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

(0) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), conf(Xs)).
conf(X) :- ','(del2(X, Z), ','(del(U, Y, Z), conf(Y))).
del2(X, Y) :- ','(del(U, X, Z), del(V, Z, Y)).
del(X1, [], X2) :- ','(!, failure(a)).
del(H, X, T) :- ','(head(X, H), tail(X, T)).
del(X, Y, .(H, T2)) :- ','(head(Y, H), ','(tail(Y, T1), del(X, T1, T2))).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X3, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X4).
head(.(H, X5), H).
tail([], []).
tail(.(X6, Xs), Xs).
p(0, 0).
p(s(X), X).
failure(b).
eq(X, X).

Queries:

goal(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: goal(T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal(T3), SCOPE: 1, CLAUSE: 0\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(s2l(T4, X9), conf(X9))\n\nT4 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: s2l(T4, X9)\n\nT4 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: conf(T5)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: s2l(T4, X9), SCOPE: 2, CLAUSE: 6 | s2l(T4, X9), SCOPE: 2, CLAUSE: 7 | ?_2\n\nT4 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_2, eq(X12, [])) | s2l(0, X9), SCOPE: 2, CLAUSE: 7 | ?_2\n\nX9 is free\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: s2l(T4, X9), SCOPE: 2, CLAUSE: 7\n\nT4 is ground\nX9 is free; \nX11 is free; \ns2l(T4, X9) !=? s2l(0, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: eq(X12, [])\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: eq(X12, []), SCOPE: 3, CLAUSE: 15\n\nX12 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(T6, X22), s2l(X22, X24))\n\nT6 is ground\nX9 is free; \nX11 is free; \nX24 is free; \nX22 is free; \ns2l(T6, X9) !=? s2l(0, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(p(T6, X22), s2l(X22, X24)), SCOPE: 4, CLAUSE: 12 | ','(p(T6, X22), s2l(X22, X24)), SCOPE: 4, CLAUSE: 13\n\nT6 is ground\nX9 is free; \nX11 is free; \nX24 is free; \nX22 is free; \ns2l(T6, X9) !=? s2l(0, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T6, X22), s2l(X22, X24)), SCOPE: 4, CLAUSE: 13\n\nT6 is ground\nX9 is free; \nX11 is free; \nX24 is free; \nX22 is free; \ns2l(T6, X9) !=? s2l(0, X11)\np(T6, X22) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: s2l(T7, X24)\n\nT7 is ground\nX9 is free; \nX11 is free; \nX24 is free; \nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\nX9 is free\nX11 is free; \nX22 is free; \nX26 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: conf(T5), SCOPE: 5, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(del2(T9, X29), ','(del(X30, X31, X29), conf(X31)))\n\nX29 is free\nX30 is free; \nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: del2(T9, X29)\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(del(X30, X31, T10), conf(X31))\n\nX30 is free\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: del2(T9, X29), SCOPE: 6, CLAUSE: 2\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(del(X36, T12, X37), del(X38, X37, X39))\n\nX39 is free\nX36 is free; \nX37 is free; \nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: del(X36, T12, X37)\n\nX36 is free\nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: del(X38, T14, X39)\n\nX39 is free\nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: del(X38, T14, X39), SCOPE: 7, CLAUSE: 3 | del(X38, T14, X39), SCOPE: 7, CLAUSE: 4 | del(X38, T14, X39), SCOPE: 7, CLAUSE: 5 | ?_7\n\nX39 is free\nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(!_7, failure(a)) | del(X38, T14, X39), SCOPE: 7, CLAUSE: 4 | del(X38, T14, X39), SCOPE: 7, CLAUSE: 5 | ?_7\n\nX39 is free\nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: del(X38, T14, X39), SCOPE: 7, CLAUSE: 4 | del(X38, T14, X39), SCOPE: 7, CLAUSE: 5\n\nX39 is free\nX38 is free; \nX42 is free; \nX43 is free; \ndel(X38, T14, X39) !=? del(X42, [], X43)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: failure(a), SCOPE: 8, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: del(X38, T14, X39), SCOPE: 7, CLAUSE: 4\n\nX39 is free\nX38 is free; \nX42 is free; \nX43 is free; \ndel(X38, T14, X39) !=? del(X42, [], X43)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: del(X38, T14, X39), SCOPE: 7, CLAUSE: 5\n\nX39 is free\nX38 is free; \nX42 is free; \nX43 is free; \ndel(X38, T14, X39) !=? del(X42, [], X43)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(head(T16, X55), tail(T16, X56))\n\nX39 is free\nX38 is free; \nX42 is free; \nX43 is free; \nX55 is free; \nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(head(T16, X55), tail(T16, X56)), SCOPE: 9, CLAUSE: 8 | ','(head(T16, X55), tail(T16, X56)), SCOPE: 9, CLAUSE: 9\n\nX55 is free\nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: tail([], X56) | ','(head(T16, X55), tail(T16, X56)), SCOPE: 9, CLAUSE: 9\n\nX55 is free\nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(head(T16, X55), tail(T16, X56)), SCOPE: 9, CLAUSE: 9\n\nX55 is free\nX56 is free; \nX58 is free; \nhead(T16, X55) !=? head([], X58)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: tail([], X56), SCOPE: 10, CLAUSE: 10 | tail([], X56), SCOPE: 10, CLAUSE: 11 | ','(head(T16, X55), tail(T16, X56)), SCOPE: 9, CLAUSE: 9\n\nX55 is free\nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true | tail([], X56), SCOPE: 10, CLAUSE: 11 | ','(head(T16, X55), tail(T16, X56)), SCOPE: 9, CLAUSE: 9\n\nX55 is free\nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: tail([], X56), SCOPE: 10, CLAUSE: 11 | ','(head(T16, X55), tail(T16, X56)), SCOPE: 9, CLAUSE: 9\n\nX55 is free\nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(head(T16, X55), tail(T16, X56)), SCOPE: 9, CLAUSE: 9\n\nX55 is free\nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: tail(.(T19, T20), X56)\n\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\nX55 is free\nX64 is free; \nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: tail(.(T19, T20), X56), SCOPE: 11, CLAUSE: 10 | tail(.(T19, T20), X56), SCOPE: 11, CLAUSE: 11\n\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: tail(.(T19, T20), X56), SCOPE: 11, CLAUSE: 11\n\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: tail(.(T25, T26), X56)\n\nX55 is free\nX56 is free; \nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\nX55 is free\nX58 is free; \nX72 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: tail(.(T25, T26), X56), SCOPE: 12, CLAUSE: 10 | tail(.(T25, T26), X56), SCOPE: 12, CLAUSE: 11\n\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: tail(.(T25, T26), X56), SCOPE: 12, CLAUSE: 11\n\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(head(T30, X88), ','(tail(T30, X86), del(X87, X86, X89)))\n\nX39 is free\nX38 is free; \nX42 is free; \nX43 is free; \nX87 is free; \nX88 is free; \nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(head(T30, X88), ','(tail(T30, X86), del(X87, X86, X89))), SCOPE: 13, CLAUSE: 8 | ','(head(T30, X88), ','(tail(T30, X86), del(X87, X86, X89))), SCOPE: 13, CLAUSE: 9\n\nX87 is free\nX88 is free; \nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(tail([], X86), del(X87, X86, X89)) | ','(head(T30, X88), ','(tail(T30, X86), del(X87, X86, X89))), SCOPE: 13, CLAUSE: 9\n\nX87 is free\nX88 is free; \nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(head(T30, X88), ','(tail(T30, X86), del(X87, X86, X89))), SCOPE: 13, CLAUSE: 9\n\nX87 is free\nX88 is free; \nX89 is free; \nX86 is free; \nX91 is free; \nhead(T30, X88) !=? head([], X91)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(tail([], X86), del(X87, X86, X89)), SCOPE: 14, CLAUSE: 10 | ','(tail([], X86), del(X87, X86, X89)), SCOPE: 14, CLAUSE: 11 | ','(head(T30, X88), ','(tail(T30, X86), del(X87, X86, X89))), SCOPE: 13, CLAUSE: 9\n\nX87 is free\nX88 is free; \nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: del(X87, [], X89) | ','(tail([], X86), del(X87, X86, X89)), SCOPE: 14, CLAUSE: 11 | ','(head(T30, X88), ','(tail(T30, X86), del(X87, X86, X89))), SCOPE: 13, CLAUSE: 9\n\nX87 is free\nX88 is free; \nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: del(X87, [], X89)\n\nX87 is free\nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(tail([], X86), del(X87, X86, X89)), SCOPE: 14, CLAUSE: 11 | ','(head(T30, X88), ','(tail(T30, X86), del(X87, X86, X89))), SCOPE: 13, CLAUSE: 9\n\nX87 is free\nX88 is free; \nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: del(X87, [], X89), SCOPE: 15, CLAUSE: 3 | del(X87, [], X89), SCOPE: 15, CLAUSE: 4 | del(X87, [], X89), SCOPE: 15, CLAUSE: 5 | ?_15\n\nX87 is free\nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(!_15, failure(a)) | del(X87, [], X89), SCOPE: 15, CLAUSE: 4 | del(X87, [], X89), SCOPE: 15, CLAUSE: 5 | ?_15\n\nX87 is free\nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: failure(a), SCOPE: 16, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(head(T30, X88), ','(tail(T30, X86), del(X87, X86, X89))), SCOPE: 13, CLAUSE: 9\n\nX87 is free\nX88 is free; \nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(tail(.(T33, T34), X86), del(X87, X86, X89))\n\nX87 is free\nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\nX88 is free\nX103 is free; \nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(tail(.(T33, T34), X86), del(X87, X86, X89)), SCOPE: 17, CLAUSE: 10 | ','(tail(.(T33, T34), X86), del(X87, X86, X89)), SCOPE: 17, CLAUSE: 11\n\nX87 is free\nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(tail(.(T33, T34), X86), del(X87, X86, X89)), SCOPE: 17, CLAUSE: 11\n\nX87 is free\nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: del(X87, T37, X89)\n\nX87 is free\nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(tail(.(T40, T41), X86), del(X87, X86, X89))\n\nX87 is free\nX88 is free; \nX89 is free; \nX86 is free; \nX91 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\nX88 is free\nX91 is free; \nX111 is free; \nX112 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(tail(.(T40, T41), X86), del(X87, X86, X89)), SCOPE: 18, CLAUSE: 10 | ','(tail(.(T40, T41), X86), del(X87, X86, X89)), SCOPE: 18, CLAUSE: 11\n\nX87 is free\nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(tail(.(T40, T41), X86), del(X87, X86, X89)), SCOPE: 18, CLAUSE: 11\n\nX87 is free\nX89 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: del(X87, T44, X89)\n\nX87 is free\nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: del(X30, X31, T10)\n\nX30 is free\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: conf(T46)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: del(X30, X31, T10), SCOPE: 19, CLAUSE: 3 | del(X30, X31, T10), SCOPE: 19, CLAUSE: 4 | del(X30, X31, T10), SCOPE: 19, CLAUSE: 5 | ?_19\n\nX30 is free\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(!_19, failure(a)) | del(X30, X31, T10), SCOPE: 19, CLAUSE: 4 | del(X30, X31, T10), SCOPE: 19, CLAUSE: 5 | ?_19\n\nX30 is free\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: failure(a), SCOPE: 20, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 85 [arrowhead = none ];
85 -> 2;

86 [label="EVAL with clause\ngoal(X8) :- ','(s2l(X8, X9), conf(X9)).\nand substitution\nT3 -> T4\nX8 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 86 [arrowhead = none ];
86 -> 3;

87 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
3 -> 87 [arrowhead = none ];
87 -> 4;

88 [label="SPLIT 2\nreplacements:\nX9 -> T5", fontsize=14, style = filled, fillcolor = violet];
3 -> 88 [arrowhead = none ];
88 -> 5;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
4 -> 89 [arrowhead = none ];
89 -> 6;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 90 [arrowhead = none ];
90 -> 18;

91 [label="EVAL with clause\ns2l(0, X11) :- ','(!, eq(X11, [])).\nand substitution\nT4 -> 0\nX9 -> X12\nX11 -> X12", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 91 [arrowhead = none ];
91 -> 7;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 92 [arrowhead = none ];
92 -> 8;

93 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 93 [arrowhead = none ];
93 -> 9;

94 [label="EVAL with clause\ns2l(X19, .(X20, X21)) :- ','(p(X19, X22), s2l(X22, X21)).\nand substitution\nT4 -> T6\nX19 -> T6\nX20 -> X23\nX21 -> X24\nX9 -> .(X23, X24)", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 94 [arrowhead = none ];
94 -> 13;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 95 [arrowhead = none ];
95 -> 10;

96 [label="EVAL with clause\neq(X14, X14).\nand substitution\nX12 -> []\nX14 -> []\nX15 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 96 [arrowhead = none ];
96 -> 11;

97 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 97 [arrowhead = none ];
97 -> 12;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 98 [arrowhead = none ];
98 -> 14;

99 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T6, X9), s2l(0, X11))", fontsize=14, style = filled, fillcolor = white];
14 -> 99 [arrowhead = none ];
99 -> 15;

100 [label="EVAL with clause\np(s(X26), X26).\nand substitution\nX26 -> T7\nT6 -> s(T7)\nX22 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 100 [arrowhead = none ];
100 -> 16;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 101 [arrowhead = none ];
101 -> 17;

102 [label="INSTANCE with matching:\nT4 -> T7\nX9 -> X24", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 102 [arrowhead = none , style=dashed];
102 -> 4[style=dashed];

103 [label="EVAL with clause\nconf(X28) :- ','(del2(X28, X29), ','(del(X30, X31, X29), conf(X31))).\nand substitution\nT5 -> T9\nX28 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 103 [arrowhead = none ];
103 -> 19;

104 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 104 [arrowhead = none ];
104 -> 20;

105 [label="SPLIT 2\nreplacements:\nX29 -> T10", fontsize=14, style = filled, fillcolor = violet];
19 -> 105 [arrowhead = none ];
105 -> 21;

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 106 [arrowhead = none ];
106 -> 22;

107 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 107 [arrowhead = none ];
107 -> 78;

108 [label="SPLIT 2\nreplacements:\nX30 -> T45\nX31 -> T46", fontsize=14, style = filled, fillcolor = violet];
21 -> 108 [arrowhead = none ];
108 -> 79;

109 [label="EVAL with clause\ndel2(X34, X35) :- ','(del(X36, X34, X37), del(X38, X37, X35)).\nand substitution\nT9 -> T12\nX34 -> T12\nX29 -> X39\nX35 -> X39\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 109 [arrowhead = none ];
109 -> 23;

110 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
23 -> 110 [arrowhead = none ];
110 -> 24;

111 [label="SPLIT 2\nreplacements:\nX36 -> T13\nX37 -> T14", fontsize=14, style = filled, fillcolor = violet];
23 -> 111 [arrowhead = none ];
111 -> 25;

112 [label="INSTANCE with matching:\nX38 -> X36\nT14 -> T12\nX39 -> X37", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 112 [arrowhead = none , style=dashed];
112 -> 25[style=dashed];

113 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 113 [arrowhead = none ];
113 -> 26;

114 [label="EVAL with clause\ndel(X42, [], X43) :- ','(!, failure(a)).\nand substitution\nX38 -> X44\nX42 -> X44\nT14 -> []\nX39 -> X45\nX43 -> X45", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 114 [arrowhead = none ];
114 -> 27;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 115 [arrowhead = none ];
115 -> 28;

116 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 116 [arrowhead = none ];
116 -> 29;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 117 [arrowhead = none ];
117 -> 32;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 118 [arrowhead = none ];
118 -> 33;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 119 [arrowhead = none ];
119 -> 30;

120 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
30 -> 120 [arrowhead = none ];
120 -> 31;

121 [label="EVAL with clause\ndel(X52, X53, X54) :- ','(head(X53, X52), tail(X53, X54)).\nand substitution\nX38 -> X55\nX52 -> X55\nT14 -> T16\nX53 -> T16\nX39 -> X56\nX54 -> X56\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 121 [arrowhead = none ];
121 -> 34;

122 [label="EVAL with clause\ndel(X82, X83, .(X84, X85)) :- ','(head(X83, X84), ','(tail(X83, X86), del(X82, X86, X85))).\nand substitution\nX38 -> X87\nX82 -> X87\nT14 -> T30\nX83 -> T30\nX84 -> X88\nX85 -> X89\nX39 -> .(X88, X89)\nT29 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 122 [arrowhead = none ];
122 -> 54;

123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 123 [arrowhead = none ];
123 -> 35;

124 [label="EVAL with clause\nhead([], X58).\nand substitution\nT16 -> []\nX55 -> X59\nX58 -> X59", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 124 [arrowhead = none ];
124 -> 36;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 125 [arrowhead = none ];
125 -> 37;

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 126 [arrowhead = none ];
126 -> 38;

127 [label="EVAL with clause\nhead(.(X72, X73), X72).\nand substitution\nX72 -> T25\nX73 -> T26\nT16 -> .(T25, T26)\nX55 -> T25\nT23 -> T25\nT24 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 127 [arrowhead = none ];
127 -> 48;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 128 [arrowhead = none ];
128 -> 49;

129 [label="EVAL with clause\ntail([], []).\nand substitution\nX56 -> []", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 129 [arrowhead = none ];
129 -> 39;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 130 [arrowhead = none ];
130 -> 40;

131 [label="BACKTRACK\nfor clause: tail(.(X6, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
40 -> 131 [arrowhead = none ];
131 -> 41;

132 [label="EVAL with clause\nhead(.(X64, X65), X64).\nand substitution\nX64 -> T19\nX65 -> T20\nT16 -> .(T19, T20)\nX55 -> T19\nT17 -> T19\nT18 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 132 [arrowhead = none ];
132 -> 42;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 133 [arrowhead = none ];
133 -> 43;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 134 [arrowhead = none ];
134 -> 44;

135 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
44 -> 135 [arrowhead = none ];
135 -> 45;

136 [label="EVAL with clause\ntail(.(X68, X69), X69).\nand substitution\nT19 -> T21\nX68 -> T21\nT20 -> T22\nX69 -> T22\nX56 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 136 [arrowhead = none ];
136 -> 46;

137 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 137 [arrowhead = none ];
137 -> 47;

138 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 138 [arrowhead = none ];
138 -> 50;

139 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
50 -> 139 [arrowhead = none ];
139 -> 51;

140 [label="EVAL with clause\ntail(.(X76, X77), X77).\nand substitution\nT25 -> T27\nX76 -> T27\nT26 -> T28\nX77 -> T28\nX56 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 140 [arrowhead = none ];
140 -> 52;

141 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 141 [arrowhead = none ];
141 -> 53;

142 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 142 [arrowhead = none ];
142 -> 55;

143 [label="EVAL with clause\nhead([], X91).\nand substitution\nT30 -> []\nX88 -> X92\nX91 -> X92", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 143 [arrowhead = none ];
143 -> 56;

144 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 144 [arrowhead = none ];
144 -> 57;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 145 [arrowhead = none ];
145 -> 58;

146 [label="EVAL with clause\nhead(.(X111, X112), X111).\nand substitution\nX111 -> T40\nX112 -> T41\nT30 -> .(T40, T41)\nX88 -> T40\nT38 -> T40\nT39 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 146 [arrowhead = none ];
146 -> 73;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 147 [arrowhead = none ];
147 -> 74;

148 [label="EVAL with clause\ntail([], []).\nand substitution\nX86 -> []", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 148 [arrowhead = none ];
148 -> 59;

149 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 149 [arrowhead = none ];
149 -> 60;

150 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 150 [arrowhead = none ];
150 -> 61;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 151 [arrowhead = none ];
151 -> 62;

152 [label="BACKTRACK\nfor clause: tail(.(X6, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
61 -> 152 [arrowhead = none ];
152 -> 67;

153 [label="EVAL with clause\ndel(X95, [], X96) :- ','(!, failure(a)).\nand substitution\nX87 -> X97\nX95 -> X97\nX89 -> X98\nX96 -> X98", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 153 [arrowhead = none ];
153 -> 63;

154 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 154 [arrowhead = none ];
154 -> 64;

155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 155 [arrowhead = none ];
155 -> 65;

156 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
65 -> 156 [arrowhead = none ];
156 -> 66;

157 [label="EVAL with clause\nhead(.(X103, X104), X103).\nand substitution\nX103 -> T33\nX104 -> T34\nT30 -> .(T33, T34)\nX88 -> T33\nT31 -> T33\nT32 -> T34", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 157 [arrowhead = none ];
157 -> 68;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 158 [arrowhead = none ];
158 -> 69;

159 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 159 [arrowhead = none ];
159 -> 70;

160 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
70 -> 160 [arrowhead = none ];
160 -> 71;

161 [label="EVAL with clause\ntail(.(X107, X108), X108).\nand substitution\nT33 -> T35\nX107 -> T35\nT34 -> T37\nX108 -> T37\nX86 -> T37\nT36 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 161 [arrowhead = none ];
161 -> 72;

162 [label="INSTANCE with matching:\nX36 -> X87\nT12 -> T37\nX37 -> X89", fontsize=14, style = filled, fillcolor = lightgrey];
72 -> 162 [arrowhead = none , style=dashed];
162 -> 24[style=dashed];

163 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
73 -> 163 [arrowhead = none ];
163 -> 75;

164 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
75 -> 164 [arrowhead = none ];
164 -> 76;

165 [label="EVAL with clause\ntail(.(X115, X116), X116).\nand substitution\nT40 -> T42\nX115 -> T42\nT41 -> T44\nX116 -> T44\nX86 -> T44\nT43 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 165 [arrowhead = none ];
165 -> 77;

166 [label="INSTANCE with matching:\nX36 -> X87\nT12 -> T44\nX37 -> X89", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 166 [arrowhead = none , style=dashed];
166 -> 24[style=dashed];

167 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
78 -> 167 [arrowhead = none ];
167 -> 80;

168 [label="INSTANCE with matching:\nT5 -> T46", fontsize=14, style = filled, fillcolor = lightgrey];
79 -> 168 [arrowhead = none , style=dashed];
168 -> 5[style=dashed];

169 [label="EVAL with clause\ndel(X119, [], X120) :- ','(!, failure(a)).\nand substitution\nX30 -> X121\nX119 -> X121\nX31 -> []\nT10 -> T47\nX120 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 169 [arrowhead = none ];
169 -> 81;

170 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 170 [arrowhead = none ];
170 -> 82;

171 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
82 -> 171 [arrowhead = none ];
171 -> 83;

172 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
83 -> 172 [arrowhead = none ];
172 -> 84;

}

(2) Obligation:

Clauses:

s2l4(0, []).
s2l4(s(T7), .(X23, X24)) :- s2l4(T7, X24).
del25(X59, [], []).
del25(T21, .(T21, T22), T22).
del25(T27, .(T27, T28), T28).
del25(X87, .(T35, T37), .(T35, X89)) :- del25(X87, T37, X89).
del25(X87, .(T42, T44), .(T42, X89)) :- del25(X87, T44, X89).
conf5(T9) :- del220(T9, X29).
del220(T12, X39) :- del25(X36, T12, X37).
del220(T12, []) :- del25(T13, T12, []).
del220(T12, T22) :- del25(T13, T12, .(T21, T22)).
del220(T12, T28) :- del25(T13, T12, .(T27, T28)).
del220(T12, .(T35, X89)) :- ','(del25(T13, T12, .(T35, T37)), del25(X87, T37, X89)).
del220(T12, .(T42, X89)) :- ','(del25(T13, T12, .(T42, T44)), del25(X87, T44, X89)).
goal1(T4) :- s2l4(T4, X9).
goal1(T4) :- ','(s2l4(T4, T9), del220(T9, X29)).

Queries:

goal1(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal1_in: (b)
s2l4_in: (b,f)
del220_in: (b,f)
del25_in: (f,b,f) (f,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(T4) → U13_g(T4, s2l4_in_ga(T4, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X23, X24)) → U1_ga(T7, X23, X24, s2l4_in_ga(T7, X24))
U1_ga(T7, X23, X24, s2l4_out_ga(T7, X24)) → s2l4_out_ga(s(T7), .(X23, X24))
U13_g(T4, s2l4_out_ga(T4, X9)) → goal1_out_g(T4)
goal1_in_g(T4) → U14_g(T4, s2l4_in_ga(T4, T9))
U14_g(T4, s2l4_out_ga(T4, T9)) → U15_g(T4, del220_in_ga(T9, X29))
del220_in_ga(T12, X39) → U5_ga(T12, X39, del25_in_aga(X36, T12, X37))
del25_in_aga(X59, [], []) → del25_out_aga(X59, [], [])
del25_in_aga(T21, .(T21, T22), T22) → del25_out_aga(T21, .(T21, T22), T22)
del25_in_aga(X87, .(T35, T37), .(T35, X89)) → U2_aga(X87, T35, T37, X89, del25_in_aga(X87, T37, X89))
del25_in_aga(X87, .(T42, T44), .(T42, X89)) → U3_aga(X87, T42, T44, X89, del25_in_aga(X87, T44, X89))
U3_aga(X87, T42, T44, X89, del25_out_aga(X87, T44, X89)) → del25_out_aga(X87, .(T42, T44), .(T42, X89))
U2_aga(X87, T35, T37, X89, del25_out_aga(X87, T37, X89)) → del25_out_aga(X87, .(T35, T37), .(T35, X89))
U5_ga(T12, X39, del25_out_aga(X36, T12, X37)) → del220_out_ga(T12, X39)
del220_in_ga(T12, []) → U6_ga(T12, del25_in_agg(T13, T12, []))
del25_in_agg(X59, [], []) → del25_out_agg(X59, [], [])
del25_in_agg(T21, .(T21, T22), T22) → del25_out_agg(T21, .(T21, T22), T22)
del25_in_agg(X87, .(T35, T37), .(T35, X89)) → U2_agg(X87, T35, T37, X89, del25_in_agg(X87, T37, X89))
del25_in_agg(X87, .(T42, T44), .(T42, X89)) → U3_agg(X87, T42, T44, X89, del25_in_agg(X87, T44, X89))
U3_agg(X87, T42, T44, X89, del25_out_agg(X87, T44, X89)) → del25_out_agg(X87, .(T42, T44), .(T42, X89))
U2_agg(X87, T35, T37, X89, del25_out_agg(X87, T37, X89)) → del25_out_agg(X87, .(T35, T37), .(T35, X89))
U6_ga(T12, del25_out_agg(T13, T12, [])) → del220_out_ga(T12, [])
del220_in_ga(T12, T22) → U7_ga(T12, T22, del25_in_aga(T13, T12, .(T21, T22)))
U7_ga(T12, T22, del25_out_aga(T13, T12, .(T21, T22))) → del220_out_ga(T12, T22)
del220_in_ga(T12, T28) → U8_ga(T12, T28, del25_in_aga(T13, T12, .(T27, T28)))
U8_ga(T12, T28, del25_out_aga(T13, T12, .(T27, T28))) → del220_out_ga(T12, T28)
del220_in_ga(T12, .(T35, X89)) → U9_ga(T12, T35, X89, del25_in_aga(T13, T12, .(T35, T37)))
U9_ga(T12, T35, X89, del25_out_aga(T13, T12, .(T35, T37))) → U10_ga(T12, T35, X89, del25_in_aga(X87, T37, X89))
U10_ga(T12, T35, X89, del25_out_aga(X87, T37, X89)) → del220_out_ga(T12, .(T35, X89))
del220_in_ga(T12, .(T42, X89)) → U11_ga(T12, T42, X89, del25_in_aga(T13, T12, .(T42, T44)))
U11_ga(T12, T42, X89, del25_out_aga(T13, T12, .(T42, T44))) → U12_ga(T12, T42, X89, del25_in_aga(X87, T44, X89))
U12_ga(T12, T42, X89, del25_out_aga(X87, T44, X89)) → del220_out_ga(T12, .(T42, X89))
U15_g(T4, del220_out_ga(T9, X29)) → goal1_out_g(T4)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
del220_in_ga(x1, x2)  =  del220_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
del25_in_aga(x1, x2, x3)  =  del25_in_aga(x2)
[]  =  []
del25_out_aga(x1, x2, x3)  =  del25_out_aga(x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del220_out_ga(x1, x2)  =  del220_out_ga
U6_ga(x1, x2)  =  U6_ga(x2)
del25_in_agg(x1, x2, x3)  =  del25_in_agg(x2, x3)
del25_out_agg(x1, x2, x3)  =  del25_out_agg
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x5)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(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_g(T4) → U13_g(T4, s2l4_in_ga(T4, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X23, X24)) → U1_ga(T7, X23, X24, s2l4_in_ga(T7, X24))
U1_ga(T7, X23, X24, s2l4_out_ga(T7, X24)) → s2l4_out_ga(s(T7), .(X23, X24))
U13_g(T4, s2l4_out_ga(T4, X9)) → goal1_out_g(T4)
goal1_in_g(T4) → U14_g(T4, s2l4_in_ga(T4, T9))
U14_g(T4, s2l4_out_ga(T4, T9)) → U15_g(T4, del220_in_ga(T9, X29))
del220_in_ga(T12, X39) → U5_ga(T12, X39, del25_in_aga(X36, T12, X37))
del25_in_aga(X59, [], []) → del25_out_aga(X59, [], [])
del25_in_aga(T21, .(T21, T22), T22) → del25_out_aga(T21, .(T21, T22), T22)
del25_in_aga(X87, .(T35, T37), .(T35, X89)) → U2_aga(X87, T35, T37, X89, del25_in_aga(X87, T37, X89))
del25_in_aga(X87, .(T42, T44), .(T42, X89)) → U3_aga(X87, T42, T44, X89, del25_in_aga(X87, T44, X89))
U3_aga(X87, T42, T44, X89, del25_out_aga(X87, T44, X89)) → del25_out_aga(X87, .(T42, T44), .(T42, X89))
U2_aga(X87, T35, T37, X89, del25_out_aga(X87, T37, X89)) → del25_out_aga(X87, .(T35, T37), .(T35, X89))
U5_ga(T12, X39, del25_out_aga(X36, T12, X37)) → del220_out_ga(T12, X39)
del220_in_ga(T12, []) → U6_ga(T12, del25_in_agg(T13, T12, []))
del25_in_agg(X59, [], []) → del25_out_agg(X59, [], [])
del25_in_agg(T21, .(T21, T22), T22) → del25_out_agg(T21, .(T21, T22), T22)
del25_in_agg(X87, .(T35, T37), .(T35, X89)) → U2_agg(X87, T35, T37, X89, del25_in_agg(X87, T37, X89))
del25_in_agg(X87, .(T42, T44), .(T42, X89)) → U3_agg(X87, T42, T44, X89, del25_in_agg(X87, T44, X89))
U3_agg(X87, T42, T44, X89, del25_out_agg(X87, T44, X89)) → del25_out_agg(X87, .(T42, T44), .(T42, X89))
U2_agg(X87, T35, T37, X89, del25_out_agg(X87, T37, X89)) → del25_out_agg(X87, .(T35, T37), .(T35, X89))
U6_ga(T12, del25_out_agg(T13, T12, [])) → del220_out_ga(T12, [])
del220_in_ga(T12, T22) → U7_ga(T12, T22, del25_in_aga(T13, T12, .(T21, T22)))
U7_ga(T12, T22, del25_out_aga(T13, T12, .(T21, T22))) → del220_out_ga(T12, T22)
del220_in_ga(T12, T28) → U8_ga(T12, T28, del25_in_aga(T13, T12, .(T27, T28)))
U8_ga(T12, T28, del25_out_aga(T13, T12, .(T27, T28))) → del220_out_ga(T12, T28)
del220_in_ga(T12, .(T35, X89)) → U9_ga(T12, T35, X89, del25_in_aga(T13, T12, .(T35, T37)))
U9_ga(T12, T35, X89, del25_out_aga(T13, T12, .(T35, T37))) → U10_ga(T12, T35, X89, del25_in_aga(X87, T37, X89))
U10_ga(T12, T35, X89, del25_out_aga(X87, T37, X89)) → del220_out_ga(T12, .(T35, X89))
del220_in_ga(T12, .(T42, X89)) → U11_ga(T12, T42, X89, del25_in_aga(T13, T12, .(T42, T44)))
U11_ga(T12, T42, X89, del25_out_aga(T13, T12, .(T42, T44))) → U12_ga(T12, T42, X89, del25_in_aga(X87, T44, X89))
U12_ga(T12, T42, X89, del25_out_aga(X87, T44, X89)) → del220_out_ga(T12, .(T42, X89))
U15_g(T4, del220_out_ga(T9, X29)) → goal1_out_g(T4)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
del220_in_ga(x1, x2)  =  del220_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
del25_in_aga(x1, x2, x3)  =  del25_in_aga(x2)
[]  =  []
del25_out_aga(x1, x2, x3)  =  del25_out_aga(x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del220_out_ga(x1, x2)  =  del220_out_ga
U6_ga(x1, x2)  =  U6_ga(x2)
del25_in_agg(x1, x2, x3)  =  del25_in_agg(x2, x3)
del25_out_agg(x1, x2, x3)  =  del25_out_agg
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x5)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(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_G(T4) → U13_G(T4, s2l4_in_ga(T4, X9))
GOAL1_IN_G(T4) → S2L4_IN_GA(T4, X9)
S2L4_IN_GA(s(T7), .(X23, X24)) → U1_GA(T7, X23, X24, s2l4_in_ga(T7, X24))
S2L4_IN_GA(s(T7), .(X23, X24)) → S2L4_IN_GA(T7, X24)
GOAL1_IN_G(T4) → U14_G(T4, s2l4_in_ga(T4, T9))
U14_G(T4, s2l4_out_ga(T4, T9)) → U15_G(T4, del220_in_ga(T9, X29))
U14_G(T4, s2l4_out_ga(T4, T9)) → DEL220_IN_GA(T9, X29)
DEL220_IN_GA(T12, X39) → U5_GA(T12, X39, del25_in_aga(X36, T12, X37))
DEL220_IN_GA(T12, X39) → DEL25_IN_AGA(X36, T12, X37)
DEL25_IN_AGA(X87, .(T35, T37), .(T35, X89)) → U2_AGA(X87, T35, T37, X89, del25_in_aga(X87, T37, X89))
DEL25_IN_AGA(X87, .(T35, T37), .(T35, X89)) → DEL25_IN_AGA(X87, T37, X89)
DEL25_IN_AGA(X87, .(T42, T44), .(T42, X89)) → U3_AGA(X87, T42, T44, X89, del25_in_aga(X87, T44, X89))
DEL220_IN_GA(T12, []) → U6_GA(T12, del25_in_agg(T13, T12, []))
DEL220_IN_GA(T12, []) → DEL25_IN_AGG(T13, T12, [])
DEL25_IN_AGG(X87, .(T35, T37), .(T35, X89)) → U2_AGG(X87, T35, T37, X89, del25_in_agg(X87, T37, X89))
DEL25_IN_AGG(X87, .(T35, T37), .(T35, X89)) → DEL25_IN_AGG(X87, T37, X89)
DEL25_IN_AGG(X87, .(T42, T44), .(T42, X89)) → U3_AGG(X87, T42, T44, X89, del25_in_agg(X87, T44, X89))
DEL220_IN_GA(T12, T22) → U7_GA(T12, T22, del25_in_aga(T13, T12, .(T21, T22)))
DEL220_IN_GA(T12, T22) → DEL25_IN_AGA(T13, T12, .(T21, T22))
DEL220_IN_GA(T12, T28) → U8_GA(T12, T28, del25_in_aga(T13, T12, .(T27, T28)))
DEL220_IN_GA(T12, .(T35, X89)) → U9_GA(T12, T35, X89, del25_in_aga(T13, T12, .(T35, T37)))
DEL220_IN_GA(T12, .(T35, X89)) → DEL25_IN_AGA(T13, T12, .(T35, T37))
U9_GA(T12, T35, X89, del25_out_aga(T13, T12, .(T35, T37))) → U10_GA(T12, T35, X89, del25_in_aga(X87, T37, X89))
U9_GA(T12, T35, X89, del25_out_aga(T13, T12, .(T35, T37))) → DEL25_IN_AGA(X87, T37, X89)
DEL220_IN_GA(T12, .(T42, X89)) → U11_GA(T12, T42, X89, del25_in_aga(T13, T12, .(T42, T44)))
U11_GA(T12, T42, X89, del25_out_aga(T13, T12, .(T42, T44))) → U12_GA(T12, T42, X89, del25_in_aga(X87, T44, X89))
U11_GA(T12, T42, X89, del25_out_aga(T13, T12, .(T42, T44))) → DEL25_IN_AGA(X87, T44, X89)

The TRS R consists of the following rules:

goal1_in_g(T4) → U13_g(T4, s2l4_in_ga(T4, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X23, X24)) → U1_ga(T7, X23, X24, s2l4_in_ga(T7, X24))
U1_ga(T7, X23, X24, s2l4_out_ga(T7, X24)) → s2l4_out_ga(s(T7), .(X23, X24))
U13_g(T4, s2l4_out_ga(T4, X9)) → goal1_out_g(T4)
goal1_in_g(T4) → U14_g(T4, s2l4_in_ga(T4, T9))
U14_g(T4, s2l4_out_ga(T4, T9)) → U15_g(T4, del220_in_ga(T9, X29))
del220_in_ga(T12, X39) → U5_ga(T12, X39, del25_in_aga(X36, T12, X37))
del25_in_aga(X59, [], []) → del25_out_aga(X59, [], [])
del25_in_aga(T21, .(T21, T22), T22) → del25_out_aga(T21, .(T21, T22), T22)
del25_in_aga(X87, .(T35, T37), .(T35, X89)) → U2_aga(X87, T35, T37, X89, del25_in_aga(X87, T37, X89))
del25_in_aga(X87, .(T42, T44), .(T42, X89)) → U3_aga(X87, T42, T44, X89, del25_in_aga(X87, T44, X89))
U3_aga(X87, T42, T44, X89, del25_out_aga(X87, T44, X89)) → del25_out_aga(X87, .(T42, T44), .(T42, X89))
U2_aga(X87, T35, T37, X89, del25_out_aga(X87, T37, X89)) → del25_out_aga(X87, .(T35, T37), .(T35, X89))
U5_ga(T12, X39, del25_out_aga(X36, T12, X37)) → del220_out_ga(T12, X39)
del220_in_ga(T12, []) → U6_ga(T12, del25_in_agg(T13, T12, []))
del25_in_agg(X59, [], []) → del25_out_agg(X59, [], [])
del25_in_agg(T21, .(T21, T22), T22) → del25_out_agg(T21, .(T21, T22), T22)
del25_in_agg(X87, .(T35, T37), .(T35, X89)) → U2_agg(X87, T35, T37, X89, del25_in_agg(X87, T37, X89))
del25_in_agg(X87, .(T42, T44), .(T42, X89)) → U3_agg(X87, T42, T44, X89, del25_in_agg(X87, T44, X89))
U3_agg(X87, T42, T44, X89, del25_out_agg(X87, T44, X89)) → del25_out_agg(X87, .(T42, T44), .(T42, X89))
U2_agg(X87, T35, T37, X89, del25_out_agg(X87, T37, X89)) → del25_out_agg(X87, .(T35, T37), .(T35, X89))
U6_ga(T12, del25_out_agg(T13, T12, [])) → del220_out_ga(T12, [])
del220_in_ga(T12, T22) → U7_ga(T12, T22, del25_in_aga(T13, T12, .(T21, T22)))
U7_ga(T12, T22, del25_out_aga(T13, T12, .(T21, T22))) → del220_out_ga(T12, T22)
del220_in_ga(T12, T28) → U8_ga(T12, T28, del25_in_aga(T13, T12, .(T27, T28)))
U8_ga(T12, T28, del25_out_aga(T13, T12, .(T27, T28))) → del220_out_ga(T12, T28)
del220_in_ga(T12, .(T35, X89)) → U9_ga(T12, T35, X89, del25_in_aga(T13, T12, .(T35, T37)))
U9_ga(T12, T35, X89, del25_out_aga(T13, T12, .(T35, T37))) → U10_ga(T12, T35, X89, del25_in_aga(X87, T37, X89))
U10_ga(T12, T35, X89, del25_out_aga(X87, T37, X89)) → del220_out_ga(T12, .(T35, X89))
del220_in_ga(T12, .(T42, X89)) → U11_ga(T12, T42, X89, del25_in_aga(T13, T12, .(T42, T44)))
U11_ga(T12, T42, X89, del25_out_aga(T13, T12, .(T42, T44))) → U12_ga(T12, T42, X89, del25_in_aga(X87, T44, X89))
U12_ga(T12, T42, X89, del25_out_aga(X87, T44, X89)) → del220_out_ga(T12, .(T42, X89))
U15_g(T4, del220_out_ga(T9, X29)) → goal1_out_g(T4)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
del220_in_ga(x1, x2)  =  del220_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
del25_in_aga(x1, x2, x3)  =  del25_in_aga(x2)
[]  =  []
del25_out_aga(x1, x2, x3)  =  del25_out_aga(x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del220_out_ga(x1, x2)  =  del220_out_ga
U6_ga(x1, x2)  =  U6_ga(x2)
del25_in_agg(x1, x2, x3)  =  del25_in_agg(x2, x3)
del25_out_agg(x1, x2, x3)  =  del25_out_agg
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x5)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x4)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U13_G(x1, x2)  =  U13_G(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U14_G(x1, x2)  =  U14_G(x2)
U15_G(x1, x2)  =  U15_G(x2)
DEL220_IN_GA(x1, x2)  =  DEL220_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
DEL25_IN_AGA(x1, x2, x3)  =  DEL25_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x5)
U6_GA(x1, x2)  =  U6_GA(x2)
DEL25_IN_AGG(x1, x2, x3)  =  DEL25_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5)  =  U2_AGG(x5)
U3_AGG(x1, x2, x3, x4, x5)  =  U3_AGG(x5)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
U12_GA(x1, x2, x3, x4)  =  U12_GA(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_G(T4) → U13_G(T4, s2l4_in_ga(T4, X9))
GOAL1_IN_G(T4) → S2L4_IN_GA(T4, X9)
S2L4_IN_GA(s(T7), .(X23, X24)) → U1_GA(T7, X23, X24, s2l4_in_ga(T7, X24))
S2L4_IN_GA(s(T7), .(X23, X24)) → S2L4_IN_GA(T7, X24)
GOAL1_IN_G(T4) → U14_G(T4, s2l4_in_ga(T4, T9))
U14_G(T4, s2l4_out_ga(T4, T9)) → U15_G(T4, del220_in_ga(T9, X29))
U14_G(T4, s2l4_out_ga(T4, T9)) → DEL220_IN_GA(T9, X29)
DEL220_IN_GA(T12, X39) → U5_GA(T12, X39, del25_in_aga(X36, T12, X37))
DEL220_IN_GA(T12, X39) → DEL25_IN_AGA(X36, T12, X37)
DEL25_IN_AGA(X87, .(T35, T37), .(T35, X89)) → U2_AGA(X87, T35, T37, X89, del25_in_aga(X87, T37, X89))
DEL25_IN_AGA(X87, .(T35, T37), .(T35, X89)) → DEL25_IN_AGA(X87, T37, X89)
DEL25_IN_AGA(X87, .(T42, T44), .(T42, X89)) → U3_AGA(X87, T42, T44, X89, del25_in_aga(X87, T44, X89))
DEL220_IN_GA(T12, []) → U6_GA(T12, del25_in_agg(T13, T12, []))
DEL220_IN_GA(T12, []) → DEL25_IN_AGG(T13, T12, [])
DEL25_IN_AGG(X87, .(T35, T37), .(T35, X89)) → U2_AGG(X87, T35, T37, X89, del25_in_agg(X87, T37, X89))
DEL25_IN_AGG(X87, .(T35, T37), .(T35, X89)) → DEL25_IN_AGG(X87, T37, X89)
DEL25_IN_AGG(X87, .(T42, T44), .(T42, X89)) → U3_AGG(X87, T42, T44, X89, del25_in_agg(X87, T44, X89))
DEL220_IN_GA(T12, T22) → U7_GA(T12, T22, del25_in_aga(T13, T12, .(T21, T22)))
DEL220_IN_GA(T12, T22) → DEL25_IN_AGA(T13, T12, .(T21, T22))
DEL220_IN_GA(T12, T28) → U8_GA(T12, T28, del25_in_aga(T13, T12, .(T27, T28)))
DEL220_IN_GA(T12, .(T35, X89)) → U9_GA(T12, T35, X89, del25_in_aga(T13, T12, .(T35, T37)))
DEL220_IN_GA(T12, .(T35, X89)) → DEL25_IN_AGA(T13, T12, .(T35, T37))
U9_GA(T12, T35, X89, del25_out_aga(T13, T12, .(T35, T37))) → U10_GA(T12, T35, X89, del25_in_aga(X87, T37, X89))
U9_GA(T12, T35, X89, del25_out_aga(T13, T12, .(T35, T37))) → DEL25_IN_AGA(X87, T37, X89)
DEL220_IN_GA(T12, .(T42, X89)) → U11_GA(T12, T42, X89, del25_in_aga(T13, T12, .(T42, T44)))
U11_GA(T12, T42, X89, del25_out_aga(T13, T12, .(T42, T44))) → U12_GA(T12, T42, X89, del25_in_aga(X87, T44, X89))
U11_GA(T12, T42, X89, del25_out_aga(T13, T12, .(T42, T44))) → DEL25_IN_AGA(X87, T44, X89)

The TRS R consists of the following rules:

goal1_in_g(T4) → U13_g(T4, s2l4_in_ga(T4, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X23, X24)) → U1_ga(T7, X23, X24, s2l4_in_ga(T7, X24))
U1_ga(T7, X23, X24, s2l4_out_ga(T7, X24)) → s2l4_out_ga(s(T7), .(X23, X24))
U13_g(T4, s2l4_out_ga(T4, X9)) → goal1_out_g(T4)
goal1_in_g(T4) → U14_g(T4, s2l4_in_ga(T4, T9))
U14_g(T4, s2l4_out_ga(T4, T9)) → U15_g(T4, del220_in_ga(T9, X29))
del220_in_ga(T12, X39) → U5_ga(T12, X39, del25_in_aga(X36, T12, X37))
del25_in_aga(X59, [], []) → del25_out_aga(X59, [], [])
del25_in_aga(T21, .(T21, T22), T22) → del25_out_aga(T21, .(T21, T22), T22)
del25_in_aga(X87, .(T35, T37), .(T35, X89)) → U2_aga(X87, T35, T37, X89, del25_in_aga(X87, T37, X89))
del25_in_aga(X87, .(T42, T44), .(T42, X89)) → U3_aga(X87, T42, T44, X89, del25_in_aga(X87, T44, X89))
U3_aga(X87, T42, T44, X89, del25_out_aga(X87, T44, X89)) → del25_out_aga(X87, .(T42, T44), .(T42, X89))
U2_aga(X87, T35, T37, X89, del25_out_aga(X87, T37, X89)) → del25_out_aga(X87, .(T35, T37), .(T35, X89))
U5_ga(T12, X39, del25_out_aga(X36, T12, X37)) → del220_out_ga(T12, X39)
del220_in_ga(T12, []) → U6_ga(T12, del25_in_agg(T13, T12, []))
del25_in_agg(X59, [], []) → del25_out_agg(X59, [], [])
del25_in_agg(T21, .(T21, T22), T22) → del25_out_agg(T21, .(T21, T22), T22)
del25_in_agg(X87, .(T35, T37), .(T35, X89)) → U2_agg(X87, T35, T37, X89, del25_in_agg(X87, T37, X89))
del25_in_agg(X87, .(T42, T44), .(T42, X89)) → U3_agg(X87, T42, T44, X89, del25_in_agg(X87, T44, X89))
U3_agg(X87, T42, T44, X89, del25_out_agg(X87, T44, X89)) → del25_out_agg(X87, .(T42, T44), .(T42, X89))
U2_agg(X87, T35, T37, X89, del25_out_agg(X87, T37, X89)) → del25_out_agg(X87, .(T35, T37), .(T35, X89))
U6_ga(T12, del25_out_agg(T13, T12, [])) → del220_out_ga(T12, [])
del220_in_ga(T12, T22) → U7_ga(T12, T22, del25_in_aga(T13, T12, .(T21, T22)))
U7_ga(T12, T22, del25_out_aga(T13, T12, .(T21, T22))) → del220_out_ga(T12, T22)
del220_in_ga(T12, T28) → U8_ga(T12, T28, del25_in_aga(T13, T12, .(T27, T28)))
U8_ga(T12, T28, del25_out_aga(T13, T12, .(T27, T28))) → del220_out_ga(T12, T28)
del220_in_ga(T12, .(T35, X89)) → U9_ga(T12, T35, X89, del25_in_aga(T13, T12, .(T35, T37)))
U9_ga(T12, T35, X89, del25_out_aga(T13, T12, .(T35, T37))) → U10_ga(T12, T35, X89, del25_in_aga(X87, T37, X89))
U10_ga(T12, T35, X89, del25_out_aga(X87, T37, X89)) → del220_out_ga(T12, .(T35, X89))
del220_in_ga(T12, .(T42, X89)) → U11_ga(T12, T42, X89, del25_in_aga(T13, T12, .(T42, T44)))
U11_ga(T12, T42, X89, del25_out_aga(T13, T12, .(T42, T44))) → U12_ga(T12, T42, X89, del25_in_aga(X87, T44, X89))
U12_ga(T12, T42, X89, del25_out_aga(X87, T44, X89)) → del220_out_ga(T12, .(T42, X89))
U15_g(T4, del220_out_ga(T9, X29)) → goal1_out_g(T4)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
del220_in_ga(x1, x2)  =  del220_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
del25_in_aga(x1, x2, x3)  =  del25_in_aga(x2)
[]  =  []
del25_out_aga(x1, x2, x3)  =  del25_out_aga(x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del220_out_ga(x1, x2)  =  del220_out_ga
U6_ga(x1, x2)  =  U6_ga(x2)
del25_in_agg(x1, x2, x3)  =  del25_in_agg(x2, x3)
del25_out_agg(x1, x2, x3)  =  del25_out_agg
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x5)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x4)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U13_G(x1, x2)  =  U13_G(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U14_G(x1, x2)  =  U14_G(x2)
U15_G(x1, x2)  =  U15_G(x2)
DEL220_IN_GA(x1, x2)  =  DEL220_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
DEL25_IN_AGA(x1, x2, x3)  =  DEL25_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x5)
U6_GA(x1, x2)  =  U6_GA(x2)
DEL25_IN_AGG(x1, x2, x3)  =  DEL25_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5)  =  U2_AGG(x5)
U3_AGG(x1, x2, x3, x4, x5)  =  U3_AGG(x5)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 24 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

DEL25_IN_AGG(X87, .(T35, T37), .(T35, X89)) → DEL25_IN_AGG(X87, T37, X89)

The TRS R consists of the following rules:

goal1_in_g(T4) → U13_g(T4, s2l4_in_ga(T4, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X23, X24)) → U1_ga(T7, X23, X24, s2l4_in_ga(T7, X24))
U1_ga(T7, X23, X24, s2l4_out_ga(T7, X24)) → s2l4_out_ga(s(T7), .(X23, X24))
U13_g(T4, s2l4_out_ga(T4, X9)) → goal1_out_g(T4)
goal1_in_g(T4) → U14_g(T4, s2l4_in_ga(T4, T9))
U14_g(T4, s2l4_out_ga(T4, T9)) → U15_g(T4, del220_in_ga(T9, X29))
del220_in_ga(T12, X39) → U5_ga(T12, X39, del25_in_aga(X36, T12, X37))
del25_in_aga(X59, [], []) → del25_out_aga(X59, [], [])
del25_in_aga(T21, .(T21, T22), T22) → del25_out_aga(T21, .(T21, T22), T22)
del25_in_aga(X87, .(T35, T37), .(T35, X89)) → U2_aga(X87, T35, T37, X89, del25_in_aga(X87, T37, X89))
del25_in_aga(X87, .(T42, T44), .(T42, X89)) → U3_aga(X87, T42, T44, X89, del25_in_aga(X87, T44, X89))
U3_aga(X87, T42, T44, X89, del25_out_aga(X87, T44, X89)) → del25_out_aga(X87, .(T42, T44), .(T42, X89))
U2_aga(X87, T35, T37, X89, del25_out_aga(X87, T37, X89)) → del25_out_aga(X87, .(T35, T37), .(T35, X89))
U5_ga(T12, X39, del25_out_aga(X36, T12, X37)) → del220_out_ga(T12, X39)
del220_in_ga(T12, []) → U6_ga(T12, del25_in_agg(T13, T12, []))
del25_in_agg(X59, [], []) → del25_out_agg(X59, [], [])
del25_in_agg(T21, .(T21, T22), T22) → del25_out_agg(T21, .(T21, T22), T22)
del25_in_agg(X87, .(T35, T37), .(T35, X89)) → U2_agg(X87, T35, T37, X89, del25_in_agg(X87, T37, X89))
del25_in_agg(X87, .(T42, T44), .(T42, X89)) → U3_agg(X87, T42, T44, X89, del25_in_agg(X87, T44, X89))
U3_agg(X87, T42, T44, X89, del25_out_agg(X87, T44, X89)) → del25_out_agg(X87, .(T42, T44), .(T42, X89))
U2_agg(X87, T35, T37, X89, del25_out_agg(X87, T37, X89)) → del25_out_agg(X87, .(T35, T37), .(T35, X89))
U6_ga(T12, del25_out_agg(T13, T12, [])) → del220_out_ga(T12, [])
del220_in_ga(T12, T22) → U7_ga(T12, T22, del25_in_aga(T13, T12, .(T21, T22)))
U7_ga(T12, T22, del25_out_aga(T13, T12, .(T21, T22))) → del220_out_ga(T12, T22)
del220_in_ga(T12, T28) → U8_ga(T12, T28, del25_in_aga(T13, T12, .(T27, T28)))
U8_ga(T12, T28, del25_out_aga(T13, T12, .(T27, T28))) → del220_out_ga(T12, T28)
del220_in_ga(T12, .(T35, X89)) → U9_ga(T12, T35, X89, del25_in_aga(T13, T12, .(T35, T37)))
U9_ga(T12, T35, X89, del25_out_aga(T13, T12, .(T35, T37))) → U10_ga(T12, T35, X89, del25_in_aga(X87, T37, X89))
U10_ga(T12, T35, X89, del25_out_aga(X87, T37, X89)) → del220_out_ga(T12, .(T35, X89))
del220_in_ga(T12, .(T42, X89)) → U11_ga(T12, T42, X89, del25_in_aga(T13, T12, .(T42, T44)))
U11_ga(T12, T42, X89, del25_out_aga(T13, T12, .(T42, T44))) → U12_ga(T12, T42, X89, del25_in_aga(X87, T44, X89))
U12_ga(T12, T42, X89, del25_out_aga(X87, T44, X89)) → del220_out_ga(T12, .(T42, X89))
U15_g(T4, del220_out_ga(T9, X29)) → goal1_out_g(T4)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
del220_in_ga(x1, x2)  =  del220_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
del25_in_aga(x1, x2, x3)  =  del25_in_aga(x2)
[]  =  []
del25_out_aga(x1, x2, x3)  =  del25_out_aga(x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del220_out_ga(x1, x2)  =  del220_out_ga
U6_ga(x1, x2)  =  U6_ga(x2)
del25_in_agg(x1, x2, x3)  =  del25_in_agg(x2, x3)
del25_out_agg(x1, x2, x3)  =  del25_out_agg
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x5)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x4)
DEL25_IN_AGG(x1, x2, x3)  =  DEL25_IN_AGG(x2, x3)

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:

DEL25_IN_AGG(X87, .(T35, T37), .(T35, X89)) → DEL25_IN_AGG(X87, T37, X89)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL25_IN_AGG(x1, x2, x3)  =  DEL25_IN_AGG(x2, x3)

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:

DEL25_IN_AGG(.(T37), .(X89)) → DEL25_IN_AGG(T37, X89)

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:

• DEL25_IN_AGG(.(T37), .(X89)) → DEL25_IN_AGG(T37, X89)
  The graph contains the following edges 1 > 1, 2 > 2

(15) TRUE

(16) Obligation:

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

DEL25_IN_AGA(X87, .(T35, T37), .(T35, X89)) → DEL25_IN_AGA(X87, T37, X89)

The TRS R consists of the following rules:

goal1_in_g(T4) → U13_g(T4, s2l4_in_ga(T4, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X23, X24)) → U1_ga(T7, X23, X24, s2l4_in_ga(T7, X24))
U1_ga(T7, X23, X24, s2l4_out_ga(T7, X24)) → s2l4_out_ga(s(T7), .(X23, X24))
U13_g(T4, s2l4_out_ga(T4, X9)) → goal1_out_g(T4)
goal1_in_g(T4) → U14_g(T4, s2l4_in_ga(T4, T9))
U14_g(T4, s2l4_out_ga(T4, T9)) → U15_g(T4, del220_in_ga(T9, X29))
del220_in_ga(T12, X39) → U5_ga(T12, X39, del25_in_aga(X36, T12, X37))
del25_in_aga(X59, [], []) → del25_out_aga(X59, [], [])
del25_in_aga(T21, .(T21, T22), T22) → del25_out_aga(T21, .(T21, T22), T22)
del25_in_aga(X87, .(T35, T37), .(T35, X89)) → U2_aga(X87, T35, T37, X89, del25_in_aga(X87, T37, X89))
del25_in_aga(X87, .(T42, T44), .(T42, X89)) → U3_aga(X87, T42, T44, X89, del25_in_aga(X87, T44, X89))
U3_aga(X87, T42, T44, X89, del25_out_aga(X87, T44, X89)) → del25_out_aga(X87, .(T42, T44), .(T42, X89))
U2_aga(X87, T35, T37, X89, del25_out_aga(X87, T37, X89)) → del25_out_aga(X87, .(T35, T37), .(T35, X89))
U5_ga(T12, X39, del25_out_aga(X36, T12, X37)) → del220_out_ga(T12, X39)
del220_in_ga(T12, []) → U6_ga(T12, del25_in_agg(T13, T12, []))
del25_in_agg(X59, [], []) → del25_out_agg(X59, [], [])
del25_in_agg(T21, .(T21, T22), T22) → del25_out_agg(T21, .(T21, T22), T22)
del25_in_agg(X87, .(T35, T37), .(T35, X89)) → U2_agg(X87, T35, T37, X89, del25_in_agg(X87, T37, X89))
del25_in_agg(X87, .(T42, T44), .(T42, X89)) → U3_agg(X87, T42, T44, X89, del25_in_agg(X87, T44, X89))
U3_agg(X87, T42, T44, X89, del25_out_agg(X87, T44, X89)) → del25_out_agg(X87, .(T42, T44), .(T42, X89))
U2_agg(X87, T35, T37, X89, del25_out_agg(X87, T37, X89)) → del25_out_agg(X87, .(T35, T37), .(T35, X89))
U6_ga(T12, del25_out_agg(T13, T12, [])) → del220_out_ga(T12, [])
del220_in_ga(T12, T22) → U7_ga(T12, T22, del25_in_aga(T13, T12, .(T21, T22)))
U7_ga(T12, T22, del25_out_aga(T13, T12, .(T21, T22))) → del220_out_ga(T12, T22)
del220_in_ga(T12, T28) → U8_ga(T12, T28, del25_in_aga(T13, T12, .(T27, T28)))
U8_ga(T12, T28, del25_out_aga(T13, T12, .(T27, T28))) → del220_out_ga(T12, T28)
del220_in_ga(T12, .(T35, X89)) → U9_ga(T12, T35, X89, del25_in_aga(T13, T12, .(T35, T37)))
U9_ga(T12, T35, X89, del25_out_aga(T13, T12, .(T35, T37))) → U10_ga(T12, T35, X89, del25_in_aga(X87, T37, X89))
U10_ga(T12, T35, X89, del25_out_aga(X87, T37, X89)) → del220_out_ga(T12, .(T35, X89))
del220_in_ga(T12, .(T42, X89)) → U11_ga(T12, T42, X89, del25_in_aga(T13, T12, .(T42, T44)))
U11_ga(T12, T42, X89, del25_out_aga(T13, T12, .(T42, T44))) → U12_ga(T12, T42, X89, del25_in_aga(X87, T44, X89))
U12_ga(T12, T42, X89, del25_out_aga(X87, T44, X89)) → del220_out_ga(T12, .(T42, X89))
U15_g(T4, del220_out_ga(T9, X29)) → goal1_out_g(T4)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
del220_in_ga(x1, x2)  =  del220_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
del25_in_aga(x1, x2, x3)  =  del25_in_aga(x2)
[]  =  []
del25_out_aga(x1, x2, x3)  =  del25_out_aga(x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del220_out_ga(x1, x2)  =  del220_out_ga
U6_ga(x1, x2)  =  U6_ga(x2)
del25_in_agg(x1, x2, x3)  =  del25_in_agg(x2, x3)
del25_out_agg(x1, x2, x3)  =  del25_out_agg
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x5)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x4)
DEL25_IN_AGA(x1, x2, x3)  =  DEL25_IN_AGA(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:

DEL25_IN_AGA(X87, .(T35, T37), .(T35, X89)) → DEL25_IN_AGA(X87, T37, X89)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL25_IN_AGA(x1, x2, x3)  =  DEL25_IN_AGA(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:

DEL25_IN_AGA(.(T37)) → DEL25_IN_AGA(T37)

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:

• DEL25_IN_AGA(.(T37)) → DEL25_IN_AGA(T37)
  The graph contains the following edges 1 > 1

(22) TRUE

(23) Obligation:

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

S2L4_IN_GA(s(T7), .(X23, X24)) → S2L4_IN_GA(T7, X24)

The TRS R consists of the following rules:

goal1_in_g(T4) → U13_g(T4, s2l4_in_ga(T4, X9))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X23, X24)) → U1_ga(T7, X23, X24, s2l4_in_ga(T7, X24))
U1_ga(T7, X23, X24, s2l4_out_ga(T7, X24)) → s2l4_out_ga(s(T7), .(X23, X24))
U13_g(T4, s2l4_out_ga(T4, X9)) → goal1_out_g(T4)
goal1_in_g(T4) → U14_g(T4, s2l4_in_ga(T4, T9))
U14_g(T4, s2l4_out_ga(T4, T9)) → U15_g(T4, del220_in_ga(T9, X29))
del220_in_ga(T12, X39) → U5_ga(T12, X39, del25_in_aga(X36, T12, X37))
del25_in_aga(X59, [], []) → del25_out_aga(X59, [], [])
del25_in_aga(T21, .(T21, T22), T22) → del25_out_aga(T21, .(T21, T22), T22)
del25_in_aga(X87, .(T35, T37), .(T35, X89)) → U2_aga(X87, T35, T37, X89, del25_in_aga(X87, T37, X89))
del25_in_aga(X87, .(T42, T44), .(T42, X89)) → U3_aga(X87, T42, T44, X89, del25_in_aga(X87, T44, X89))
U3_aga(X87, T42, T44, X89, del25_out_aga(X87, T44, X89)) → del25_out_aga(X87, .(T42, T44), .(T42, X89))
U2_aga(X87, T35, T37, X89, del25_out_aga(X87, T37, X89)) → del25_out_aga(X87, .(T35, T37), .(T35, X89))
U5_ga(T12, X39, del25_out_aga(X36, T12, X37)) → del220_out_ga(T12, X39)
del220_in_ga(T12, []) → U6_ga(T12, del25_in_agg(T13, T12, []))
del25_in_agg(X59, [], []) → del25_out_agg(X59, [], [])
del25_in_agg(T21, .(T21, T22), T22) → del25_out_agg(T21, .(T21, T22), T22)
del25_in_agg(X87, .(T35, T37), .(T35, X89)) → U2_agg(X87, T35, T37, X89, del25_in_agg(X87, T37, X89))
del25_in_agg(X87, .(T42, T44), .(T42, X89)) → U3_agg(X87, T42, T44, X89, del25_in_agg(X87, T44, X89))
U3_agg(X87, T42, T44, X89, del25_out_agg(X87, T44, X89)) → del25_out_agg(X87, .(T42, T44), .(T42, X89))
U2_agg(X87, T35, T37, X89, del25_out_agg(X87, T37, X89)) → del25_out_agg(X87, .(T35, T37), .(T35, X89))
U6_ga(T12, del25_out_agg(T13, T12, [])) → del220_out_ga(T12, [])
del220_in_ga(T12, T22) → U7_ga(T12, T22, del25_in_aga(T13, T12, .(T21, T22)))
U7_ga(T12, T22, del25_out_aga(T13, T12, .(T21, T22))) → del220_out_ga(T12, T22)
del220_in_ga(T12, T28) → U8_ga(T12, T28, del25_in_aga(T13, T12, .(T27, T28)))
U8_ga(T12, T28, del25_out_aga(T13, T12, .(T27, T28))) → del220_out_ga(T12, T28)
del220_in_ga(T12, .(T35, X89)) → U9_ga(T12, T35, X89, del25_in_aga(T13, T12, .(T35, T37)))
U9_ga(T12, T35, X89, del25_out_aga(T13, T12, .(T35, T37))) → U10_ga(T12, T35, X89, del25_in_aga(X87, T37, X89))
U10_ga(T12, T35, X89, del25_out_aga(X87, T37, X89)) → del220_out_ga(T12, .(T35, X89))
del220_in_ga(T12, .(T42, X89)) → U11_ga(T12, T42, X89, del25_in_aga(T13, T12, .(T42, T44)))
U11_ga(T12, T42, X89, del25_out_aga(T13, T12, .(T42, T44))) → U12_ga(T12, T42, X89, del25_in_aga(X87, T44, X89))
U12_ga(T12, T42, X89, del25_out_aga(X87, T44, X89)) → del220_out_ga(T12, .(T42, X89))
U15_g(T4, del220_out_ga(T9, X29)) → goal1_out_g(T4)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
del220_in_ga(x1, x2)  =  del220_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
del25_in_aga(x1, x2, x3)  =  del25_in_aga(x2)
[]  =  []
del25_out_aga(x1, x2, x3)  =  del25_out_aga(x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del220_out_ga(x1, x2)  =  del220_out_ga
U6_ga(x1, x2)  =  U6_ga(x2)
del25_in_agg(x1, x2, x3)  =  del25_in_agg(x2, x3)
del25_out_agg(x1, x2, x3)  =  del25_out_agg
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x5)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x4)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

S2L4_IN_GA(s(T7), .(X23, X24)) → S2L4_IN_GA(T7, X24)

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

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

S2L4_IN_GA(s(T7)) → S2L4_IN_GA(T7)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• S2L4_IN_GA(s(T7)) → S2L4_IN_GA(T7)
  The graph contains the following edges 1 > 1

(29) TRUE