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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: goal(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(s2l(T7, X11), applast(X11, T10, T11))\n\nT7 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(s2l(T7, X11), applast(X11, T10, T11)), SCOPE: 2, CLAUSE: 6 | ','(s2l(T7, X11), applast(X11, T10, T11)), SCOPE: 2, CLAUSE: 7\n\nT7 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_2, ','(eq(X17, []), applast(X17, T10, T11))) | ','(s2l(0, X11), applast(X11, T10, T11)), SCOPE: 2, CLAUSE: 7\n\nX11 is free\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(s2l(T7, X11), applast(X11, T10, T11)), SCOPE: 2, CLAUSE: 7\n\nT7 is ground\nX11 is free; \ns2l(T7, X11) !=? s2l(0, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(eq(X17, []), applast(X17, T10, T11))\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(eq(X17, []), applast(X17, T10, T11)), SCOPE: 3, CLAUSE: 14\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: applast([], T10, T11)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: applast([], T10, T11), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(app([], .(T27, []), X51), last(T28, X51))\n\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app([], .(T27, []), X51)\n\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: last(T31, T30)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app([], .(T27, []), X51), SCOPE: 5, CLAUSE: 4 | app([], .(T27, []), X51), SCOPE: 5, CLAUSE: 5\n\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(!_5, eq(.(T37, []), X66)) | app([], .(T27, []), X51), SCOPE: 5, CLAUSE: 5\n\nX51 is free\nX66 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: eq(.(T37, []), X66)\n\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: eq(.(T37, []), X66), SCOPE: 6, CLAUSE: 14\n\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: last(T31, T30), SCOPE: 7, CLAUSE: 2 | last(T31, T30), SCOPE: 7, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: !_7 | last(T31, T30), SCOPE: 7, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: last(T31, T30), SCOPE: 7, CLAUSE: 3\n\nlast(T31, T30) !=? last(X73, .(X73, []))", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(tail(T54, X83), last(T55, X83))\n\nX83 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(tail(T54, X83), last(T55, X83)), SCOPE: 8, CLAUSE: 10 | ','(tail(T54, X83), last(T55, X83)), SCOPE: 8, CLAUSE: 11\n\nX83 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(tail(T54, X83), last(T55, X83)), SCOPE: 8, CLAUSE: 10\n\nX83 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(tail(T54, X83), last(T55, X83)), SCOPE: 8, CLAUSE: 11\n\nX83 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: last(T57, [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: last(T57, []), SCOPE: 9, CLAUSE: 2 | last(T57, []), SCOPE: 9, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: last(T57, []), SCOPE: 9, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(tail([], X98), last(T66, X98))\n\nX98 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(tail([], X98), last(T66, X98)), SCOPE: 10, CLAUSE: 10 | ','(tail([], X98), last(T66, X98)), SCOPE: 10, CLAUSE: 11\n\nX98 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(tail([], X98), last(T66, X98)), SCOPE: 10, CLAUSE: 10\n\nX98 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(tail([], X98), last(T66, X98)), SCOPE: 10, CLAUSE: 11\n\nX98 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: last(T66, [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: last(T80, T81)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(p(T85, X139), ','(s2l(X139, X141), applast(.(X140, X141), T10, T11)))\n\nT85 is ground\nX140 is free; \nX141 is free; \nX139 is free; \ns2l(T85, X11) !=? s2l(0, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(p(T85, X139), ','(s2l(X139, X141), applast(.(X140, X141), T10, T11))), SCOPE: 11, CLAUSE: 12 | ','(p(T85, X139), ','(s2l(X139, X141), applast(.(X140, X141), T10, T11))), SCOPE: 11, CLAUSE: 13\n\nT85 is ground\nX140 is free; \nX141 is free; \nX139 is free; \ns2l(T85, X11) !=? s2l(0, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(p(T85, X139), ','(s2l(X139, X141), applast(.(X140, X141), T10, T11))), SCOPE: 11, CLAUSE: 13\n\nT85 is ground\nX140 is free; \nX141 is free; \nX139 is free; \ns2l(T85, X11) !=? s2l(0, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(s2l(T88, X141), applast(.(X140, X141), T10, T11))\n\nT88 is ground\nX140 is free; \nX141 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: s2l(T88, X141)\n\nT88 is ground\nX141 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: applast(.(X140, T89), T10, T11)\n\nX140 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: s2l(T88, X141), SCOPE: 12, CLAUSE: 6 | s2l(T88, X141), SCOPE: 12, CLAUSE: 7\n\nT88 is ground\nX141 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(!_12, eq(X157, [])) | s2l(0, X141), SCOPE: 12, CLAUSE: 7\n\nX141 is free\nX157 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: s2l(T88, X141), SCOPE: 12, CLAUSE: 7\n\nT88 is ground\nX141 is free; \ns2l(T88, X141) !=? s2l(0, X156)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: eq(X157, [])\n\nX157 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: eq(X157, []), SCOPE: 13, CLAUSE: 14\n\nX157 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(p(T92, X181), s2l(X181, X183))\n\nT92 is ground\nX183 is free; \nX181 is free; \ns2l(T92, X141) !=? s2l(0, X156)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(p(T92, X181), s2l(X181, X183)), SCOPE: 14, CLAUSE: 12 | ','(p(T92, X181), s2l(X181, X183)), SCOPE: 14, CLAUSE: 13\n\nT92 is ground\nX183 is free; \nX181 is free; \ns2l(T92, X141) !=? s2l(0, X156)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(p(T92, X181), s2l(X181, X183)), SCOPE: 14, CLAUSE: 13\n\nT92 is ground\nX183 is free; \nX181 is free; \ns2l(T92, X141) !=? s2l(0, X156)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: s2l(T95, X183)\n\nT95 is ground\nX183 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: applast(.(X140, T89), T10, T11), SCOPE: 15, CLAUSE: 1\n\nX140 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(app(.(X220, T113), .(T114, []), X219), last(T115, X219))\n\nX220 is free\nX219 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: app(.(X220, T113), .(T114, []), X219)\n\nX220 is free\nX219 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: last(T120, T119)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: app(.(X220, T113), .(T114, []), X219), SCOPE: 16, CLAUSE: 4 | app(.(X220, T113), .(T114, []), X219), SCOPE: 16, CLAUSE: 5\n\nX220 is free\nX219 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: app(.(X220, T113), .(T114, []), X219), SCOPE: 16, CLAUSE: 5\n\nX220 is free\nX219 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(head(.(X266, T132), X267), ','(tail(.(X266, T132), X265), app(X265, .(T133, []), X268)))\n\nX266 is free\nX267 is free; \nX268 is free; \nX265 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(head(.(X266, T132), X267), ','(tail(.(X266, T132), X265), app(X265, .(T133, []), X268))), SCOPE: 17, CLAUSE: 8 | ','(head(.(X266, T132), X267), ','(tail(.(X266, T132), X265), app(X265, .(T133, []), X268))), SCOPE: 17, CLAUSE: 9\n\nX266 is free\nX267 is free; \nX268 is free; \nX265 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(head(.(X266, T132), X267), ','(tail(.(X266, T132), X265), app(X265, .(T133, []), X268))), SCOPE: 17, CLAUSE: 9\n\nX266 is free\nX267 is free; \nX268 is free; \nX265 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(tail(.(X284, T141), X265), app(X265, .(T142, []), X268))\n\nX268 is free\nX265 is free; \nX284 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(tail(.(X284, T141), X265), app(X265, .(T142, []), X268)), SCOPE: 18, CLAUSE: 10 | ','(tail(.(X284, T141), X265), app(X265, .(T142, []), X268)), SCOPE: 18, CLAUSE: 11\n\nX268 is free\nX265 is free; \nX284 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(tail(.(X284, T141), X265), app(X265, .(T142, []), X268)), SCOPE: 18, CLAUSE: 11\n\nX268 is free\nX265 is free; \nX284 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: app(T150, .(T151, []), X268)\n\nX268 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: app(T150, .(T151, []), X268), SCOPE: 19, CLAUSE: 4 | app(T150, .(T151, []), X268), SCOPE: 19, CLAUSE: 5\n\nX268 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(!_19, eq(.(T156, []), X311)) | app(T150, .(T151, []), X268), SCOPE: 19, CLAUSE: 5\n\nX268 is free\nX311 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: app(T150, .(T151, []), X268), SCOPE: 19, CLAUSE: 5\n\nX268 is free\napp(T150, .(T151, []), X268) !=? app([], X309, X310)", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: eq(.(T156, []), X311)\n\nX311 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: eq(.(T156, []), X311), SCOPE: 20, CLAUSE: 14\n\nX311 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(head(T169, X338), ','(tail(T169, X337), app(X337, .(T170, []), X339)))\n\nX338 is free\nX339 is free; \nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(head(T169, X338), ','(tail(T169, X337), app(X337, .(T170, []), X339))), SCOPE: 21, CLAUSE: 8 | ','(head(T169, X338), ','(tail(T169, X337), app(X337, .(T170, []), X339))), SCOPE: 21, CLAUSE: 9\n\nX338 is free\nX339 is free; \nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(head(T169, X338), ','(tail(T169, X337), app(X337, .(T170, []), X339))), SCOPE: 21, CLAUSE: 8\n\nX338 is free\nX339 is free; \nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(head(T169, X338), ','(tail(T169, X337), app(X337, .(T170, []), X339))), SCOPE: 21, CLAUSE: 9\n\nX338 is free\nX339 is free; \nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(tail([], X337), app(X337, .(T172, []), X339))\n\nX339 is free\nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(tail([], X337), app(X337, .(T172, []), X339)), SCOPE: 22, CLAUSE: 10 | ','(tail([], X337), app(X337, .(T172, []), X339)), SCOPE: 22, CLAUSE: 11\n\nX339 is free\nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(tail([], X337), app(X337, .(T172, []), X339)), SCOPE: 22, CLAUSE: 10\n\nX339 is free\nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(tail([], X337), app(X337, .(T172, []), X339)), SCOPE: 22, CLAUSE: 11\n\nX339 is free\nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: app([], .(T172, []), X339)\n\nX339 is free", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(tail(.(T183, T184), X337), app(X337, .(T185, []), X339))\n\nX339 is free\nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(tail(.(T183, T184), X337), app(X337, .(T185, []), X339)), SCOPE: 23, CLAUSE: 10 | ','(tail(.(T183, T184), X337), app(X337, .(T185, []), X339)), SCOPE: 23, CLAUSE: 11\n\nX339 is free\nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(tail(.(T183, T184), X337), app(X337, .(T185, []), X339)), SCOPE: 23, CLAUSE: 11\n\nX339 is free\nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: app(T196, .(T197, []), X339)\n\nX339 is free", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 96 [arrowhead = none ];
96 -> 2;

97 [label="ONLY EVAL with clause\ngoal(X8, X9, X10) :- ','(s2l(X8, X11), applast(X11, X9, X10)).\nand substitution\nT1 -> T7\nX8 -> T7\nT2 -> T10\nX9 -> T10\nT3 -> T11\nX10 -> T11\nT8 -> T10\nT9 -> T11", fontsize=14, style = filled, fillcolor = white];
2 -> 97 [arrowhead = none ];
97 -> 3;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 98 [arrowhead = none ];
98 -> 4;

99 [label="EVAL with clause\ns2l(0, X16) :- ','(!_2, eq(X16, [])).\nand substitution\nT7 -> 0\nX11 -> X17\nX16 -> X17", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 99 [arrowhead = none ];
99 -> 5;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 100 [arrowhead = none ];
100 -> 6;

101 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 101 [arrowhead = none ];
101 -> 7;

102 [label="ONLY EVAL with clause\ns2l(X136, .(X137, X138)) :- ','(p(X136, X139), s2l(X139, X138)).\nand substitution\nT7 -> T85\nX136 -> T85\nX137 -> X140\nX138 -> X141\nX11 -> .(X140, X141)", fontsize=14, style = filled, fillcolor = white];
6 -> 102 [arrowhead = none ];
102 -> 41;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 103 [arrowhead = none ];
103 -> 8;

104 [label="ONLY EVAL with clause\neq(X32, X32).\nand substitution\nX17 -> []\nX32 -> []\nX33 -> []", fontsize=14, style = filled, fillcolor = white];
8 -> 104 [arrowhead = none ];
104 -> 9;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 105 [arrowhead = none ];
105 -> 10;

106 [label="ONLY EVAL with clause\napplast(X48, X49, X50) :- ','(app(X48, .(X49, []), X51), last(X50, X51)).\nand substitution\nX48 -> []\nT10 -> T27\nX49 -> T27\nT11 -> T28\nX50 -> T28\nT25 -> T27\nT26 -> T28", fontsize=14, style = filled, fillcolor = white];
10 -> 106 [arrowhead = none ];
106 -> 11;

107 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 107 [arrowhead = none ];
107 -> 12;

108 [label="SPLIT 2\nreplacements:\nX51 -> T30\nT28 -> T31", fontsize=14, style = filled, fillcolor = violet];
11 -> 108 [arrowhead = none ];
108 -> 13;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 109 [arrowhead = none ];
109 -> 14;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 110 [arrowhead = none ];
110 -> 20;

111 [label="ONLY EVAL with clause\napp([], X64, X65) :- ','(!_5, eq(X64, X65)).\nand substitution\nT27 -> T37\nX64 -> .(T37, [])\nX51 -> X66\nX65 -> X66\nT36 -> T37", fontsize=14, style = filled, fillcolor = white];
14 -> 111 [arrowhead = none ];
111 -> 15;

112 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 112 [arrowhead = none ];
112 -> 16;

113 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 113 [arrowhead = none ];
113 -> 17;

114 [label="ONLY EVAL with clause\neq(X69, X69).\nand substitution\nT37 -> T40\nX69 -> .(T40, [])\nX66 -> .(T40, [])", fontsize=14, style = filled, fillcolor = white];
17 -> 114 [arrowhead = none ];
114 -> 18;

115 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 115 [arrowhead = none ];
115 -> 19;

116 [label="EVAL with clause\nlast(X73, .(X73, [])) :- !_7.\nand substitution\nT31 -> T44\nX73 -> T44\nT30 -> .(T44, [])", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 116 [arrowhead = none ];
116 -> 21;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 117 [arrowhead = none ];
117 -> 22;

118 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 118 [arrowhead = none ];
118 -> 23;

119 [label="ONLY EVAL with clause\nlast(X81, X82) :- ','(tail(X82, X83), last(X81, X83)).\nand substitution\nT31 -> T55\nX81 -> T55\nT30 -> T54\nX82 -> T54\nT53 -> T54\nT52 -> T55", fontsize=14, style = filled, fillcolor = white];
22 -> 119 [arrowhead = none ];
119 -> 25;

120 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 120 [arrowhead = none ];
120 -> 24;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 121 [arrowhead = none ];
121 -> 26;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 122 [arrowhead = none ];
122 -> 27;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 123 [arrowhead = none ];
123 -> 28;

124 [label="EVAL with clause\ntail([], []).\nand substitution\nT54 -> []\nX83 -> []\nT55 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 124 [arrowhead = none ];
124 -> 29;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 125 [arrowhead = none ];
125 -> 30;

126 [label="EVAL with clause\ntail(.(X118, X119), X119).\nand substitution\nX118 -> T78\nX119 -> T81\nT54 -> .(T78, T81)\nX83 -> T81\nT55 -> T80\nT79 -> T81", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 126 [arrowhead = none ];
126 -> 39;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 127 [arrowhead = none ];
127 -> 40;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 128 [arrowhead = none ];
128 -> 31;

129 [label="BACKTRACK\nfor clause: last(X, .(X, [])) :- !because of non-unification", fontsize=14, style = filled, fillcolor = white];
31 -> 129 [arrowhead = none ];
129 -> 32;

130 [label="ONLY EVAL with clause\nlast(X96, X97) :- ','(tail(X97, X98), last(X96, X98)).\nand substitution\nT57 -> T66\nX96 -> T66\nX97 -> []\nT65 -> T66", fontsize=14, style = filled, fillcolor = white];
32 -> 130 [arrowhead = none ];
130 -> 33;

131 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 131 [arrowhead = none ];
131 -> 34;

132 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 132 [arrowhead = none ];
132 -> 35;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 133 [arrowhead = none ];
133 -> 36;

134 [label="ONLY EVAL with clause\ntail([], []).\nand substitution\nX98 -> []", fontsize=14, style = filled, fillcolor = white];
35 -> 134 [arrowhead = none ];
134 -> 37;

135 [label="BACKTRACK\nfor clause: tail(.(X4, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
36 -> 135 [arrowhead = none ];
135 -> 38;

136 [label="INSTANCE with matching:\nT57 -> T66", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 136 [arrowhead = none , style=dashed];
136 -> 29[style=dashed];

137 [label="INSTANCE with matching:\nT31 -> T80\nT30 -> T81", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 137 [arrowhead = none , style=dashed];
137 -> 13[style=dashed];

138 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 138 [arrowhead = none ];
138 -> 42;

139 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T85, X11), s2l(0, X16))", fontsize=14, style = filled, fillcolor = white];
42 -> 139 [arrowhead = none ];
139 -> 43;

140 [label="EVAL with clause\np(s(X146), X146).\nand substitution\nX146 -> T88\nT85 -> s(T88)\nX139 -> T88", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 140 [arrowhead = none ];
140 -> 44;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 141 [arrowhead = none ];
141 -> 45;

142 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
44 -> 142 [arrowhead = none ];
142 -> 46;

143 [label="SPLIT 2\nreplacements:\nX141 -> T89", fontsize=14, style = filled, fillcolor = violet];
44 -> 143 [arrowhead = none ];
143 -> 47;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 144 [arrowhead = none ];
144 -> 48;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 145 [arrowhead = none ];
145 -> 60;

146 [label="EVAL with clause\ns2l(0, X156) :- ','(!_12, eq(X156, [])).\nand substitution\nT88 -> 0\nX141 -> X157\nX156 -> X157", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 146 [arrowhead = none ];
146 -> 49;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 147 [arrowhead = none ];
147 -> 50;

148 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 148 [arrowhead = none ];
148 -> 51;

149 [label="ONLY EVAL with clause\ns2l(X178, .(X179, X180)) :- ','(p(X178, X181), s2l(X181, X180)).\nand substitution\nT88 -> T92\nX178 -> T92\nX179 -> X182\nX180 -> X183\nX141 -> .(X182, X183)", fontsize=14, style = filled, fillcolor = white];
50 -> 149 [arrowhead = none ];
149 -> 55;

150 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 150 [arrowhead = none ];
150 -> 52;

151 [label="ONLY EVAL with clause\neq(X163, X163).\nand substitution\nX157 -> []\nX163 -> []\nX164 -> []", fontsize=14, style = filled, fillcolor = white];
52 -> 151 [arrowhead = none ];
151 -> 53;

152 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 152 [arrowhead = none ];
152 -> 54;

153 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 153 [arrowhead = none ];
153 -> 56;

154 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T92, X141), s2l(0, X156))", fontsize=14, style = filled, fillcolor = white];
56 -> 154 [arrowhead = none ];
154 -> 57;

155 [label="EVAL with clause\np(s(X188), X188).\nand substitution\nX188 -> T95\nT92 -> s(T95)\nX181 -> T95", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 155 [arrowhead = none ];
155 -> 58;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 156 [arrowhead = none ];
156 -> 59;

157 [label="INSTANCE with matching:\nT88 -> T95\nX141 -> X183", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 157 [arrowhead = none , style=dashed];
157 -> 46[style=dashed];

158 [label="ONLY EVAL with clause\napplast(X216, X217, X218) :- ','(app(X216, .(X217, []), X219), last(X218, X219)).\nand substitution\nX140 -> X220\nT89 -> T113\nX216 -> .(X220, T113)\nT10 -> T114\nX217 -> T114\nT11 -> T115\nX218 -> T115\nT110 -> T113\nT111 -> T114\nT112 -> T115", fontsize=14, style = filled, fillcolor = white];
60 -> 158 [arrowhead = none ];
158 -> 61;

159 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
61 -> 159 [arrowhead = none ];
159 -> 62;

160 [label="SPLIT 2\nreplacements:\nX220 -> T118\nX219 -> T119\nT115 -> T120", fontsize=14, style = filled, fillcolor = violet];
61 -> 160 [arrowhead = none ];
160 -> 63;

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 161 [arrowhead = none ];
161 -> 64;

162 [label="INSTANCE with matching:\nT31 -> T120\nT30 -> T119", fontsize=14, style = filled, fillcolor = lightgrey];
63 -> 162 [arrowhead = none , style=dashed];
162 -> 13[style=dashed];

163 [label="BACKTRACK\nfor clause: app([], Y, Z) :- ','(!, eq(Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = white];
64 -> 163 [arrowhead = none ];
163 -> 65;

164 [label="ONLY EVAL with clause\napp(X261, X262, .(X263, X264)) :- ','(head(X261, X263), ','(tail(X261, X265), app(X265, X262, X264))).\nand substitution\nX220 -> X266\nT113 -> T132\nX261 -> .(X266, T132)\nT114 -> T133\nX262 -> .(T133, [])\nX263 -> X267\nX264 -> X268\nX219 -> .(X267, X268)\nT130 -> T132\nT131 -> T133", fontsize=14, style = filled, fillcolor = white];
65 -> 164 [arrowhead = none ];
164 -> 66;

165 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 165 [arrowhead = none ];
165 -> 67;

166 [label="BACKTRACK\nfor clause: head([], X2)because of non-unification", fontsize=14, style = filled, fillcolor = white];
67 -> 166 [arrowhead = none ];
166 -> 68;

167 [label="ONLY EVAL with clause\nhead(.(X282, X283), X282).\nand substitution\nX266 -> X284\nX282 -> X284\nT132 -> T141\nX283 -> T141\nX267 -> X284\nT140 -> T141\nT133 -> T142", fontsize=14, style = filled, fillcolor = white];
68 -> 167 [arrowhead = none ];
167 -> 69;

168 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 168 [arrowhead = none ];
168 -> 70;

169 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
70 -> 169 [arrowhead = none ];
169 -> 71;

170 [label="ONLY EVAL with clause\ntail(.(X297, X298), X298).\nand substitution\nX284 -> X299\nX297 -> X299\nT141 -> T150\nX298 -> T150\nX265 -> T150\nT149 -> T150\nT142 -> T151", fontsize=14, style = filled, fillcolor = white];
71 -> 170 [arrowhead = none ];
170 -> 72;

171 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 171 [arrowhead = none ];
171 -> 73;

172 [label="EVAL with clause\napp([], X309, X310) :- ','(!_19, eq(X309, X310)).\nand substitution\nT150 -> []\nT151 -> T156\nX309 -> .(T156, [])\nX268 -> X311\nX310 -> X311\nT155 -> T156", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 172 [arrowhead = none ];
172 -> 74;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 173 [arrowhead = none ];
173 -> 75;

174 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 174 [arrowhead = none ];
174 -> 76;

175 [label="ONLY EVAL with clause\napp(X333, X334, .(X335, X336)) :- ','(head(X333, X335), ','(tail(X333, X337), app(X337, X334, X336))).\nand substitution\nT150 -> T169\nX333 -> T169\nT151 -> T170\nX334 -> .(T170, [])\nX335 -> X338\nX336 -> X339\nX268 -> .(X338, X339)\nT167 -> T169\nT168 -> T170", fontsize=14, style = filled, fillcolor = white];
75 -> 175 [arrowhead = none ];
175 -> 80;

176 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
76 -> 176 [arrowhead = none ];
176 -> 77;

177 [label="ONLY EVAL with clause\neq(X316, X316).\nand substitution\nT156 -> T159\nX316 -> .(T159, [])\nX311 -> .(T159, [])", fontsize=14, style = filled, fillcolor = white];
77 -> 177 [arrowhead = none ];
177 -> 78;

178 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 178 [arrowhead = none ];
178 -> 79;

179 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 179 [arrowhead = none ];
179 -> 81;

180 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 180 [arrowhead = none ];
180 -> 82;

181 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 181 [arrowhead = none ];
181 -> 83;

182 [label="EVAL with clause\nhead([], X351).\nand substitution\nT169 -> []\nX338 -> X352\nX351 -> X352\nT170 -> T172", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 182 [arrowhead = none ];
182 -> 84;

183 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 183 [arrowhead = none ];
183 -> 85;

184 [label="EVAL with clause\nhead(.(X372, X373), X372).\nand substitution\nX372 -> T183\nX373 -> T184\nT169 -> .(T183, T184)\nX338 -> T183\nT181 -> T183\nT182 -> T184\nT170 -> T185", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 184 [arrowhead = none ];
184 -> 91;

185 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 185 [arrowhead = none ];
185 -> 92;

186 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
84 -> 186 [arrowhead = none ];
186 -> 86;

187 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 187 [arrowhead = none ];
187 -> 87;

188 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 188 [arrowhead = none ];
188 -> 88;

189 [label="ONLY EVAL with clause\ntail([], []).\nand substitution\nX337 -> []", fontsize=14, style = filled, fillcolor = white];
87 -> 189 [arrowhead = none ];
189 -> 89;

190 [label="BACKTRACK\nfor clause: tail(.(X4, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
88 -> 190 [arrowhead = none ];
190 -> 90;

191 [label="INSTANCE with matching:\nT27 -> T172\nX51 -> X339", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 191 [arrowhead = none , style=dashed];
191 -> 12[style=dashed];

192 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
91 -> 192 [arrowhead = none ];
192 -> 93;

193 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
93 -> 193 [arrowhead = none ];
193 -> 94;

194 [label="ONLY EVAL with clause\ntail(.(X386, X387), X387).\nand substitution\nT183 -> T194\nX386 -> T194\nT184 -> T196\nX387 -> T196\nX337 -> T196\nT195 -> T196\nT185 -> T197", fontsize=14, style = filled, fillcolor = white];
94 -> 194 [arrowhead = none ];
194 -> 95;

195 [label="INSTANCE with matching:\nT150 -> T196\nT151 -> T197\nX268 -> X339", fontsize=14, style = filled, fillcolor = lightgrey];
95 -> 195 [arrowhead = none , style=dashed];
195 -> 72[style=dashed];

}

(2) Obligation:

Triples:

last29(T66) :- last29(T66).
last13(T57, []) :- last29(T57).
last13(T80, .(T78, T81)) :- last13(T80, T81).
s2l46(s(T95), .(X182, X183)) :- s2l46(T95, X183).
app72([], T172, .(X352, X339)) :- app12(T172, X339).
app72(.(T194, T196), T197, .(T194, X339)) :- app72(T196, T197, X339).
goal1(0, T27, T28) :- app12(T27, X51).
goal1(0, T27, T31) :- ','(appc12(T27, T30), last13(T31, T30)).
goal1(s(T88), T10, T11) :- s2l46(T88, X141).
goal1(s(T88), T151, T115) :- ','(s2lc46(T88, T150), app72(T150, T151, X268)).
goal1(s(T88), T114, T120) :- ','(s2lc46(T88, T113), ','(appc62(T118, T113, T114, T119), last13(T120, T119))).

Clauses:

lastc29(T66) :- lastc29(T66).
lastc13(T44, .(T44, [])).
lastc13(T57, []) :- lastc29(T57).
lastc13(T80, .(T78, T81)) :- lastc13(T80, T81).
s2lc46(0, []).
s2lc46(s(T95), .(X182, X183)) :- s2lc46(T95, X183).
appc12(T40, .(T40, [])).
appc72([], T159, .(T159, [])).
appc72([], T172, .(X352, X339)) :- appc12(T172, X339).
appc72(.(T194, T196), T197, .(T194, X339)) :- appc72(T196, T197, X339).
appc62(X299, T150, T151, .(X299, X268)) :- appc72(T150, T151, X268).

Afs:

goal1(x1, x2, x3)  =  goal1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

last29(T66) :- last29(T66).
last13(T57, []) :- last29(T57).
last13(T80, .(T78, T81)) :- last13(T80, T81).
s2l46(s(T95), .(X182, X183)) :- s2l46(T95, X183).
app72(.(T194, T196), T197, .(T194, X339)) :- app72(T196, T197, X339).
goal1(0, T27, T31) :- ','(appc12(T27, T30), last13(T31, T30)).
goal1(s(T88), T10, T11) :- s2l46(T88, X141).
goal1(s(T88), T151, T115) :- ','(s2lc46(T88, T150), app72(T150, T151, X268)).
goal1(s(T88), T114, T120) :- ','(s2lc46(T88, T113), ','(appc62(T118, T113, T114, T119), last13(T120, T119))).

Clauses:

lastc29(T66) :- lastc29(T66).
lastc13(T44, .(T44, [])).
lastc13(T57, []) :- lastc29(T57).
lastc13(T80, .(T78, T81)) :- lastc13(T80, T81).
s2lc46(0, []).
s2lc46(s(T95), .(X182, X183)) :- s2lc46(T95, X183).
appc12(T40, .(T40, [])).
appc72([], T159, .(T159, [])).
appc72([], T172, .(X352, X339)) :- appc12(T172, X339).
appc72(.(T194, T196), T197, .(T194, X339)) :- appc72(T196, T197, X339).
appc62(X299, T150, T151, .(X299, X268)) :- appc72(T150, T151, X268).

Afs:

goal1(x1, x2, x3)  =  goal1(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal1_in: (b,f,f)
last13_in: (f,b)
last29_in: (f)
s2l46_in: (b,f)
s2lc46_in: (b,f)
app72_in: (b,f,f)
appc62_in: (f,b,f,f)
appc72_in: (b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_GAA(0, T27, T31) → U6_GAA(T27, T31, appc12_in_aa(T27, T30))
U6_GAA(T27, T31, appc12_out_aa(T27, T30)) → U7_GAA(T27, T31, last13_in_ag(T31, T30))
U6_GAA(T27, T31, appc12_out_aa(T27, T30)) → LAST13_IN_AG(T31, T30)
LAST13_IN_AG(T57, []) → U2_AG(T57, last29_in_a(T57))
LAST13_IN_AG(T57, []) → LAST29_IN_A(T57)
LAST29_IN_A(T66) → U1_A(T66, last29_in_a(T66))
LAST29_IN_A(T66) → LAST29_IN_A(T66)
LAST13_IN_AG(T80, .(T78, T81)) → U3_AG(T80, T78, T81, last13_in_ag(T80, T81))
LAST13_IN_AG(T80, .(T78, T81)) → LAST13_IN_AG(T80, T81)
GOAL1_IN_GAA(s(T88), T10, T11) → U8_GAA(T88, T10, T11, s2l46_in_ga(T88, X141))
GOAL1_IN_GAA(s(T88), T10, T11) → S2L46_IN_GA(T88, X141)
S2L46_IN_GA(s(T95), .(X182, X183)) → U4_GA(T95, X182, X183, s2l46_in_ga(T95, X183))
S2L46_IN_GA(s(T95), .(X182, X183)) → S2L46_IN_GA(T95, X183)
GOAL1_IN_GAA(s(T88), T151, T115) → U9_GAA(T88, T151, T115, s2lc46_in_ga(T88, T150))
U9_GAA(T88, T151, T115, s2lc46_out_ga(T88, T150)) → U10_GAA(T88, T151, T115, app72_in_gaa(T150, T151, X268))
U9_GAA(T88, T151, T115, s2lc46_out_ga(T88, T150)) → APP72_IN_GAA(T150, T151, X268)
APP72_IN_GAA(.(T194, T196), T197, .(T194, X339)) → U5_GAA(T194, T196, T197, X339, app72_in_gaa(T196, T197, X339))
APP72_IN_GAA(.(T194, T196), T197, .(T194, X339)) → APP72_IN_GAA(T196, T197, X339)
GOAL1_IN_GAA(s(T88), T114, T120) → U11_GAA(T88, T114, T120, s2lc46_in_ga(T88, T113))
U11_GAA(T88, T114, T120, s2lc46_out_ga(T88, T113)) → U12_GAA(T88, T114, T120, appc62_in_agaa(T118, T113, T114, T119))
U12_GAA(T88, T114, T120, appc62_out_agaa(T118, T113, T114, T119)) → U13_GAA(T88, T114, T120, last13_in_ag(T120, T119))
U12_GAA(T88, T114, T120, appc62_out_agaa(T118, T113, T114, T119)) → LAST13_IN_AG(T120, T119)

The TRS R consists of the following rules:

appc12_in_aa(T40, .(T40, [])) → appc12_out_aa(T40, .(T40, []))
s2lc46_in_ga(0, []) → s2lc46_out_ga(0, [])
s2lc46_in_ga(s(T95), .(X182, X183)) → U18_ga(T95, X182, X183, s2lc46_in_ga(T95, X183))
U18_ga(T95, X182, X183, s2lc46_out_ga(T95, X183)) → s2lc46_out_ga(s(T95), .(X182, X183))
appc62_in_agaa(X299, T150, T151, .(X299, X268)) → U21_agaa(X299, T150, T151, X268, appc72_in_gaa(T150, T151, X268))
appc72_in_gaa([], T159, .(T159, [])) → appc72_out_gaa([], T159, .(T159, []))
appc72_in_gaa([], T172, .(X352, X339)) → U19_gaa(T172, X352, X339, appc12_in_aa(T172, X339))
U19_gaa(T172, X352, X339, appc12_out_aa(T172, X339)) → appc72_out_gaa([], T172, .(X352, X339))
appc72_in_gaa(.(T194, T196), T197, .(T194, X339)) → U20_gaa(T194, T196, T197, X339, appc72_in_gaa(T196, T197, X339))
U20_gaa(T194, T196, T197, X339, appc72_out_gaa(T196, T197, X339)) → appc72_out_gaa(.(T194, T196), T197, .(T194, X339))
U21_agaa(X299, T150, T151, X268, appc72_out_gaa(T150, T151, X268)) → appc62_out_agaa(X299, T150, T151, .(X299, X268))

The argument filtering Pi contains the following mapping:
0  =  0
appc12_in_aa(x1, x2)  =  appc12_in_aa
appc12_out_aa(x1, x2)  =  appc12_out_aa(x2)
.(x1, x2)  =  .(x2)
last13_in_ag(x1, x2)  =  last13_in_ag(x2)
[]  =  []
last29_in_a(x1)  =  last29_in_a
s(x1)  =  s(x1)
s2l46_in_ga(x1, x2)  =  s2l46_in_ga(x1)
s2lc46_in_ga(x1, x2)  =  s2lc46_in_ga(x1)
s2lc46_out_ga(x1, x2)  =  s2lc46_out_ga(x1, x2)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x4)
app72_in_gaa(x1, x2, x3)  =  app72_in_gaa(x1)
appc62_in_agaa(x1, x2, x3, x4)  =  appc62_in_agaa(x2)
U21_agaa(x1, x2, x3, x4, x5)  =  U21_agaa(x2, x5)
appc72_in_gaa(x1, x2, x3)  =  appc72_in_gaa(x1)
appc72_out_gaa(x1, x2, x3)  =  appc72_out_gaa(x1, x3)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x2, x5)
appc62_out_agaa(x1, x2, x3, x4)  =  appc62_out_agaa(x2, x4)
GOAL1_IN_GAA(x1, x2, x3)  =  GOAL1_IN_GAA(x1)
U6_GAA(x1, x2, x3)  =  U6_GAA(x3)
U7_GAA(x1, x2, x3)  =  U7_GAA(x3)
LAST13_IN_AG(x1, x2)  =  LAST13_IN_AG(x2)
U2_AG(x1, x2)  =  U2_AG(x2)
LAST29_IN_A(x1)  =  LAST29_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
S2L46_IN_GA(x1, x2)  =  S2L46_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
APP72_IN_GAA(x1, x2, x3)  =  APP72_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5)  =  U5_GAA(x2, x5)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x4)
U12_GAA(x1, x2, x3, x4)  =  U12_GAA(x1, x4)
U13_GAA(x1, x2, x3, x4)  =  U13_GAA(x1, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

GOAL1_IN_GAA(0, T27, T31) → U6_GAA(T27, T31, appc12_in_aa(T27, T30))
U6_GAA(T27, T31, appc12_out_aa(T27, T30)) → U7_GAA(T27, T31, last13_in_ag(T31, T30))
U6_GAA(T27, T31, appc12_out_aa(T27, T30)) → LAST13_IN_AG(T31, T30)
LAST13_IN_AG(T57, []) → U2_AG(T57, last29_in_a(T57))
LAST13_IN_AG(T57, []) → LAST29_IN_A(T57)
LAST29_IN_A(T66) → U1_A(T66, last29_in_a(T66))
LAST29_IN_A(T66) → LAST29_IN_A(T66)
LAST13_IN_AG(T80, .(T78, T81)) → U3_AG(T80, T78, T81, last13_in_ag(T80, T81))
LAST13_IN_AG(T80, .(T78, T81)) → LAST13_IN_AG(T80, T81)
GOAL1_IN_GAA(s(T88), T10, T11) → U8_GAA(T88, T10, T11, s2l46_in_ga(T88, X141))
GOAL1_IN_GAA(s(T88), T10, T11) → S2L46_IN_GA(T88, X141)
S2L46_IN_GA(s(T95), .(X182, X183)) → U4_GA(T95, X182, X183, s2l46_in_ga(T95, X183))
S2L46_IN_GA(s(T95), .(X182, X183)) → S2L46_IN_GA(T95, X183)
GOAL1_IN_GAA(s(T88), T151, T115) → U9_GAA(T88, T151, T115, s2lc46_in_ga(T88, T150))
U9_GAA(T88, T151, T115, s2lc46_out_ga(T88, T150)) → U10_GAA(T88, T151, T115, app72_in_gaa(T150, T151, X268))
U9_GAA(T88, T151, T115, s2lc46_out_ga(T88, T150)) → APP72_IN_GAA(T150, T151, X268)
APP72_IN_GAA(.(T194, T196), T197, .(T194, X339)) → U5_GAA(T194, T196, T197, X339, app72_in_gaa(T196, T197, X339))
APP72_IN_GAA(.(T194, T196), T197, .(T194, X339)) → APP72_IN_GAA(T196, T197, X339)
GOAL1_IN_GAA(s(T88), T114, T120) → U11_GAA(T88, T114, T120, s2lc46_in_ga(T88, T113))
U11_GAA(T88, T114, T120, s2lc46_out_ga(T88, T113)) → U12_GAA(T88, T114, T120, appc62_in_agaa(T118, T113, T114, T119))
U12_GAA(T88, T114, T120, appc62_out_agaa(T118, T113, T114, T119)) → U13_GAA(T88, T114, T120, last13_in_ag(T120, T119))
U12_GAA(T88, T114, T120, appc62_out_agaa(T118, T113, T114, T119)) → LAST13_IN_AG(T120, T119)

The TRS R consists of the following rules:

appc12_in_aa(T40, .(T40, [])) → appc12_out_aa(T40, .(T40, []))
s2lc46_in_ga(0, []) → s2lc46_out_ga(0, [])
s2lc46_in_ga(s(T95), .(X182, X183)) → U18_ga(T95, X182, X183, s2lc46_in_ga(T95, X183))
U18_ga(T95, X182, X183, s2lc46_out_ga(T95, X183)) → s2lc46_out_ga(s(T95), .(X182, X183))
appc62_in_agaa(X299, T150, T151, .(X299, X268)) → U21_agaa(X299, T150, T151, X268, appc72_in_gaa(T150, T151, X268))
appc72_in_gaa([], T159, .(T159, [])) → appc72_out_gaa([], T159, .(T159, []))
appc72_in_gaa([], T172, .(X352, X339)) → U19_gaa(T172, X352, X339, appc12_in_aa(T172, X339))
U19_gaa(T172, X352, X339, appc12_out_aa(T172, X339)) → appc72_out_gaa([], T172, .(X352, X339))
appc72_in_gaa(.(T194, T196), T197, .(T194, X339)) → U20_gaa(T194, T196, T197, X339, appc72_in_gaa(T196, T197, X339))
U20_gaa(T194, T196, T197, X339, appc72_out_gaa(T196, T197, X339)) → appc72_out_gaa(.(T194, T196), T197, .(T194, X339))
U21_agaa(X299, T150, T151, X268, appc72_out_gaa(T150, T151, X268)) → appc62_out_agaa(X299, T150, T151, .(X299, X268))

The argument filtering Pi contains the following mapping:
0  =  0
appc12_in_aa(x1, x2)  =  appc12_in_aa
appc12_out_aa(x1, x2)  =  appc12_out_aa(x2)
.(x1, x2)  =  .(x2)
last13_in_ag(x1, x2)  =  last13_in_ag(x2)
[]  =  []
last29_in_a(x1)  =  last29_in_a
s(x1)  =  s(x1)
s2l46_in_ga(x1, x2)  =  s2l46_in_ga(x1)
s2lc46_in_ga(x1, x2)  =  s2lc46_in_ga(x1)
s2lc46_out_ga(x1, x2)  =  s2lc46_out_ga(x1, x2)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x4)
app72_in_gaa(x1, x2, x3)  =  app72_in_gaa(x1)
appc62_in_agaa(x1, x2, x3, x4)  =  appc62_in_agaa(x2)
U21_agaa(x1, x2, x3, x4, x5)  =  U21_agaa(x2, x5)
appc72_in_gaa(x1, x2, x3)  =  appc72_in_gaa(x1)
appc72_out_gaa(x1, x2, x3)  =  appc72_out_gaa(x1, x3)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x2, x5)
appc62_out_agaa(x1, x2, x3, x4)  =  appc62_out_agaa(x2, x4)
GOAL1_IN_GAA(x1, x2, x3)  =  GOAL1_IN_GAA(x1)
U6_GAA(x1, x2, x3)  =  U6_GAA(x3)
U7_GAA(x1, x2, x3)  =  U7_GAA(x3)
LAST13_IN_AG(x1, x2)  =  LAST13_IN_AG(x2)
U2_AG(x1, x2)  =  U2_AG(x2)
LAST29_IN_A(x1)  =  LAST29_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
S2L46_IN_GA(x1, x2)  =  S2L46_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
APP72_IN_GAA(x1, x2, x3)  =  APP72_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5)  =  U5_GAA(x2, x5)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x4)
U12_GAA(x1, x2, x3, x4)  =  U12_GAA(x1, x4)
U13_GAA(x1, x2, x3, x4)  =  U13_GAA(x1, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 18 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP72_IN_GAA(.(T194, T196), T197, .(T194, X339)) → APP72_IN_GAA(T196, T197, X339)

The TRS R consists of the following rules:

appc12_in_aa(T40, .(T40, [])) → appc12_out_aa(T40, .(T40, []))
s2lc46_in_ga(0, []) → s2lc46_out_ga(0, [])
s2lc46_in_ga(s(T95), .(X182, X183)) → U18_ga(T95, X182, X183, s2lc46_in_ga(T95, X183))
U18_ga(T95, X182, X183, s2lc46_out_ga(T95, X183)) → s2lc46_out_ga(s(T95), .(X182, X183))
appc62_in_agaa(X299, T150, T151, .(X299, X268)) → U21_agaa(X299, T150, T151, X268, appc72_in_gaa(T150, T151, X268))
appc72_in_gaa([], T159, .(T159, [])) → appc72_out_gaa([], T159, .(T159, []))
appc72_in_gaa([], T172, .(X352, X339)) → U19_gaa(T172, X352, X339, appc12_in_aa(T172, X339))
U19_gaa(T172, X352, X339, appc12_out_aa(T172, X339)) → appc72_out_gaa([], T172, .(X352, X339))
appc72_in_gaa(.(T194, T196), T197, .(T194, X339)) → U20_gaa(T194, T196, T197, X339, appc72_in_gaa(T196, T197, X339))
U20_gaa(T194, T196, T197, X339, appc72_out_gaa(T196, T197, X339)) → appc72_out_gaa(.(T194, T196), T197, .(T194, X339))
U21_agaa(X299, T150, T151, X268, appc72_out_gaa(T150, T151, X268)) → appc62_out_agaa(X299, T150, T151, .(X299, X268))

The argument filtering Pi contains the following mapping:
0  =  0
appc12_in_aa(x1, x2)  =  appc12_in_aa
appc12_out_aa(x1, x2)  =  appc12_out_aa(x2)
.(x1, x2)  =  .(x2)
[]  =  []
s(x1)  =  s(x1)
s2lc46_in_ga(x1, x2)  =  s2lc46_in_ga(x1)
s2lc46_out_ga(x1, x2)  =  s2lc46_out_ga(x1, x2)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x4)
appc62_in_agaa(x1, x2, x3, x4)  =  appc62_in_agaa(x2)
U21_agaa(x1, x2, x3, x4, x5)  =  U21_agaa(x2, x5)
appc72_in_gaa(x1, x2, x3)  =  appc72_in_gaa(x1)
appc72_out_gaa(x1, x2, x3)  =  appc72_out_gaa(x1, x3)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x2, x5)
appc62_out_agaa(x1, x2, x3, x4)  =  appc62_out_agaa(x2, x4)
APP72_IN_GAA(x1, x2, x3)  =  APP72_IN_GAA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APP72_IN_GAA(.(T194, T196), T197, .(T194, X339)) → APP72_IN_GAA(T196, T197, X339)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP72_IN_GAA(x1, x2, x3)  =  APP72_IN_GAA(x1)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APP72_IN_GAA(.(T196)) → APP72_IN_GAA(T196)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• APP72_IN_GAA(.(T196)) → APP72_IN_GAA(T196)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

S2L46_IN_GA(s(T95), .(X182, X183)) → S2L46_IN_GA(T95, X183)

The TRS R consists of the following rules:

appc12_in_aa(T40, .(T40, [])) → appc12_out_aa(T40, .(T40, []))
s2lc46_in_ga(0, []) → s2lc46_out_ga(0, [])
s2lc46_in_ga(s(T95), .(X182, X183)) → U18_ga(T95, X182, X183, s2lc46_in_ga(T95, X183))
U18_ga(T95, X182, X183, s2lc46_out_ga(T95, X183)) → s2lc46_out_ga(s(T95), .(X182, X183))
appc62_in_agaa(X299, T150, T151, .(X299, X268)) → U21_agaa(X299, T150, T151, X268, appc72_in_gaa(T150, T151, X268))
appc72_in_gaa([], T159, .(T159, [])) → appc72_out_gaa([], T159, .(T159, []))
appc72_in_gaa([], T172, .(X352, X339)) → U19_gaa(T172, X352, X339, appc12_in_aa(T172, X339))
U19_gaa(T172, X352, X339, appc12_out_aa(T172, X339)) → appc72_out_gaa([], T172, .(X352, X339))
appc72_in_gaa(.(T194, T196), T197, .(T194, X339)) → U20_gaa(T194, T196, T197, X339, appc72_in_gaa(T196, T197, X339))
U20_gaa(T194, T196, T197, X339, appc72_out_gaa(T196, T197, X339)) → appc72_out_gaa(.(T194, T196), T197, .(T194, X339))
U21_agaa(X299, T150, T151, X268, appc72_out_gaa(T150, T151, X268)) → appc62_out_agaa(X299, T150, T151, .(X299, X268))

The argument filtering Pi contains the following mapping:
0  =  0
appc12_in_aa(x1, x2)  =  appc12_in_aa
appc12_out_aa(x1, x2)  =  appc12_out_aa(x2)
.(x1, x2)  =  .(x2)
[]  =  []
s(x1)  =  s(x1)
s2lc46_in_ga(x1, x2)  =  s2lc46_in_ga(x1)
s2lc46_out_ga(x1, x2)  =  s2lc46_out_ga(x1, x2)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x4)
appc62_in_agaa(x1, x2, x3, x4)  =  appc62_in_agaa(x2)
U21_agaa(x1, x2, x3, x4, x5)  =  U21_agaa(x2, x5)
appc72_in_gaa(x1, x2, x3)  =  appc72_in_gaa(x1)
appc72_out_gaa(x1, x2, x3)  =  appc72_out_gaa(x1, x3)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x2, x5)
appc62_out_agaa(x1, x2, x3, x4)  =  appc62_out_agaa(x2, x4)
S2L46_IN_GA(x1, x2)  =  S2L46_IN_GA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

S2L46_IN_GA(s(T95), .(X182, X183)) → S2L46_IN_GA(T95, X183)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
s(x1)  =  s(x1)
S2L46_IN_GA(x1, x2)  =  S2L46_IN_GA(x1)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

S2L46_IN_GA(s(T95)) → S2L46_IN_GA(T95)

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:

• S2L46_IN_GA(s(T95)) → S2L46_IN_GA(T95)
  The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

LAST29_IN_A(T66) → LAST29_IN_A(T66)

The TRS R consists of the following rules:

appc12_in_aa(T40, .(T40, [])) → appc12_out_aa(T40, .(T40, []))
s2lc46_in_ga(0, []) → s2lc46_out_ga(0, [])
s2lc46_in_ga(s(T95), .(X182, X183)) → U18_ga(T95, X182, X183, s2lc46_in_ga(T95, X183))
U18_ga(T95, X182, X183, s2lc46_out_ga(T95, X183)) → s2lc46_out_ga(s(T95), .(X182, X183))
appc62_in_agaa(X299, T150, T151, .(X299, X268)) → U21_agaa(X299, T150, T151, X268, appc72_in_gaa(T150, T151, X268))
appc72_in_gaa([], T159, .(T159, [])) → appc72_out_gaa([], T159, .(T159, []))
appc72_in_gaa([], T172, .(X352, X339)) → U19_gaa(T172, X352, X339, appc12_in_aa(T172, X339))
U19_gaa(T172, X352, X339, appc12_out_aa(T172, X339)) → appc72_out_gaa([], T172, .(X352, X339))
appc72_in_gaa(.(T194, T196), T197, .(T194, X339)) → U20_gaa(T194, T196, T197, X339, appc72_in_gaa(T196, T197, X339))
U20_gaa(T194, T196, T197, X339, appc72_out_gaa(T196, T197, X339)) → appc72_out_gaa(.(T194, T196), T197, .(T194, X339))
U21_agaa(X299, T150, T151, X268, appc72_out_gaa(T150, T151, X268)) → appc62_out_agaa(X299, T150, T151, .(X299, X268))

The argument filtering Pi contains the following mapping:
0  =  0
appc12_in_aa(x1, x2)  =  appc12_in_aa
appc12_out_aa(x1, x2)  =  appc12_out_aa(x2)
.(x1, x2)  =  .(x2)
[]  =  []
s(x1)  =  s(x1)
s2lc46_in_ga(x1, x2)  =  s2lc46_in_ga(x1)
s2lc46_out_ga(x1, x2)  =  s2lc46_out_ga(x1, x2)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x4)
appc62_in_agaa(x1, x2, x3, x4)  =  appc62_in_agaa(x2)
U21_agaa(x1, x2, x3, x4, x5)  =  U21_agaa(x2, x5)
appc72_in_gaa(x1, x2, x3)  =  appc72_in_gaa(x1)
appc72_out_gaa(x1, x2, x3)  =  appc72_out_gaa(x1, x3)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x2, x5)
appc62_out_agaa(x1, x2, x3, x4)  =  appc62_out_agaa(x2, x4)
LAST29_IN_A(x1)  =  LAST29_IN_A

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:

LAST29_IN_A(T66) → LAST29_IN_A(T66)

R is empty.
The argument filtering Pi contains the following mapping:
LAST29_IN_A(x1)  =  LAST29_IN_A

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:

LAST29_IN_A → LAST29_IN_A

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

(28) 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 = LAST29_IN_A evaluates to t =LAST29_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 LAST29_IN_A to LAST29_IN_A.

(29) NO

(30) Obligation:

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

LAST13_IN_AG(T80, .(T78, T81)) → LAST13_IN_AG(T80, T81)

The TRS R consists of the following rules:

appc12_in_aa(T40, .(T40, [])) → appc12_out_aa(T40, .(T40, []))
s2lc46_in_ga(0, []) → s2lc46_out_ga(0, [])
s2lc46_in_ga(s(T95), .(X182, X183)) → U18_ga(T95, X182, X183, s2lc46_in_ga(T95, X183))
U18_ga(T95, X182, X183, s2lc46_out_ga(T95, X183)) → s2lc46_out_ga(s(T95), .(X182, X183))
appc62_in_agaa(X299, T150, T151, .(X299, X268)) → U21_agaa(X299, T150, T151, X268, appc72_in_gaa(T150, T151, X268))
appc72_in_gaa([], T159, .(T159, [])) → appc72_out_gaa([], T159, .(T159, []))
appc72_in_gaa([], T172, .(X352, X339)) → U19_gaa(T172, X352, X339, appc12_in_aa(T172, X339))
U19_gaa(T172, X352, X339, appc12_out_aa(T172, X339)) → appc72_out_gaa([], T172, .(X352, X339))
appc72_in_gaa(.(T194, T196), T197, .(T194, X339)) → U20_gaa(T194, T196, T197, X339, appc72_in_gaa(T196, T197, X339))
U20_gaa(T194, T196, T197, X339, appc72_out_gaa(T196, T197, X339)) → appc72_out_gaa(.(T194, T196), T197, .(T194, X339))
U21_agaa(X299, T150, T151, X268, appc72_out_gaa(T150, T151, X268)) → appc62_out_agaa(X299, T150, T151, .(X299, X268))

The argument filtering Pi contains the following mapping:
0  =  0
appc12_in_aa(x1, x2)  =  appc12_in_aa
appc12_out_aa(x1, x2)  =  appc12_out_aa(x2)
.(x1, x2)  =  .(x2)
[]  =  []
s(x1)  =  s(x1)
s2lc46_in_ga(x1, x2)  =  s2lc46_in_ga(x1)
s2lc46_out_ga(x1, x2)  =  s2lc46_out_ga(x1, x2)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x4)
appc62_in_agaa(x1, x2, x3, x4)  =  appc62_in_agaa(x2)
U21_agaa(x1, x2, x3, x4, x5)  =  U21_agaa(x2, x5)
appc72_in_gaa(x1, x2, x3)  =  appc72_in_gaa(x1)
appc72_out_gaa(x1, x2, x3)  =  appc72_out_gaa(x1, x3)
U19_gaa(x1, x2, x3, x4)  =  U19_gaa(x4)
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x2, x5)
appc62_out_agaa(x1, x2, x3, x4)  =  appc62_out_agaa(x2, x4)
LAST13_IN_AG(x1, x2)  =  LAST13_IN_AG(x2)

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:

LAST13_IN_AG(T80, .(T78, T81)) → LAST13_IN_AG(T80, T81)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
LAST13_IN_AG(x1, x2)  =  LAST13_IN_AG(x2)

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:

LAST13_IN_AG(.(T81)) → LAST13_IN_AG(T81)

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:

• LAST13_IN_AG(.(T81)) → LAST13_IN_AG(T81)
  The graph contains the following edges 1 > 1

(36) YES