Left Termination of the query pattern preorder_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(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).

Queries:

preorder(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

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(T3, -(T5, []))\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: pdl(T3, -(T5, [])), SCOPE: 2, CLAUSE: 1 | pdl(T3, -(T5, [])), SCOPE: 2, CLAUSE: 2 | ?_2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_2, eq(-(T7, []), -(X14, X14))) | pdl(nil, -(T5, [])), SCOPE: 2, CLAUSE: 2 | ?_2\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: pdl(T3, -(T5, [])), SCOPE: 2, CLAUSE: 2\n\nT3 is ground\nX13 is free; \npdl(T3, -(T5, [])) !=? pdl(nil, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: eq(-(T7, []), -(X14, X14))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: eq(-(T7, []), -(X14, X14)), SCOPE: 3, CLAUSE: 9\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\nX14 is free\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(value(T9, T12), ','(left(T9, X25), ','(right(T9, X26), ','(pdl(X25, -(T13, X27)), pdl(X26, -(X27, []))))))\n\nT9 is ground\nX13 is free; \nX25 is free; \nX26 is free; \nX27 is free; \npdl(T9, -(T5, [])) !=? pdl(nil, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\nX13 is free\nX21 is free; \nX22 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(value(T9, T12), ','(left(T9, X25), ','(right(T9, X26), ','(pdl(X25, -(T13, X27)), pdl(X26, -(X27, [])))))), SCOPE: 4, CLAUSE: 7 | ','(value(T9, T12), ','(left(T9, X25), ','(right(T9, X26), ','(pdl(X25, -(T13, X27)), pdl(X26, -(X27, [])))))), SCOPE: 4, CLAUSE: 8\n\nT9 is ground\nX13 is free; \nX25 is free; \nX26 is free; \nX27 is free; \npdl(T9, -(T5, [])) !=? pdl(nil, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(value(T9, T12), ','(left(T9, X25), ','(right(T9, X26), ','(pdl(X25, -(T13, X27)), pdl(X26, -(X27, [])))))), SCOPE: 4, CLAUSE: 8\n\nT9 is ground\nX13 is free; \nX25 is free; \nX26 is free; \nX27 is free; \nX29 is free; \npdl(T9, -(T5, [])) !=? pdl(nil, X13)\nvalue(T9, T12) !=? value(nil, X29)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(left(tree(T14, T15, T16), X25), ','(right(tree(T14, T15, T16), X26), ','(pdl(X25, -(T17, X27)), pdl(X26, -(X27, [])))))\n\nT14 is ground\nT15 is ground\nT16 is ground\nX13 is free; \nX25 is free; \nX26 is free; \nX27 is free; \nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\nX13 is free\nX29 is free; \nX33 is free; \nX34 is free; \nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(left(tree(T14, T15, T16), X25), ','(right(tree(T14, T15, T16), X26), ','(pdl(X25, -(T17, X27)), pdl(X26, -(X27, []))))), SCOPE: 5, CLAUSE: 3 | ','(left(tree(T14, T15, T16), X25), ','(right(tree(T14, T15, T16), X26), ','(pdl(X25, -(T17, X27)), pdl(X26, -(X27, []))))), SCOPE: 5, CLAUSE: 4\n\nT14 is ground\nT15 is ground\nT16 is ground\nX25 is free; \nX26 is free; \nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(left(tree(T14, T15, T16), X25), ','(right(tree(T14, T15, T16), X26), ','(pdl(X25, -(T17, X27)), pdl(X26, -(X27, []))))), SCOPE: 5, CLAUSE: 4\n\nT14 is ground\nT15 is ground\nT16 is ground\nX25 is free; \nX26 is free; \nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(right(tree(T18, T19, T20), X26), ','(pdl(T18, -(T17, X27)), pdl(X26, -(X27, []))))\n\nT18 is ground\nT19 is ground\nT20 is ground\nX26 is free; \nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(right(tree(T18, T19, T20), X26), ','(pdl(T18, -(T17, X27)), pdl(X26, -(X27, [])))), SCOPE: 6, CLAUSE: 5 | ','(right(tree(T18, T19, T20), X26), ','(pdl(T18, -(T17, X27)), pdl(X26, -(X27, [])))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nT19 is ground\nT20 is ground\nX26 is free; \nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(right(tree(T18, T19, T20), X26), ','(pdl(T18, -(T17, X27)), pdl(X26, -(X27, [])))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nT19 is ground\nT20 is ground\nX26 is free; \nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(pdl(T21, -(T17, X27)), pdl(T23, -(X27, [])))\n\nT21 is ground\nT23 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: pdl(T21, -(T17, X27))\n\nT21 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: pdl(T23, -(T24, []))\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: pdl(T21, -(T17, X27)), SCOPE: 7, CLAUSE: 1 | pdl(T21, -(T17, X27)), SCOPE: 7, CLAUSE: 2 | ?_7\n\nT21 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(!_7, eq(-(T26, X51), -(X50, X50))) | pdl(nil, -(T17, X27)), SCOPE: 7, CLAUSE: 2 | ?_7\n\nX27 is free\nX51 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: pdl(T21, -(T17, X27)), SCOPE: 7, CLAUSE: 2\n\nT21 is ground\nX27 is free; \nX49 is free; \npdl(T21, -(T17, X27)) !=? pdl(nil, X49)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: eq(-(T26, X51), -(X50, X50))\n\nX51 is free\nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: eq(-(T26, X51), -(X50, X50)), SCOPE: 8, CLAUSE: 9\n\nX51 is free\nX50 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(T28, T31), ','(left(T28, X63), ','(right(T28, X64), ','(pdl(X63, -(T32, X65)), pdl(X64, -(X65, X66))))))\n\nT28 is ground\nX27 is free; \nX49 is free; \nX66 is free; \nX63 is free; \nX64 is free; \nX65 is free; \npdl(T28, -(T17, X27)) !=? pdl(nil, X49)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\nX27 is free\nX49 is free; \nX59 is free; \nX60 is free; \nX61 is free; \nX62 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(value(T28, T31), ','(left(T28, X63), ','(right(T28, X64), ','(pdl(X63, -(T32, X65)), pdl(X64, -(X65, X66)))))), SCOPE: 9, CLAUSE: 7 | ','(value(T28, T31), ','(left(T28, X63), ','(right(T28, X64), ','(pdl(X63, -(T32, X65)), pdl(X64, -(X65, X66)))))), SCOPE: 9, CLAUSE: 8\n\nT28 is ground\nX27 is free; \nX49 is free; \nX66 is free; \nX63 is free; \nX64 is free; \nX65 is free; \npdl(T28, -(T17, X27)) !=? pdl(nil, X49)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(value(T28, T31), ','(left(T28, X63), ','(right(T28, X64), ','(pdl(X63, -(T32, X65)), pdl(X64, -(X65, X66)))))), SCOPE: 9, CLAUSE: 8\n\nT28 is ground\nX27 is free; \nX49 is free; \nX66 is free; \nX63 is free; \nX64 is free; \nX65 is free; \nX68 is free; \npdl(T28, -(T17, X27)) !=? pdl(nil, X49)\nvalue(T28, T31) !=? value(nil, X68)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(left(tree(T33, T34, T35), X63), ','(right(tree(T33, T34, T35), X64), ','(pdl(X63, -(T36, X65)), pdl(X64, -(X65, X66)))))\n\nT33 is ground\nT34 is ground\nT35 is ground\nX27 is free; \nX49 is free; \nX66 is free; \nX63 is free; \nX64 is free; \nX65 is free; \nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\nX27 is free\nX49 is free; \nX68 is free; \nX72 is free; \nX73 is free; \nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(left(tree(T33, T34, T35), X63), ','(right(tree(T33, T34, T35), X64), ','(pdl(X63, -(T36, X65)), pdl(X64, -(X65, X66))))), SCOPE: 10, CLAUSE: 3 | ','(left(tree(T33, T34, T35), X63), ','(right(tree(T33, T34, T35), X64), ','(pdl(X63, -(T36, X65)), pdl(X64, -(X65, X66))))), SCOPE: 10, CLAUSE: 4\n\nT33 is ground\nT34 is ground\nT35 is ground\nX66 is free; \nX63 is free; \nX64 is free; \nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(left(tree(T33, T34, T35), X63), ','(right(tree(T33, T34, T35), X64), ','(pdl(X63, -(T36, X65)), pdl(X64, -(X65, X66))))), SCOPE: 10, CLAUSE: 4\n\nT33 is ground\nT34 is ground\nT35 is ground\nX66 is free; \nX63 is free; \nX64 is free; \nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(right(tree(T37, T38, T39), X64), ','(pdl(T37, -(T36, X65)), pdl(X64, -(X65, X66))))\n\nT37 is ground\nT38 is ground\nT39 is ground\nX66 is free; \nX64 is free; \nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(right(tree(T37, T38, T39), X64), ','(pdl(T37, -(T36, X65)), pdl(X64, -(X65, X66)))), SCOPE: 11, CLAUSE: 5 | ','(right(tree(T37, T38, T39), X64), ','(pdl(T37, -(T36, X65)), pdl(X64, -(X65, X66)))), SCOPE: 11, CLAUSE: 6\n\nT37 is ground\nT38 is ground\nT39 is ground\nX66 is free; \nX64 is free; \nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(right(tree(T37, T38, T39), X64), ','(pdl(T37, -(T36, X65)), pdl(X64, -(X65, X66)))), SCOPE: 11, CLAUSE: 6\n\nT37 is ground\nT38 is ground\nT39 is ground\nX66 is free; \nX64 is free; \nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(pdl(T40, -(T36, X65)), pdl(T42, -(X65, X66)))\n\nT40 is ground\nT42 is ground\nX66 is free; \nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: pdl(T40, -(T36, X65))\n\nT40 is ground\nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: pdl(T42, -(T43, X66))\n\nT42 is ground\nX66 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="EVAL with clause\npreorder(X10, X11) :- pdl(X10, -(X11, [])).\nand substitution\nT1 -> T3\nX10 -> T3\nT2 -> T5\nX11 -> T5\nT4 -> T5", 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, X13) :- ','(!, eq(X13, -(X14, X14))).\nand substitution\nT3 -> nil\nT5 -> T7\nX13 -> -(T7, [])\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 50 [arrowhead = none ];
50 -> 5;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X21, -(.(X22, X23), X24)) :- ','(value(X21, X22), ','(left(X21, X25), ','(right(X21, X26), ','(pdl(X25, -(X23, X27)), pdl(X26, -(X27, X24)))))).\nand substitution\nT3 -> T9\nX21 -> T9\nX22 -> T12\nX23 -> T13\nT5 -> .(T12, T13)\nX24 -> []\nT10 -> T12\nT11 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 53 [arrowhead = none ];
53 -> 12;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X16, X16).\nand substitution\nT7 -> []\nX16 -> -([], [])\nX14 -> []\nT8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 56 [arrowhead = none ];
56 -> 9;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(T9, -(T5, [])), pdl(nil, X13))", fontsize=14, style = filled, fillcolor = white];
14 -> 60 [arrowhead = none ];
60 -> 15;

61 [label="EVAL with clause\nvalue(tree(X33, X34, X35), X34).\nand substitution\nX33 -> T14\nX34 -> T15\nX35 -> T16\nT9 -> tree(T14, T15, T16)\nT12 -> T15\nT13 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 61 [arrowhead = none ];
61 -> 16;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
18 -> 64 [arrowhead = none ];
64 -> 19;

65 [label="EVAL with clause\nleft(tree(X39, X40, X41), X39).\nand substitution\nT14 -> T18\nX39 -> T18\nT15 -> T19\nX40 -> T19\nT16 -> T20\nX41 -> T20\nX25 -> T18", 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 = white];
21 -> 67 [arrowhead = none ];
67 -> 22;

68 [label="EVAL with clause\nright(tree(X45, X46, X47), X47).\nand substitution\nT18 -> T21\nX45 -> T21\nT19 -> T22\nX46 -> T22\nT20 -> T23\nX47 -> T23\nX26 -> T23", 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\nreplacements:\nX27 -> T24", 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:\nT3 -> T23\nT5 -> T24", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 72 [arrowhead = none , style=dashed];
72 -> 3[style=dashed];

73 [label="EVAL with clause\npdl(nil, X49) :- ','(!, eq(X49, -(X50, X50))).\nand substitution\nT21 -> nil\nT17 -> T26\nX27 -> X51\nX49 -> -(T26, X51)\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 73 [arrowhead = none ];
73 -> 27;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X59, -(.(X60, X61), X62)) :- ','(value(X59, X60), ','(left(X59, X63), ','(right(X59, X64), ','(pdl(X63, -(X61, X65)), pdl(X64, -(X65, X62)))))).\nand substitution\nT21 -> T28\nX59 -> T28\nX60 -> T31\nX61 -> T32\nT17 -> .(T31, T32)\nX27 -> X66\nX62 -> X66\nT29 -> T31\nT30 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 76 [arrowhead = none ];
76 -> 33;

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

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

79 [label="EVAL with clause\neq(X53, X53).\nand substitution\nT26 -> T27\nX51 -> T27\nX53 -> -(T27, T27)\nX50 -> T27\nX54 -> T27", 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(T28, -(T17, X27)), pdl(nil, X49))", fontsize=14, style = filled, fillcolor = white];
35 -> 82 [arrowhead = none ];
82 -> 36;

83 [label="EVAL with clause\nvalue(tree(X72, X73, X74), X73).\nand substitution\nX72 -> T33\nX73 -> T34\nX74 -> T35\nT28 -> tree(T33, T34, T35)\nT31 -> T34\nT32 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 83 [arrowhead = none ];
83 -> 37;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
39 -> 86 [arrowhead = none ];
86 -> 40;

87 [label="EVAL with clause\nleft(tree(X78, X79, X80), X78).\nand substitution\nT33 -> T37\nX78 -> T37\nT34 -> T38\nX79 -> T38\nT35 -> T39\nX80 -> T39\nX63 -> T37", 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 = white];
42 -> 89 [arrowhead = none ];
89 -> 43;

90 [label="EVAL with clause\nright(tree(X84, X85, X86), X86).\nand substitution\nT37 -> T40\nX84 -> T40\nT38 -> T41\nX85 -> T41\nT39 -> T42\nX86 -> T42\nX64 -> T42", 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\nreplacements:\nX65 -> T43", fontsize=14, style = filled, fillcolor = violet];
44 -> 92 [arrowhead = none ];
92 -> 46;

93 [label="INSTANCE with matching:\nT21 -> T40\nT17 -> T36\nX27 -> X65", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 93 [arrowhead = none , style=dashed];
93 -> 24[style=dashed];

94 [label="INSTANCE with matching:\nT21 -> T42\nT17 -> T43\nX27 -> X66", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 94 [arrowhead = none , style=dashed];
94 -> 24[style=dashed];

}

(2) Obligation:

Clauses:

pdl3(nil, []).
pdl3(tree(T21, T22, T23), .(T22, T17)) :- pdl24(T21, T17, X27).
pdl3(tree(T21, T22, T23), .(T22, T17)) :- ','(pdl24(T21, T17, T24), pdl3(T23, T24)).
pdl24(nil, T27, T27).
pdl24(tree(T40, T41, T42), .(T41, T36), X66) :- pdl24(T40, T36, X65).
pdl24(tree(T40, T41, T42), .(T41, T36), X66) :- ','(pdl24(T40, T36, T43), pdl24(T42, T43, X66)).
preorder1(nil, []).
preorder1(tree(T21, T22, T23), .(T22, T17)) :- pdl24(T21, T17, X27).
preorder1(tree(T21, T22, T23), .(T22, T17)) :- ','(pdl24(T21, T17, T24), pdl3(T23, T24)).

Queries:

preorder1(g,a).

(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:
preorder1_in: (b,f)
pdl24_in: (b,f,f)
pdl3_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

preorder1_in_ga(nil, []) → preorder1_out_ga(nil, [])
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U7_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
pdl24_in_gaa(nil, T27, T27) → pdl24_out_gaa(nil, T27, T27)
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U4_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, X65))
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U5_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U5_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → U6_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T42, T43, X66))
U6_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T42, T43, X66)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U4_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, X65)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U7_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U8_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U8_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U9_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
pdl3_in_ga(nil, []) → pdl3_out_ga(nil, [])
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U1_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
U1_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U2_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U2_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U3_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
U3_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
U9_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))

The argument filtering Pi contains the following mapping:
preorder1_in_ga(x1, x2)  =  preorder1_in_ga(x1)
nil  =  nil
preorder1_out_ga(x1, x2)  =  preorder1_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
pdl24_in_gaa(x1, x2, x3)  =  pdl24_in_gaa(x1)
pdl24_out_gaa(x1, x2, x3)  =  pdl24_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)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
pdl3_in_ga(x1, x2)  =  pdl3_in_ga(x1)
pdl3_out_ga(x1, x2)  =  pdl3_out_ga
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

preorder1_in_ga(nil, []) → preorder1_out_ga(nil, [])
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U7_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
pdl24_in_gaa(nil, T27, T27) → pdl24_out_gaa(nil, T27, T27)
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U4_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, X65))
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U5_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U5_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → U6_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T42, T43, X66))
U6_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T42, T43, X66)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U4_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, X65)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U7_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U8_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U8_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U9_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
pdl3_in_ga(nil, []) → pdl3_out_ga(nil, [])
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U1_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
U1_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U2_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U2_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U3_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
U3_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
U9_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))

The argument filtering Pi contains the following mapping:
preorder1_in_ga(x1, x2)  =  preorder1_in_ga(x1)
nil  =  nil
preorder1_out_ga(x1, x2)  =  preorder1_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
pdl24_in_gaa(x1, x2, x3)  =  pdl24_in_gaa(x1)
pdl24_out_gaa(x1, x2, x3)  =  pdl24_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)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
pdl3_in_ga(x1, x2)  =  pdl3_in_ga(x1)
pdl3_out_ga(x1, x2)  =  pdl3_out_ga
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)

(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:

PREORDER1_IN_GA(tree(T21, T22, T23), .(T22, T17)) → U7_GA(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
PREORDER1_IN_GA(tree(T21, T22, T23), .(T22, T17)) → PDL24_IN_GAA(T21, T17, X27)
PDL24_IN_GAA(tree(T40, T41, T42), .(T41, T36), X66) → U4_GAA(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, X65))
PDL24_IN_GAA(tree(T40, T41, T42), .(T41, T36), X66) → PDL24_IN_GAA(T40, T36, X65)
PDL24_IN_GAA(tree(T40, T41, T42), .(T41, T36), X66) → U5_GAA(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U5_GAA(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → U6_GAA(T40, T41, T42, T36, X66, pdl24_in_gaa(T42, T43, X66))
U5_GAA(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → PDL24_IN_GAA(T42, T43, X66)
PREORDER1_IN_GA(tree(T21, T22, T23), .(T22, T17)) → U8_GA(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U8_GA(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U9_GA(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
U8_GA(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → PDL3_IN_GA(T23, T24)
PDL3_IN_GA(tree(T21, T22, T23), .(T22, T17)) → U1_GA(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
PDL3_IN_GA(tree(T21, T22, T23), .(T22, T17)) → PDL24_IN_GAA(T21, T17, X27)
PDL3_IN_GA(tree(T21, T22, T23), .(T22, T17)) → U2_GA(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U2_GA(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U3_GA(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
U2_GA(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → PDL3_IN_GA(T23, T24)

The TRS R consists of the following rules:

preorder1_in_ga(nil, []) → preorder1_out_ga(nil, [])
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U7_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
pdl24_in_gaa(nil, T27, T27) → pdl24_out_gaa(nil, T27, T27)
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U4_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, X65))
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U5_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U5_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → U6_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T42, T43, X66))
U6_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T42, T43, X66)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U4_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, X65)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U7_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U8_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U8_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U9_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
pdl3_in_ga(nil, []) → pdl3_out_ga(nil, [])
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U1_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
U1_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U2_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U2_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U3_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
U3_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
U9_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))

The argument filtering Pi contains the following mapping:
preorder1_in_ga(x1, x2)  =  preorder1_in_ga(x1)
nil  =  nil
preorder1_out_ga(x1, x2)  =  preorder1_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
pdl24_in_gaa(x1, x2, x3)  =  pdl24_in_gaa(x1)
pdl24_out_gaa(x1, x2, x3)  =  pdl24_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)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
pdl3_in_ga(x1, x2)  =  pdl3_in_ga(x1)
pdl3_out_ga(x1, x2)  =  pdl3_out_ga
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
PREORDER1_IN_GA(x1, x2)  =  PREORDER1_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
PDL24_IN_GAA(x1, x2, x3)  =  PDL24_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)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
PDL3_IN_GA(x1, x2)  =  PDL3_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
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:

PREORDER1_IN_GA(tree(T21, T22, T23), .(T22, T17)) → U7_GA(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
PREORDER1_IN_GA(tree(T21, T22, T23), .(T22, T17)) → PDL24_IN_GAA(T21, T17, X27)
PDL24_IN_GAA(tree(T40, T41, T42), .(T41, T36), X66) → U4_GAA(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, X65))
PDL24_IN_GAA(tree(T40, T41, T42), .(T41, T36), X66) → PDL24_IN_GAA(T40, T36, X65)
PDL24_IN_GAA(tree(T40, T41, T42), .(T41, T36), X66) → U5_GAA(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U5_GAA(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → U6_GAA(T40, T41, T42, T36, X66, pdl24_in_gaa(T42, T43, X66))
U5_GAA(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → PDL24_IN_GAA(T42, T43, X66)
PREORDER1_IN_GA(tree(T21, T22, T23), .(T22, T17)) → U8_GA(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U8_GA(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U9_GA(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
U8_GA(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → PDL3_IN_GA(T23, T24)
PDL3_IN_GA(tree(T21, T22, T23), .(T22, T17)) → U1_GA(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
PDL3_IN_GA(tree(T21, T22, T23), .(T22, T17)) → PDL24_IN_GAA(T21, T17, X27)
PDL3_IN_GA(tree(T21, T22, T23), .(T22, T17)) → U2_GA(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U2_GA(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U3_GA(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
U2_GA(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → PDL3_IN_GA(T23, T24)

The TRS R consists of the following rules:

preorder1_in_ga(nil, []) → preorder1_out_ga(nil, [])
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U7_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
pdl24_in_gaa(nil, T27, T27) → pdl24_out_gaa(nil, T27, T27)
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U4_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, X65))
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U5_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U5_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → U6_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T42, T43, X66))
U6_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T42, T43, X66)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U4_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, X65)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U7_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U8_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U8_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U9_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
pdl3_in_ga(nil, []) → pdl3_out_ga(nil, [])
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U1_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
U1_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U2_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U2_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U3_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
U3_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
U9_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))

The argument filtering Pi contains the following mapping:
preorder1_in_ga(x1, x2)  =  preorder1_in_ga(x1)
nil  =  nil
preorder1_out_ga(x1, x2)  =  preorder1_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
pdl24_in_gaa(x1, x2, x3)  =  pdl24_in_gaa(x1)
pdl24_out_gaa(x1, x2, x3)  =  pdl24_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)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
pdl3_in_ga(x1, x2)  =  pdl3_in_ga(x1)
pdl3_out_ga(x1, x2)  =  pdl3_out_ga
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
PREORDER1_IN_GA(x1, x2)  =  PREORDER1_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
PDL24_IN_GAA(x1, x2, x3)  =  PDL24_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)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
PDL3_IN_GA(x1, x2)  =  PDL3_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 10 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

PDL24_IN_GAA(tree(T40, T41, T42), .(T41, T36), X66) → U5_GAA(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U5_GAA(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → PDL24_IN_GAA(T42, T43, X66)
PDL24_IN_GAA(tree(T40, T41, T42), .(T41, T36), X66) → PDL24_IN_GAA(T40, T36, X65)

The TRS R consists of the following rules:

preorder1_in_ga(nil, []) → preorder1_out_ga(nil, [])
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U7_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
pdl24_in_gaa(nil, T27, T27) → pdl24_out_gaa(nil, T27, T27)
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U4_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, X65))
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U5_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U5_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → U6_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T42, T43, X66))
U6_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T42, T43, X66)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U4_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, X65)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U7_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U8_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U8_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U9_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
pdl3_in_ga(nil, []) → pdl3_out_ga(nil, [])
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U1_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
U1_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U2_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U2_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U3_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
U3_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
U9_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))

The argument filtering Pi contains the following mapping:
preorder1_in_ga(x1, x2)  =  preorder1_in_ga(x1)
nil  =  nil
preorder1_out_ga(x1, x2)  =  preorder1_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
pdl24_in_gaa(x1, x2, x3)  =  pdl24_in_gaa(x1)
pdl24_out_gaa(x1, x2, x3)  =  pdl24_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)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
pdl3_in_ga(x1, x2)  =  pdl3_in_ga(x1)
pdl3_out_ga(x1, x2)  =  pdl3_out_ga
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
PDL24_IN_GAA(x1, x2, x3)  =  PDL24_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:

PDL24_IN_GAA(tree(T40, T41, T42), .(T41, T36), X66) → U5_GAA(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U5_GAA(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → PDL24_IN_GAA(T42, T43, X66)
PDL24_IN_GAA(tree(T40, T41, T42), .(T41, T36), X66) → PDL24_IN_GAA(T40, T36, X65)

The TRS R consists of the following rules:

pdl24_in_gaa(nil, T27, T27) → pdl24_out_gaa(nil, T27, T27)
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U4_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, X65))
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U5_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U4_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, X65)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U5_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → U6_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T42, T43, X66))
U6_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T42, T43, X66)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)

The argument filtering Pi contains the following mapping:
nil  =  nil
tree(x1, x2, x3)  =  tree(x1, x2, x3)
pdl24_in_gaa(x1, x2, x3)  =  pdl24_in_gaa(x1)
pdl24_out_gaa(x1, x2, x3)  =  pdl24_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)
PDL24_IN_GAA(x1, x2, x3)  =  PDL24_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:

PDL24_IN_GAA(tree(T40, T41, T42)) → U5_GAA(T42, pdl24_in_gaa(T40))
U5_GAA(T42, pdl24_out_gaa) → PDL24_IN_GAA(T42)
PDL24_IN_GAA(tree(T40, T41, T42)) → PDL24_IN_GAA(T40)

The TRS R consists of the following rules:

pdl24_in_gaa(nil) → pdl24_out_gaa
pdl24_in_gaa(tree(T40, T41, T42)) → U4_gaa(pdl24_in_gaa(T40))
pdl24_in_gaa(tree(T40, T41, T42)) → U5_gaa(T42, pdl24_in_gaa(T40))
U4_gaa(pdl24_out_gaa) → pdl24_out_gaa
U5_gaa(T42, pdl24_out_gaa) → U6_gaa(pdl24_in_gaa(T42))
U6_gaa(pdl24_out_gaa) → pdl24_out_gaa

The set Q consists of the following terms:

pdl24_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(T42, pdl24_out_gaa) → PDL24_IN_GAA(T42)
  The graph contains the following edges 1 >= 1

• PDL24_IN_GAA(tree(T40, T41, T42)) → PDL24_IN_GAA(T40)
  The graph contains the following edges 1 > 1

• PDL24_IN_GAA(tree(T40, T41, T42)) → U5_GAA(T42, pdl24_in_gaa(T40))
  The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

PDL3_IN_GA(tree(T21, T22, T23), .(T22, T17)) → U2_GA(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U2_GA(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → PDL3_IN_GA(T23, T24)

The TRS R consists of the following rules:

preorder1_in_ga(nil, []) → preorder1_out_ga(nil, [])
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U7_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
pdl24_in_gaa(nil, T27, T27) → pdl24_out_gaa(nil, T27, T27)
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U4_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, X65))
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U5_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U5_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → U6_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T42, T43, X66))
U6_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T42, T43, X66)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U4_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, X65)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U7_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))
preorder1_in_ga(tree(T21, T22, T23), .(T22, T17)) → U8_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U8_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U9_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
pdl3_in_ga(nil, []) → pdl3_out_ga(nil, [])
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U1_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, X27))
U1_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, X27)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
pdl3_in_ga(tree(T21, T22, T23), .(T22, T17)) → U2_ga(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U2_ga(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → U3_ga(T21, T22, T23, T17, pdl3_in_ga(T23, T24))
U3_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → pdl3_out_ga(tree(T21, T22, T23), .(T22, T17))
U9_ga(T21, T22, T23, T17, pdl3_out_ga(T23, T24)) → preorder1_out_ga(tree(T21, T22, T23), .(T22, T17))

The argument filtering Pi contains the following mapping:
preorder1_in_ga(x1, x2)  =  preorder1_in_ga(x1)
nil  =  nil
preorder1_out_ga(x1, x2)  =  preorder1_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
pdl24_in_gaa(x1, x2, x3)  =  pdl24_in_gaa(x1)
pdl24_out_gaa(x1, x2, x3)  =  pdl24_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)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
pdl3_in_ga(x1, x2)  =  pdl3_in_ga(x1)
pdl3_out_ga(x1, x2)  =  pdl3_out_ga
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
PDL3_IN_GA(x1, x2)  =  PDL3_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

PDL3_IN_GA(tree(T21, T22, T23), .(T22, T17)) → U2_GA(T21, T22, T23, T17, pdl24_in_gaa(T21, T17, T24))
U2_GA(T21, T22, T23, T17, pdl24_out_gaa(T21, T17, T24)) → PDL3_IN_GA(T23, T24)

The TRS R consists of the following rules:

pdl24_in_gaa(nil, T27, T27) → pdl24_out_gaa(nil, T27, T27)
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U4_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, X65))
pdl24_in_gaa(tree(T40, T41, T42), .(T41, T36), X66) → U5_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T40, T36, T43))
U4_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, X65)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)
U5_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T40, T36, T43)) → U6_gaa(T40, T41, T42, T36, X66, pdl24_in_gaa(T42, T43, X66))
U6_gaa(T40, T41, T42, T36, X66, pdl24_out_gaa(T42, T43, X66)) → pdl24_out_gaa(tree(T40, T41, T42), .(T41, T36), X66)

The argument filtering Pi contains the following mapping:
nil  =  nil
tree(x1, x2, x3)  =  tree(x1, x2, x3)
pdl24_in_gaa(x1, x2, x3)  =  pdl24_in_gaa(x1)
pdl24_out_gaa(x1, x2, x3)  =  pdl24_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)
PDL3_IN_GA(x1, x2)  =  PDL3_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:

PDL3_IN_GA(tree(T21, T22, T23)) → U2_GA(T23, pdl24_in_gaa(T21))
U2_GA(T23, pdl24_out_gaa) → PDL3_IN_GA(T23)

The TRS R consists of the following rules:

pdl24_in_gaa(nil) → pdl24_out_gaa
pdl24_in_gaa(tree(T40, T41, T42)) → U4_gaa(pdl24_in_gaa(T40))
pdl24_in_gaa(tree(T40, T41, T42)) → U5_gaa(T42, pdl24_in_gaa(T40))
U4_gaa(pdl24_out_gaa) → pdl24_out_gaa
U5_gaa(T42, pdl24_out_gaa) → U6_gaa(pdl24_in_gaa(T42))
U6_gaa(pdl24_out_gaa) → pdl24_out_gaa

The set Q consists of the following terms:

pdl24_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(T23, pdl24_out_gaa) → PDL3_IN_GA(T23)
  The graph contains the following edges 1 >= 1

• PDL3_IN_GA(tree(T21, T22, T23)) → U2_GA(T23, pdl24_in_gaa(T21))
  The graph contains the following edges 1 > 1

(22) TRUE