, app20_out_gaa(T15, T16, X35)) → goal1_out_gaa(T4, T16, T17)
goal1_in_gaa(T4, T16, T49) → U11_gaa(T4, T16, T49, s2l4_in_ga(T4, T15))
U11_gaa(T4, T16, T49, s2l4_out_ga(T4, T15)) → U12_gaa(T4, T16, T49, app20_in_gag(T15, T16, .(T49, [])))
app20_in_gag([], T22, .(T22, [])) → app20_out_gag([], T22, .(T22, []))
app20_in_gag([], T30, .(X56, .(T30, []))) → app20_out_gag([], T30, .(X56, .(T30, [])))
app20_in_gag(.(T36, T38), T39, .(T36, X53)) → U2_gag(T36, T38, T39, X53, app20_in_gag(T38, T39, X53))
app20_in_gag(.(T45, T47), T48, .(T45, X53)) → U3_gag(T45, T47, T48, X53, app20_in_gag(T47, T48, X53))
U3_gag(T45, T47, T48, X53, app20_out_gag(T47, T48, X53)) → app20_out_gag(.(T45, T47), T48, .(T45, X53))
U2_gag(T36, T38, T39, X53, app20_out_gag(T38, T39, X53)) → app20_out_gag(.(T36, T38), T39, .(T36, X53))
U12_gaa(T4, T16, T49, app20_out_gag(T15, T16, .(T49, []))) → goal1_out_gaa(T4, T16, T49)
goal1_in_gaa(T4, T16, T56) → U13_gaa(T4, T16, T56, s2l4_in_ga(T4, T15))
U13_gaa(T4, T16, T56, s2l4_out_ga(T4, T15)) → U14_gaa(T4, T16, T56, app20_in_gag(T15, T16, []))
U14_gaa(T4, T16, T56, app20_out_gag(T15, T16, [])) → U15_gaa(T4, T16, T56, last63_in_a(T56))
last63_in_a(T56) → U4_a(T56, last63_in_a(T56))
U4_a(T56, last63_out_a(T56)) → last63_out_a(T56)
U15_gaa(T4, T16, T56, last63_out_a(T56)) → goal1_out_gaa(T4, T16, T56)
goal1_in_gaa(T4, T16, T59) → U16_gaa(T4, T16, T59, s2l4_in_ga(T4, T15))
U16_gaa(T4, T16, T59, s2l4_out_ga(T4, T15)) → U17_gaa(T4, T16, T59, app20_in_gaa(T15, T16, .(T57, T60)))
U17_gaa(T4, T16, T59, app20_out_gaa(T15, T16, .(T57, T60))) → U18_gaa(T4, T16, T59, last21_in_ag(T59, T60))
last21_in_ag(T49, .(T49, [])) → last21_out_ag(T49, .(T49, []))
last21_in_ag(T56, []) → U5_ag(T56, last63_in_a(T56))
U5_ag(T56, last63_out_a(T56)) → last21_out_ag(T56, [])
last21_in_ag(T59, .(T57, T60)) → U6_ag(T59, T57, T60, last21_in_ag(T59, T60))
last21_in_ag(T63, .(T61, T64)) → U7_ag(T63, T61, T64, last21_in_ag(T63, T64))
U7_ag(T63, T61, T64, last21_out_ag(T63, T64)) → last21_out_ag(T63, .(T61, T64))
U6_ag(T59, T57, T60, last21_out_ag(T59, T60)) → last21_out_ag(T59, .(T57, T60))
U18_gaa(T4, T16, T59, last21_out_ag(T59, T60)) → goal1_out_gaa(T4, T16, T59)
goal1_in_gaa(T4, T16, T63) → U19_gaa(T4, T16, T63, s2l4_in_ga(T4, T15))
U19_gaa(T4, T16, T63, s2l4_out_ga(T4, T15)) → U20_gaa(T4, T16, T63, app20_in_gaa(T15, T16, .(T61, T64)))
U20_gaa(T4, T16, T63, app20_out_gaa(T15, T16, .(T61, T64))) → U21_gaa(T4, T16, T63, last21_in_ag(T63, T64))
U21_gaa(T4, T16, T63, last21_out_ag(T63, T64)) → goal1_out_gaa(T4, T16, T63)

The argument filtering Pi contains the following mapping:
goal1_in_gaa(x1, x2, x3)  =  goal1_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
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_gaa(x1, x2, x3)  =  goal1_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
app20_in_gaa(x1, x2, x3)  =  app20_in_gaa(x1)
[]  =  []
app20_out_gaa(x1, x2, x3)  =  app20_out_gaa(x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x4)
app20_in_gag(x1, x2, x3)  =  app20_in_gag(x1, x3)
app20_out_gag(x1, x2, x3)  =  app20_out_gag
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U13_gaa(x1, x2, x3, x4)  =  U13_gaa(x4)
U14_gaa(x1, x2, x3, x4)  =  U14_gaa(x4)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
last63_in_a(x1)  =  last63_in_a
U4_a(x1, x2)  =  U4_a(x2)
last63_out_a(x1)  =  last63_out_a(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x4)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x4)
U18_gaa(x1, x2, x3, x4)  =  U18_gaa(x4)
last21_in_ag(x1, x2)  =  last21_in_ag(x2)
last21_out_ag(x1, x2)  =  last21_out_ag
U5_ag(x1, x2)  =  U5_ag(x2)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4)  =  U20_gaa(x4)
U21_gaa(x1, x2, x3, x4)  =  U21_gaa(x4)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)

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

(79) UsableRulesProof (EQUIVALENT transformation)

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

(80) Obligation:

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

S2L4_IN_GA(s(T11), .(X25, X26)) → S2L4_IN_GA(T11, 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

(81) PiDPToQDPProof (SOUND transformation)

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

(82) Obligation:

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

S2L4_IN_GA(s(T11)) → S2L4_IN_GA(T11)

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

(83) 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(T11)) → S2L4_IN_GA(T11)
  The graph contains the following edges 1 > 1

(84) TRUE