Left Termination proof of /tmp/tmpT17gCB/btree2.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

↳3 PrologToPiTRSProof (⇐)

↳4 PiTRS

↳5 DependencyPairsProof (⇔)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

            ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                    ↳13 QDP

                        ↳14 QDPSizeChangeProof (⇔)

                            ↳15 TRUE

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

            ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                    ↳20 QDP

                        ↳21 QDPSizeChangeProof (⇔)

                            ↳22 TRUE

(0) Obligation:

Clauses:

goal(X) :- ','(s2t(X, T), tree(T)).
tree(nil).
tree(X) :- ','(no(empty(X)), ','(left(T, L), ','(right(T, R), ','(tree(L), tree(R))))).
s2t(0, nil).
s2t(X, node(T, X1, T)) :- ','(no(zero(X)), ','(p(X, P), s2t(P, T))).
s2t(X, node(nil, X2, T)) :- ','(no(zero(X)), ','(p(X, P), s2t(P, T))).
s2t(X, node(T, X3, nil)) :- ','(no(zero(X)), ','(p(X, P), s2t(P, T))).
s2t(X, node(nil, X4, nil)) :- no(zero(X)).
left(nil, nil).
left(node(L, X5, X6), L).
right(nil, nil).
right(node(X7, X8, R), R).
p(0, 0).
p(s(X), X).
empty(nil).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X9).
failure(b).

Queries:

goal(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: goal(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(s2t(T2, X12), tree(X12))\n\nT2 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: s2t(T2, X12)\n\nT2 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: tree(T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: s2t(T2, X12), SCOPE: 2, CLAUSE: 3 | s2t(T2, X12), SCOPE: 2, CLAUSE: 4 | s2t(T2, X12), SCOPE: 2, CLAUSE: 5 | s2t(T2, X12), SCOPE: 2, CLAUSE: 6 | s2t(T2, X12), SCOPE: 2, CLAUSE: 7\n\nT2 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: s2t(T2, X12), SCOPE: 2, CLAUSE: 3\n\nT2 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: s2t(T2, X12), SCOPE: 2, CLAUSE: 4 | s2t(T2, X12), SCOPE: 2, CLAUSE: 5 | s2t(T2, X12), SCOPE: 2, CLAUSE: 6 | s2t(T2, X12), SCOPE: 2, CLAUSE: 7\n\nT2 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: s2t(T2, X12), SCOPE: 2, CLAUSE: 4\n\nT2 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: s2t(T2, X12), SCOPE: 2, CLAUSE: 5 | s2t(T2, X12), SCOPE: 2, CLAUSE: 6 | s2t(T2, X12), SCOPE: 2, CLAUSE: 7\n\nT2 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(no(zero(T4)), ','(p(T4, X22), s2t(X22, X23)))\n\nT4 is ground\nX23 is free; \nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(no(zero(T4)), ','(p(T4, X22), s2t(X22, X23))), SCOPE: 3, CLAUSE: 16 | ','(no(zero(T4)), ','(p(T4, X22), s2t(X22, X23))), SCOPE: 3, CLAUSE: 17 | ?_3\n\nT4 is ground\nX23 is free; \nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(zero(T5), ','(!_3, ','(failure(a), ','(p(T5, X22), s2t(X22, X23))))) | ','(no(zero(T5)), ','(p(T5, X22), s2t(X22, X23))), SCOPE: 3, CLAUSE: 17 | ?_3\n\nT5 is ground\nX23 is free; \nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(zero(T5), ','(!_3, ','(failure(a), ','(p(T5, X22), s2t(X22, X23))))), SCOPE: 4, CLAUSE: 15 | ','(no(zero(T5)), ','(p(T5, X22), s2t(X22, X23))), SCOPE: 3, CLAUSE: 17 | ?_3\n\nT5 is ground\nX23 is free; \nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(!_3, ','(failure(a), ','(p(0, X22), s2t(X22, X23)))) | ','(no(zero(0)), ','(p(0, X22), s2t(X22, X23))), SCOPE: 3, CLAUSE: 17 | ?_3\n\nX23 is free\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(no(zero(T5)), ','(p(T5, X22), s2t(X22, X23))), SCOPE: 3, CLAUSE: 17\n\nT5 is ground\nX23 is free; \nX22 is free; \nzero(T5) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(failure(a), ','(p(0, X22), s2t(X22, X23)))\n\nX23 is free\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(failure(a), ','(p(0, X22), s2t(X22, X23))), SCOPE: 5, CLAUSE: 18\n\nX23 is free\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(p(T6, X22), s2t(X22, X23))\n\nT6 is ground\nX23 is free; \nX22 is free; \nzero(T6) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(T6, X22), s2t(X22, X23)), SCOPE: 6, CLAUSE: 12 | ','(p(T6, X22), s2t(X22, X23)), SCOPE: 6, CLAUSE: 13\n\nT6 is ground\nX23 is free; \nX22 is free; \nzero(T6) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(p(T6, X22), s2t(X22, X23)), SCOPE: 6, CLAUSE: 13\n\nT6 is ground\nX23 is free; \nX22 is free; \nzero(T6) !=? zero(0)\np(T6, X22) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: s2t(T7, X23)\n\nT7 is ground\nX23 is free; \nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\nX22 is free\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: s2t(T2, X12), SCOPE: 2, CLAUSE: 5\n\nT2 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: s2t(T2, X12), SCOPE: 2, CLAUSE: 6 | s2t(T2, X12), SCOPE: 2, CLAUSE: 7\n\nT2 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(no(zero(T8)), ','(p(T8, X40), s2t(X40, X42)))\n\nT8 is ground\nX42 is free; \nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(no(zero(T8)), ','(p(T8, X40), s2t(X40, X42))), SCOPE: 7, CLAUSE: 16 | ','(no(zero(T8)), ','(p(T8, X40), s2t(X40, X42))), SCOPE: 7, CLAUSE: 17 | ?_7\n\nT8 is ground\nX42 is free; \nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(zero(T9), ','(!_7, ','(failure(a), ','(p(T9, X40), s2t(X40, X42))))) | ','(no(zero(T9)), ','(p(T9, X40), s2t(X40, X42))), SCOPE: 7, CLAUSE: 17 | ?_7\n\nT9 is ground\nX42 is free; \nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(zero(T9), ','(!_7, ','(failure(a), ','(p(T9, X40), s2t(X40, X42))))), SCOPE: 8, CLAUSE: 15 | ','(no(zero(T9)), ','(p(T9, X40), s2t(X40, X42))), SCOPE: 7, CLAUSE: 17 | ?_7\n\nT9 is ground\nX42 is free; \nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(!_7, ','(failure(a), ','(p(0, X40), s2t(X40, X42)))) | ','(no(zero(0)), ','(p(0, X40), s2t(X40, X42))), SCOPE: 7, CLAUSE: 17 | ?_7\n\nX42 is free\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(no(zero(T9)), ','(p(T9, X40), s2t(X40, X42))), SCOPE: 7, CLAUSE: 17\n\nT9 is ground\nX42 is free; \nX40 is free; \nzero(T9) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(failure(a), ','(p(0, X40), s2t(X40, X42)))\n\nX42 is free\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(failure(a), ','(p(0, X40), s2t(X40, X42))), SCOPE: 9, CLAUSE: 18\n\nX42 is free\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(p(T10, X40), s2t(X40, X42))\n\nT10 is ground\nX42 is free; \nX40 is free; \nzero(T10) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(p(T10, X40), s2t(X40, X42)), SCOPE: 10, CLAUSE: 12 | ','(p(T10, X40), s2t(X40, X42)), SCOPE: 10, CLAUSE: 13\n\nT10 is ground\nX42 is free; \nX40 is free; \nzero(T10) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(p(T10, X40), s2t(X40, X42)), SCOPE: 10, CLAUSE: 13\n\nT10 is ground\nX42 is free; \nX40 is free; \nzero(T10) !=? zero(0)\np(T10, X40) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: s2t(T11, X42)\n\nT11 is ground\nX42 is free; \nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\nX40 is free\nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: s2t(T2, X12), SCOPE: 2, CLAUSE: 6\n\nT2 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: s2t(T2, X12), SCOPE: 2, CLAUSE: 7\n\nT2 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(no(zero(T12)), ','(p(T12, X58), s2t(X58, X59)))\n\nT12 is ground\nX59 is free; \nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(no(zero(T12)), ','(p(T12, X58), s2t(X58, X59))), SCOPE: 11, CLAUSE: 16 | ','(no(zero(T12)), ','(p(T12, X58), s2t(X58, X59))), SCOPE: 11, CLAUSE: 17 | ?_11\n\nT12 is ground\nX59 is free; \nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(zero(T13), ','(!_11, ','(failure(a), ','(p(T13, X58), s2t(X58, X59))))) | ','(no(zero(T13)), ','(p(T13, X58), s2t(X58, X59))), SCOPE: 11, CLAUSE: 17 | ?_11\n\nT13 is ground\nX59 is free; \nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(zero(T13), ','(!_11, ','(failure(a), ','(p(T13, X58), s2t(X58, X59))))), SCOPE: 12, CLAUSE: 15 | ','(no(zero(T13)), ','(p(T13, X58), s2t(X58, X59))), SCOPE: 11, CLAUSE: 17 | ?_11\n\nT13 is ground\nX59 is free; \nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(!_11, ','(failure(a), ','(p(0, X58), s2t(X58, X59)))) | ','(no(zero(0)), ','(p(0, X58), s2t(X58, X59))), SCOPE: 11, CLAUSE: 17 | ?_11\n\nX59 is free\nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(no(zero(T13)), ','(p(T13, X58), s2t(X58, X59))), SCOPE: 11, CLAUSE: 17\n\nT13 is ground\nX59 is free; \nX58 is free; \nzero(T13) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(failure(a), ','(p(0, X58), s2t(X58, X59)))\n\nX59 is free\nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(failure(a), ','(p(0, X58), s2t(X58, X59))), SCOPE: 13, CLAUSE: 18\n\nX59 is free\nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(p(T14, X58), s2t(X58, X59))\n\nT14 is ground\nX59 is free; \nX58 is free; \nzero(T14) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(p(T14, X58), s2t(X58, X59)), SCOPE: 14, CLAUSE: 12 | ','(p(T14, X58), s2t(X58, X59)), SCOPE: 14, CLAUSE: 13\n\nT14 is ground\nX59 is free; \nX58 is free; \nzero(T14) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(p(T14, X58), s2t(X58, X59)), SCOPE: 14, CLAUSE: 13\n\nT14 is ground\nX59 is free; \nX58 is free; \nzero(T14) !=? zero(0)\np(T14, X58) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: s2t(T15, X59)\n\nT15 is ground\nX59 is free; \nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\nX58 is free\nX66 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: no(zero(T16))\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: no(zero(T16)), SCOPE: 15, CLAUSE: 16 | no(zero(T16)), SCOPE: 15, CLAUSE: 17 | ?_15\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(zero(T17), ','(!_15, failure(a))) | no(zero(T17)), SCOPE: 15, CLAUSE: 17 | ?_15\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(zero(T17), ','(!_15, failure(a))), SCOPE: 16, CLAUSE: 15 | no(zero(T17)), SCOPE: 15, CLAUSE: 17 | ?_15\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(!_15, failure(a)) | no(zero(0)), SCOPE: 15, CLAUSE: 17 | ?_15\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: no(zero(T17)), SCOPE: 15, CLAUSE: 17\n\nT17 is ground\nzero(T17) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: failure(a), SCOPE: 17, CLAUSE: 18\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: tree(T3), SCOPE: 18, CLAUSE: 1 | tree(T3), SCOPE: 18, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: tree(T3), SCOPE: 18, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: tree(T3), SCOPE: 18, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(no(empty(T20)), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80)))))\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(no(empty(T20)), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80))))), SCOPE: 19, CLAUSE: 16 | ','(no(empty(T20)), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80))))), SCOPE: 19, CLAUSE: 17 | ?_19\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(empty(T22), ','(!_19, ','(failure(a), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80))))))) | ','(no(empty(T20)), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80))))), SCOPE: 19, CLAUSE: 17 | ?_19\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(empty(T22), ','(!_19, ','(failure(a), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80))))))), SCOPE: 20, CLAUSE: 14 | ','(no(empty(T20)), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80))))), SCOPE: 19, CLAUSE: 17 | ?_19\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(!_19, ','(failure(a), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80)))))) | ','(no(empty(T20)), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80))))), SCOPE: 19, CLAUSE: 17 | ?_19\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(no(empty(T20)), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80))))), SCOPE: 19, CLAUSE: 17\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(failure(a), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80)))))\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(failure(a), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80))))), SCOPE: 21, CLAUSE: 18\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80))))\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80)))), SCOPE: 22, CLAUSE: 8 | ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80)))), SCOPE: 22, CLAUSE: 9\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(right(nil, X80), ','(tree(nil), tree(X80))) | ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80)))), SCOPE: 22, CLAUSE: 9\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(right(nil, X80), ','(tree(nil), tree(X80))), SCOPE: 23, CLAUSE: 10 | ','(right(nil, X80), ','(tree(nil), tree(X80))), SCOPE: 23, CLAUSE: 11 | ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80)))), SCOPE: 22, CLAUSE: 9\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(tree(nil), tree(nil)) | ','(right(nil, X80), ','(tree(nil), tree(X80))), SCOPE: 23, CLAUSE: 11 | ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80)))), SCOPE: 22, CLAUSE: 9\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(tree(nil), tree(nil))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(right(nil, X80), ','(tree(nil), tree(X80))), SCOPE: 23, CLAUSE: 11 | ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80)))), SCOPE: 22, CLAUSE: 9\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: tree(nil)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: tree(nil)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: tree(nil), SCOPE: 24, CLAUSE: 1 | tree(nil), SCOPE: 24, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: tree(nil), SCOPE: 24, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: tree(nil), SCOPE: 24, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: ','(no(empty(nil)), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89)))))\n\nX87 is free\nX88 is free; \nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(no(empty(nil)), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89))))), SCOPE: 25, CLAUSE: 16 | ','(no(empty(nil)), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89))))), SCOPE: 25, CLAUSE: 17 | ?_25\n\nX87 is free\nX88 is free; \nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: ','(empty(nil), ','(!_25, ','(failure(a), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89))))))) | ','(no(empty(nil)), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89))))), SCOPE: 25, CLAUSE: 17 | ?_25\n\nX87 is free\nX88 is free; \nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: ','(empty(nil), ','(!_25, ','(failure(a), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89))))))), SCOPE: 26, CLAUSE: 14 | ','(no(empty(nil)), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89))))), SCOPE: 25, CLAUSE: 17 | ?_25\n\nX87 is free\nX88 is free; \nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: ','(!_25, ','(failure(a), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89)))))) | ','(no(empty(nil)), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89))))), SCOPE: 25, CLAUSE: 17 | ?_25\n\nX87 is free\nX88 is free; \nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: ','(failure(a), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89)))))\n\nX87 is free\nX88 is free; \nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: ','(failure(a), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89))))), SCOPE: 27, CLAUSE: 18\n\nX87 is free\nX88 is free; \nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80)))), SCOPE: 22, CLAUSE: 9\n\nX78 is free\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: ','(right(node(X101, X102, X103), X80), ','(tree(X101), tree(X80)))\n\nX80 is free\nX101 is free; \nX102 is free; \nX103 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: ','(right(node(X101, X102, X103), X80), ','(tree(X101), tree(X80))), SCOPE: 28, CLAUSE: 10 | ','(right(node(X101, X102, X103), X80), ','(tree(X101), tree(X80))), SCOPE: 28, CLAUSE: 11\n\nX80 is free\nX101 is free; \nX102 is free; \nX103 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: ','(right(node(X101, X102, X103), X80), ','(tree(X101), tree(X80))), SCOPE: 28, CLAUSE: 11\n\nX80 is free\nX101 is free; \nX102 is free; \nX103 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: ','(tree(X110), tree(X112))\n\nX110 is free\nX112 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: tree(X110)\n\nX110 is free", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: tree(X112)\n\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: tree(X112), SCOPE: 29, CLAUSE: 1 | tree(X112), SCOPE: 29, CLAUSE: 2\n\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: tree(X112), SCOPE: 29, CLAUSE: 1\n\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: tree(X112), SCOPE: 29, CLAUSE: 2\n\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(no(empty(X118)), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117)))))\n\nX118 is free\nX115 is free; \nX116 is free; \nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: ','(no(empty(X118)), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117))))), SCOPE: 30, CLAUSE: 16 | ','(no(empty(X118)), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117))))), SCOPE: 30, CLAUSE: 17 | ?_30\n\nX118 is free\nX115 is free; \nX116 is free; \nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: ','(empty(X121), ','(!_30, ','(failure(a), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117))))))) | ','(no(empty(X118)), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117))))), SCOPE: 30, CLAUSE: 17 | ?_30\n\nX118 is free\nX115 is free; \nX116 is free; \nX117 is free; \nX121 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: ','(empty(X121), ','(!_30, ','(failure(a), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117))))))), SCOPE: 31, CLAUSE: 14 | ','(no(empty(X118)), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117))))), SCOPE: 30, CLAUSE: 17 | ?_30\n\nX118 is free\nX115 is free; \nX116 is free; \nX117 is free; \nX121 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: ','(!_30, ','(failure(a), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117)))))) | ','(no(empty(X118)), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117))))), SCOPE: 30, CLAUSE: 17 | ?_30\n\nX118 is free\nX115 is free; \nX116 is free; \nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: ','(failure(a), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117)))))\n\nX115 is free\nX116 is free; \nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: ','(failure(a), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117))))), SCOPE: 32, CLAUSE: 18\n\nX115 is free\nX116 is free; \nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 128 [arrowhead = none ];
128 -> 2;

129 [label="EVAL with clause\ngoal(X11) :- ','(s2t(X11, X12), tree(X12)).\nand substitution\nT1 -> T2\nX11 -> T2", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 129 [arrowhead = none ];
129 -> 3;

130 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
3 -> 130 [arrowhead = none ];
130 -> 4;

131 [label="SPLIT 2\nreplacements:\nX12 -> T3", fontsize=14, style = filled, fillcolor = violet];
3 -> 131 [arrowhead = none ];
131 -> 5;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
4 -> 132 [arrowhead = none ];
132 -> 6;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 133 [arrowhead = none ];
133 -> 71;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 134 [arrowhead = none ];
134 -> 7;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 135 [arrowhead = none ];
135 -> 8;

136 [label="EVAL with clause\ns2t(0, nil).\nand substitution\nT2 -> 0\nX12 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 136 [arrowhead = none ];
136 -> 9;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 137 [arrowhead = none ];
137 -> 10;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 138 [arrowhead = none ];
138 -> 12;

139 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 139 [arrowhead = none ];
139 -> 13;

140 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 140 [arrowhead = none ];
140 -> 11;

141 [label="EVAL with clause\ns2t(X19, node(X20, X21, X20)) :- ','(no(zero(X19)), ','(p(X19, X22), s2t(X22, X20))).\nand substitution\nT2 -> T4\nX19 -> T4\nX20 -> X23\nX21 -> X24\nX12 -> node(X23, X24, X23)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 141 [arrowhead = none ];
141 -> 14;

142 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 142 [arrowhead = none ];
142 -> 28;

143 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 143 [arrowhead = none ];
143 -> 29;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 144 [arrowhead = none ];
144 -> 15;

145 [label="EVAL with clause\nno(X26) :- ','(X26, ','(!, failure(a))).\nand substitution\nT4 -> T5\nX26 -> zero(T5)", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 145 [arrowhead = none ];
145 -> 16;

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 146 [arrowhead = none ];
146 -> 17;

147 [label="EVAL with clause\nzero(0).\nand substitution\nT5 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 147 [arrowhead = none ];
147 -> 18;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 148 [arrowhead = none ];
148 -> 19;

149 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 149 [arrowhead = none ];
149 -> 20;

150 [label="EVAL with clause\nno(X28).\nand substitution\nT5 -> T6\nX28 -> zero(T6)", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 150 [arrowhead = none ];
150 -> 23;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 151 [arrowhead = none ];
151 -> 21;

152 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
21 -> 152 [arrowhead = none ];
152 -> 22;

153 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 153 [arrowhead = none ];
153 -> 24;

154 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T6), zero(0))", fontsize=14, style = filled, fillcolor = white];
24 -> 154 [arrowhead = none ];
154 -> 25;

155 [label="EVAL with clause\np(s(X30), X30).\nand substitution\nX30 -> T7\nT6 -> s(T7)\nX22 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 155 [arrowhead = none ];
155 -> 26;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 156 [arrowhead = none ];
156 -> 27;

157 [label="INSTANCE with matching:\nT2 -> T7\nX12 -> X23", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 157 [arrowhead = none , style=dashed];
157 -> 4[style=dashed];

158 [label="EVAL with clause\ns2t(X37, node(nil, X38, X39)) :- ','(no(zero(X37)), ','(p(X37, X40), s2t(X40, X39))).\nand substitution\nT2 -> T8\nX37 -> T8\nX38 -> X41\nX39 -> X42\nX12 -> node(nil, X41, X42)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 158 [arrowhead = none ];
158 -> 30;

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 159 [arrowhead = none ];
159 -> 44;

160 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 160 [arrowhead = none ];
160 -> 45;

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 161 [arrowhead = none ];
161 -> 31;

162 [label="EVAL with clause\nno(X44) :- ','(X44, ','(!, failure(a))).\nand substitution\nT8 -> T9\nX44 -> zero(T9)", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 162 [arrowhead = none ];
162 -> 32;

163 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 163 [arrowhead = none ];
163 -> 33;

164 [label="EVAL with clause\nzero(0).\nand substitution\nT9 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 164 [arrowhead = none ];
164 -> 34;

165 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 165 [arrowhead = none ];
165 -> 35;

166 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 166 [arrowhead = none ];
166 -> 36;

167 [label="EVAL with clause\nno(X46).\nand substitution\nT9 -> T10\nX46 -> zero(T10)", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 167 [arrowhead = none ];
167 -> 39;

168 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 168 [arrowhead = none ];
168 -> 37;

169 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
37 -> 169 [arrowhead = none ];
169 -> 38;

170 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 170 [arrowhead = none ];
170 -> 40;

171 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T10), zero(0))", fontsize=14, style = filled, fillcolor = white];
40 -> 171 [arrowhead = none ];
171 -> 41;

172 [label="EVAL with clause\np(s(X48), X48).\nand substitution\nX48 -> T11\nT10 -> s(T11)\nX40 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 172 [arrowhead = none ];
172 -> 42;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 173 [arrowhead = none ];
173 -> 43;

174 [label="INSTANCE with matching:\nT2 -> T11\nX12 -> X42", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 174 [arrowhead = none , style=dashed];
174 -> 4[style=dashed];

175 [label="EVAL with clause\ns2t(X55, node(X56, X57, nil)) :- ','(no(zero(X55)), ','(p(X55, X58), s2t(X58, X56))).\nand substitution\nT2 -> T12\nX55 -> T12\nX56 -> X59\nX57 -> X60\nX12 -> node(X59, X60, nil)", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 175 [arrowhead = none ];
175 -> 46;

176 [label="EVAL with clause\ns2t(X69, node(nil, X70, nil)) :- no(zero(X69)).\nand substitution\nT2 -> T16\nX69 -> T16\nX70 -> X71\nX12 -> node(nil, X71, nil)", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 176 [arrowhead = none ];
176 -> 60;

177 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 177 [arrowhead = none ];
177 -> 47;

178 [label="EVAL with clause\nno(X62) :- ','(X62, ','(!, failure(a))).\nand substitution\nT12 -> T13\nX62 -> zero(T13)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 178 [arrowhead = none ];
178 -> 48;

179 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 179 [arrowhead = none ];
179 -> 49;

180 [label="EVAL with clause\nzero(0).\nand substitution\nT13 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 180 [arrowhead = none ];
180 -> 50;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 181 [arrowhead = none ];
181 -> 51;

182 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 182 [arrowhead = none ];
182 -> 52;

183 [label="EVAL with clause\nno(X64).\nand substitution\nT13 -> T14\nX64 -> zero(T14)", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 183 [arrowhead = none ];
183 -> 55;

184 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 184 [arrowhead = none ];
184 -> 53;

185 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
53 -> 185 [arrowhead = none ];
185 -> 54;

186 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 186 [arrowhead = none ];
186 -> 56;

187 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T14), zero(0))", fontsize=14, style = filled, fillcolor = white];
56 -> 187 [arrowhead = none ];
187 -> 57;

188 [label="EVAL with clause\np(s(X66), X66).\nand substitution\nX66 -> T15\nT14 -> s(T15)\nX58 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 188 [arrowhead = none ];
188 -> 58;

189 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 189 [arrowhead = none ];
189 -> 59;

190 [label="INSTANCE with matching:\nT2 -> T15\nX12 -> X59", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 190 [arrowhead = none , style=dashed];
190 -> 4[style=dashed];

191 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 191 [arrowhead = none ];
191 -> 61;

192 [label="EVAL with clause\nno(X73) :- ','(X73, ','(!, failure(a))).\nand substitution\nT16 -> T17\nX73 -> zero(T17)", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 192 [arrowhead = none ];
192 -> 62;

193 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 193 [arrowhead = none ];
193 -> 63;

194 [label="EVAL with clause\nzero(0).\nand substitution\nT17 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 194 [arrowhead = none ];
194 -> 64;

195 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 195 [arrowhead = none ];
195 -> 65;

196 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 196 [arrowhead = none ];
196 -> 66;

197 [label="EVAL with clause\nno(X75).\nand substitution\nT17 -> T18\nX75 -> zero(T18)", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 197 [arrowhead = none ];
197 -> 69;

198 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 198 [arrowhead = none ];
198 -> 67;

199 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
67 -> 199 [arrowhead = none ];
199 -> 68;

200 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 200 [arrowhead = none ];
200 -> 70;

201 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 201 [arrowhead = none ];
201 -> 72;

202 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 202 [arrowhead = none ];
202 -> 73;

203 [label="EVAL with clause\ntree(nil).\nand substitution\nT3 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 203 [arrowhead = none ];
203 -> 74;

204 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 204 [arrowhead = none ];
204 -> 75;

205 [label="EVAL with clause\ntree(X77) :- ','(no(empty(X77)), ','(left(X78, X79), ','(right(X78, X80), ','(tree(X79), tree(X80))))).\nand substitution\nT3 -> T20\nX77 -> T20\nT19 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 205 [arrowhead = none ];
205 -> 77;

206 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 206 [arrowhead = none ];
206 -> 76;

207 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 207 [arrowhead = none ];
207 -> 78;

208 [label="EVAL with clause\nno(X82) :- ','(X82, ','(!, failure(a))).\nand substitution\nT20 -> T22\nX82 -> empty(T22)\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 208 [arrowhead = none ];
208 -> 79;

209 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
79 -> 209 [arrowhead = none ];
209 -> 80;

210 [label="EVAL with clause\nempty(nil).\nand substitution\nT22 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 210 [arrowhead = none ];
210 -> 81;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 211 [arrowhead = none ];
211 -> 82;

212 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 212 [arrowhead = none ];
212 -> 83;

213 [label="EVAL with clause\nno(X84).\nand substitution\nT20 -> T23\nX84 -> empty(T23)", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 213 [arrowhead = none ];
213 -> 86;

214 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
83 -> 214 [arrowhead = none ];
214 -> 84;

215 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
84 -> 215 [arrowhead = none ];
215 -> 85;

216 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
86 -> 216 [arrowhead = none ];
216 -> 87;

217 [label="EVAL with clause\nleft(nil, nil).\nand substitution\nX78 -> nil\nX79 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 217 [arrowhead = none ];
217 -> 88;

218 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
88 -> 218 [arrowhead = none ];
218 -> 89;

219 [label="EVAL with clause\nright(nil, nil).\nand substitution\nX80 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 219 [arrowhead = none ];
219 -> 90;

220 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 220 [arrowhead = none ];
220 -> 91;

221 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 221 [arrowhead = none ];
221 -> 92;

222 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
91 -> 222 [arrowhead = none ];
222 -> 93;

223 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
91 -> 223 [arrowhead = none ];
223 -> 94;

224 [label="BACKTRACK\nfor clause: right(node(X7, X8, R), R)because of non-unification", fontsize=14, style = filled, fillcolor = white];
92 -> 224 [arrowhead = none ];
224 -> 108;

225 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 225 [arrowhead = none , style=dashed];
225 -> 94[style=dashed];

226 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
94 -> 226 [arrowhead = none ];
226 -> 95;

227 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
95 -> 227 [arrowhead = none ];
227 -> 96;

228 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
95 -> 228 [arrowhead = none ];
228 -> 97;

229 [label="EVAL with clause\ntree(nil).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 229 [arrowhead = none ];
229 -> 98;

230 [label="EVAL with clause\ntree(X86) :- ','(no(empty(X86)), ','(left(X87, X88), ','(right(X87, X89), ','(tree(X88), tree(X89))))).\nand substitution\nX86 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 230 [arrowhead = none ];
230 -> 100;

231 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
98 -> 231 [arrowhead = none ];
231 -> 99;

232 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
100 -> 232 [arrowhead = none ];
232 -> 101;

233 [label="EVAL with clause\nno(X91) :- ','(X91, ','(!, failure(a))).\nand substitution\nX91 -> empty(nil)", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 233 [arrowhead = none ];
233 -> 102;

234 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
102 -> 234 [arrowhead = none ];
234 -> 103;

235 [label="EVAL with clause\nempty(nil).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 235 [arrowhead = none ];
235 -> 104;

236 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
104 -> 236 [arrowhead = none ];
236 -> 105;

237 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
105 -> 237 [arrowhead = none ];
237 -> 106;

238 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
106 -> 238 [arrowhead = none ];
238 -> 107;

239 [label="EVAL with clause\nleft(node(X98, X99, X100), X98).\nand substitution\nX98 -> X101\nX99 -> X102\nX100 -> X103\nX78 -> node(X101, X102, X103)\nX79 -> X101", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 239 [arrowhead = none ];
239 -> 109;

240 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
109 -> 240 [arrowhead = none ];
240 -> 110;

241 [label="BACKTRACK\nfor clause: right(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = white];
110 -> 241 [arrowhead = none ];
241 -> 111;

242 [label="EVAL with clause\nright(node(X107, X108, X109), X109).\nand substitution\nX101 -> X110\nX107 -> X110\nX102 -> X111\nX108 -> X111\nX103 -> X112\nX109 -> X112\nX80 -> X112", fontsize=14, style = filled, fillcolor = lightblue];
111 -> 242 [arrowhead = none ];
242 -> 112;

243 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
112 -> 243 [arrowhead = none ];
243 -> 113;

244 [label="SPLIT 2\nreplacements:\nX110 -> T24", fontsize=14, style = filled, fillcolor = violet];
112 -> 244 [arrowhead = none ];
244 -> 114;

245 [label="INSTANCE with matching:\nX112 -> X110", fontsize=14, style = filled, fillcolor = lightgrey];
113 -> 245 [arrowhead = none , style=dashed];
245 -> 114[style=dashed];

246 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
114 -> 246 [arrowhead = none ];
246 -> 115;

247 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
115 -> 247 [arrowhead = none ];
247 -> 116;

248 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
115 -> 248 [arrowhead = none ];
248 -> 117;

249 [label="EVAL with clause\ntree(nil).\nand substitution\nX112 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
116 -> 249 [arrowhead = none ];
249 -> 118;

250 [label="EVAL with clause\ntree(X114) :- ','(no(empty(X114)), ','(left(X115, X116), ','(right(X115, X117), ','(tree(X116), tree(X117))))).\nand substitution\nX112 -> X118\nX114 -> X118", fontsize=14, style = filled, fillcolor = lightblue];
117 -> 250 [arrowhead = none ];
250 -> 120;

251 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
118 -> 251 [arrowhead = none ];
251 -> 119;

252 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
120 -> 252 [arrowhead = none ];
252 -> 121;

253 [label="EVAL with clause\nno(X120) :- ','(X120, ','(!, failure(a))).\nand substitution\nX118 -> X121\nX120 -> empty(X121)", fontsize=14, style = filled, fillcolor = lightblue];
121 -> 253 [arrowhead = none ];
253 -> 122;

254 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
122 -> 254 [arrowhead = none ];
254 -> 123;

255 [label="EVAL with clause\nempty(nil).\nand substitution\nX121 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 255 [arrowhead = none ];
255 -> 124;

256 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
124 -> 256 [arrowhead = none ];
256 -> 125;

257 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
125 -> 257 [arrowhead = none ];
257 -> 126;

258 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
126 -> 258 [arrowhead = none ];
258 -> 127;

}

(2) Obligation:

Clauses:

s2t4(0, nil).
s2t4(s(T7), node(X23, X24, X23)) :- s2t4(T7, X23).
s2t4(s(T11), node(nil, X41, X42)) :- s2t4(T11, X42).
s2t4(s(T15), node(X59, X60, nil)) :- s2t4(T15, X59).
s2t4(T18, node(nil, X71, nil)).
tree94.
tree114(nil).
goal1(T2) :- s2t4(T2, X12).
goal1(T2) :- s2t4(T2, nil).
goal1(T2) :- ','(s2t4(T2, T23), tree94).
goal1(T2) :- ','(s2t4(T2, T23), tree94).
goal1(T2) :- ','(s2t4(T2, T23), tree114(X110)).
goal1(T2) :- ','(s2t4(T2, T23), tree114(T24)).

Queries:

goal1(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal1_in: (b)
s2t4_in: (b,f) (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2t4_in_ga(T2, X12))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T7), node(X23, X24, X23)) → U1_ga(T7, X23, X24, s2t4_in_ga(T7, X23))
s2t4_in_ga(s(T11), node(nil, X41, X42)) → U2_ga(T11, X41, X42, s2t4_in_ga(T11, X42))
s2t4_in_ga(s(T15), node(X59, X60, nil)) → U3_ga(T15, X59, X60, s2t4_in_ga(T15, X59))
s2t4_in_ga(T18, node(nil, X71, nil)) → s2t4_out_ga(T18, node(nil, X71, nil))
U3_ga(T15, X59, X60, s2t4_out_ga(T15, X59)) → s2t4_out_ga(s(T15), node(X59, X60, nil))
U2_ga(T11, X41, X42, s2t4_out_ga(T11, X42)) → s2t4_out_ga(s(T11), node(nil, X41, X42))
U1_ga(T7, X23, X24, s2t4_out_ga(T7, X23)) → s2t4_out_ga(s(T7), node(X23, X24, X23))
U4_g(T2, s2t4_out_ga(T2, X12)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T7), node(X23, X24, X23)) → U1_gg(T7, X23, X24, s2t4_in_gg(T7, X23))
s2t4_in_gg(s(T11), node(nil, X41, X42)) → U2_gg(T11, X41, X42, s2t4_in_gg(T11, X42))
s2t4_in_gg(s(T15), node(X59, X60, nil)) → U3_gg(T15, X59, X60, s2t4_in_gg(T15, X59))
s2t4_in_gg(T18, node(nil, X71, nil)) → s2t4_out_gg(T18, node(nil, X71, nil))
U3_gg(T15, X59, X60, s2t4_out_gg(T15, X59)) → s2t4_out_gg(s(T15), node(X59, X60, nil))
U2_gg(T11, X41, X42, s2t4_out_gg(T11, X42)) → s2t4_out_gg(s(T11), node(nil, X41, X42))
U1_gg(T7, X23, X24, s2t4_out_gg(T7, X23)) → s2t4_out_gg(s(T7), node(X23, X24, X23))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T23))
U6_g(T2, s2t4_out_ga(T2, T23)) → U7_g(T2, tree94_in_)
tree94_in_ → tree94_out_
U7_g(T2, tree94_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U8_g(T2, tree114_in_a(X110))
tree114_in_a(nil) → tree114_out_a(nil)
U8_g(T2, tree114_out_a(X110)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U9_g(T2, tree114_in_a(T24))
U9_g(T2, tree114_out_a(T24)) → goal1_out_g(T2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goal1_in_g(T2) → U4_g(T2, s2t4_in_ga(T2, X12))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T7), node(X23, X24, X23)) → U1_ga(T7, X23, X24, s2t4_in_ga(T7, X23))
s2t4_in_ga(s(T11), node(nil, X41, X42)) → U2_ga(T11, X41, X42, s2t4_in_ga(T11, X42))
s2t4_in_ga(s(T15), node(X59, X60, nil)) → U3_ga(T15, X59, X60, s2t4_in_ga(T15, X59))
s2t4_in_ga(T18, node(nil, X71, nil)) → s2t4_out_ga(T18, node(nil, X71, nil))
U3_ga(T15, X59, X60, s2t4_out_ga(T15, X59)) → s2t4_out_ga(s(T15), node(X59, X60, nil))
U2_ga(T11, X41, X42, s2t4_out_ga(T11, X42)) → s2t4_out_ga(s(T11), node(nil, X41, X42))
U1_ga(T7, X23, X24, s2t4_out_ga(T7, X23)) → s2t4_out_ga(s(T7), node(X23, X24, X23))
U4_g(T2, s2t4_out_ga(T2, X12)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T7), node(X23, X24, X23)) → U1_gg(T7, X23, X24, s2t4_in_gg(T7, X23))
s2t4_in_gg(s(T11), node(nil, X41, X42)) → U2_gg(T11, X41, X42, s2t4_in_gg(T11, X42))
s2t4_in_gg(s(T15), node(X59, X60, nil)) → U3_gg(T15, X59, X60, s2t4_in_gg(T15, X59))
s2t4_in_gg(T18, node(nil, X71, nil)) → s2t4_out_gg(T18, node(nil, X71, nil))
U3_gg(T15, X59, X60, s2t4_out_gg(T15, X59)) → s2t4_out_gg(s(T15), node(X59, X60, nil))
U2_gg(T11, X41, X42, s2t4_out_gg(T11, X42)) → s2t4_out_gg(s(T11), node(nil, X41, X42))
U1_gg(T7, X23, X24, s2t4_out_gg(T7, X23)) → s2t4_out_gg(s(T7), node(X23, X24, X23))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T23))
U6_g(T2, s2t4_out_ga(T2, T23)) → U7_g(T2, tree94_in_)
tree94_in_ → tree94_out_
U7_g(T2, tree94_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U8_g(T2, tree114_in_a(X110))
tree114_in_a(nil) → tree114_out_a(nil)
U8_g(T2, tree114_out_a(X110)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U9_g(T2, tree114_in_a(T24))
U9_g(T2, tree114_out_a(T24)) → goal1_out_g(T2)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_G(T2) → U4_G(T2, s2t4_in_ga(T2, X12))
GOAL1_IN_G(T2) → S2T4_IN_GA(T2, X12)
S2T4_IN_GA(s(T7), node(X23, X24, X23)) → U1_GA(T7, X23, X24, s2t4_in_ga(T7, X23))
S2T4_IN_GA(s(T7), node(X23, X24, X23)) → S2T4_IN_GA(T7, X23)
S2T4_IN_GA(s(T11), node(nil, X41, X42)) → U2_GA(T11, X41, X42, s2t4_in_ga(T11, X42))
S2T4_IN_GA(s(T11), node(nil, X41, X42)) → S2T4_IN_GA(T11, X42)
S2T4_IN_GA(s(T15), node(X59, X60, nil)) → U3_GA(T15, X59, X60, s2t4_in_ga(T15, X59))
S2T4_IN_GA(s(T15), node(X59, X60, nil)) → S2T4_IN_GA(T15, X59)
GOAL1_IN_G(T2) → U5_G(T2, s2t4_in_gg(T2, nil))
GOAL1_IN_G(T2) → S2T4_IN_GG(T2, nil)
S2T4_IN_GG(s(T7), node(X23, X24, X23)) → U1_GG(T7, X23, X24, s2t4_in_gg(T7, X23))
S2T4_IN_GG(s(T7), node(X23, X24, X23)) → S2T4_IN_GG(T7, X23)
S2T4_IN_GG(s(T11), node(nil, X41, X42)) → U2_GG(T11, X41, X42, s2t4_in_gg(T11, X42))
S2T4_IN_GG(s(T11), node(nil, X41, X42)) → S2T4_IN_GG(T11, X42)
S2T4_IN_GG(s(T15), node(X59, X60, nil)) → U3_GG(T15, X59, X60, s2t4_in_gg(T15, X59))
S2T4_IN_GG(s(T15), node(X59, X60, nil)) → S2T4_IN_GG(T15, X59)
GOAL1_IN_G(T2) → U6_G(T2, s2t4_in_ga(T2, T23))
U6_G(T2, s2t4_out_ga(T2, T23)) → U7_G(T2, tree94_in_)
U6_G(T2, s2t4_out_ga(T2, T23)) → TREE94_IN_
U6_G(T2, s2t4_out_ga(T2, T23)) → U8_G(T2, tree114_in_a(X110))
U6_G(T2, s2t4_out_ga(T2, T23)) → TREE114_IN_A(X110)
U6_G(T2, s2t4_out_ga(T2, T23)) → U9_G(T2, tree114_in_a(T24))

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2t4_in_ga(T2, X12))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T7), node(X23, X24, X23)) → U1_ga(T7, X23, X24, s2t4_in_ga(T7, X23))
s2t4_in_ga(s(T11), node(nil, X41, X42)) → U2_ga(T11, X41, X42, s2t4_in_ga(T11, X42))
s2t4_in_ga(s(T15), node(X59, X60, nil)) → U3_ga(T15, X59, X60, s2t4_in_ga(T15, X59))
s2t4_in_ga(T18, node(nil, X71, nil)) → s2t4_out_ga(T18, node(nil, X71, nil))
U3_ga(T15, X59, X60, s2t4_out_ga(T15, X59)) → s2t4_out_ga(s(T15), node(X59, X60, nil))
U2_ga(T11, X41, X42, s2t4_out_ga(T11, X42)) → s2t4_out_ga(s(T11), node(nil, X41, X42))
U1_ga(T7, X23, X24, s2t4_out_ga(T7, X23)) → s2t4_out_ga(s(T7), node(X23, X24, X23))
U4_g(T2, s2t4_out_ga(T2, X12)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T7), node(X23, X24, X23)) → U1_gg(T7, X23, X24, s2t4_in_gg(T7, X23))
s2t4_in_gg(s(T11), node(nil, X41, X42)) → U2_gg(T11, X41, X42, s2t4_in_gg(T11, X42))
s2t4_in_gg(s(T15), node(X59, X60, nil)) → U3_gg(T15, X59, X60, s2t4_in_gg(T15, X59))
s2t4_in_gg(T18, node(nil, X71, nil)) → s2t4_out_gg(T18, node(nil, X71, nil))
U3_gg(T15, X59, X60, s2t4_out_gg(T15, X59)) → s2t4_out_gg(s(T15), node(X59, X60, nil))
U2_gg(T11, X41, X42, s2t4_out_gg(T11, X42)) → s2t4_out_gg(s(T11), node(nil, X41, X42))
U1_gg(T7, X23, X24, s2t4_out_gg(T7, X23)) → s2t4_out_gg(s(T7), node(X23, X24, X23))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T23))
U6_g(T2, s2t4_out_ga(T2, T23)) → U7_g(T2, tree94_in_)
tree94_in_ → tree94_out_
U7_g(T2, tree94_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U8_g(T2, tree114_in_a(X110))
tree114_in_a(nil) → tree114_out_a(nil)
U8_g(T2, tree114_out_a(X110)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U9_g(T2, tree114_in_a(T24))
U9_g(T2, tree114_out_a(T24)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1) = goal1_in_g(x1)
U4_g(x1, x2) = U4_g(x2)
s2t4_in_ga(x1, x2) = s2t4_in_ga(x1)
0 = 0
s2t4_out_ga(x1, x2) = s2t4_out_ga(x2)
s(x1) = s(x1)
U1_ga(x1, x2, x3, x4) = U1_ga(x4)
U2_ga(x1, x2, x3, x4) = U2_ga(x4)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
node(x1, x2, x3) = node(x1, x3)
goal1_out_g(x1) = goal1_out_g
U5_g(x1, x2) = U5_g(x2)
s2t4_in_gg(x1, x2) = s2t4_in_gg(x1, x2)
nil = nil
s2t4_out_gg(x1, x2) = s2t4_out_gg
U1_gg(x1, x2, x3, x4) = U1_gg(x4)
U2_gg(x1, x2, x3, x4) = U2_gg(x4)
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
U6_g(x1, x2) = U6_g(x2)
U7_g(x1, x2) = U7_g(x2)
tree94_in_ = tree94_in_
tree94_out_ = tree94_out_
U8_g(x1, x2) = U8_g(x2)
tree114_in_a(x1) = tree114_in_a
tree114_out_a(x1) = tree114_out_a(x1)
U9_g(x1, x2) = U9_g(x2)
GOAL1_IN_G(x1) = GOAL1_IN_G(x1)
U4_G(x1, x2) = U4_G(x2)
S2T4_IN_GA(x1, x2) = S2T4_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x4)
U2_GA(x1, x2, x3, x4) = U2_GA(x4)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
U5_G(x1, x2) = U5_G(x2)
S2T4_IN_GG(x1, x2) = S2T4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4) = U1_GG(x4)
U2_GG(x1, x2, x3, x4) = U2_GG(x4)
U3_GG(x1, x2, x3, x4) = U3_GG(x4)
U6_G(x1, x2) = U6_G(x2)
U7_G(x1, x2) = U7_G(x2)
TREE94_IN_ = TREE94_IN_
U8_G(x1, x2) = U8_G(x2)
TREE114_IN_A(x1) = TREE114_IN_A
U9_G(x1, x2) = U9_G(x2)

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

(6) Obligation:

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

GOAL1_IN_G(T2) → U4_G(T2, s2t4_in_ga(T2, X12))
GOAL1_IN_G(T2) → S2T4_IN_GA(T2, X12)
S2T4_IN_GA(s(T7), node(X23, X24, X23)) → U1_GA(T7, X23, X24, s2t4_in_ga(T7, X23))
S2T4_IN_GA(s(T7), node(X23, X24, X23)) → S2T4_IN_GA(T7, X23)
S2T4_IN_GA(s(T11), node(nil, X41, X42)) → U2_GA(T11, X41, X42, s2t4_in_ga(T11, X42))
S2T4_IN_GA(s(T11), node(nil, X41, X42)) → S2T4_IN_GA(T11, X42)
S2T4_IN_GA(s(T15), node(X59, X60, nil)) → U3_GA(T15, X59, X60, s2t4_in_ga(T15, X59))
S2T4_IN_GA(s(T15), node(X59, X60, nil)) → S2T4_IN_GA(T15, X59)
GOAL1_IN_G(T2) → U5_G(T2, s2t4_in_gg(T2, nil))
GOAL1_IN_G(T2) → S2T4_IN_GG(T2, nil)
S2T4_IN_GG(s(T7), node(X23, X24, X23)) → U1_GG(T7, X23, X24, s2t4_in_gg(T7, X23))
S2T4_IN_GG(s(T7), node(X23, X24, X23)) → S2T4_IN_GG(T7, X23)
S2T4_IN_GG(s(T11), node(nil, X41, X42)) → U2_GG(T11, X41, X42, s2t4_in_gg(T11, X42))
S2T4_IN_GG(s(T11), node(nil, X41, X42)) → S2T4_IN_GG(T11, X42)
S2T4_IN_GG(s(T15), node(X59, X60, nil)) → U3_GG(T15, X59, X60, s2t4_in_gg(T15, X59))
S2T4_IN_GG(s(T15), node(X59, X60, nil)) → S2T4_IN_GG(T15, X59)
GOAL1_IN_G(T2) → U6_G(T2, s2t4_in_ga(T2, T23))
U6_G(T2, s2t4_out_ga(T2, T23)) → U7_G(T2, tree94_in_)
U6_G(T2, s2t4_out_ga(T2, T23)) → TREE94_IN_
U6_G(T2, s2t4_out_ga(T2, T23)) → U8_G(T2, tree114_in_a(X110))
U6_G(T2, s2t4_out_ga(T2, T23)) → TREE114_IN_A(X110)
U6_G(T2, s2t4_out_ga(T2, T23)) → U9_G(T2, tree114_in_a(T24))

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2t4_in_ga(T2, X12))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T7), node(X23, X24, X23)) → U1_ga(T7, X23, X24, s2t4_in_ga(T7, X23))
s2t4_in_ga(s(T11), node(nil, X41, X42)) → U2_ga(T11, X41, X42, s2t4_in_ga(T11, X42))
s2t4_in_ga(s(T15), node(X59, X60, nil)) → U3_ga(T15, X59, X60, s2t4_in_ga(T15, X59))
s2t4_in_ga(T18, node(nil, X71, nil)) → s2t4_out_ga(T18, node(nil, X71, nil))
U3_ga(T15, X59, X60, s2t4_out_ga(T15, X59)) → s2t4_out_ga(s(T15), node(X59, X60, nil))
U2_ga(T11, X41, X42, s2t4_out_ga(T11, X42)) → s2t4_out_ga(s(T11), node(nil, X41, X42))
U1_ga(T7, X23, X24, s2t4_out_ga(T7, X23)) → s2t4_out_ga(s(T7), node(X23, X24, X23))
U4_g(T2, s2t4_out_ga(T2, X12)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T7), node(X23, X24, X23)) → U1_gg(T7, X23, X24, s2t4_in_gg(T7, X23))
s2t4_in_gg(s(T11), node(nil, X41, X42)) → U2_gg(T11, X41, X42, s2t4_in_gg(T11, X42))
s2t4_in_gg(s(T15), node(X59, X60, nil)) → U3_gg(T15, X59, X60, s2t4_in_gg(T15, X59))
s2t4_in_gg(T18, node(nil, X71, nil)) → s2t4_out_gg(T18, node(nil, X71, nil))
U3_gg(T15, X59, X60, s2t4_out_gg(T15, X59)) → s2t4_out_gg(s(T15), node(X59, X60, nil))
U2_gg(T11, X41, X42, s2t4_out_gg(T11, X42)) → s2t4_out_gg(s(T11), node(nil, X41, X42))
U1_gg(T7, X23, X24, s2t4_out_gg(T7, X23)) → s2t4_out_gg(s(T7), node(X23, X24, X23))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T23))
U6_g(T2, s2t4_out_ga(T2, T23)) → U7_g(T2, tree94_in_)
tree94_in_ → tree94_out_
U7_g(T2, tree94_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U8_g(T2, tree114_in_a(X110))
tree114_in_a(nil) → tree114_out_a(nil)
U8_g(T2, tree114_out_a(X110)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U9_g(T2, tree114_in_a(T24))
U9_g(T2, tree114_out_a(T24)) → goal1_out_g(T2)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 16 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

S2T4_IN_GG(s(T11), node(nil, X41, X42)) → S2T4_IN_GG(T11, X42)
S2T4_IN_GG(s(T7), node(X23, X24, X23)) → S2T4_IN_GG(T7, X23)
S2T4_IN_GG(s(T15), node(X59, X60, nil)) → S2T4_IN_GG(T15, X59)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2t4_in_ga(T2, X12))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T7), node(X23, X24, X23)) → U1_ga(T7, X23, X24, s2t4_in_ga(T7, X23))
s2t4_in_ga(s(T11), node(nil, X41, X42)) → U2_ga(T11, X41, X42, s2t4_in_ga(T11, X42))
s2t4_in_ga(s(T15), node(X59, X60, nil)) → U3_ga(T15, X59, X60, s2t4_in_ga(T15, X59))
s2t4_in_ga(T18, node(nil, X71, nil)) → s2t4_out_ga(T18, node(nil, X71, nil))
U3_ga(T15, X59, X60, s2t4_out_ga(T15, X59)) → s2t4_out_ga(s(T15), node(X59, X60, nil))
U2_ga(T11, X41, X42, s2t4_out_ga(T11, X42)) → s2t4_out_ga(s(T11), node(nil, X41, X42))
U1_ga(T7, X23, X24, s2t4_out_ga(T7, X23)) → s2t4_out_ga(s(T7), node(X23, X24, X23))
U4_g(T2, s2t4_out_ga(T2, X12)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T7), node(X23, X24, X23)) → U1_gg(T7, X23, X24, s2t4_in_gg(T7, X23))
s2t4_in_gg(s(T11), node(nil, X41, X42)) → U2_gg(T11, X41, X42, s2t4_in_gg(T11, X42))
s2t4_in_gg(s(T15), node(X59, X60, nil)) → U3_gg(T15, X59, X60, s2t4_in_gg(T15, X59))
s2t4_in_gg(T18, node(nil, X71, nil)) → s2t4_out_gg(T18, node(nil, X71, nil))
U3_gg(T15, X59, X60, s2t4_out_gg(T15, X59)) → s2t4_out_gg(s(T15), node(X59, X60, nil))
U2_gg(T11, X41, X42, s2t4_out_gg(T11, X42)) → s2t4_out_gg(s(T11), node(nil, X41, X42))
U1_gg(T7, X23, X24, s2t4_out_gg(T7, X23)) → s2t4_out_gg(s(T7), node(X23, X24, X23))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T23))
U6_g(T2, s2t4_out_ga(T2, T23)) → U7_g(T2, tree94_in_)
tree94_in_ → tree94_out_
U7_g(T2, tree94_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U8_g(T2, tree114_in_a(X110))
tree114_in_a(nil) → tree114_out_a(nil)
U8_g(T2, tree114_out_a(X110)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U9_g(T2, tree114_in_a(T24))
U9_g(T2, tree114_out_a(T24)) → goal1_out_g(T2)

(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:

S2T4_IN_GG(s(T11), node(nil, X41, X42)) → S2T4_IN_GG(T11, X42)
S2T4_IN_GG(s(T7), node(X23, X24, X23)) → S2T4_IN_GG(T7, X23)
S2T4_IN_GG(s(T15), node(X59, X60, nil)) → S2T4_IN_GG(T15, X59)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
node(x1, x2, x3) = node(x1, x3)
nil = nil
S2T4_IN_GG(x1, x2) = S2T4_IN_GG(x1, 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:

S2T4_IN_GG(s(T11), node(nil, X42)) → S2T4_IN_GG(T11, X42)
S2T4_IN_GG(s(T7), node(X23, X23)) → S2T4_IN_GG(T7, X23)
S2T4_IN_GG(s(T15), node(X59, nil)) → S2T4_IN_GG(T15, X59)

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:

S2T4_IN_GG(s(T11), node(nil, X42)) → S2T4_IN_GG(T11, X42)
The graph contains the following edges 1 > 1, 2 > 2
S2T4_IN_GG(s(T7), node(X23, X23)) → S2T4_IN_GG(T7, X23)
The graph contains the following edges 1 > 1, 2 > 2
S2T4_IN_GG(s(T15), node(X59, nil)) → S2T4_IN_GG(T15, X59)
The graph contains the following edges 1 > 1, 2 > 2

(15) TRUE

(16) Obligation:

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

S2T4_IN_GA(s(T11), node(nil, X41, X42)) → S2T4_IN_GA(T11, X42)
S2T4_IN_GA(s(T7), node(X23, X24, X23)) → S2T4_IN_GA(T7, X23)
S2T4_IN_GA(s(T15), node(X59, X60, nil)) → S2T4_IN_GA(T15, X59)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2t4_in_ga(T2, X12))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T7), node(X23, X24, X23)) → U1_ga(T7, X23, X24, s2t4_in_ga(T7, X23))
s2t4_in_ga(s(T11), node(nil, X41, X42)) → U2_ga(T11, X41, X42, s2t4_in_ga(T11, X42))
s2t4_in_ga(s(T15), node(X59, X60, nil)) → U3_ga(T15, X59, X60, s2t4_in_ga(T15, X59))
s2t4_in_ga(T18, node(nil, X71, nil)) → s2t4_out_ga(T18, node(nil, X71, nil))
U3_ga(T15, X59, X60, s2t4_out_ga(T15, X59)) → s2t4_out_ga(s(T15), node(X59, X60, nil))
U2_ga(T11, X41, X42, s2t4_out_ga(T11, X42)) → s2t4_out_ga(s(T11), node(nil, X41, X42))
U1_ga(T7, X23, X24, s2t4_out_ga(T7, X23)) → s2t4_out_ga(s(T7), node(X23, X24, X23))
U4_g(T2, s2t4_out_ga(T2, X12)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T7), node(X23, X24, X23)) → U1_gg(T7, X23, X24, s2t4_in_gg(T7, X23))
s2t4_in_gg(s(T11), node(nil, X41, X42)) → U2_gg(T11, X41, X42, s2t4_in_gg(T11, X42))
s2t4_in_gg(s(T15), node(X59, X60, nil)) → U3_gg(T15, X59, X60, s2t4_in_gg(T15, X59))
s2t4_in_gg(T18, node(nil, X71, nil)) → s2t4_out_gg(T18, node(nil, X71, nil))
U3_gg(T15, X59, X60, s2t4_out_gg(T15, X59)) → s2t4_out_gg(s(T15), node(X59, X60, nil))
U2_gg(T11, X41, X42, s2t4_out_gg(T11, X42)) → s2t4_out_gg(s(T11), node(nil, X41, X42))
U1_gg(T7, X23, X24, s2t4_out_gg(T7, X23)) → s2t4_out_gg(s(T7), node(X23, X24, X23))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T23))
U6_g(T2, s2t4_out_ga(T2, T23)) → U7_g(T2, tree94_in_)
tree94_in_ → tree94_out_
U7_g(T2, tree94_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U8_g(T2, tree114_in_a(X110))
tree114_in_a(nil) → tree114_out_a(nil)
U8_g(T2, tree114_out_a(X110)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T23)) → U9_g(T2, tree114_in_a(T24))
U9_g(T2, tree114_out_a(T24)) → goal1_out_g(T2)

(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:

S2T4_IN_GA(s(T11), node(nil, X41, X42)) → S2T4_IN_GA(T11, X42)
S2T4_IN_GA(s(T7), node(X23, X24, X23)) → S2T4_IN_GA(T7, X23)
S2T4_IN_GA(s(T15), node(X59, X60, nil)) → S2T4_IN_GA(T15, X59)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
node(x1, x2, x3) = node(x1, x3)
nil = nil
S2T4_IN_GA(x1, x2) = S2T4_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:

S2T4_IN_GA(s(T11)) → S2T4_IN_GA(T11)

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:

S2T4_IN_GA(s(T11)) → S2T4_IN_GA(T11)
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) TRUE

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) TRUE