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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT 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(T5, X9), conf(X9))\n\nT5 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(s2l(T5, X9), conf(X9)), SCOPE: 2, CLAUSE: 6 | ','(s2l(T5, X9), conf(X9)), SCOPE: 2, CLAUSE: 7\n\nT5 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_2, ','(eq(X15, []), conf(X15))) | ','(s2l(0, X9), conf(X9)), SCOPE: 2, CLAUSE: 7\n\nX9 is free\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(s2l(T5, X9), conf(X9)), SCOPE: 2, CLAUSE: 7\n\nT5 is ground\nX9 is free; \ns2l(T5, X9) !=? s2l(0, X14)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(eq(X15, []), conf(X15))\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(eq(X15, []), conf(X15)), SCOPE: 3, CLAUSE: 15\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: conf([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: conf([]), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(del2([], X48), ','(del(X49, X50, X48), conf(X50)))\n\nX48 is free\nX49 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: del2([], X48)\n\nX48 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(del(X49, X50, T6), conf(X50))\n\nX49 is free\nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: del2([], X48), SCOPE: 5, CLAUSE: 2\n\nX48 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(del(X78, [], X79), del(X80, X79, X81))\n\nX81 is free\nX78 is free; \nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: del(X78, [], X79)\n\nX78 is free\nX79 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: del(X80, T8, X81)\n\nX81 is free\nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: del(X78, [], X79), SCOPE: 6, CLAUSE: 3 | del(X78, [], X79), SCOPE: 6, CLAUSE: 4 | del(X78, [], X79), SCOPE: 6, CLAUSE: 5\n\nX78 is free\nX79 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(!_6, failure(a)) | del(X78, [], X79), SCOPE: 6, CLAUSE: 4 | del(X78, [], X79), SCOPE: 6, CLAUSE: 5\n\nX78 is free\nX79 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: failure(a), SCOPE: 7, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: del(X80, T8, X81), SCOPE: 8, CLAUSE: 3 | del(X80, T8, X81), SCOPE: 8, CLAUSE: 4 | del(X80, T8, X81), SCOPE: 8, CLAUSE: 5\n\nX81 is free\nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(!_8, failure(a)) | del(X80, T8, X81), SCOPE: 8, CLAUSE: 4 | del(X80, T8, X81), SCOPE: 8, CLAUSE: 5\n\nX81 is free\nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: del(X80, T8, X81), SCOPE: 8, CLAUSE: 4 | del(X80, T8, X81), SCOPE: 8, CLAUSE: 5\n\nX81 is free\nX80 is free; \ndel(X80, T8, X81) !=? del(X114, [], X115)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: failure(a), SCOPE: 9, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: del(X80, T8, X81), SCOPE: 8, CLAUSE: 4\n\nX81 is free\nX80 is free; \ndel(X80, T8, X81) !=? del(X114, [], X115)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: del(X80, T8, X81), SCOPE: 8, CLAUSE: 5\n\nX81 is free\nX80 is free; \ndel(X80, T8, X81) !=? del(X114, [], X115)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(head(T16, X143), tail(T16, X144))\n\nX143 is free\nX144 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(head(T16, X143), tail(T16, X144)), SCOPE: 10, CLAUSE: 8 | ','(head(T16, X143), tail(T16, X144)), SCOPE: 10, CLAUSE: 9\n\nX143 is free\nX144 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(head(T16, X143), tail(T16, X144)), SCOPE: 10, CLAUSE: 8\n\nX143 is free\nX144 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(head(T16, X143), tail(T16, X144)), SCOPE: 10, CLAUSE: 9\n\nX143 is free\nX144 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: tail([], X144)\n\nX144 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: tail([], X144), SCOPE: 11, CLAUSE: 10 | tail([], X144), SCOPE: 11, CLAUSE: 11\n\nX144 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: tail([], X144), SCOPE: 11, CLAUSE: 10\n\nX144 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: tail([], X144), SCOPE: 11, CLAUSE: 11\n\nX144 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: tail(.(T23, T24), X144)\n\nX144 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: tail(.(T23, T24), X144), SCOPE: 12, CLAUSE: 10 | tail(.(T23, T24), X144), SCOPE: 12, CLAUSE: 11\n\nX144 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: tail(.(T23, T24), X144), SCOPE: 12, CLAUSE: 11\n\nX144 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(head(T35, X197), ','(tail(T35, X195), del(X196, X195, X198)))\n\nX196 is free\nX197 is free; \nX198 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(head(T35, X197), ','(tail(T35, X195), del(X196, X195, X198))), SCOPE: 13, CLAUSE: 8 | ','(head(T35, X197), ','(tail(T35, X195), del(X196, X195, X198))), SCOPE: 13, CLAUSE: 9\n\nX196 is free\nX197 is free; \nX198 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(head(T35, X197), ','(tail(T35, X195), del(X196, X195, X198))), SCOPE: 13, CLAUSE: 8\n\nX196 is free\nX197 is free; \nX198 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(head(T35, X197), ','(tail(T35, X195), del(X196, X195, X198))), SCOPE: 13, CLAUSE: 9\n\nX196 is free\nX197 is free; \nX198 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(tail([], X195), del(X196, X195, X198))\n\nX196 is free\nX198 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(tail([], X195), del(X196, X195, X198)), SCOPE: 14, CLAUSE: 10 | ','(tail([], X195), del(X196, X195, X198)), SCOPE: 14, CLAUSE: 11\n\nX196 is free\nX198 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(tail([], X195), del(X196, X195, X198)), SCOPE: 14, CLAUSE: 10\n\nX196 is free\nX198 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(tail([], X195), del(X196, X195, X198)), SCOPE: 14, CLAUSE: 11\n\nX196 is free\nX198 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: del(X196, [], X198)\n\nX196 is free\nX198 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(tail(.(T42, T43), X195), del(X196, X195, X198))\n\nX196 is free\nX198 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(tail(.(T42, T43), X195), del(X196, X195, X198)), SCOPE: 15, CLAUSE: 10 | ','(tail(.(T42, T43), X195), del(X196, X195, X198)), SCOPE: 15, CLAUSE: 11\n\nX196 is free\nX198 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(tail(.(T42, T43), X195), del(X196, X195, X198)), SCOPE: 15, CLAUSE: 11\n\nX196 is free\nX198 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: del(X196, T51, X198)\n\nX196 is free\nX198 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: del(X49, X50, T6)\n\nX49 is free\nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: conf(T54)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: del(X49, X50, T6), SCOPE: 16, CLAUSE: 3 | del(X49, X50, T6), SCOPE: 16, CLAUSE: 4 | del(X49, X50, T6), SCOPE: 16, CLAUSE: 5\n\nX49 is free\nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(!_16, failure(a)) | del(X49, X50, T6), SCOPE: 16, CLAUSE: 4 | del(X49, X50, T6), SCOPE: 16, CLAUSE: 5\n\nX49 is free\nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: failure(a), SCOPE: 17, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: conf(T54), SCOPE: 18, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(del2(T65, X289), ','(del(X290, X291, X289), conf(X291)))\n\nX289 is free\nX290 is free; \nX291 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: del2(T65, X289)\n\nX289 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(del(X290, X291, T67), conf(X291))\n\nX290 is free\nX291 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: del2(T65, X289), SCOPE: 19, CLAUSE: 2\n\nX289 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(del(X319, T73, X320), del(X321, X320, X322))\n\nX322 is free\nX319 is free; \nX320 is free; \nX321 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: del(X319, T73, X320)\n\nX319 is free\nX320 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: del(X321, T75, X322)\n\nX322 is free\nX321 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(p(T79, X354), ','(s2l(X354, X356), conf(.(X355, X356))))\n\nT79 is ground\nX355 is free; \nX356 is free; \nX354 is free; \ns2l(T79, X9) !=? s2l(0, X14)", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(p(T79, X354), ','(s2l(X354, X356), conf(.(X355, X356)))), SCOPE: 20, CLAUSE: 12 | ','(p(T79, X354), ','(s2l(X354, X356), conf(.(X355, X356)))), SCOPE: 20, CLAUSE: 13\n\nT79 is ground\nX355 is free; \nX356 is free; \nX354 is free; \ns2l(T79, X9) !=? s2l(0, X14)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(p(T79, X354), ','(s2l(X354, X356), conf(.(X355, X356)))), SCOPE: 20, CLAUSE: 13\n\nT79 is ground\nX355 is free; \nX356 is free; \nX354 is free; \ns2l(T79, X9) !=? s2l(0, X14)", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(s2l(T82, X356), conf(.(X355, X356)))\n\nT82 is ground\nX355 is free; \nX356 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: s2l(T82, X356)\n\nT82 is ground\nX356 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: conf(.(X355, T83))\n\nX355 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: s2l(T82, X356), SCOPE: 21, CLAUSE: 6 | s2l(T82, X356), SCOPE: 21, CLAUSE: 7\n\nT82 is ground\nX356 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(!_21, eq(X372, [])) | s2l(0, X356), SCOPE: 21, CLAUSE: 7\n\nX356 is free\nX372 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: s2l(T82, X356), SCOPE: 21, CLAUSE: 7\n\nT82 is ground\nX356 is free; \ns2l(T82, X356) !=? s2l(0, X371)", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: eq(X372, [])\n\nX372 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: eq(X372, []), SCOPE: 22, CLAUSE: 15\n\nX372 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(p(T86, X396), s2l(X396, X398))\n\nT86 is ground\nX398 is free; \nX396 is free; \ns2l(T86, X356) !=? s2l(0, X371)", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(p(T86, X396), s2l(X396, X398)), SCOPE: 23, CLAUSE: 12 | ','(p(T86, X396), s2l(X396, X398)), SCOPE: 23, CLAUSE: 13\n\nT86 is ground\nX398 is free; \nX396 is free; \ns2l(T86, X356) !=? s2l(0, X371)", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(p(T86, X396), s2l(X396, X398)), SCOPE: 23, CLAUSE: 13\n\nT86 is ground\nX398 is free; \nX396 is free; \ns2l(T86, X356) !=? s2l(0, X371)", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: s2l(T89, X398)\n\nT89 is ground\nX398 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 99 [arrowhead = none ];
99 -> 2;

100 [label="ONLY EVAL with clause\ngoal(X8) :- ','(s2l(X8, X9), conf(X9)).\nand substitution\nT3 -> T5\nX8 -> T5", fontsize=14, style = filled, fillcolor = white];
2 -> 100 [arrowhead = none ];
100 -> 3;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 101 [arrowhead = none ];
101 -> 4;

102 [label="EVAL with clause\ns2l(0, X14) :- ','(!_2, eq(X14, [])).\nand substitution\nT5 -> 0\nX9 -> X15\nX14 -> X15", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 102 [arrowhead = none ];
102 -> 5;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 103 [arrowhead = none ];
103 -> 6;

104 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 104 [arrowhead = none ];
104 -> 7;

105 [label="ONLY EVAL with clause\ns2l(X351, .(X352, X353)) :- ','(p(X351, X354), s2l(X354, X353)).\nand substitution\nT5 -> T79\nX351 -> T79\nX352 -> X355\nX353 -> X356\nX9 -> .(X355, X356)", fontsize=14, style = filled, fillcolor = white];
6 -> 105 [arrowhead = none ];
105 -> 80;

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 106 [arrowhead = none ];
106 -> 8;

107 [label="ONLY EVAL with clause\neq(X30, X30).\nand substitution\nX15 -> []\nX30 -> []\nX31 -> []", fontsize=14, style = filled, fillcolor = white];
8 -> 107 [arrowhead = none ];
107 -> 9;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 108 [arrowhead = none ];
108 -> 10;

109 [label="ONLY EVAL with clause\nconf(X47) :- ','(del2(X47, X48), ','(del(X49, X50, X48), conf(X50))).\nand substitution\nX47 -> []", fontsize=14, style = filled, fillcolor = white];
10 -> 109 [arrowhead = none ];
109 -> 11;

110 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 110 [arrowhead = none ];
110 -> 12;

111 [label="SPLIT 2\nreplacements:\nX48 -> T6", fontsize=14, style = filled, fillcolor = violet];
11 -> 111 [arrowhead = none ];
111 -> 13;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 112 [arrowhead = none ];
112 -> 14;

113 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 113 [arrowhead = none ];
113 -> 65;

114 [label="SPLIT 2\nreplacements:\nX49 -> T53\nX50 -> T54", fontsize=14, style = filled, fillcolor = violet];
13 -> 114 [arrowhead = none ];
114 -> 66;

115 [label="ONLY EVAL with clause\ndel2(X76, X77) :- ','(del(X78, X76, X79), del(X80, X79, X77)).\nand substitution\nX76 -> []\nX48 -> X81\nX77 -> X81", fontsize=14, style = filled, fillcolor = white];
14 -> 115 [arrowhead = none ];
115 -> 15;

116 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 116 [arrowhead = none ];
116 -> 16;

117 [label="SPLIT 2\nreplacements:\nX78 -> T7\nX79 -> T8", fontsize=14, style = filled, fillcolor = violet];
15 -> 117 [arrowhead = none ];
117 -> 17;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 118 [arrowhead = none ];
118 -> 18;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 119 [arrowhead = none ];
119 -> 23;

120 [label="ONLY EVAL with clause\ndel(X98, [], X99) :- ','(!_6, failure(a)).\nand substitution\nX78 -> X100\nX98 -> X100\nX79 -> X101\nX99 -> X101", fontsize=14, style = filled, fillcolor = white];
18 -> 120 [arrowhead = none ];
120 -> 19;

121 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 121 [arrowhead = none ];
121 -> 20;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 122 [arrowhead = none ];
122 -> 21;

123 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
21 -> 123 [arrowhead = none ];
123 -> 22;

124 [label="EVAL with clause\ndel(X114, [], X115) :- ','(!_8, failure(a)).\nand substitution\nX80 -> X116\nX114 -> X116\nT8 -> []\nX81 -> X117\nX115 -> X117", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 124 [arrowhead = none ];
124 -> 24;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 125 [arrowhead = none ];
125 -> 25;

126 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 126 [arrowhead = none ];
126 -> 26;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 127 [arrowhead = none ];
127 -> 29;

128 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 128 [arrowhead = none ];
128 -> 30;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 129 [arrowhead = none ];
129 -> 27;

130 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 130 [arrowhead = none ];
130 -> 28;

131 [label="ONLY EVAL with clause\ndel(X140, X141, X142) :- ','(head(X141, X140), tail(X141, X142)).\nand substitution\nX80 -> X143\nX140 -> X143\nT8 -> T16\nX141 -> T16\nX81 -> X144\nX142 -> X144\nT15 -> T16", fontsize=14, style = filled, fillcolor = white];
29 -> 131 [arrowhead = none ];
131 -> 31;

132 [label="ONLY EVAL with clause\ndel(X191, X192, .(X193, X194)) :- ','(head(X192, X193), ','(tail(X192, X195), del(X191, X195, X194))).\nand substitution\nX80 -> X196\nX191 -> X196\nT8 -> T35\nX192 -> T35\nX193 -> X197\nX194 -> X198\nX81 -> .(X197, X198)\nT34 -> T35", fontsize=14, style = filled, fillcolor = white];
30 -> 132 [arrowhead = none ];
132 -> 49;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 133 [arrowhead = none ];
133 -> 32;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 134 [arrowhead = none ];
134 -> 33;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 135 [arrowhead = none ];
135 -> 34;

136 [label="EVAL with clause\nhead([], X153).\nand substitution\nT16 -> []\nX143 -> X154\nX153 -> X154", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 136 [arrowhead = none ];
136 -> 35;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 137 [arrowhead = none ];
137 -> 36;

138 [label="EVAL with clause\nhead(.(X162, X163), X162).\nand substitution\nX162 -> T23\nX163 -> T24\nT16 -> .(T23, T24)\nX143 -> T23\nT21 -> T23\nT22 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 138 [arrowhead = none ];
138 -> 43;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 139 [arrowhead = none ];
139 -> 44;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 140 [arrowhead = none ];
140 -> 37;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 141 [arrowhead = none ];
141 -> 38;

142 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 142 [arrowhead = none ];
142 -> 39;

143 [label="ONLY EVAL with clause\ntail([], []).\nand substitution\nX144 -> []", fontsize=14, style = filled, fillcolor = white];
38 -> 143 [arrowhead = none ];
143 -> 40;

144 [label="BACKTRACK\nfor clause: tail(.(X6, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
39 -> 144 [arrowhead = none ];
144 -> 42;

145 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 145 [arrowhead = none ];
145 -> 41;

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 146 [arrowhead = none ];
146 -> 45;

147 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
45 -> 147 [arrowhead = none ];
147 -> 46;

148 [label="ONLY EVAL with clause\ntail(.(X170, X171), X171).\nand substitution\nT23 -> T29\nX170 -> T29\nT24 -> T30\nX171 -> T30\nX144 -> T30", fontsize=14, style = filled, fillcolor = white];
46 -> 148 [arrowhead = none ];
148 -> 47;

149 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 149 [arrowhead = none ];
149 -> 48;

150 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 150 [arrowhead = none ];
150 -> 50;

151 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 151 [arrowhead = none ];
151 -> 51;

152 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 152 [arrowhead = none ];
152 -> 52;

153 [label="EVAL with clause\nhead([], X211).\nand substitution\nT35 -> []\nX197 -> X212\nX211 -> X212", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 153 [arrowhead = none ];
153 -> 53;

154 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 154 [arrowhead = none ];
154 -> 54;

155 [label="EVAL with clause\nhead(.(X236, X237), X236).\nand substitution\nX236 -> T42\nX237 -> T43\nT35 -> .(T42, T43)\nX197 -> T42\nT40 -> T42\nT41 -> T43", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 155 [arrowhead = none ];
155 -> 60;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 156 [arrowhead = none ];
156 -> 61;

157 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 157 [arrowhead = none ];
157 -> 55;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 158 [arrowhead = none ];
158 -> 56;

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 159 [arrowhead = none ];
159 -> 57;

160 [label="ONLY EVAL with clause\ntail([], []).\nand substitution\nX195 -> []", fontsize=14, style = filled, fillcolor = white];
56 -> 160 [arrowhead = none ];
160 -> 58;

161 [label="BACKTRACK\nfor clause: tail(.(X6, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
57 -> 161 [arrowhead = none ];
161 -> 59;

162 [label="INSTANCE with matching:\nX78 -> X196\nX79 -> X198", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 162 [arrowhead = none , style=dashed];
162 -> 16[style=dashed];

163 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 163 [arrowhead = none ];
163 -> 62;

164 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
62 -> 164 [arrowhead = none ];
164 -> 63;

165 [label="ONLY EVAL with clause\ntail(.(X252, X253), X253).\nand substitution\nT42 -> T49\nX252 -> T49\nT43 -> T51\nX253 -> T51\nX195 -> T51\nT50 -> T51", fontsize=14, style = filled, fillcolor = white];
63 -> 165 [arrowhead = none ];
165 -> 64;

166 [label="INSTANCE with matching:\nX80 -> X196\nT8 -> T51\nX81 -> X198", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 166 [arrowhead = none , style=dashed];
166 -> 17[style=dashed];

167 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 167 [arrowhead = none ];
167 -> 67;

168 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 168 [arrowhead = none ];
168 -> 72;

169 [label="ONLY EVAL with clause\ndel(X270, [], X271) :- ','(!_16, failure(a)).\nand substitution\nX49 -> X272\nX270 -> X272\nX50 -> []\nT6 -> T58\nX271 -> T58", fontsize=14, style = filled, fillcolor = white];
67 -> 169 [arrowhead = none ];
169 -> 68;

170 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 170 [arrowhead = none ];
170 -> 69;

171 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 171 [arrowhead = none ];
171 -> 70;

172 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
70 -> 172 [arrowhead = none ];
172 -> 71;

173 [label="ONLY EVAL with clause\nconf(X288) :- ','(del2(X288, X289), ','(del(X290, X291, X289), conf(X291))).\nand substitution\nT54 -> T65\nX288 -> T65\nT64 -> T65", fontsize=14, style = filled, fillcolor = white];
72 -> 173 [arrowhead = none ];
173 -> 73;

174 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
73 -> 174 [arrowhead = none ];
174 -> 74;

175 [label="SPLIT 2\nreplacements:\nX289 -> T67", fontsize=14, style = filled, fillcolor = violet];
73 -> 175 [arrowhead = none ];
175 -> 75;

176 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
74 -> 176 [arrowhead = none ];
176 -> 76;

177 [label="INSTANCE with matching:\nX49 -> X290\nX50 -> X291\nT6 -> T67", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 177 [arrowhead = none , style=dashed];
177 -> 13[style=dashed];

178 [label="ONLY EVAL with clause\ndel2(X317, X318) :- ','(del(X319, X317, X320), del(X321, X320, X318)).\nand substitution\nT65 -> T73\nX317 -> T73\nX289 -> X322\nX318 -> X322\nT72 -> T73", fontsize=14, style = filled, fillcolor = white];
76 -> 178 [arrowhead = none ];
178 -> 77;

179 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
77 -> 179 [arrowhead = none ];
179 -> 78;

180 [label="SPLIT 2\nreplacements:\nX319 -> T74\nX320 -> T75", fontsize=14, style = filled, fillcolor = violet];
77 -> 180 [arrowhead = none ];
180 -> 79;

181 [label="INSTANCE with matching:\nX80 -> X319\nT8 -> T73\nX81 -> X320", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 181 [arrowhead = none , style=dashed];
181 -> 17[style=dashed];

182 [label="INSTANCE with matching:\nX80 -> X321\nT8 -> T75\nX81 -> X322", fontsize=14, style = filled, fillcolor = lightgrey];
79 -> 182 [arrowhead = none , style=dashed];
182 -> 17[style=dashed];

183 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 183 [arrowhead = none ];
183 -> 81;

184 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T79, X9), s2l(0, X14))", fontsize=14, style = filled, fillcolor = white];
81 -> 184 [arrowhead = none ];
184 -> 82;

185 [label="EVAL with clause\np(s(X361), X361).\nand substitution\nX361 -> T82\nT79 -> s(T82)\nX354 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 185 [arrowhead = none ];
185 -> 83;

186 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 186 [arrowhead = none ];
186 -> 84;

187 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
83 -> 187 [arrowhead = none ];
187 -> 85;

188 [label="SPLIT 2\nreplacements:\nX356 -> T83", fontsize=14, style = filled, fillcolor = violet];
83 -> 188 [arrowhead = none ];
188 -> 86;

189 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 189 [arrowhead = none ];
189 -> 87;

190 [label="INSTANCE with matching:\nT54 -> .(X355, T83)", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 190 [arrowhead = none , style=dashed];
190 -> 66[style=dashed];

191 [label="EVAL with clause\ns2l(0, X371) :- ','(!_21, eq(X371, [])).\nand substitution\nT82 -> 0\nX356 -> X372\nX371 -> X372", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 191 [arrowhead = none ];
191 -> 88;

192 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 192 [arrowhead = none ];
192 -> 89;

193 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 193 [arrowhead = none ];
193 -> 90;

194 [label="ONLY EVAL with clause\ns2l(X393, .(X394, X395)) :- ','(p(X393, X396), s2l(X396, X395)).\nand substitution\nT82 -> T86\nX393 -> T86\nX394 -> X397\nX395 -> X398\nX356 -> .(X397, X398)", fontsize=14, style = filled, fillcolor = white];
89 -> 194 [arrowhead = none ];
194 -> 94;

195 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
90 -> 195 [arrowhead = none ];
195 -> 91;

196 [label="ONLY EVAL with clause\neq(X378, X378).\nand substitution\nX372 -> []\nX378 -> []\nX379 -> []", fontsize=14, style = filled, fillcolor = white];
91 -> 196 [arrowhead = none ];
196 -> 92;

197 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 197 [arrowhead = none ];
197 -> 93;

198 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
94 -> 198 [arrowhead = none ];
198 -> 95;

199 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T86, X356), s2l(0, X371))", fontsize=14, style = filled, fillcolor = white];
95 -> 199 [arrowhead = none ];
199 -> 96;

200 [label="EVAL with clause\np(s(X403), X403).\nand substitution\nX403 -> T89\nT86 -> s(T89)\nX396 -> T89", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 200 [arrowhead = none ];
200 -> 97;

201 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 201 [arrowhead = none ];
201 -> 98;

202 [label="INSTANCE with matching:\nT82 -> T89\nX356 -> X398", fontsize=14, style = filled, fillcolor = lightgrey];
97 -> 202 [arrowhead = none , style=dashed];
202 -> 85[style=dashed];

}

(2) Obligation:

Triples:

del17(X196, [], .(X212, X198)) :- del16(X196, X198).
del17(X196, .(T49, T51), .(T49, X198)) :- del17(X196, T51, X198).
p13(T53, T54, T6) :- ','(delc65(T53, T54, T6), conf66(T54)).
conf66(T73) :- del17(X319, T73, X320).
conf66(T73) :- ','(delc17(T74, T73, T75), del17(X321, T75, X322)).
conf66(T65) :- ','(del2c74(T65, T67), p13(X290, X291, T67)).
s2l85(s(T89), .(X397, X398)) :- s2l85(T89, X398).
goal1(0) :- del16(X78, X79).
goal1(0) :- ','(delc16(T7, T8), del17(X80, T8, X81)).
goal1(0) :- ','(del2c12(T6), p13(X49, X50, T6)).
goal1(s(T82)) :- s2l85(T82, X356).
goal1(s(T82)) :- ','(s2lc85(T82, T83), conf66(.(X355, T83))).

Clauses:

delc17(X154, [], []).
delc17(T29, .(T29, T30), T30).
delc17(X196, [], .(X212, X198)) :- delc16(X196, X198).
delc17(X196, .(T49, T51), .(T49, X198)) :- delc17(X196, T51, X198).
qc13(T53, T54, T6) :- ','(delc65(T53, T54, T6), confc66(T54)).
confc66(T65) :- ','(del2c74(T65, T67), qc13(X290, X291, T67)).
s2lc85(0, []).
s2lc85(s(T89), .(X397, X398)) :- s2lc85(T89, X398).
del2c12(X81) :- ','(delc16(T7, T8), delc17(X80, T8, X81)).
del2c74(T73, X322) :- ','(delc17(T74, T73, T75), delc17(X321, T75, X322)).

Afs:

goal1(x1)  =  goal1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

del17(X196, .(T49, T51), .(T49, X198)) :- del17(X196, T51, X198).
conf66(T73) :- del17(X319, T73, X320).
conf66(T73) :- ','(delc17(T74, T73, T75), del17(X321, T75, X322)).
s2l85(s(T89), .(X397, X398)) :- s2l85(T89, X398).
goal1(s(T82)) :- s2l85(T82, X356).
goal1(s(T82)) :- ','(s2lc85(T82, T83), conf66(.(X355, T83))).

Clauses:

delc17(X154, [], []).
delc17(T29, .(T29, T30), T30).
delc17(X196, .(T49, T51), .(T49, X198)) :- delc17(X196, T51, X198).
s2lc85(0, []).
s2lc85(s(T89), .(X397, X398)) :- s2lc85(T89, X398).
del2c74(T73, X322) :- ','(delc17(T74, T73, T75), delc17(X321, T75, X322)).

Afs:

goal1(x1)  =  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)
s2l85_in: (b,f)
s2lc85_in: (b,f)
conf66_in: (b)
del17_in: (f,b,f)
delc17_in: (f,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_G(s(T82)) → U6_G(T82, s2l85_in_ga(T82, X356))
GOAL1_IN_G(s(T82)) → S2L85_IN_GA(T82, X356)
S2L85_IN_GA(s(T89), .(X397, X398)) → U5_GA(T89, X397, X398, s2l85_in_ga(T89, X398))
S2L85_IN_GA(s(T89), .(X397, X398)) → S2L85_IN_GA(T89, X398)
GOAL1_IN_G(s(T82)) → U7_G(T82, s2lc85_in_ga(T82, T83))
U7_G(T82, s2lc85_out_ga(T82, T83)) → U8_G(T82, conf66_in_g(.(X355, T83)))
U7_G(T82, s2lc85_out_ga(T82, T83)) → CONF66_IN_G(.(X355, T83))
CONF66_IN_G(T73) → U2_G(T73, del17_in_aga(X319, T73, X320))
CONF66_IN_G(T73) → DEL17_IN_AGA(X319, T73, X320)
DEL17_IN_AGA(X196, .(T49, T51), .(T49, X198)) → U1_AGA(X196, T49, T51, X198, del17_in_aga(X196, T51, X198))
DEL17_IN_AGA(X196, .(T49, T51), .(T49, X198)) → DEL17_IN_AGA(X196, T51, X198)
CONF66_IN_G(T73) → U3_G(T73, delc17_in_aga(T74, T73, T75))
U3_G(T73, delc17_out_aga(T74, T73, T75)) → U4_G(T73, del17_in_aga(X321, T75, X322))
U3_G(T73, delc17_out_aga(T74, T73, T75)) → DEL17_IN_AGA(X321, T75, X322)

The TRS R consists of the following rules:

s2lc85_in_ga(0, []) → s2lc85_out_ga(0, [])
s2lc85_in_ga(s(T89), .(X397, X398)) → U11_ga(T89, X397, X398, s2lc85_in_ga(T89, X398))
U11_ga(T89, X397, X398, s2lc85_out_ga(T89, X398)) → s2lc85_out_ga(s(T89), .(X397, X398))
delc17_in_aga(X154, [], []) → delc17_out_aga(X154, [], [])
delc17_in_aga(T29, .(T29, T30), T30) → delc17_out_aga(T29, .(T29, T30), T30)
delc17_in_aga(X196, .(T49, T51), .(T49, X198)) → U10_aga(X196, T49, T51, X198, delc17_in_aga(X196, T51, X198))
U10_aga(X196, T49, T51, X198, delc17_out_aga(X196, T51, X198)) → delc17_out_aga(X196, .(T49, T51), .(T49, X198))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2l85_in_ga(x1, x2)  =  s2l85_in_ga(x1)
.(x1, x2)  =  .(x2)
s2lc85_in_ga(x1, x2)  =  s2lc85_in_ga(x1)
0  =  0
s2lc85_out_ga(x1, x2)  =  s2lc85_out_ga(x1, x2)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
conf66_in_g(x1)  =  conf66_in_g(x1)
del17_in_aga(x1, x2, x3)  =  del17_in_aga(x2)
delc17_in_aga(x1, x2, x3)  =  delc17_in_aga(x2)
[]  =  []
delc17_out_aga(x1, x2, x3)  =  delc17_out_aga(x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U6_G(x1, x2)  =  U6_G(x1, x2)
S2L85_IN_GA(x1, x2)  =  S2L85_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U7_G(x1, x2)  =  U7_G(x1, x2)
U8_G(x1, x2)  =  U8_G(x1, x2)
CONF66_IN_G(x1)  =  CONF66_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
DEL17_IN_AGA(x1, x2, x3)  =  DEL17_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
U3_G(x1, x2)  =  U3_G(x1, x2)
U4_G(x1, x2)  =  U4_G(x1, x2)

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_G(s(T82)) → U6_G(T82, s2l85_in_ga(T82, X356))
GOAL1_IN_G(s(T82)) → S2L85_IN_GA(T82, X356)
S2L85_IN_GA(s(T89), .(X397, X398)) → U5_GA(T89, X397, X398, s2l85_in_ga(T89, X398))
S2L85_IN_GA(s(T89), .(X397, X398)) → S2L85_IN_GA(T89, X398)
GOAL1_IN_G(s(T82)) → U7_G(T82, s2lc85_in_ga(T82, T83))
U7_G(T82, s2lc85_out_ga(T82, T83)) → U8_G(T82, conf66_in_g(.(X355, T83)))
U7_G(T82, s2lc85_out_ga(T82, T83)) → CONF66_IN_G(.(X355, T83))
CONF66_IN_G(T73) → U2_G(T73, del17_in_aga(X319, T73, X320))
CONF66_IN_G(T73) → DEL17_IN_AGA(X319, T73, X320)
DEL17_IN_AGA(X196, .(T49, T51), .(T49, X198)) → U1_AGA(X196, T49, T51, X198, del17_in_aga(X196, T51, X198))
DEL17_IN_AGA(X196, .(T49, T51), .(T49, X198)) → DEL17_IN_AGA(X196, T51, X198)
CONF66_IN_G(T73) → U3_G(T73, delc17_in_aga(T74, T73, T75))
U3_G(T73, delc17_out_aga(T74, T73, T75)) → U4_G(T73, del17_in_aga(X321, T75, X322))
U3_G(T73, delc17_out_aga(T74, T73, T75)) → DEL17_IN_AGA(X321, T75, X322)

The TRS R consists of the following rules:

s2lc85_in_ga(0, []) → s2lc85_out_ga(0, [])
s2lc85_in_ga(s(T89), .(X397, X398)) → U11_ga(T89, X397, X398, s2lc85_in_ga(T89, X398))
U11_ga(T89, X397, X398, s2lc85_out_ga(T89, X398)) → s2lc85_out_ga(s(T89), .(X397, X398))
delc17_in_aga(X154, [], []) → delc17_out_aga(X154, [], [])
delc17_in_aga(T29, .(T29, T30), T30) → delc17_out_aga(T29, .(T29, T30), T30)
delc17_in_aga(X196, .(T49, T51), .(T49, X198)) → U10_aga(X196, T49, T51, X198, delc17_in_aga(X196, T51, X198))
U10_aga(X196, T49, T51, X198, delc17_out_aga(X196, T51, X198)) → delc17_out_aga(X196, .(T49, T51), .(T49, X198))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2l85_in_ga(x1, x2)  =  s2l85_in_ga(x1)
.(x1, x2)  =  .(x2)
s2lc85_in_ga(x1, x2)  =  s2lc85_in_ga(x1)
0  =  0
s2lc85_out_ga(x1, x2)  =  s2lc85_out_ga(x1, x2)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
conf66_in_g(x1)  =  conf66_in_g(x1)
del17_in_aga(x1, x2, x3)  =  del17_in_aga(x2)
delc17_in_aga(x1, x2, x3)  =  delc17_in_aga(x2)
[]  =  []
delc17_out_aga(x1, x2, x3)  =  delc17_out_aga(x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U6_G(x1, x2)  =  U6_G(x1, x2)
S2L85_IN_GA(x1, x2)  =  S2L85_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U7_G(x1, x2)  =  U7_G(x1, x2)
U8_G(x1, x2)  =  U8_G(x1, x2)
CONF66_IN_G(x1)  =  CONF66_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
DEL17_IN_AGA(x1, x2, x3)  =  DEL17_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
U3_G(x1, x2)  =  U3_G(x1, x2)
U4_G(x1, x2)  =  U4_G(x1, x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 12 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

DEL17_IN_AGA(X196, .(T49, T51), .(T49, X198)) → DEL17_IN_AGA(X196, T51, X198)

The TRS R consists of the following rules:

s2lc85_in_ga(0, []) → s2lc85_out_ga(0, [])
s2lc85_in_ga(s(T89), .(X397, X398)) → U11_ga(T89, X397, X398, s2lc85_in_ga(T89, X398))
U11_ga(T89, X397, X398, s2lc85_out_ga(T89, X398)) → s2lc85_out_ga(s(T89), .(X397, X398))
delc17_in_aga(X154, [], []) → delc17_out_aga(X154, [], [])
delc17_in_aga(T29, .(T29, T30), T30) → delc17_out_aga(T29, .(T29, T30), T30)
delc17_in_aga(X196, .(T49, T51), .(T49, X198)) → U10_aga(X196, T49, T51, X198, delc17_in_aga(X196, T51, X198))
U10_aga(X196, T49, T51, X198, delc17_out_aga(X196, T51, X198)) → delc17_out_aga(X196, .(T49, T51), .(T49, X198))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lc85_in_ga(x1, x2)  =  s2lc85_in_ga(x1)
0  =  0
s2lc85_out_ga(x1, x2)  =  s2lc85_out_ga(x1, x2)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
delc17_in_aga(x1, x2, x3)  =  delc17_in_aga(x2)
[]  =  []
delc17_out_aga(x1, x2, x3)  =  delc17_out_aga(x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
DEL17_IN_AGA(x1, x2, x3)  =  DEL17_IN_AGA(x2)

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:

DEL17_IN_AGA(X196, .(T49, T51), .(T49, X198)) → DEL17_IN_AGA(X196, T51, X198)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL17_IN_AGA(x1, x2, x3)  =  DEL17_IN_AGA(x2)

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:

DEL17_IN_AGA(.(T51)) → DEL17_IN_AGA(T51)

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:

• DEL17_IN_AGA(.(T51)) → DEL17_IN_AGA(T51)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

S2L85_IN_GA(s(T89), .(X397, X398)) → S2L85_IN_GA(T89, X398)

The TRS R consists of the following rules:

s2lc85_in_ga(0, []) → s2lc85_out_ga(0, [])
s2lc85_in_ga(s(T89), .(X397, X398)) → U11_ga(T89, X397, X398, s2lc85_in_ga(T89, X398))
U11_ga(T89, X397, X398, s2lc85_out_ga(T89, X398)) → s2lc85_out_ga(s(T89), .(X397, X398))
delc17_in_aga(X154, [], []) → delc17_out_aga(X154, [], [])
delc17_in_aga(T29, .(T29, T30), T30) → delc17_out_aga(T29, .(T29, T30), T30)
delc17_in_aga(X196, .(T49, T51), .(T49, X198)) → U10_aga(X196, T49, T51, X198, delc17_in_aga(X196, T51, X198))
U10_aga(X196, T49, T51, X198, delc17_out_aga(X196, T51, X198)) → delc17_out_aga(X196, .(T49, T51), .(T49, X198))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lc85_in_ga(x1, x2)  =  s2lc85_in_ga(x1)
0  =  0
s2lc85_out_ga(x1, x2)  =  s2lc85_out_ga(x1, x2)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
delc17_in_aga(x1, x2, x3)  =  delc17_in_aga(x2)
[]  =  []
delc17_out_aga(x1, x2, x3)  =  delc17_out_aga(x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
S2L85_IN_GA(x1, x2)  =  S2L85_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:

S2L85_IN_GA(s(T89), .(X397, X398)) → S2L85_IN_GA(T89, X398)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L85_IN_GA(x1, x2)  =  S2L85_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:

S2L85_IN_GA(s(T89)) → S2L85_IN_GA(T89)

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:

• S2L85_IN_GA(s(T89)) → S2L85_IN_GA(T89)
  The graph contains the following edges 1 > 1

(22) YES