Left Termination proof of /tmp/tmp1kBbOO/preorder.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 88 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 14 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 60 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:

preorder(X, Y) :- pdl(X, -(Y, [])).
pdl(nil, Y) :- ','(!, eq(Y, -(X, X))).
pdl(T, -(.(X, Xs), Zs)) :- ','(value(T, X), ','(left(T, L), ','(right(T, R), ','(pdl(L, -(Xs, Ys)), pdl(R, -(Ys, Zs)))))).
left(nil, nil).
left(tree(L, X1, X2), L).
right(nil, nil).
right(tree(X3, X4, R), R).
value(nil, X5).
value(tree(X6, X, X7), X).
eq(X, X).

Query: preorder(g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: preorder(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: preorder(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: pdl(T5, -(T7, []))\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: pdl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 1 | pdl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_2, eq(-(T11, []), -(X16, X16))) | pdl(nil, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: pdl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nX15 is free\npdl(T5, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: eq(-(T11, []), -(X16, X16))\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: eq(-(T11, []), -(X16, X16)), SCOPE: 3, CLAUSE: 9\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, []))))))\n\nT21 is ground\nX15 is free\nX35 is free\nX36 is free\nX37 is free\npdl(T21, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, [])))))), SCOPE: 4, CLAUSE: 7 | ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, [])))))), SCOPE: 4, CLAUSE: 8\n\nT21 is ground\nX15 is free\nX35 is free\nX36 is free\nX37 is free\npdl(T21, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, [])))))), SCOPE: 4, CLAUSE: 8\n\nT21 is ground\nX15 is free\nX35 is free\nX36 is free\nX37 is free\npdl(T21, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, [])))))\n\nT34 is ground\nT35 is ground\nT36 is ground\nX35 is free\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, []))))), SCOPE: 5, CLAUSE: 3 | ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, []))))), SCOPE: 5, CLAUSE: 4\n\nT34 is ground\nT35 is ground\nT36 is ground\nX35 is free\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, []))))), SCOPE: 5, CLAUSE: 4\n\nT34 is ground\nT35 is ground\nT36 is ground\nX35 is free\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, []))))\n\nT46 is ground\nT47 is ground\nT48 is ground\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, [])))), SCOPE: 6, CLAUSE: 5 | ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, [])))), SCOPE: 6, CLAUSE: 6\n\nT46 is ground\nT47 is ground\nT48 is ground\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, [])))), SCOPE: 6, CLAUSE: 6\n\nT46 is ground\nT47 is ground\nT48 is ground\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(pdl(T57, -(T37, X37)), pdl(T59, -(X37, [])))\n\nT57 is ground\nT59 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: pdl(T57, -(T37, X37))\n\nT57 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: pdl(T59, -(T61, []))\n\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: pdl(T57, -(T37, X37)), SCOPE: 7, CLAUSE: 1 | pdl(T57, -(T37, X37)), SCOPE: 7, CLAUSE: 2\n\nT57 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(!_7, eq(-(T66, X94), -(X93, X93))) | pdl(nil, -(T37, X37)), SCOPE: 7, CLAUSE: 2\n\nX37 is free\nX94 is free\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: pdl(T57, -(T37, X37)), SCOPE: 7, CLAUSE: 2\n\nT57 is ground\nX37 is free\nX92 is free\npdl(T57, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: eq(-(T66, X94), -(X93, X93))\n\nX94 is free\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: eq(-(T66, X94), -(X93, X93)), SCOPE: 8, CLAUSE: 9\n\nX94 is free\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121))))))\n\nT76 is ground\nX37 is free\nX92 is free\nX121 is free\nX118 is free\nX119 is free\nX120 is free\npdl(T76, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121)))))), SCOPE: 9, CLAUSE: 7 | ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121)))))), SCOPE: 9, CLAUSE: 8\n\nT76 is ground\nX37 is free\nX92 is free\nX121 is free\nX118 is free\nX119 is free\nX120 is free\npdl(T76, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121)))))), SCOPE: 9, CLAUSE: 8\n\nT76 is ground\nX37 is free\nX92 is free\nX121 is free\nX118 is free\nX119 is free\nX120 is free\npdl(T76, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121)))))\n\nT89 is ground\nT90 is ground\nT91 is ground\nX121 is free\nX118 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121))))), SCOPE: 10, CLAUSE: 3 | ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121))))), SCOPE: 10, CLAUSE: 4\n\nT89 is ground\nT90 is ground\nT91 is ground\nX121 is free\nX118 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121))))), SCOPE: 10, CLAUSE: 4\n\nT89 is ground\nT90 is ground\nT91 is ground\nX121 is free\nX118 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121))))\n\nT101 is ground\nT102 is ground\nT103 is ground\nX121 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121)))), SCOPE: 11, CLAUSE: 5 | ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121)))), SCOPE: 11, CLAUSE: 6\n\nT101 is ground\nT102 is ground\nT103 is ground\nX121 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121)))), SCOPE: 11, CLAUSE: 6\n\nT101 is ground\nT102 is ground\nT103 is ground\nX121 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(pdl(T112, -(T92, X120)), pdl(T114, -(X120, X121)))\n\nT112 is ground\nT114 is ground\nX121 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: pdl(T112, -(T92, X120))\n\nT112 is ground\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: pdl(T114, -(T116, X121))\n\nT114 is ground\nX121 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 47 [arrowhead = none ];
47 -> 2;

48 [label="ONLY EVAL with clause\npreorder(X10, X11) :- pdl(X10, -(X11, [])).\nand substitutionT1 -> T5,\nX10 -> T5,\nT2 -> T7,\nX11 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 48 [arrowhead = none ];
48 -> 3;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 49 [arrowhead = none ];
49 -> 4;

50 [label="EVAL with clause\npdl(nil, X15) :- ','(!_2, eq(X15, -(X16, X16))).\nand substitutionT5 -> nil,\nT7 -> T11,\nX15 -> -(T11, []),\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 50 [arrowhead = none ];
50 -> 5;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 51 [arrowhead = none ];
51 -> 6;

52 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 52 [arrowhead = none ];
52 -> 7;

53 [label="EVAL with clause\npdl(X31, -(.(X32, X33), X34)) :- ','(value(X31, X32), ','(left(X31, X35), ','(right(X31, X36), ','(pdl(X35, -(X33, X37)), pdl(X36, -(X37, X34)))))).\nand substitutionT5 -> T21,\nX31 -> T21,\nX32 -> T24,\nX33 -> T25,\nT7 -> .(T24, T25),\nX34 -> [],\nT22 -> T24,\nT23 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 53 [arrowhead = none ];
53 -> 12;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 54 [arrowhead = none ];
54 -> 13;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 55 [arrowhead = none ];
55 -> 8;

56 [label="EVAL with clause\neq(X19, X19).\nand substitutionT11 -> [],\nX19 -> -([], []),\nX16 -> [],\nT14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 56 [arrowhead = none ];
56 -> 9;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 57 [arrowhead = none ];
57 -> 10;

58 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 58 [arrowhead = none ];
58 -> 11;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 59 [arrowhead = none ];
59 -> 14;

60 [label="BACKTRACK\nfor clause: value(nil, X5)\nwith clash: (pdl(T21, -(T7, [])), pdl(nil, X15))", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 60 [arrowhead = none ];
60 -> 15;

61 [label="EVAL with clause\nvalue(tree(X48, X49, X50), X49).\nand substitutionX48 -> T34,\nX49 -> T35,\nX50 -> T36,\nT21 -> tree(T34, T35, T36),\nT24 -> T35,\nT25 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 61 [arrowhead = none ];
61 -> 16;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 62 [arrowhead = none ];
62 -> 17;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 63 [arrowhead = none ];
63 -> 18;

64 [label="BACKTRACK\nfor clause: left(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 64 [arrowhead = none ];
64 -> 19;

65 [label="ONLY EVAL with clause\nleft(tree(X63, X64, X65), X63).\nand substitutionT34 -> T46,\nX63 -> T46,\nT35 -> T47,\nX64 -> T47,\nT36 -> T48,\nX65 -> T48,\nX35 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 65 [arrowhead = none ];
65 -> 20;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 66 [arrowhead = none ];
66 -> 21;

67 [label="BACKTRACK\nfor clause: right(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 67 [arrowhead = none ];
67 -> 22;

68 [label="ONLY EVAL with clause\nright(tree(X78, X79, X80), X80).\nand substitutionT46 -> T57,\nX78 -> T57,\nT47 -> T58,\nX79 -> T58,\nT48 -> T59,\nX80 -> T59,\nX36 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 68 [arrowhead = none ];
68 -> 23;

69 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
23 -> 69 [arrowhead = none ];
69 -> 24;

70 [label="SPLIT 2\nnew knowledge:\nT57 is ground\nreplacements:X37 -> T61", fontsize=14, style = filled, fillcolor = violet];
23 -> 70 [arrowhead = none ];
70 -> 25;

71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 71 [arrowhead = none ];
71 -> 26;

72 [label="INSTANCE with matching:\nT5 -> T59\nT7 -> T61", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 72 [arrowhead = none , style=dashed];
72 -> 3[style=dashed];

73 [label="EVAL with clause\npdl(nil, X92) :- ','(!_7, eq(X92, -(X93, X93))).\nand substitutionT57 -> nil,\nT37 -> T66,\nX37 -> X94,\nX92 -> -(T66, X94),\nT65 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 73 [arrowhead = none ];
73 -> 27;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 74 [arrowhead = none ];
74 -> 28;

75 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 75 [arrowhead = none ];
75 -> 29;

76 [label="EVAL with clause\npdl(X114, -(.(X115, X116), X117)) :- ','(value(X114, X115), ','(left(X114, X118), ','(right(X114, X119), ','(pdl(X118, -(X116, X120)), pdl(X119, -(X120, X117)))))).\nand substitutionT57 -> T76,\nX114 -> T76,\nX115 -> T79,\nX116 -> T80,\nT37 -> .(T79, T80),\nX37 -> X121,\nX117 -> X121,\nT77 -> T79,\nT78 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 76 [arrowhead = none ];
76 -> 33;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 77 [arrowhead = none ];
77 -> 34;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 78 [arrowhead = none ];
78 -> 30;

79 [label="ONLY EVAL with clause\neq(X99, X99).\nand substitutionT66 -> T69,\nX94 -> T69,\nX99 -> -(T69, T69),\nX93 -> T69,\nX100 -> T69", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 79 [arrowhead = none ];
79 -> 31;

80 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 80 [arrowhead = none ];
80 -> 32;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 81 [arrowhead = none ];
81 -> 35;

82 [label="BACKTRACK\nfor clause: value(nil, X5)\nwith clash: (pdl(T76, -(T37, X37)), pdl(nil, X92))", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 82 [arrowhead = none ];
82 -> 36;

83 [label="EVAL with clause\nvalue(tree(X132, X133, X134), X133).\nand substitutionX132 -> T89,\nX133 -> T90,\nX134 -> T91,\nT76 -> tree(T89, T90, T91),\nT79 -> T90,\nT80 -> T92", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 83 [arrowhead = none ];
83 -> 37;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 84 [arrowhead = none ];
84 -> 38;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 85 [arrowhead = none ];
85 -> 39;

86 [label="BACKTRACK\nfor clause: left(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 86 [arrowhead = none ];
86 -> 40;

87 [label="ONLY EVAL with clause\nleft(tree(X147, X148, X149), X147).\nand substitutionT89 -> T101,\nX147 -> T101,\nT90 -> T102,\nX148 -> T102,\nT91 -> T103,\nX149 -> T103,\nX118 -> T101", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 87 [arrowhead = none ];
87 -> 41;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 88 [arrowhead = none ];
88 -> 42;

89 [label="BACKTRACK\nfor clause: right(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 89 [arrowhead = none ];
89 -> 43;

90 [label="ONLY EVAL with clause\nright(tree(X162, X163, X164), X164).\nand substitutionT101 -> T112,\nX162 -> T112,\nT102 -> T113,\nX163 -> T113,\nT103 -> T114,\nX164 -> T114,\nX119 -> T114", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 90 [arrowhead = none ];
90 -> 44;

91 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
44 -> 91 [arrowhead = none ];
91 -> 45;

92 [label="SPLIT 2\nnew knowledge:\nT112 is ground\nreplacements:X120 -> T116", fontsize=14, style = filled, fillcolor = violet];
44 -> 92 [arrowhead = none ];
92 -> 46;

93 [label="INSTANCE with matching:\nT57 -> T112\nT37 -> T92\nX37 -> X120", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 93 [arrowhead = none , style=dashed];
93 -> 24[style=dashed];

94 [label="INSTANCE with matching:\nT57 -> T114\nT37 -> T116\nX37 -> X121", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 94 [arrowhead = none , style=dashed];
94 -> 24[style=dashed];

}

(2) Obligation:

Clauses:

pdlA(nil, []).
pdlA(tree(T57, T58, T59), .(T58, T37)) :- pdlB(T57, T37, X37).
pdlA(tree(T57, T58, T59), .(T58, T37)) :- ','(pdlB(T57, T37, T61), pdlA(T59, T61)).
pdlB(nil, T69, T69).
pdlB(tree(T112, T113, T114), .(T113, T92), X121) :- pdlB(T112, T92, X120).
pdlB(tree(T112, T113, T114), .(T113, T92), X121) :- ','(pdlB(T112, T92, T116), pdlB(T114, T116, X121)).
preorderC(T5, T7) :- pdlA(T5, T7).

Query: preorderC(g,a)

(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:
preorderC_in: (b,f)
pdlA_in: (b,f)
pdlB_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

preorderC_in_ga(T5, T7) → U7_ga(T5, T7, pdlA_in_ga(T5, T7))
pdlA_in_ga(nil, []) → pdlA_out_ga(nil, [])
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U1_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, X37))
pdlB_in_gaa(nil, T69, T69) → pdlB_out_gaa(nil, T69, T69)
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U4_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, X120))
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U5_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U5_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → U6_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T114, T116, X121))
U6_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T114, T116, X121)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U4_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, X120)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U1_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, X37)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U2_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, T61))
U2_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → U3_ga(T57, T58, T59, T37, pdlA_in_ga(T59, T61))
U3_ga(T57, T58, T59, T37, pdlA_out_ga(T59, T61)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
U7_ga(T5, T7, pdlA_out_ga(T5, T7)) → preorderC_out_ga(T5, T7)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

preorderC_in_ga(T5, T7) → U7_ga(T5, T7, pdlA_in_ga(T5, T7))
pdlA_in_ga(nil, []) → pdlA_out_ga(nil, [])
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U1_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, X37))
pdlB_in_gaa(nil, T69, T69) → pdlB_out_gaa(nil, T69, T69)
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U4_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, X120))
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U5_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U5_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → U6_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T114, T116, X121))
U6_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T114, T116, X121)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U4_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, X120)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U1_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, X37)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U2_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, T61))
U2_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → U3_ga(T57, T58, T59, T37, pdlA_in_ga(T59, T61))
U3_ga(T57, T58, T59, T37, pdlA_out_ga(T59, T61)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
U7_ga(T5, T7, pdlA_out_ga(T5, T7)) → preorderC_out_ga(T5, T7)

(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:

PREORDERC_IN_GA(T5, T7) → U7_GA(T5, T7, pdlA_in_ga(T5, T7))
PREORDERC_IN_GA(T5, T7) → PDLA_IN_GA(T5, T7)
PDLA_IN_GA(tree(T57, T58, T59), .(T58, T37)) → U1_GA(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, X37))
PDLA_IN_GA(tree(T57, T58, T59), .(T58, T37)) → PDLB_IN_GAA(T57, T37, X37)
PDLB_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → U4_GAA(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, X120))
PDLB_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → PDLB_IN_GAA(T112, T92, X120)
PDLB_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → U5_GAA(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U5_GAA(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → U6_GAA(T112, T113, T114, T92, X121, pdlB_in_gaa(T114, T116, X121))
U5_GAA(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → PDLB_IN_GAA(T114, T116, X121)
PDLA_IN_GA(tree(T57, T58, T59), .(T58, T37)) → U2_GA(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, T61))
U2_GA(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → U3_GA(T57, T58, T59, T37, pdlA_in_ga(T59, T61))
U2_GA(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → PDLA_IN_GA(T59, T61)

The TRS R consists of the following rules:

preorderC_in_ga(T5, T7) → U7_ga(T5, T7, pdlA_in_ga(T5, T7))
pdlA_in_ga(nil, []) → pdlA_out_ga(nil, [])
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U1_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, X37))
pdlB_in_gaa(nil, T69, T69) → pdlB_out_gaa(nil, T69, T69)
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U4_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, X120))
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U5_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U5_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → U6_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T114, T116, X121))
U6_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T114, T116, X121)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U4_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, X120)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U1_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, X37)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U2_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, T61))
U2_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → U3_ga(T57, T58, T59, T37, pdlA_in_ga(T59, T61))
U3_ga(T57, T58, T59, T37, pdlA_out_ga(T59, T61)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
U7_ga(T5, T7, pdlA_out_ga(T5, T7)) → preorderC_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderC_in_ga(x1, x2) = preorderC_in_ga(x1)
U7_ga(x1, x2, x3) = U7_ga(x3)
pdlA_in_ga(x1, x2) = pdlA_in_ga(x1)
nil = nil
pdlA_out_ga(x1, x2) = pdlA_out_ga
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
pdlB_in_gaa(x1, x2, x3) = pdlB_in_gaa(x1)
pdlB_out_gaa(x1, x2, x3) = pdlB_out_gaa
U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x6)
U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x6)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5)
preorderC_out_ga(x1, x2) = preorderC_out_ga
PREORDERC_IN_GA(x1, x2) = PREORDERC_IN_GA(x1)
U7_GA(x1, x2, x3) = U7_GA(x3)
PDLA_IN_GA(x1, x2) = PDLA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5)
PDLB_IN_GAA(x1, x2, x3) = PDLB_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x6)
U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x3, x6)
U6_GAA(x1, x2, x3, x4, x5, x6) = U6_GAA(x6)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x3, x5)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5)

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

(6) Obligation:

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

PREORDERC_IN_GA(T5, T7) → U7_GA(T5, T7, pdlA_in_ga(T5, T7))
PREORDERC_IN_GA(T5, T7) → PDLA_IN_GA(T5, T7)
PDLA_IN_GA(tree(T57, T58, T59), .(T58, T37)) → U1_GA(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, X37))
PDLA_IN_GA(tree(T57, T58, T59), .(T58, T37)) → PDLB_IN_GAA(T57, T37, X37)
PDLB_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → U4_GAA(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, X120))
PDLB_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → PDLB_IN_GAA(T112, T92, X120)
PDLB_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → U5_GAA(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U5_GAA(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → U6_GAA(T112, T113, T114, T92, X121, pdlB_in_gaa(T114, T116, X121))
U5_GAA(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → PDLB_IN_GAA(T114, T116, X121)
PDLA_IN_GA(tree(T57, T58, T59), .(T58, T37)) → U2_GA(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, T61))
U2_GA(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → U3_GA(T57, T58, T59, T37, pdlA_in_ga(T59, T61))
U2_GA(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → PDLA_IN_GA(T59, T61)

The TRS R consists of the following rules:

preorderC_in_ga(T5, T7) → U7_ga(T5, T7, pdlA_in_ga(T5, T7))
pdlA_in_ga(nil, []) → pdlA_out_ga(nil, [])
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U1_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, X37))
pdlB_in_gaa(nil, T69, T69) → pdlB_out_gaa(nil, T69, T69)
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U4_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, X120))
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U5_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U5_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → U6_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T114, T116, X121))
U6_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T114, T116, X121)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U4_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, X120)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U1_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, X37)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U2_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, T61))
U2_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → U3_ga(T57, T58, T59, T37, pdlA_in_ga(T59, T61))
U3_ga(T57, T58, T59, T37, pdlA_out_ga(T59, T61)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
U7_ga(T5, T7, pdlA_out_ga(T5, T7)) → preorderC_out_ga(T5, T7)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

PDLB_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → U5_GAA(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U5_GAA(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → PDLB_IN_GAA(T114, T116, X121)
PDLB_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → PDLB_IN_GAA(T112, T92, X120)

The TRS R consists of the following rules:

preorderC_in_ga(T5, T7) → U7_ga(T5, T7, pdlA_in_ga(T5, T7))
pdlA_in_ga(nil, []) → pdlA_out_ga(nil, [])
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U1_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, X37))
pdlB_in_gaa(nil, T69, T69) → pdlB_out_gaa(nil, T69, T69)
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U4_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, X120))
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U5_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U5_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → U6_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T114, T116, X121))
U6_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T114, T116, X121)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U4_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, X120)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U1_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, X37)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U2_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, T61))
U2_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → U3_ga(T57, T58, T59, T37, pdlA_in_ga(T59, T61))
U3_ga(T57, T58, T59, T37, pdlA_out_ga(T59, T61)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
U7_ga(T5, T7, pdlA_out_ga(T5, T7)) → preorderC_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderC_in_ga(x1, x2) = preorderC_in_ga(x1)
U7_ga(x1, x2, x3) = U7_ga(x3)
pdlA_in_ga(x1, x2) = pdlA_in_ga(x1)
nil = nil
pdlA_out_ga(x1, x2) = pdlA_out_ga
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
pdlB_in_gaa(x1, x2, x3) = pdlB_in_gaa(x1)
pdlB_out_gaa(x1, x2, x3) = pdlB_out_gaa
U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x6)
U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x6)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5)
preorderC_out_ga(x1, x2) = preorderC_out_ga
PDLB_IN_GAA(x1, x2, x3) = PDLB_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x3, x6)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

PDLB_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → U5_GAA(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U5_GAA(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → PDLB_IN_GAA(T114, T116, X121)
PDLB_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → PDLB_IN_GAA(T112, T92, X120)

The TRS R consists of the following rules:

pdlB_in_gaa(nil, T69, T69) → pdlB_out_gaa(nil, T69, T69)
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U4_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, X120))
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U5_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U4_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, X120)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U5_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → U6_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T114, T116, X121))
U6_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T114, T116, X121)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)

The argument filtering Pi contains the following mapping:
nil = nil
tree(x1, x2, x3) = tree(x1, x2, x3)
pdlB_in_gaa(x1, x2, x3) = pdlB_in_gaa(x1)
pdlB_out_gaa(x1, x2, x3) = pdlB_out_gaa
U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x6)
U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x6)
PDLB_IN_GAA(x1, x2, x3) = PDLB_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x3, x6)

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:

PDLB_IN_GAA(tree(T112, T113, T114)) → U5_GAA(T114, pdlB_in_gaa(T112))
U5_GAA(T114, pdlB_out_gaa) → PDLB_IN_GAA(T114)
PDLB_IN_GAA(tree(T112, T113, T114)) → PDLB_IN_GAA(T112)

The TRS R consists of the following rules:

pdlB_in_gaa(nil) → pdlB_out_gaa
pdlB_in_gaa(tree(T112, T113, T114)) → U4_gaa(pdlB_in_gaa(T112))
pdlB_in_gaa(tree(T112, T113, T114)) → U5_gaa(T114, pdlB_in_gaa(T112))
U4_gaa(pdlB_out_gaa) → pdlB_out_gaa
U5_gaa(T114, pdlB_out_gaa) → U6_gaa(pdlB_in_gaa(T114))
U6_gaa(pdlB_out_gaa) → pdlB_out_gaa

The set Q consists of the following terms:

pdlB_in_gaa(x0)
U4_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0)

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:

U5_GAA(T114, pdlB_out_gaa) → PDLB_IN_GAA(T114)
The graph contains the following edges 1 >= 1
PDLB_IN_GAA(tree(T112, T113, T114)) → PDLB_IN_GAA(T112)
The graph contains the following edges 1 > 1
PDLB_IN_GAA(tree(T112, T113, T114)) → U5_GAA(T114, pdlB_in_gaa(T112))
The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

PDLA_IN_GA(tree(T57, T58, T59), .(T58, T37)) → U2_GA(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, T61))
U2_GA(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → PDLA_IN_GA(T59, T61)

The TRS R consists of the following rules:

preorderC_in_ga(T5, T7) → U7_ga(T5, T7, pdlA_in_ga(T5, T7))
pdlA_in_ga(nil, []) → pdlA_out_ga(nil, [])
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U1_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, X37))
pdlB_in_gaa(nil, T69, T69) → pdlB_out_gaa(nil, T69, T69)
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U4_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, X120))
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U5_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U5_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → U6_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T114, T116, X121))
U6_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T114, T116, X121)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U4_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, X120)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U1_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, X37)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
pdlA_in_ga(tree(T57, T58, T59), .(T58, T37)) → U2_ga(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, T61))
U2_ga(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → U3_ga(T57, T58, T59, T37, pdlA_in_ga(T59, T61))
U3_ga(T57, T58, T59, T37, pdlA_out_ga(T59, T61)) → pdlA_out_ga(tree(T57, T58, T59), .(T58, T37))
U7_ga(T5, T7, pdlA_out_ga(T5, T7)) → preorderC_out_ga(T5, T7)

(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:

PDLA_IN_GA(tree(T57, T58, T59), .(T58, T37)) → U2_GA(T57, T58, T59, T37, pdlB_in_gaa(T57, T37, T61))
U2_GA(T57, T58, T59, T37, pdlB_out_gaa(T57, T37, T61)) → PDLA_IN_GA(T59, T61)

The TRS R consists of the following rules:

pdlB_in_gaa(nil, T69, T69) → pdlB_out_gaa(nil, T69, T69)
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U4_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, X120))
pdlB_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U5_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T112, T92, T116))
U4_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, X120)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U5_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T112, T92, T116)) → U6_gaa(T112, T113, T114, T92, X121, pdlB_in_gaa(T114, T116, X121))
U6_gaa(T112, T113, T114, T92, X121, pdlB_out_gaa(T114, T116, X121)) → pdlB_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)

The argument filtering Pi contains the following mapping:
nil = nil
tree(x1, x2, x3) = tree(x1, x2, x3)
pdlB_in_gaa(x1, x2, x3) = pdlB_in_gaa(x1)
pdlB_out_gaa(x1, x2, x3) = pdlB_out_gaa
U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x6)
U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x6)
PDLA_IN_GA(x1, x2) = PDLA_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x3, x5)

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:

PDLA_IN_GA(tree(T57, T58, T59)) → U2_GA(T59, pdlB_in_gaa(T57))
U2_GA(T59, pdlB_out_gaa) → PDLA_IN_GA(T59)

The TRS R consists of the following rules:

pdlB_in_gaa(nil) → pdlB_out_gaa
pdlB_in_gaa(tree(T112, T113, T114)) → U4_gaa(pdlB_in_gaa(T112))
pdlB_in_gaa(tree(T112, T113, T114)) → U5_gaa(T114, pdlB_in_gaa(T112))
U4_gaa(pdlB_out_gaa) → pdlB_out_gaa
U5_gaa(T114, pdlB_out_gaa) → U6_gaa(pdlB_in_gaa(T114))
U6_gaa(pdlB_out_gaa) → pdlB_out_gaa

The set Q consists of the following terms:

pdlB_in_gaa(x0)
U4_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0)

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:

U2_GA(T59, pdlB_out_gaa) → PDLA_IN_GA(T59)
The graph contains the following edges 1 >= 1
PDLA_IN_GA(tree(T57, T58, T59)) → U2_GA(T59, pdlB_in_gaa(T57))
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