Runtime Complexity of the query pattern
preorder(g,a)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Clauses:

preorder(T, Xs) :- preorder_dl(T, -(Xs, [])).
preorder_dl(nil, -(X, X)).
preorder_dl(tree(L, X, R), -(.(X, Xs), Zs)) :- ','(preorder_dl(L, -(Xs, Ys)), preorder_dl(R, -(Ys, Zs))).

Query: preorder(g,a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

preorder_dl(nil, -(X, X)).
preorder(T, Xs) :- preorder_dl(T, -(Xs, [])).
preorder_dl(tree(L, X, R), -(.(X, Xs), Zs)) :- ','(preorder_dl(L, -(Xs, Ys)), preorder_dl(R, -(Ys, Zs))).

Query: preorder(g,a)

(3) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: preorder(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: preorder(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: preorder_dl(T10, -(T12, []))\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: preorder_dl(T10, -(T12, [])), SCOPE: 2, CLAUSE: 0 | preorder_dl(T10, -(T12, [])), SCOPE: 2, CLAUSE: 2\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true | preorder_dl(nil, -(T12, [])), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: preorder_dl(T10, -(T12, [])), SCOPE: 2, CLAUSE: 2\n\nT10 is ground\nX13 is free\npreorder_dl(T10, -(T12, [])) !=? preorder_dl(nil, -(X13, X13))", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: preorder_dl(nil, -(T12, [])), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(preorder_dl(T26, -(T30, X35)), preorder_dl(T28, -(X35, [])))\n\nT26 is ground\nT28 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: preorder_dl(T26, -(T30, X35))\n\nT26 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: preorder_dl(T28, -(T33, []))\n\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: preorder_dl(T26, -(T30, X35)), SCOPE: 3, CLAUSE: 0 | preorder_dl(T26, -(T30, X35)), SCOPE: 3, CLAUSE: 2\n\nT26 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: true | preorder_dl(nil, -(T30, X35)), SCOPE: 3, CLAUSE: 2\n\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: preorder_dl(T26, -(T30, X35)), SCOPE: 3, CLAUSE: 2\n\nT26 is ground\nX35 is free\nX42 is free\npreorder_dl(T26, -(T30, X35)) !=? preorder_dl(nil, -(X42, X42))", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: preorder_dl(nil, -(T30, X35)), SCOPE: 3, CLAUSE: 2\n\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(preorder_dl(T47, -(T51, X66)), preorder_dl(T49, -(X66, X67)))\n\nT47 is ground\nT49 is ground\nX67 is free\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: preorder_dl(T47, -(T51, X66))\n\nT47 is ground\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: preorder_dl(T49, -(T54, X67))\n\nT49 is ground\nX67 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 43 [arrowhead = none ];
43 -> 4;

44 [label="ONLY EVAL with clause\npreorder(X7, X8) :- preorder_dl(X7, -(X8, [])).\nand substitutionT1 -> T10,\nX7 -> T10,\nT2 -> T12,\nX8 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 44 [arrowhead = none ];
44 -> 7;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 45 [arrowhead = none ];
45 -> 14;

46 [label="EVAL with clause\npreorder_dl(nil, -(X13, X13)).\nand substitutionT10 -> nil,\nT12 -> [],\nX13 -> [],\nT17 -> []", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 46 [arrowhead = none ];
46 -> 17;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 47 [arrowhead = none ];
47 -> 18;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 48 [arrowhead = none ];
48 -> 19;

49 [label="EVAL with clause\npreorder_dl(tree(X30, X31, X32), -(.(X31, X33), X34)) :- ','(preorder_dl(X30, -(X33, X35)), preorder_dl(X32, -(X35, X34))).\nand substitutionX30 -> T26,\nX31 -> T27,\nX32 -> T28,\nT10 -> tree(T26, T27, T28),\nX33 -> T30,\nT12 -> .(T27, T30),\nX34 -> [],\nT29 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 49 [arrowhead = none ];
49 -> 26;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 50 [arrowhead = none ];
50 -> 27;

51 [label="BACKTRACK\nfor clause: preorder_dl(tree(L, X, R), -(.(X, Xs), Zs)) :- ','(preorder_dl(L, -(Xs, Ys)), preorder_dl(R, -(Ys, Zs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 51 [arrowhead = none ];
51 -> 20;

52 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
26 -> 52 [arrowhead = none ];
52 -> 28;

53 [label="SPLIT 2\nnew knowledge:\nT26 is ground\nreplacements:X35 -> T33", fontsize=14, style = filled, fillcolor = violet];
26 -> 53 [arrowhead = none ];
53 -> 29;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 54 [arrowhead = none ];
54 -> 32;

55 [label="INSTANCE with matching:\nT10 -> T28\nT12 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 55 [arrowhead = none , style=dashed];
55 -> 7[style=dashed];

56 [label="EVAL with clause\npreorder_dl(nil, -(X42, X42)).\nand substitutionT26 -> nil,\nT30 -> T38,\nX42 -> T38,\nX35 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 56 [arrowhead = none ];
56 -> 35;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 57 [arrowhead = none ];
57 -> 36;

58 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 58 [arrowhead = none ];
58 -> 37;

59 [label="EVAL with clause\npreorder_dl(tree(X61, X62, X63), -(.(X62, X64), X65)) :- ','(preorder_dl(X61, -(X64, X66)), preorder_dl(X63, -(X66, X65))).\nand substitutionX61 -> T47,\nX62 -> T48,\nX63 -> T49,\nT26 -> tree(T47, T48, T49),\nX64 -> T51,\nT30 -> .(T48, T51),\nX35 -> X67,\nX65 -> X67,\nT50 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 59 [arrowhead = none ];
59 -> 39;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 60 [arrowhead = none ];
60 -> 40;

61 [label="BACKTRACK\nfor clause: preorder_dl(tree(L, X, R), -(.(X, Xs), Zs)) :- ','(preorder_dl(L, -(Xs, Ys)), preorder_dl(R, -(Ys, Zs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 61 [arrowhead = none ];
61 -> 38;

62 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
39 -> 62 [arrowhead = none ];
62 -> 41;

63 [label="SPLIT 2\nnew knowledge:\nT47 is ground\nreplacements:X66 -> T54", fontsize=14, style = filled, fillcolor = violet];
39 -> 63 [arrowhead = none ];
63 -> 42;

64 [label="INSTANCE with matching:\nT26 -> T47\nT30 -> T51\nX35 -> X66", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 64 [arrowhead = none , style=dashed];
64 -> 28[style=dashed];

65 [label="INSTANCE with matching:\nT26 -> T49\nT30 -> T54\nX35 -> X67", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 65 [arrowhead = none , style=dashed];
65 -> 28[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f2_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f26_in(z0, z2), tree(z0, z1, z2))
U2(f26_out1, tree(z0, z1, z2)) → f7_out1
f28_in(nil) → f28_out1
f28_in(tree(z0, z1, z2)) → U3(f39_in(z0, z2), tree(z0, z1, z2))
U3(f39_out1, tree(z0, z1, z2)) → f28_out1
f26_in(z0, z1) → U4(f28_in(z0), z0, z1)
U4(f28_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f26_out1
f39_in(z0, z1) → U6(f28_in(z0), z0, z1)
U6(f28_out1, z0, z1) → U7(f28_in(z1), z0, z1)
U7(f28_out1, z0, z1) → f39_out1

Tuples:

F2_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(U2'(f26_in(z0, z2), tree(z0, z1, z2)), F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(U3'(f39_in(z0, z2), tree(z0, z1, z2)), F39_IN(z0, z2))
F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
U4'(f28_out1, z0, z1) → c9(U5'(f7_in(z1), z0, z1), F7_IN(z1))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
U6'(f28_out1, z0, z1) → c12(U7'(f28_in(z1), z0, z1), F28_IN(z1))

S tuples:

F2_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(U2'(f26_in(z0, z2), tree(z0, z1, z2)), F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(U3'(f39_in(z0, z2), tree(z0, z1, z2)), F39_IN(z0, z2))
F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
U4'(f28_out1, z0, z1) → c9(U5'(f7_in(z1), z0, z1), F7_IN(z1))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
U6'(f28_out1, z0, z1) → c12(U7'(f28_in(z1), z0, z1), F28_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, f28_in, U3, f26_in, U4, U5, f39_in, U6, U7

Defined Pair Symbols:

F2_IN, F7_IN, F28_IN, F26_IN, U4', F39_IN, U6'

Compound Symbols:

c, c3, c6, c8, c9, c11, c12

(5) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f2_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f26_in(z0, z2), tree(z0, z1, z2))
U2(f26_out1, tree(z0, z1, z2)) → f7_out1
f28_in(nil) → f28_out1
f28_in(tree(z0, z1, z2)) → U3(f39_in(z0, z2), tree(z0, z1, z2))
U3(f39_out1, tree(z0, z1, z2)) → f28_out1
f26_in(z0, z1) → U4(f28_in(z0), z0, z1)
U4(f28_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f26_out1
f39_in(z0, z1) → U6(f28_in(z0), z0, z1)
U6(f28_out1, z0, z1) → U7(f28_in(z1), z0, z1)
U7(f28_out1, z0, z1) → f39_out1

Tuples:

F7_IN(tree(z0, z1, z2)) → c3(U2'(f26_in(z0, z2), tree(z0, z1, z2)), F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(U3'(f39_in(z0, z2), tree(z0, z1, z2)), F39_IN(z0, z2))
F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
U4'(f28_out1, z0, z1) → c9(U5'(f7_in(z1), z0, z1), F7_IN(z1))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
U6'(f28_out1, z0, z1) → c12(U7'(f28_in(z1), z0, z1), F28_IN(z1))
F2_IN(z0) → c1(U1'(f7_in(z0), z0))
F2_IN(z0) → c1(F7_IN(z0))

S tuples:

F7_IN(tree(z0, z1, z2)) → c3(U2'(f26_in(z0, z2), tree(z0, z1, z2)), F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(U3'(f39_in(z0, z2), tree(z0, z1, z2)), F39_IN(z0, z2))
F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
U4'(f28_out1, z0, z1) → c9(U5'(f7_in(z1), z0, z1), F7_IN(z1))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
U6'(f28_out1, z0, z1) → c12(U7'(f28_in(z1), z0, z1), F28_IN(z1))
F2_IN(z0) → c1(U1'(f7_in(z0), z0))
F2_IN(z0) → c1(F7_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, f28_in, U3, f26_in, U4, U5, f39_in, U6, U7

Defined Pair Symbols:

F7_IN, F28_IN, F26_IN, U4', F39_IN, U6', F2_IN

Compound Symbols:

c3, c6, c8, c9, c11, c12, c1

(7) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 5 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f2_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f26_in(z0, z2), tree(z0, z1, z2))
U2(f26_out1, tree(z0, z1, z2)) → f7_out1
f28_in(nil) → f28_out1
f28_in(tree(z0, z1, z2)) → U3(f39_in(z0, z2), tree(z0, z1, z2))
U3(f39_out1, tree(z0, z1, z2)) → f28_out1
f26_in(z0, z1) → U4(f28_in(z0), z0, z1)
U4(f28_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f26_out1
f39_in(z0, z1) → U6(f28_in(z0), z0, z1)
U6(f28_out1, z0, z1) → U7(f28_in(z1), z0, z1)
U7(f28_out1, z0, z1) → f39_out1

Tuples:

F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
F2_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))
F2_IN(z0) → c1

S tuples:

F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
F2_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))
F2_IN(z0) → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, f28_in, U3, f26_in, U4, U5, f39_in, U6, U7

Defined Pair Symbols:

F26_IN, F39_IN, F2_IN, F7_IN, F28_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F2_IN(z0) → c1(F7_IN(z0))
F2_IN(z0) → c1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f2_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f26_in(z0, z2), tree(z0, z1, z2))
U2(f26_out1, tree(z0, z1, z2)) → f7_out1
f28_in(nil) → f28_out1
f28_in(tree(z0, z1, z2)) → U3(f39_in(z0, z2), tree(z0, z1, z2))
U3(f39_out1, tree(z0, z1, z2)) → f28_out1
f26_in(z0, z1) → U4(f28_in(z0), z0, z1)
U4(f28_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f26_out1
f39_in(z0, z1) → U6(f28_in(z0), z0, z1)
U6(f28_out1, z0, z1) → U7(f28_in(z1), z0, z1)
U7(f28_out1, z0, z1) → f39_out1

Tuples:

F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
F2_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))
F2_IN(z0) → c1

S tuples:

F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))

K tuples:

F2_IN(z0) → c1(F7_IN(z0))
F2_IN(z0) → c1

Defined Rule Symbols:

f2_in, U1, f7_in, U2, f28_in, U3, f26_in, U4, U5, f39_in, U6, U7

Defined Pair Symbols:

F26_IN, F39_IN, F2_IN, F7_IN, F28_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))

We considered the (Usable) Rules:

f28_in(nil) → f28_out1
f28_in(tree(z0, z1, z2)) → U3(f39_in(z0, z2), tree(z0, z1, z2))
f39_in(z0, z1) → U6(f28_in(z0), z0, z1)
U3(f39_out1, tree(z0, z1, z2)) → f28_out1
U6(f28_out1, z0, z1) → U7(f28_in(z1), z0, z1)
U7(f28_out1, z0, z1) → f39_out1

And the Tuples:

F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
F2_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))
F2_IN(z0) → c1

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F26_IN(x₁, x₂)) = [2]x₂   
POL(F28_IN(x₁)) = 0   
POL(F2_IN(x₁)) = [1] + [2]x₁   
POL(F39_IN(x₁, x₂)) = 0   
POL(F7_IN(x₁)) = [2]x₁   
POL(U3(x₁, x₂)) = 0   
POL(U4'(x₁, x₂, x₃)) = [2]x₃   
POL(U6(x₁, x₂, x₃)) = [2]x₃   
POL(U6'(x₁, x₂, x₃)) = 0   
POL(U7(x₁, x₂, x₃)) = x₃   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f28_in(x₁)) = 0   
POL(f28_out1) = 0   
POL(f39_in(x₁, x₂)) = [2] + [2]x₂   
POL(f39_out1) = 0   
POL(nil) = 0   
POL(tree(x₁, x₂, x₃)) = [2] + x₃   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f2_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f26_in(z0, z2), tree(z0, z1, z2))
U2(f26_out1, tree(z0, z1, z2)) → f7_out1
f28_in(nil) → f28_out1
f28_in(tree(z0, z1, z2)) → U3(f39_in(z0, z2), tree(z0, z1, z2))
U3(f39_out1, tree(z0, z1, z2)) → f28_out1
f26_in(z0, z1) → U4(f28_in(z0), z0, z1)
U4(f28_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f26_out1
f39_in(z0, z1) → U6(f28_in(z0), z0, z1)
U6(f28_out1, z0, z1) → U7(f28_in(z1), z0, z1)
U7(f28_out1, z0, z1) → f39_out1

Tuples:

F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
F2_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))
F2_IN(z0) → c1

S tuples:

F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))

K tuples:

F2_IN(z0) → c1(F7_IN(z0))
F2_IN(z0) → c1
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))

Defined Rule Symbols:

f2_in, U1, f7_in, U2, f28_in, U3, f26_in, U4, U5, f39_in, U6, U7

Defined Pair Symbols:

F26_IN, F39_IN, F2_IN, F7_IN, F28_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(13) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f2_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f26_in(z0, z2), tree(z0, z1, z2))
U2(f26_out1, tree(z0, z1, z2)) → f7_out1
f28_in(nil) → f28_out1
f28_in(tree(z0, z1, z2)) → U3(f39_in(z0, z2), tree(z0, z1, z2))
U3(f39_out1, tree(z0, z1, z2)) → f28_out1
f26_in(z0, z1) → U4(f28_in(z0), z0, z1)
U4(f28_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f26_out1
f39_in(z0, z1) → U6(f28_in(z0), z0, z1)
U6(f28_out1, z0, z1) → U7(f28_in(z1), z0, z1)
U7(f28_out1, z0, z1) → f39_out1

Tuples:

F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
F2_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))
F2_IN(z0) → c1

S tuples:

F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))

K tuples:

F2_IN(z0) → c1(F7_IN(z0))
F2_IN(z0) → c1
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))
F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))

Defined Rule Symbols:

f2_in, U1, f7_in, U2, f28_in, U3, f26_in, U4, U5, f39_in, U6, U7

Defined Pair Symbols:

F26_IN, F39_IN, F2_IN, F7_IN, F28_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))

We considered the (Usable) Rules:

f28_in(nil) → f28_out1
f28_in(tree(z0, z1, z2)) → U3(f39_in(z0, z2), tree(z0, z1, z2))
f39_in(z0, z1) → U6(f28_in(z0), z0, z1)
U3(f39_out1, tree(z0, z1, z2)) → f28_out1
U6(f28_out1, z0, z1) → U7(f28_in(z1), z0, z1)
U7(f28_out1, z0, z1) → f39_out1

And the Tuples:

F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
F2_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))
F2_IN(z0) → c1

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F26_IN(x₁, x₂)) = [2] + [3]x₁ + [3]x₂   
POL(F28_IN(x₁)) = x₁   
POL(F2_IN(x₁)) = [3]x₁   
POL(F39_IN(x₁, x₂)) = x₁ + x₂   
POL(F7_IN(x₁)) = [3]x₁   
POL(U3(x₁, x₂)) = 0   
POL(U4'(x₁, x₂, x₃)) = [3]x₃   
POL(U6(x₁, x₂, x₃)) = [2] + [2]x₂   
POL(U6'(x₁, x₂, x₃)) = x₃   
POL(U7(x₁, x₂, x₃)) = [2] + x₂   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f28_in(x₁)) = 0   
POL(f28_out1) = 0   
POL(f39_in(x₁, x₂)) = [2] + [2]x₁   
POL(f39_out1) = [2]   
POL(nil) = 0   
POL(tree(x₁, x₂, x₃)) = [1] + x₁ + x₃   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f2_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f26_in(z0, z2), tree(z0, z1, z2))
U2(f26_out1, tree(z0, z1, z2)) → f7_out1
f28_in(nil) → f28_out1
f28_in(tree(z0, z1, z2)) → U3(f39_in(z0, z2), tree(z0, z1, z2))
U3(f39_out1, tree(z0, z1, z2)) → f28_out1
f26_in(z0, z1) → U4(f28_in(z0), z0, z1)
U4(f28_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f26_out1
f39_in(z0, z1) → U6(f28_in(z0), z0, z1)
U6(f28_out1, z0, z1) → U7(f28_in(z1), z0, z1)
U7(f28_out1, z0, z1) → f39_out1

Tuples:

F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
F2_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))
F2_IN(z0) → c1

S tuples:

F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))

K tuples:

F2_IN(z0) → c1(F7_IN(z0))
F2_IN(z0) → c1
F7_IN(tree(z0, z1, z2)) → c3(F26_IN(z0, z2))
F26_IN(z0, z1) → c8(U4'(f28_in(z0), z0, z1), F28_IN(z0))
U4'(f28_out1, z0, z1) → c9(F7_IN(z1))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))

Defined Rule Symbols:

f2_in, U1, f7_in, U2, f28_in, U3, f26_in, U4, U5, f39_in, U6, U7

Defined Pair Symbols:

F26_IN, F39_IN, F2_IN, F7_IN, F28_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(17) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F39_IN(z0, z1) → c11(U6'(f28_in(z0), z0, z1), F28_IN(z0))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))
U6'(f28_out1, z0, z1) → c12(F28_IN(z1))
F28_IN(tree(z0, z1, z2)) → c6(F39_IN(z0, z2))

Now S is empty

(18) BOUNDS(O(1), O(1))

(19) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation 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];
3 [label="3: preorder(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: preorder_dl(T5, -(T7, []))\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: preorder_dl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 0 | preorder_dl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true | preorder_dl(nil, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: preorder_dl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nX7 is free\npreorder_dl(T5, -(T7, [])) !=? preorder_dl(nil, -(X7, X7))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: preorder_dl(nil, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(preorder_dl(T19, -(T23, X29)), preorder_dl(T21, -(X29, [])))\n\nT19 is ground\nT21 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: preorder_dl(T19, -(T23, X29))\n\nT19 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: preorder_dl(T21, -(T26, []))\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: preorder_dl(T19, -(T23, X29)), SCOPE: 3, CLAUSE: 0 | preorder_dl(T19, -(T23, X29)), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true | preorder_dl(nil, -(T23, X29)), SCOPE: 3, CLAUSE: 2\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: preorder_dl(T19, -(T23, X29)), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nX29 is free\nX36 is free\npreorder_dl(T19, -(T23, X29)) !=? preorder_dl(nil, -(X36, X36))", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: preorder_dl(nil, -(T23, X29)), SCOPE: 3, CLAUSE: 2\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(preorder_dl(T40, -(T44, X60)), preorder_dl(T42, -(X60, X61)))\n\nT40 is ground\nT42 is ground\nX61 is free\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: preorder_dl(T40, -(T44, X60))\n\nT40 is ground\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: preorder_dl(T42, -(T47, X61))\n\nT42 is ground\nX61 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 35 [arrowhead = none ];
35 -> 3;

36 [label="ONLY EVAL with clause\npreorder(X3, X4) :- preorder_dl(X3, -(X4, [])).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 36 [arrowhead = none ];
36 -> 5;

37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 37 [arrowhead = none ];
37 -> 6;

38 [label="EVAL with clause\npreorder_dl(nil, -(X7, X7)).\nand substitutionT5 -> nil,\nT7 -> [],\nX7 -> [],\nT10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 38 [arrowhead = none ];
38 -> 8;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 39 [arrowhead = none ];
39 -> 9;

40 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 40 [arrowhead = none ];
40 -> 10;

41 [label="EVAL with clause\npreorder_dl(tree(X24, X25, X26), -(.(X25, X27), X28)) :- ','(preorder_dl(X24, -(X27, X29)), preorder_dl(X26, -(X29, X28))).\nand substitutionX24 -> T19,\nX25 -> T20,\nX26 -> T21,\nT5 -> tree(T19, T20, T21),\nX27 -> T23,\nT7 -> .(T20, T23),\nX28 -> [],\nT22 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 41 [arrowhead = none ];
41 -> 12;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 42 [arrowhead = none ];
42 -> 13;

43 [label="BACKTRACK\nfor clause: preorder_dl(tree(L, X, R), -(.(X, Xs), Zs)) :- ','(preorder_dl(L, -(Xs, Ys)), preorder_dl(R, -(Ys, Zs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 43 [arrowhead = none ];
43 -> 11;

44 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
12 -> 44 [arrowhead = none ];
44 -> 15;

45 [label="SPLIT 2\nnew knowledge:\nT19 is ground\nreplacements:X29 -> T26", fontsize=14, style = filled, fillcolor = violet];
12 -> 45 [arrowhead = none ];
45 -> 16;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 46 [arrowhead = none ];
46 -> 21;

47 [label="INSTANCE with matching:\nT5 -> T21\nT7 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 47 [arrowhead = none , style=dashed];
47 -> 5[style=dashed];

48 [label="EVAL with clause\npreorder_dl(nil, -(X36, X36)).\nand substitutionT19 -> nil,\nT23 -> T31,\nX36 -> T31,\nX29 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 48 [arrowhead = none ];
48 -> 22;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 49 [arrowhead = none ];
49 -> 23;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 50 [arrowhead = none ];
50 -> 24;

51 [label="EVAL with clause\npreorder_dl(tree(X55, X56, X57), -(.(X56, X58), X59)) :- ','(preorder_dl(X55, -(X58, X60)), preorder_dl(X57, -(X60, X59))).\nand substitutionX55 -> T40,\nX56 -> T41,\nX57 -> T42,\nT19 -> tree(T40, T41, T42),\nX58 -> T44,\nT23 -> .(T41, T44),\nX29 -> X61,\nX59 -> X61,\nT43 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 51 [arrowhead = none ];
51 -> 30;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 52 [arrowhead = none ];
52 -> 31;

53 [label="BACKTRACK\nfor clause: preorder_dl(tree(L, X, R), -(.(X, Xs), Zs)) :- ','(preorder_dl(L, -(Xs, Ys)), preorder_dl(R, -(Ys, Zs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 53 [arrowhead = none ];
53 -> 25;

54 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
30 -> 54 [arrowhead = none ];
54 -> 33;

55 [label="SPLIT 2\nnew knowledge:\nT40 is ground\nreplacements:X60 -> T47", fontsize=14, style = filled, fillcolor = violet];
30 -> 55 [arrowhead = none ];
55 -> 34;

56 [label="INSTANCE with matching:\nT19 -> T40\nT23 -> T44\nX29 -> X60", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 56 [arrowhead = none , style=dashed];
56 -> 15[style=dashed];

57 [label="INSTANCE with matching:\nT19 -> T42\nT23 -> T47\nX29 -> X61", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 57 [arrowhead = none , style=dashed];
57 -> 15[style=dashed];

}

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f1_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f12_in(z0, z2), tree(z0, z1, z2))
U2(f12_out1, tree(z0, z1, z2)) → f5_out1
f15_in(nil) → f15_out1
f15_in(tree(z0, z1, z2)) → U3(f30_in(z0, z2), tree(z0, z1, z2))
U3(f30_out1, tree(z0, z1, z2)) → f15_out1
f12_in(z0, z1) → U4(f15_in(z0), z0, z1)
U4(f15_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f12_out1
f30_in(z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → U7(f15_in(z1), z0, z1)
U7(f15_out1, z0, z1) → f30_out1

Tuples:

F1_IN(z0) → c(U1'(f5_in(z0), z0), F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(U2'(f12_in(z0, z2), tree(z0, z1, z2)), F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(U3'(f30_in(z0, z2), tree(z0, z1, z2)), F30_IN(z0, z2))
F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
U4'(f15_out1, z0, z1) → c9(U5'(f5_in(z1), z0, z1), F5_IN(z1))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
U6'(f15_out1, z0, z1) → c12(U7'(f15_in(z1), z0, z1), F15_IN(z1))

S tuples:

F1_IN(z0) → c(U1'(f5_in(z0), z0), F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(U2'(f12_in(z0, z2), tree(z0, z1, z2)), F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(U3'(f30_in(z0, z2), tree(z0, z1, z2)), F30_IN(z0, z2))
F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
U4'(f15_out1, z0, z1) → c9(U5'(f5_in(z1), z0, z1), F5_IN(z1))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
U6'(f15_out1, z0, z1) → c12(U7'(f15_in(z1), z0, z1), F15_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f5_in, U2, f15_in, U3, f12_in, U4, U5, f30_in, U6, U7

Defined Pair Symbols:

F1_IN, F5_IN, F15_IN, F12_IN, U4', F30_IN, U6'

Compound Symbols:

c, c3, c6, c8, c9, c11, c12

(21) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f1_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f12_in(z0, z2), tree(z0, z1, z2))
U2(f12_out1, tree(z0, z1, z2)) → f5_out1
f15_in(nil) → f15_out1
f15_in(tree(z0, z1, z2)) → U3(f30_in(z0, z2), tree(z0, z1, z2))
U3(f30_out1, tree(z0, z1, z2)) → f15_out1
f12_in(z0, z1) → U4(f15_in(z0), z0, z1)
U4(f15_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f12_out1
f30_in(z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → U7(f15_in(z1), z0, z1)
U7(f15_out1, z0, z1) → f30_out1

Tuples:

F5_IN(tree(z0, z1, z2)) → c3(U2'(f12_in(z0, z2), tree(z0, z1, z2)), F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(U3'(f30_in(z0, z2), tree(z0, z1, z2)), F30_IN(z0, z2))
F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
U4'(f15_out1, z0, z1) → c9(U5'(f5_in(z1), z0, z1), F5_IN(z1))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
U6'(f15_out1, z0, z1) → c12(U7'(f15_in(z1), z0, z1), F15_IN(z1))
F1_IN(z0) → c1(U1'(f5_in(z0), z0))
F1_IN(z0) → c1(F5_IN(z0))

S tuples:

F5_IN(tree(z0, z1, z2)) → c3(U2'(f12_in(z0, z2), tree(z0, z1, z2)), F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(U3'(f30_in(z0, z2), tree(z0, z1, z2)), F30_IN(z0, z2))
F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
U4'(f15_out1, z0, z1) → c9(U5'(f5_in(z1), z0, z1), F5_IN(z1))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
U6'(f15_out1, z0, z1) → c12(U7'(f15_in(z1), z0, z1), F15_IN(z1))
F1_IN(z0) → c1(U1'(f5_in(z0), z0))
F1_IN(z0) → c1(F5_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f5_in, U2, f15_in, U3, f12_in, U4, U5, f30_in, U6, U7

Defined Pair Symbols:

F5_IN, F15_IN, F12_IN, U4', F30_IN, U6', F1_IN

Compound Symbols:

c3, c6, c8, c9, c11, c12, c1

(23) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 5 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f1_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f12_in(z0, z2), tree(z0, z1, z2))
U2(f12_out1, tree(z0, z1, z2)) → f5_out1
f15_in(nil) → f15_out1
f15_in(tree(z0, z1, z2)) → U3(f30_in(z0, z2), tree(z0, z1, z2))
U3(f30_out1, tree(z0, z1, z2)) → f15_out1
f12_in(z0, z1) → U4(f15_in(z0), z0, z1)
U4(f15_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f12_out1
f30_in(z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → U7(f15_in(z1), z0, z1)
U7(f15_out1, z0, z1) → f30_out1

Tuples:

F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
F1_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))
F1_IN(z0) → c1

S tuples:

F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
F1_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))
F1_IN(z0) → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, f5_in, U2, f15_in, U3, f12_in, U4, U5, f30_in, U6, U7

Defined Pair Symbols:

F12_IN, F30_IN, F1_IN, F5_IN, F15_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(25) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F1_IN(z0) → c1(F5_IN(z0))
F1_IN(z0) → c1

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f1_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f12_in(z0, z2), tree(z0, z1, z2))
U2(f12_out1, tree(z0, z1, z2)) → f5_out1
f15_in(nil) → f15_out1
f15_in(tree(z0, z1, z2)) → U3(f30_in(z0, z2), tree(z0, z1, z2))
U3(f30_out1, tree(z0, z1, z2)) → f15_out1
f12_in(z0, z1) → U4(f15_in(z0), z0, z1)
U4(f15_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f12_out1
f30_in(z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → U7(f15_in(z1), z0, z1)
U7(f15_out1, z0, z1) → f30_out1

Tuples:

F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
F1_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))
F1_IN(z0) → c1

S tuples:

F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))

K tuples:

F1_IN(z0) → c1(F5_IN(z0))
F1_IN(z0) → c1

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f15_in, U3, f12_in, U4, U5, f30_in, U6, U7

Defined Pair Symbols:

F12_IN, F30_IN, F1_IN, F5_IN, F15_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))

We considered the (Usable) Rules:

f15_in(nil) → f15_out1
f15_in(tree(z0, z1, z2)) → U3(f30_in(z0, z2), tree(z0, z1, z2))
f30_in(z0, z1) → U6(f15_in(z0), z0, z1)
U3(f30_out1, tree(z0, z1, z2)) → f15_out1
U6(f15_out1, z0, z1) → U7(f15_in(z1), z0, z1)
U7(f15_out1, z0, z1) → f30_out1

And the Tuples:

F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
F1_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))
F1_IN(z0) → c1

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F12_IN(x₁, x₂)) = [2]x₂   
POL(F15_IN(x₁)) = 0   
POL(F1_IN(x₁)) = [1] + [2]x₁   
POL(F30_IN(x₁, x₂)) = 0   
POL(F5_IN(x₁)) = [2]x₁   
POL(U3(x₁, x₂)) = 0   
POL(U4'(x₁, x₂, x₃)) = [2]x₃   
POL(U6(x₁, x₂, x₃)) = [2]x₃   
POL(U6'(x₁, x₂, x₃)) = 0   
POL(U7(x₁, x₂, x₃)) = x₃   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f15_in(x₁)) = 0   
POL(f15_out1) = 0   
POL(f30_in(x₁, x₂)) = [2] + [2]x₂   
POL(f30_out1) = 0   
POL(nil) = 0   
POL(tree(x₁, x₂, x₃)) = [2] + x₃   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f1_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f12_in(z0, z2), tree(z0, z1, z2))
U2(f12_out1, tree(z0, z1, z2)) → f5_out1
f15_in(nil) → f15_out1
f15_in(tree(z0, z1, z2)) → U3(f30_in(z0, z2), tree(z0, z1, z2))
U3(f30_out1, tree(z0, z1, z2)) → f15_out1
f12_in(z0, z1) → U4(f15_in(z0), z0, z1)
U4(f15_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f12_out1
f30_in(z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → U7(f15_in(z1), z0, z1)
U7(f15_out1, z0, z1) → f30_out1

Tuples:

F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
F1_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))
F1_IN(z0) → c1

S tuples:

F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))

K tuples:

F1_IN(z0) → c1(F5_IN(z0))
F1_IN(z0) → c1
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f15_in, U3, f12_in, U4, U5, f30_in, U6, U7

Defined Pair Symbols:

F12_IN, F30_IN, F1_IN, F5_IN, F15_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(29) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f1_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f12_in(z0, z2), tree(z0, z1, z2))
U2(f12_out1, tree(z0, z1, z2)) → f5_out1
f15_in(nil) → f15_out1
f15_in(tree(z0, z1, z2)) → U3(f30_in(z0, z2), tree(z0, z1, z2))
U3(f30_out1, tree(z0, z1, z2)) → f15_out1
f12_in(z0, z1) → U4(f15_in(z0), z0, z1)
U4(f15_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f12_out1
f30_in(z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → U7(f15_in(z1), z0, z1)
U7(f15_out1, z0, z1) → f30_out1

Tuples:

F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
F1_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))
F1_IN(z0) → c1

S tuples:

F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))

K tuples:

F1_IN(z0) → c1(F5_IN(z0))
F1_IN(z0) → c1
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))
F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f15_in, U3, f12_in, U4, U5, f30_in, U6, U7

Defined Pair Symbols:

F12_IN, F30_IN, F1_IN, F5_IN, F15_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(31) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))

We considered the (Usable) Rules:

f15_in(nil) → f15_out1
f15_in(tree(z0, z1, z2)) → U3(f30_in(z0, z2), tree(z0, z1, z2))
f30_in(z0, z1) → U6(f15_in(z0), z0, z1)
U3(f30_out1, tree(z0, z1, z2)) → f15_out1
U6(f15_out1, z0, z1) → U7(f15_in(z1), z0, z1)
U7(f15_out1, z0, z1) → f30_out1

And the Tuples:

F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
F1_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))
F1_IN(z0) → c1

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F12_IN(x₁, x₂)) = [2] + [3]x₁ + [3]x₂   
POL(F15_IN(x₁)) = x₁   
POL(F1_IN(x₁)) = [3]x₁   
POL(F30_IN(x₁, x₂)) = x₁ + x₂   
POL(F5_IN(x₁)) = [3]x₁   
POL(U3(x₁, x₂)) = 0   
POL(U4'(x₁, x₂, x₃)) = [3]x₃   
POL(U6(x₁, x₂, x₃)) = [2] + [2]x₂   
POL(U6'(x₁, x₂, x₃)) = x₃   
POL(U7(x₁, x₂, x₃)) = [2] + x₂   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f15_in(x₁)) = 0   
POL(f15_out1) = 0   
POL(f30_in(x₁, x₂)) = [2] + [2]x₁   
POL(f30_out1) = [2]   
POL(nil) = 0   
POL(tree(x₁, x₂, x₃)) = [1] + x₁ + x₃   

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f1_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f12_in(z0, z2), tree(z0, z1, z2))
U2(f12_out1, tree(z0, z1, z2)) → f5_out1
f15_in(nil) → f15_out1
f15_in(tree(z0, z1, z2)) → U3(f30_in(z0, z2), tree(z0, z1, z2))
U3(f30_out1, tree(z0, z1, z2)) → f15_out1
f12_in(z0, z1) → U4(f15_in(z0), z0, z1)
U4(f15_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f12_out1
f30_in(z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → U7(f15_in(z1), z0, z1)
U7(f15_out1, z0, z1) → f30_out1

Tuples:

F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
F1_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))
F1_IN(z0) → c1

S tuples:

F30_IN(z0, z1) → c11(U6'(f15_in(z0), z0, z1), F15_IN(z0))
U6'(f15_out1, z0, z1) → c12(F15_IN(z1))

K tuples:

F1_IN(z0) → c1(F5_IN(z0))
F1_IN(z0) → c1
F5_IN(tree(z0, z1, z2)) → c3(F12_IN(z0, z2))
F12_IN(z0, z1) → c8(U4'(f15_in(z0), z0, z1), F15_IN(z0))
U4'(f15_out1, z0, z1) → c9(F5_IN(z1))
F15_IN(tree(z0, z1, z2)) → c6(F30_IN(z0, z2))

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f15_in, U3, f12_in, U4, U5, f30_in, U6, U7

Defined Pair Symbols:

F12_IN, F30_IN, F1_IN, F5_IN, F15_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1