Left Termination proof of /tmp/tmppI2q5m/confdel.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 183 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 34 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 69 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇒, 0 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇒, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

(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).

Query: goal(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

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\nX14 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([], X47), ','(del(X48, X49, X47), conf(X49)))\n\nX47 is free\nX48 is free\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: del2([], X47)\n\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(del(X48, X49, T6), conf(X49))\n\nX48 is free\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: del2([], X47), SCOPE: 5, CLAUSE: 2\n\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(del(X77, [], X78), del(X79, X78, X80))\n\nX80 is free\nX77 is free\nX78 is free\nX79 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: del(X77, [], X78)\n\nX77 is free\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: del(X79, T8, X80)\n\nX80 is free\nX79 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: del(X77, [], X78), SCOPE: 6, CLAUSE: 3 | del(X77, [], X78), SCOPE: 6, CLAUSE: 4 | del(X77, [], X78), SCOPE: 6, CLAUSE: 5\n\nX77 is free\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(!_6, failure(a)) | del(X77, [], X78), SCOPE: 6, CLAUSE: 4 | del(X77, [], X78), SCOPE: 6, CLAUSE: 5\n\nX77 is free\nX78 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(X79, T8, X80), SCOPE: 8, CLAUSE: 3 | del(X79, T8, X80), SCOPE: 8, CLAUSE: 4 | del(X79, T8, X80), SCOPE: 8, CLAUSE: 5\n\nX80 is free\nX79 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(!_8, failure(a)) | del(X79, T8, X80), SCOPE: 8, CLAUSE: 4 | del(X79, T8, X80), SCOPE: 8, CLAUSE: 5\n\nX80 is free\nX79 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: del(X79, T8, X80), SCOPE: 8, CLAUSE: 4 | del(X79, T8, X80), SCOPE: 8, CLAUSE: 5\n\nX80 is free\nX79 is free\nX113 is free\nX114 is free\ndel(X79, T8, X80) !=? del(X113, [], X114)", 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(X79, T8, X80), SCOPE: 8, CLAUSE: 4\n\nX80 is free\nX79 is free\nX113 is free\nX114 is free\ndel(X79, T8, X80) !=? del(X113, [], X114)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: del(X79, T8, X80), SCOPE: 8, CLAUSE: 5\n\nX80 is free\nX79 is free\nX113 is free\nX114 is free\ndel(X79, T8, X80) !=? del(X113, [], X114)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(head(T16, X140), tail(T16, X141))\n\nX140 is free\nX141 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(head(T16, X140), tail(T16, X141)), SCOPE: 10, CLAUSE: 8 | ','(head(T16, X140), tail(T16, X141)), SCOPE: 10, CLAUSE: 9\n\nX140 is free\nX141 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(head(T16, X140), tail(T16, X141)), SCOPE: 10, CLAUSE: 8\n\nX140 is free\nX141 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(head(T16, X140), tail(T16, X141)), SCOPE: 10, CLAUSE: 9\n\nX140 is free\nX141 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: tail([], X141)\n\nX141 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: tail([], X141), SCOPE: 11, CLAUSE: 10 | tail([], X141), SCOPE: 11, CLAUSE: 11\n\nX141 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: tail([], X141), SCOPE: 11, CLAUSE: 10\n\nX141 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: tail([], X141), SCOPE: 11, CLAUSE: 11\n\nX141 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), X141)\n\nX141 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), X141), SCOPE: 12, CLAUSE: 10 | tail(.(T23, T24), X141), SCOPE: 12, CLAUSE: 11\n\nX141 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: tail(.(T23, T24), X141), SCOPE: 12, CLAUSE: 11\n\nX141 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, X191), ','(tail(T35, X189), del(X190, X189, X192)))\n\nX190 is free\nX191 is free\nX192 is free\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(head(T35, X191), ','(tail(T35, X189), del(X190, X189, X192))), SCOPE: 13, CLAUSE: 8 | ','(head(T35, X191), ','(tail(T35, X189), del(X190, X189, X192))), SCOPE: 13, CLAUSE: 9\n\nX190 is free\nX191 is free\nX192 is free\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(head(T35, X191), ','(tail(T35, X189), del(X190, X189, X192))), SCOPE: 13, CLAUSE: 8\n\nX190 is free\nX191 is free\nX192 is free\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(head(T35, X191), ','(tail(T35, X189), del(X190, X189, X192))), SCOPE: 13, CLAUSE: 9\n\nX190 is free\nX191 is free\nX192 is free\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(tail([], X189), del(X190, X189, X192))\n\nX190 is free\nX192 is free\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(tail([], X189), del(X190, X189, X192)), SCOPE: 14, CLAUSE: 10 | ','(tail([], X189), del(X190, X189, X192)), SCOPE: 14, CLAUSE: 11\n\nX190 is free\nX192 is free\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(tail([], X189), del(X190, X189, X192)), SCOPE: 14, CLAUSE: 10\n\nX190 is free\nX192 is free\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(tail([], X189), del(X190, X189, X192)), SCOPE: 14, CLAUSE: 11\n\nX190 is free\nX192 is free\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: del(X190, [], X192)\n\nX190 is free\nX192 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), X189), del(X190, X189, X192))\n\nX190 is free\nX192 is free\nX189 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), X189), del(X190, X189, X192)), SCOPE: 15, CLAUSE: 10 | ','(tail(.(T42, T43), X189), del(X190, X189, X192)), SCOPE: 15, CLAUSE: 11\n\nX190 is free\nX192 is free\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(tail(.(T42, T43), X189), del(X190, X189, X192)), SCOPE: 15, CLAUSE: 11\n\nX190 is free\nX192 is free\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: del(X190, T51, X192)\n\nX190 is free\nX192 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: del(X48, X49, T6)\n\nX48 is free\nX49 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(X48, X49, T6), SCOPE: 16, CLAUSE: 3 | del(X48, X49, T6), SCOPE: 16, CLAUSE: 4 | del(X48, X49, T6), SCOPE: 16, CLAUSE: 5\n\nX48 is free\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(!_16, failure(a)) | del(X48, X49, T6), SCOPE: 16, CLAUSE: 4 | del(X48, X49, T6), SCOPE: 16, CLAUSE: 5\n\nX48 is free\nX49 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, X280), ','(del(X281, X282, X280), conf(X282)))\n\nX280 is free\nX281 is free\nX282 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: del2(T65, X280)\n\nX280 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(del(X281, X282, T67), conf(X282))\n\nX281 is free\nX282 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: del2(T65, X280), SCOPE: 19, CLAUSE: 2\n\nX280 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(del(X310, T73, X311), del(X312, X311, X313))\n\nX313 is free\nX310 is free\nX311 is free\nX312 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: del(X310, T73, X311)\n\nX310 is free\nX311 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: del(X312, T75, X313)\n\nX313 is free\nX312 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(','(p(T79, X345), s2l(X345, X347)), conf(.(X346, X347)))\n\nT79 is ground\nX9 is free\nX14 is free\nX346 is free\nX347 is free\nX345 is free\ns2l(T79, X9) !=? s2l(0, X14)", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(','(p(T79, X345), s2l(X345, X347)), conf(.(X346, X347))), SCOPE: 20, CLAUSE: 12 | ','(','(p(T79, X345), s2l(X345, X347)), conf(.(X346, X347))), SCOPE: 20, CLAUSE: 13\n\nT79 is ground\nX9 is free\nX14 is free\nX346 is free\nX347 is free\nX345 is free\ns2l(T79, X9) !=? s2l(0, X14)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(','(p(T79, X345), s2l(X345, X347)), conf(.(X346, X347))), SCOPE: 20, CLAUSE: 13\n\nT79 is ground\nX9 is free\nX14 is free\nX346 is free\nX347 is free\nX345 is free\ns2l(T79, X9) !=? s2l(0, X14)", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(s2l(T82, X347), conf(.(X346, X347)))\n\nT82 is ground\nX346 is free\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: s2l(T82, X347)\n\nT82 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: conf(.(X346, T83))\n\nX346 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: s2l(T82, X347), SCOPE: 21, CLAUSE: 6 | s2l(T82, X347), SCOPE: 21, CLAUSE: 7\n\nT82 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(!_21, eq(X362, [])) | s2l(0, X347), SCOPE: 21, CLAUSE: 7\n\nX347 is free\nX362 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: s2l(T82, X347), SCOPE: 21, CLAUSE: 7\n\nT82 is ground\nX347 is free\nX361 is free\ns2l(T82, X347) !=? s2l(0, X361)", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: eq(X362, [])\n\nX362 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: eq(X362, []), SCOPE: 22, CLAUSE: 15\n\nX362 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, X385), s2l(X385, X387))\n\nT86 is ground\nX347 is free\nX361 is free\nX387 is free\nX385 is free\ns2l(T86, X347) !=? s2l(0, X361)", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(p(T86, X385), s2l(X385, X387)), SCOPE: 23, CLAUSE: 12 | ','(p(T86, X385), s2l(X385, X387)), SCOPE: 23, CLAUSE: 13\n\nT86 is ground\nX347 is free\nX361 is free\nX387 is free\nX385 is free\ns2l(T86, X347) !=? s2l(0, X361)", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(p(T86, X385), s2l(X385, X387)), SCOPE: 23, CLAUSE: 13\n\nT86 is ground\nX347 is free\nX361 is free\nX387 is free\nX385 is free\ns2l(T86, X347) !=? s2l(0, X361)", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: s2l(T89, X387)\n\nT89 is ground\nX387 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 substitutionT3 -> T5,\nX8 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
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 substitutionT5 -> 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 = lightgreen];
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(X342, .(X343, X344)) :- ','(p(X342, X345), s2l(X345, X344)).\nand substitutionT5 -> T79,\nX342 -> T79,\nX343 -> X346,\nX344 -> X347,\nX9 -> .(X346, X347)", fontsize=14, style = filled, fillcolor = lightblue];
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(X29, X29).\nand substitutionX15 -> [],\nX29 -> [],\nX30 -> []", fontsize=14, style = filled, fillcolor = lightblue];
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(X46) :- ','(del2(X46, X47), ','(del(X48, X49, X47), conf(X49))).\nand substitutionX46 -> []", fontsize=14, style = filled, fillcolor = lightblue];
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:X47 -> 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:X48 -> T53,\nX49 -> T54", fontsize=14, style = filled, fillcolor = violet];
13 -> 114 [arrowhead = none ];
114 -> 66;

115 [label="ONLY EVAL with clause\ndel2(X75, X76) :- ','(del(X77, X75, X78), del(X79, X78, X76)).\nand substitutionX75 -> [],\nX47 -> X80,\nX76 -> X80", fontsize=14, style = filled, fillcolor = lightblue];
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:X77 -> T7,\nX78 -> 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(X97, [], X98) :- ','(!_6, failure(a)).\nand substitutionX77 -> X99,\nX97 -> X99,\nX78 -> X100,\nX98 -> X100", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
21 -> 123 [arrowhead = none ];
123 -> 22;

124 [label="EVAL with clause\ndel(X113, [], X114) :- ','(!_8, failure(a)).\nand substitutionX79 -> X115,\nX113 -> X115,\nT8 -> [],\nX80 -> X116,\nX114 -> X116", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 124 [arrowhead = none ];
124 -> 24;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
27 -> 130 [arrowhead = none ];
130 -> 28;

131 [label="ONLY EVAL with clause\ndel(X137, X138, X139) :- ','(head(X138, X137), tail(X138, X139)).\nand substitutionX79 -> X140,\nX137 -> X140,\nT8 -> T16,\nX138 -> T16,\nX80 -> X141,\nX139 -> X141,\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 131 [arrowhead = none ];
131 -> 31;

132 [label="ONLY EVAL with clause\ndel(X185, X186, .(X187, X188)) :- ','(head(X186, X187), ','(tail(X186, X189), del(X185, X189, X188))).\nand substitutionX79 -> X190,\nX185 -> X190,\nT8 -> T35,\nX186 -> T35,\nX187 -> X191,\nX188 -> X192,\nX80 -> .(X191, X192),\nT34 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
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([], X150).\nand substitutionT16 -> [],\nX140 -> X151,\nX150 -> X151", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 136 [arrowhead = none ];
136 -> 35;

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

138 [label="EVAL with clause\nhead(.(X158, X159), X158).\nand substitutionX158 -> T23,\nX159 -> T24,\nT16 -> .(T23, T24),\nX140 -> 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 = lightgreen];
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 substitutionX141 -> []", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 143 [arrowhead = none ];
143 -> 40;

144 [label="BACKTRACK\nfor clause: tail(.(X6, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
45 -> 147 [arrowhead = none ];
147 -> 46;

148 [label="ONLY EVAL with clause\ntail(.(X164, X165), X165).\nand substitutionT23 -> T29,\nX164 -> T29,\nT24 -> T30,\nX165 -> T30,\nX141 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
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([], X205).\nand substitutionT35 -> [],\nX191 -> X206,\nX205 -> X206", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 153 [arrowhead = none ];
153 -> 53;

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

155 [label="EVAL with clause\nhead(.(X229, X230), X229).\nand substitutionX229 -> T42,\nX230 -> T43,\nT35 -> .(T42, T43),\nX191 -> 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 = lightgreen];
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 substitutionX189 -> []", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 160 [arrowhead = none ];
160 -> 58;

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

162 [label="INSTANCE with matching:\nX77 -> X190\nX78 -> X192", 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 = lightgreen];
62 -> 164 [arrowhead = none ];
164 -> 63;

165 [label="ONLY EVAL with clause\ntail(.(X243, X244), X244).\nand substitutionT42 -> T49,\nX243 -> T49,\nT43 -> T51,\nX244 -> T51,\nX189 -> T51,\nT50 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 165 [arrowhead = none ];
165 -> 64;

166 [label="INSTANCE with matching:\nX79 -> X190\nT8 -> T51\nX80 -> X192", 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(X261, [], X262) :- ','(!_16, failure(a)).\nand substitutionX48 -> X263,\nX261 -> X263,\nX49 -> [],\nT6 -> T58,\nX262 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
70 -> 172 [arrowhead = none ];
172 -> 71;

173 [label="ONLY EVAL with clause\nconf(X279) :- ','(del2(X279, X280), ','(del(X281, X282, X280), conf(X282))).\nand substitutionT54 -> T65,\nX279 -> T65,\nT64 -> T65", fontsize=14, style = filled, fillcolor = lightblue];
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:X280 -> 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:\nX48 -> X281\nX49 -> X282\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(X308, X309) :- ','(del(X310, X308, X311), del(X312, X311, X309)).\nand substitutionT65 -> T73,\nX308 -> T73,\nX280 -> X313,\nX309 -> X313,\nT72 -> T73", fontsize=14, style = filled, fillcolor = lightblue];
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:X310 -> T74,\nX311 -> T75", fontsize=14, style = filled, fillcolor = violet];
77 -> 180 [arrowhead = none ];
180 -> 79;

181 [label="INSTANCE with matching:\nX79 -> X310\nT8 -> T73\nX80 -> X311", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 181 [arrowhead = none , style=dashed];
181 -> 17[style=dashed];

182 [label="INSTANCE with matching:\nX79 -> X312\nT8 -> T75\nX80 -> X313", 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 = lightgreen];
81 -> 184 [arrowhead = none ];
184 -> 82;

185 [label="EVAL with clause\np(s(X352), X352).\nand substitutionX352 -> T82,\nT79 -> s(T82),\nX345 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 185 [arrowhead = none ];
185 -> 83;

186 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nnew knowledge:\nT82 is ground\nreplacements:X347 -> 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 -> .(X346, T83)", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 190 [arrowhead = none , style=dashed];
190 -> 66[style=dashed];

191 [label="EVAL with clause\ns2l(0, X361) :- ','(!_21, eq(X361, [])).\nand substitutionT82 -> 0,\nX347 -> X362,\nX361 -> X362", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 191 [arrowhead = none ];
191 -> 88;

192 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X382, .(X383, X384)) :- ','(p(X382, X385), s2l(X385, X384)).\nand substitutionT82 -> T86,\nX382 -> T86,\nX383 -> X386,\nX384 -> X387,\nX347 -> .(X386, X387)", fontsize=14, style = filled, fillcolor = lightblue];
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(X367, X367).\nand substitutionX362 -> [],\nX367 -> [],\nX368 -> []", fontsize=14, style = filled, fillcolor = lightblue];
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, X347), s2l(0, X361))", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 199 [arrowhead = none ];
199 -> 96;

200 [label="EVAL with clause\np(s(X392), X392).\nand substitutionX392 -> T89,\nT86 -> s(T89),\nX385 -> T89", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 200 [arrowhead = none ];
200 -> 97;

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

202 [label="INSTANCE with matching:\nT82 -> T89\nX347 -> X387", fontsize=14, style = filled, fillcolor = lightgrey];
97 -> 202 [arrowhead = none , style=dashed];
202 -> 85[style=dashed];

}

(2) Obligation:

Clauses:

delA(X151, [], []).
delA(T29, .(T29, T30), T30).
delA(X190, .(T49, T51), .(T49, X192)) :- delA(X190, T51, X192).
confE(T65) :- del2F(T65, X280).
s2lG(0, []).
s2lG(s(T89), .(X386, X387)) :- s2lG(T89, X387).
del2F(T73, X313) :- delA(X310, T73, X311).
del2F(T73, X313) :- ','(delA(T74, T73, T75), delA(X312, T75, X313)).
goalI(s(T82)) :- s2lG(T82, X347).
goalI(s(T82)) :- ','(s2lG(T82, T83), confE(.(X346, T83))).

Query: goalI(g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
goalI_in: (b)
s2lG_in: (b,f)
confE_in: (b)
del2F_in: (b,f)
delA_in: (f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goalI_in_g(s(T82)) → U7_g(T82, s2lG_in_ga(T82, X347))
s2lG_in_ga(0, []) → s2lG_out_ga(0, [])
s2lG_in_ga(s(T89), .(X386, X387)) → U3_ga(T89, X386, X387, s2lG_in_ga(T89, X387))
U3_ga(T89, X386, X387, s2lG_out_ga(T89, X387)) → s2lG_out_ga(s(T89), .(X386, X387))
U7_g(T82, s2lG_out_ga(T82, X347)) → goalI_out_g(s(T82))
goalI_in_g(s(T82)) → U8_g(T82, s2lG_in_ga(T82, T83))
U8_g(T82, s2lG_out_ga(T82, T83)) → U9_g(T82, confE_in_g(.(X346, T83)))
confE_in_g(T65) → U2_g(T65, del2F_in_ga(T65, X280))
del2F_in_ga(T73, X313) → U4_ga(T73, X313, delA_in_aga(X310, T73, X311))
delA_in_aga(X151, [], []) → delA_out_aga(X151, [], [])
delA_in_aga(T29, .(T29, T30), T30) → delA_out_aga(T29, .(T29, T30), T30)
delA_in_aga(X190, .(T49, T51), .(T49, X192)) → U1_aga(X190, T49, T51, X192, delA_in_aga(X190, T51, X192))
U1_aga(X190, T49, T51, X192, delA_out_aga(X190, T51, X192)) → delA_out_aga(X190, .(T49, T51), .(T49, X192))
U4_ga(T73, X313, delA_out_aga(X310, T73, X311)) → del2F_out_ga(T73, X313)
del2F_in_ga(T73, X313) → U5_ga(T73, X313, delA_in_aga(T74, T73, T75))
U5_ga(T73, X313, delA_out_aga(T74, T73, T75)) → U6_ga(T73, X313, delA_in_aga(X312, T75, X313))
U6_ga(T73, X313, delA_out_aga(X312, T75, X313)) → del2F_out_ga(T73, X313)
U2_g(T65, del2F_out_ga(T65, X280)) → confE_out_g(T65)
U9_g(T82, confE_out_g(.(X346, T83))) → goalI_out_g(s(T82))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goalI_in_g(s(T82)) → U7_g(T82, s2lG_in_ga(T82, X347))
s2lG_in_ga(0, []) → s2lG_out_ga(0, [])
s2lG_in_ga(s(T89), .(X386, X387)) → U3_ga(T89, X386, X387, s2lG_in_ga(T89, X387))
U3_ga(T89, X386, X387, s2lG_out_ga(T89, X387)) → s2lG_out_ga(s(T89), .(X386, X387))
U7_g(T82, s2lG_out_ga(T82, X347)) → goalI_out_g(s(T82))
goalI_in_g(s(T82)) → U8_g(T82, s2lG_in_ga(T82, T83))
U8_g(T82, s2lG_out_ga(T82, T83)) → U9_g(T82, confE_in_g(.(X346, T83)))
confE_in_g(T65) → U2_g(T65, del2F_in_ga(T65, X280))
del2F_in_ga(T73, X313) → U4_ga(T73, X313, delA_in_aga(X310, T73, X311))
delA_in_aga(X151, [], []) → delA_out_aga(X151, [], [])
delA_in_aga(T29, .(T29, T30), T30) → delA_out_aga(T29, .(T29, T30), T30)
delA_in_aga(X190, .(T49, T51), .(T49, X192)) → U1_aga(X190, T49, T51, X192, delA_in_aga(X190, T51, X192))
U1_aga(X190, T49, T51, X192, delA_out_aga(X190, T51, X192)) → delA_out_aga(X190, .(T49, T51), .(T49, X192))
U4_ga(T73, X313, delA_out_aga(X310, T73, X311)) → del2F_out_ga(T73, X313)
del2F_in_ga(T73, X313) → U5_ga(T73, X313, delA_in_aga(T74, T73, T75))
U5_ga(T73, X313, delA_out_aga(T74, T73, T75)) → U6_ga(T73, X313, delA_in_aga(X312, T75, X313))
U6_ga(T73, X313, delA_out_aga(X312, T75, X313)) → del2F_out_ga(T73, X313)
U2_g(T65, del2F_out_ga(T65, X280)) → confE_out_g(T65)
U9_g(T82, confE_out_g(.(X346, T83))) → goalI_out_g(s(T82))

(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:

GOALI_IN_G(s(T82)) → U7_G(T82, s2lG_in_ga(T82, X347))
GOALI_IN_G(s(T82)) → S2LG_IN_GA(T82, X347)
S2LG_IN_GA(s(T89), .(X386, X387)) → U3_GA(T89, X386, X387, s2lG_in_ga(T89, X387))
S2LG_IN_GA(s(T89), .(X386, X387)) → S2LG_IN_GA(T89, X387)
GOALI_IN_G(s(T82)) → U8_G(T82, s2lG_in_ga(T82, T83))
U8_G(T82, s2lG_out_ga(T82, T83)) → U9_G(T82, confE_in_g(.(X346, T83)))
U8_G(T82, s2lG_out_ga(T82, T83)) → CONFE_IN_G(.(X346, T83))
CONFE_IN_G(T65) → U2_G(T65, del2F_in_ga(T65, X280))
CONFE_IN_G(T65) → DEL2F_IN_GA(T65, X280)
DEL2F_IN_GA(T73, X313) → U4_GA(T73, X313, delA_in_aga(X310, T73, X311))
DEL2F_IN_GA(T73, X313) → DELA_IN_AGA(X310, T73, X311)
DELA_IN_AGA(X190, .(T49, T51), .(T49, X192)) → U1_AGA(X190, T49, T51, X192, delA_in_aga(X190, T51, X192))
DELA_IN_AGA(X190, .(T49, T51), .(T49, X192)) → DELA_IN_AGA(X190, T51, X192)
DEL2F_IN_GA(T73, X313) → U5_GA(T73, X313, delA_in_aga(T74, T73, T75))
U5_GA(T73, X313, delA_out_aga(T74, T73, T75)) → U6_GA(T73, X313, delA_in_aga(X312, T75, X313))
U5_GA(T73, X313, delA_out_aga(T74, T73, T75)) → DELA_IN_AGA(X312, T75, X313)

The TRS R consists of the following rules:

goalI_in_g(s(T82)) → U7_g(T82, s2lG_in_ga(T82, X347))
s2lG_in_ga(0, []) → s2lG_out_ga(0, [])
s2lG_in_ga(s(T89), .(X386, X387)) → U3_ga(T89, X386, X387, s2lG_in_ga(T89, X387))
U3_ga(T89, X386, X387, s2lG_out_ga(T89, X387)) → s2lG_out_ga(s(T89), .(X386, X387))
U7_g(T82, s2lG_out_ga(T82, X347)) → goalI_out_g(s(T82))
goalI_in_g(s(T82)) → U8_g(T82, s2lG_in_ga(T82, T83))
U8_g(T82, s2lG_out_ga(T82, T83)) → U9_g(T82, confE_in_g(.(X346, T83)))
confE_in_g(T65) → U2_g(T65, del2F_in_ga(T65, X280))
del2F_in_ga(T73, X313) → U4_ga(T73, X313, delA_in_aga(X310, T73, X311))
delA_in_aga(X151, [], []) → delA_out_aga(X151, [], [])
delA_in_aga(T29, .(T29, T30), T30) → delA_out_aga(T29, .(T29, T30), T30)
delA_in_aga(X190, .(T49, T51), .(T49, X192)) → U1_aga(X190, T49, T51, X192, delA_in_aga(X190, T51, X192))
U1_aga(X190, T49, T51, X192, delA_out_aga(X190, T51, X192)) → delA_out_aga(X190, .(T49, T51), .(T49, X192))
U4_ga(T73, X313, delA_out_aga(X310, T73, X311)) → del2F_out_ga(T73, X313)
del2F_in_ga(T73, X313) → U5_ga(T73, X313, delA_in_aga(T74, T73, T75))
U5_ga(T73, X313, delA_out_aga(T74, T73, T75)) → U6_ga(T73, X313, delA_in_aga(X312, T75, X313))
U6_ga(T73, X313, delA_out_aga(X312, T75, X313)) → del2F_out_ga(T73, X313)
U2_g(T65, del2F_out_ga(T65, X280)) → confE_out_g(T65)
U9_g(T82, confE_out_g(.(X346, T83))) → goalI_out_g(s(T82))

The argument filtering Pi contains the following mapping:
goalI_in_g(x1) = goalI_in_g(x1)
s(x1) = s(x1)
U7_g(x1, x2) = U7_g(x2)
s2lG_in_ga(x1, x2) = s2lG_in_ga(x1)
0 = 0
s2lG_out_ga(x1, x2) = s2lG_out_ga(x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
.(x1, x2) = .(x2)
goalI_out_g(x1) = goalI_out_g
U8_g(x1, x2) = U8_g(x2)
U9_g(x1, x2) = U9_g(x2)
confE_in_g(x1) = confE_in_g(x1)
U2_g(x1, x2) = U2_g(x2)
del2F_in_ga(x1, x2) = del2F_in_ga(x1)
U4_ga(x1, x2, x3) = U4_ga(x3)
delA_in_aga(x1, x2, x3) = delA_in_aga(x2)
[] = []
delA_out_aga(x1, x2, x3) = delA_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x5)
del2F_out_ga(x1, x2) = del2F_out_ga
U5_ga(x1, x2, x3) = U5_ga(x3)
U6_ga(x1, x2, x3) = U6_ga(x3)
confE_out_g(x1) = confE_out_g
GOALI_IN_G(x1) = GOALI_IN_G(x1)
U7_G(x1, x2) = U7_G(x2)
S2LG_IN_GA(x1, x2) = S2LG_IN_GA(x1)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
U8_G(x1, x2) = U8_G(x2)
U9_G(x1, x2) = U9_G(x2)
CONFE_IN_G(x1) = CONFE_IN_G(x1)
U2_G(x1, x2) = U2_G(x2)
DEL2F_IN_GA(x1, x2) = DEL2F_IN_GA(x1)
U4_GA(x1, x2, x3) = U4_GA(x3)
DELA_IN_AGA(x1, x2, x3) = DELA_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x5)
U5_GA(x1, x2, x3) = U5_GA(x3)
U6_GA(x1, x2, x3) = U6_GA(x3)

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

(6) Obligation:

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

GOALI_IN_G(s(T82)) → U7_G(T82, s2lG_in_ga(T82, X347))
GOALI_IN_G(s(T82)) → S2LG_IN_GA(T82, X347)
S2LG_IN_GA(s(T89), .(X386, X387)) → U3_GA(T89, X386, X387, s2lG_in_ga(T89, X387))
S2LG_IN_GA(s(T89), .(X386, X387)) → S2LG_IN_GA(T89, X387)
GOALI_IN_G(s(T82)) → U8_G(T82, s2lG_in_ga(T82, T83))
U8_G(T82, s2lG_out_ga(T82, T83)) → U9_G(T82, confE_in_g(.(X346, T83)))
U8_G(T82, s2lG_out_ga(T82, T83)) → CONFE_IN_G(.(X346, T83))
CONFE_IN_G(T65) → U2_G(T65, del2F_in_ga(T65, X280))
CONFE_IN_G(T65) → DEL2F_IN_GA(T65, X280)
DEL2F_IN_GA(T73, X313) → U4_GA(T73, X313, delA_in_aga(X310, T73, X311))
DEL2F_IN_GA(T73, X313) → DELA_IN_AGA(X310, T73, X311)
DELA_IN_AGA(X190, .(T49, T51), .(T49, X192)) → U1_AGA(X190, T49, T51, X192, delA_in_aga(X190, T51, X192))
DELA_IN_AGA(X190, .(T49, T51), .(T49, X192)) → DELA_IN_AGA(X190, T51, X192)
DEL2F_IN_GA(T73, X313) → U5_GA(T73, X313, delA_in_aga(T74, T73, T75))
U5_GA(T73, X313, delA_out_aga(T74, T73, T75)) → U6_GA(T73, X313, delA_in_aga(X312, T75, X313))
U5_GA(T73, X313, delA_out_aga(T74, T73, T75)) → DELA_IN_AGA(X312, T75, X313)

The TRS R consists of the following rules:

goalI_in_g(s(T82)) → U7_g(T82, s2lG_in_ga(T82, X347))
s2lG_in_ga(0, []) → s2lG_out_ga(0, [])
s2lG_in_ga(s(T89), .(X386, X387)) → U3_ga(T89, X386, X387, s2lG_in_ga(T89, X387))
U3_ga(T89, X386, X387, s2lG_out_ga(T89, X387)) → s2lG_out_ga(s(T89), .(X386, X387))
U7_g(T82, s2lG_out_ga(T82, X347)) → goalI_out_g(s(T82))
goalI_in_g(s(T82)) → U8_g(T82, s2lG_in_ga(T82, T83))
U8_g(T82, s2lG_out_ga(T82, T83)) → U9_g(T82, confE_in_g(.(X346, T83)))
confE_in_g(T65) → U2_g(T65, del2F_in_ga(T65, X280))
del2F_in_ga(T73, X313) → U4_ga(T73, X313, delA_in_aga(X310, T73, X311))
delA_in_aga(X151, [], []) → delA_out_aga(X151, [], [])
delA_in_aga(T29, .(T29, T30), T30) → delA_out_aga(T29, .(T29, T30), T30)
delA_in_aga(X190, .(T49, T51), .(T49, X192)) → U1_aga(X190, T49, T51, X192, delA_in_aga(X190, T51, X192))
U1_aga(X190, T49, T51, X192, delA_out_aga(X190, T51, X192)) → delA_out_aga(X190, .(T49, T51), .(T49, X192))
U4_ga(T73, X313, delA_out_aga(X310, T73, X311)) → del2F_out_ga(T73, X313)
del2F_in_ga(T73, X313) → U5_ga(T73, X313, delA_in_aga(T74, T73, T75))
U5_ga(T73, X313, delA_out_aga(T74, T73, T75)) → U6_ga(T73, X313, delA_in_aga(X312, T75, X313))
U6_ga(T73, X313, delA_out_aga(X312, T75, X313)) → del2F_out_ga(T73, X313)
U2_g(T65, del2F_out_ga(T65, X280)) → confE_out_g(T65)
U9_g(T82, confE_out_g(.(X346, T83))) → goalI_out_g(s(T82))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 14 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

DELA_IN_AGA(X190, .(T49, T51), .(T49, X192)) → DELA_IN_AGA(X190, T51, X192)

The TRS R consists of the following rules:

goalI_in_g(s(T82)) → U7_g(T82, s2lG_in_ga(T82, X347))
s2lG_in_ga(0, []) → s2lG_out_ga(0, [])
s2lG_in_ga(s(T89), .(X386, X387)) → U3_ga(T89, X386, X387, s2lG_in_ga(T89, X387))
U3_ga(T89, X386, X387, s2lG_out_ga(T89, X387)) → s2lG_out_ga(s(T89), .(X386, X387))
U7_g(T82, s2lG_out_ga(T82, X347)) → goalI_out_g(s(T82))
goalI_in_g(s(T82)) → U8_g(T82, s2lG_in_ga(T82, T83))
U8_g(T82, s2lG_out_ga(T82, T83)) → U9_g(T82, confE_in_g(.(X346, T83)))
confE_in_g(T65) → U2_g(T65, del2F_in_ga(T65, X280))
del2F_in_ga(T73, X313) → U4_ga(T73, X313, delA_in_aga(X310, T73, X311))
delA_in_aga(X151, [], []) → delA_out_aga(X151, [], [])
delA_in_aga(T29, .(T29, T30), T30) → delA_out_aga(T29, .(T29, T30), T30)
delA_in_aga(X190, .(T49, T51), .(T49, X192)) → U1_aga(X190, T49, T51, X192, delA_in_aga(X190, T51, X192))
U1_aga(X190, T49, T51, X192, delA_out_aga(X190, T51, X192)) → delA_out_aga(X190, .(T49, T51), .(T49, X192))
U4_ga(T73, X313, delA_out_aga(X310, T73, X311)) → del2F_out_ga(T73, X313)
del2F_in_ga(T73, X313) → U5_ga(T73, X313, delA_in_aga(T74, T73, T75))
U5_ga(T73, X313, delA_out_aga(T74, T73, T75)) → U6_ga(T73, X313, delA_in_aga(X312, T75, X313))
U6_ga(T73, X313, delA_out_aga(X312, T75, X313)) → del2F_out_ga(T73, X313)
U2_g(T65, del2F_out_ga(T65, X280)) → confE_out_g(T65)
U9_g(T82, confE_out_g(.(X346, T83))) → goalI_out_g(s(T82))

(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:

DELA_IN_AGA(X190, .(T49, T51), .(T49, X192)) → DELA_IN_AGA(X190, T51, X192)

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

DELA_IN_AGA(.(T51)) → DELA_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:

DELA_IN_AGA(.(T51)) → DELA_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:

S2LG_IN_GA(s(T89), .(X386, X387)) → S2LG_IN_GA(T89, X387)

The TRS R consists of the following rules:

goalI_in_g(s(T82)) → U7_g(T82, s2lG_in_ga(T82, X347))
s2lG_in_ga(0, []) → s2lG_out_ga(0, [])
s2lG_in_ga(s(T89), .(X386, X387)) → U3_ga(T89, X386, X387, s2lG_in_ga(T89, X387))
U3_ga(T89, X386, X387, s2lG_out_ga(T89, X387)) → s2lG_out_ga(s(T89), .(X386, X387))
U7_g(T82, s2lG_out_ga(T82, X347)) → goalI_out_g(s(T82))
goalI_in_g(s(T82)) → U8_g(T82, s2lG_in_ga(T82, T83))
U8_g(T82, s2lG_out_ga(T82, T83)) → U9_g(T82, confE_in_g(.(X346, T83)))
confE_in_g(T65) → U2_g(T65, del2F_in_ga(T65, X280))
del2F_in_ga(T73, X313) → U4_ga(T73, X313, delA_in_aga(X310, T73, X311))
delA_in_aga(X151, [], []) → delA_out_aga(X151, [], [])
delA_in_aga(T29, .(T29, T30), T30) → delA_out_aga(T29, .(T29, T30), T30)
delA_in_aga(X190, .(T49, T51), .(T49, X192)) → U1_aga(X190, T49, T51, X192, delA_in_aga(X190, T51, X192))
U1_aga(X190, T49, T51, X192, delA_out_aga(X190, T51, X192)) → delA_out_aga(X190, .(T49, T51), .(T49, X192))
U4_ga(T73, X313, delA_out_aga(X310, T73, X311)) → del2F_out_ga(T73, X313)
del2F_in_ga(T73, X313) → U5_ga(T73, X313, delA_in_aga(T74, T73, T75))
U5_ga(T73, X313, delA_out_aga(T74, T73, T75)) → U6_ga(T73, X313, delA_in_aga(X312, T75, X313))
U6_ga(T73, X313, delA_out_aga(X312, T75, X313)) → del2F_out_ga(T73, X313)
U2_g(T65, del2F_out_ga(T65, X280)) → confE_out_g(T65)
U9_g(T82, confE_out_g(.(X346, T83))) → goalI_out_g(s(T82))

(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:

S2LG_IN_GA(s(T89), .(X386, X387)) → S2LG_IN_GA(T89, X387)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
.(x1, x2) = .(x2)
S2LG_IN_GA(x1, x2) = S2LG_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:

S2LG_IN_GA(s(T89)) → S2LG_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:

S2LG_IN_GA(s(T89)) → S2LG_IN_GA(T89)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES